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The vortex flow of dilute polymer solutions

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The vortex flow of dilute polymer solutions
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Chiou, Chii-Shyoung, 1948-
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vi, 298 leaves : ill. ; 28 cm.

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Boundary layers ( jstor )
Drag reduction ( jstor )
Dyes ( jstor )
Flow distribution ( jstor )
pH ( jstor )
Polymers ( jstor )
Velocity ( jstor )
Velocity distribution ( jstor )
Viscosity ( jstor )
Vortices ( jstor )
Polymerization ( lcsh )
Polymers ( lcsh )
Vortex-motion ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis--University of Florida.
Bibliography:
Includes bibliographical references (leaves 292-297).
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Typescript.
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Vita.
Statement of Responsibility:
by Chii-Shyoung Chiou.

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THE VORTEX FLOW OF DILUTE POLYMER SOLUTIONS


By

CHII-SHYOUNG CHIOU

















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA


1976
















ACKNOWLEDGMENTS


I wish to express my gratitude and appreciation to

my advisor, Professor R. J. Gordon, for his guidance and

instruction throughout the course of this work. I would

also like to thank the other members of my committee,

Professor R. W. Fahien, Professor R. D. Walter, Professor

D. W. Kirmse, and Professor U. H. Kurzweg for their par-

ticipation. I am especially grateful for the numerous

hours of consultation Professor Kurzweg has provided on

the work covered here and related problems.

Many thanks are extended to my colleagues, Dr. C.

Balakrishnan, M. C. Johnson, D. White, and F. J. Consoli,

for their help and suggestions. The help given by the

Chemical Engineering Shop personnel, particularly by Mr.

M. R. Jones and Mr. R. L. Baxley, in the design and fabrica-

tion of the experimental equipment, is greatly appreciated.

I am also grateful to the Chemical Engineering Depart-

ment for financial support and the National Science Founda-

tion (Grant GK-31590) for partial support of this research.

Finally, my wife,Mei,has provided the love and under-

standing that have made the completion of this work possible.

















TABLE OF CONTENTS


Page


ACKNOWLEDGMENTS . .

ABSTRACT . .

CHAPTER

I INTRODUCTION .

1.1 Preliminary Remarks .
1.2 Mechanisms of Drag Reduction .
1.3 Elongational Flows of Dilute Polymer
Solutions .
1.4 Vortex Inhibition Phenomenon .
1.5 Scope of Problem .

II BACKGROUND .


2.1 General . .
2.2 A Description of the Vortex Flow with an
Air Core Present .
2.3 Classical Treatment of Vortex Flows .
2.4 Previous Investigations on Real Vortices
2.5 Viscoelasticity Considerations .


III


EXPERIMENTAL .


3.1 General Description of Experiments .
3.2 Vortex Chamber and Flow System .
3.3 Optical Assembly .
3.4 Tracer Particles .
3.5 Experimental Fluids .
3.6 Procedures .
3.7 Velocity Data Analysis .
3.8 Concentric-Cylinder Viscometer for Very
Low Shear Rates .

IV EXPERIMENTAL RESULTS .

4.1 General Flow Pattern .
4.2 Physical Properties of Test Fluids .
4.3 Flow Field Measurements--Water Vortices
4.4 Influence of Newtonian Viscosity on Velo-
city Distributions .


iii


v .


. 1

. 1
S 2

S. 5
S 9
. 17


S. 43


43
44
46
49
49
52
54

63

68

68
83
99

107










CHAPTER Page

IV 4.5 Flow Field Measurements of Polymer Solu-
tions ... 114
4.6 Comparison of Experimental Results be-
tween Those of Elastic and Inelastic
Fluids at Equivalent Shear Viscosities .132
4.7 Conformational Studies .. .141

V THEORETICAL ANALYSIS ... .164

5.1 Fundamental Equations .. .165
5.2 Nonsteady-state Solution of Equations of
Motion (Following Dergarbedian) ..... 168
5.3 Steady-state Solutions for Viscous Vortex
Flows ................. 180
5.4 The Application of Lewellen's Asymptotic
Solution to our Laboratory Vortices 199
5.5 Theoretical Analysis for Viscoelastic
Fluids ... .. .. 215
5.6 A Proposed Model for the Vortex Inhibi-
tion Process .. 233

VI DISCUSSION OF RESULTS .. .249

6.1 Air Core Formation ......... .249
6.2 Significance of Secondary Flow in Bottom
Boundary Layer 253
6.3 Zero-shear Viscosities of the Polymer
Solutions ... 260
6.4 Theoretical Analysis on Vortex Flows with
Polymer Additives ... 262
6.5 Dependence of Drag Reduction and Vortex
Inhibition on Polymer Conformation .271
6.6 Conclusions and Recommendations ... .273

APPENDIX

A DERIVATION OF EQUATIONS (3-8) THROUGH (3-10) 277

B ORDER-OF-MAGNITUDE ANALYSIS OF VORTEX FLOWS .281

C PREDICTION OF MAXWELL MODEL--RECONSIDERATION
IN A TRANSIENT VORTEX FLOW .. .283

D STABILITY OF VORTEX FLOWS (NEWTONIAN) 286

BIBLIOGRAPHY . ... .. 292

BIOGRAPHICAL SKETCH . .. .298










Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy



THE VORTEX FLOW OF DILUTE POLYMER SOLUTIONS

By

Chii-Shyoung Chiou

December, 1976

Chairman: Ronald J. Gordon
Major Department: Chemical Engineering

When minute quantities of various high molecular

weight polymers are added to a solvent, the tendency of

the solvent to form an air core or vortex as it drains

from a large tank, is severely inhibited. This effect,

which is referred to as "vortex inhibition," correlates

extremely well with the drag-reducing ability of the

polymers. It would appear that the two phenomena are

closely related, and a thorough understanding of vortex

inhibition may shed light on the mechanism of drag reduc-

tion.

In this work, a steady three-dimensional vortex flow

was used as an approximation to the actual vortex inhibition

experiment. The fluid enters tangentially, spirals radially

inward and exits axially at the bottom of a tank. The

three-dimensional velocity distribution within the vortex

was determined using a photographic technique. It was

found that the shearing nature of the flow field is much

more important than its stretching nature which only exists

in the very near vicinity of the vortex axis.










Flow field measurements show that the presence of the

polymer causes a dramatic change in the originally steady

flow field, leading to a strong flow fluctuation near the

vortex axis. It is the flow fluctuation which breaks up

the concentrated vortex core and results in the vortex in-

hibition.

A theoretical analysis of the problem using the equa-

tions of motion and constitutive theories of dilute polymer

solutions in terms of experimentally supported kinematics

shows that the normal stress difference, (T 0-Trr)--the

tension along the lines of flow in the circumferential

direction--is of significance in suppressing the vortex

formation.

The roles of polymeric additives in the mechanism of

vortex inhibition and drag reduction were closely examined.

The results of this work offer considerable support to

theories that explain these phenomena as a consequence of

the viscoelastic behavior of polymer solutions in time-

varying shear fields.
















CHAPTER I

INTRODUCTION



1.1 Preliminary Remarks

Since the drag reduction phenomenon was first reported

by Toms in 1948, an intensive interest in this topic has

developed. Especially in recent years, extensive research

has been conducted by a large number of investigators.

Drag reduction is defined by Savins (1964) as the reduction

in pressure drop in turbulent pipe flow following addition

of various substances such as some high molecular weight

polymers, soaps (Agoston, 1954) or suspended particles

(Bobkowicz, 1965; Zandi, 1967). Drag reduction occurs in

turbulent flow and is, therefore, of great potential value

to industry, where most flows are turbulent. Promising ap-

plications of drag reduction are in long crude petroleum

pipelines, in brine disposal pipelines, in sewer disposal

systems, in fire fighting, and in reducing drag in the flow

of fluids past submerged vessels.

The recent development of drag reduction has provided

a fairly comprehensive picture of the drag reduction phe-

nomenon. Many important aspects of drag reduction have been

identified, such as its dependence on the average molecular

weight of the polymer, polymer concentration, and the wall










shear stress. In the last few years, the study of new

polymer-solvent pairs has either added new insight to our

knowledge or has been used in solving practical problems; the

development of new flow-visualization technique has aided

in understanding the influence of drag-reducing polymers

on turbulence in near-wall regions; the study of polymer

degradation has shown that some highly effective drag-

reducing polymers appear to break up with mechanical shear

and thus lose their effectiveness. However, in spite of

the considerable accomplishments, the actual mechanisms

which govern the magnitude and character of the drag reduc-

tion phenomenon are still unresolved.



1.2 Mechanisms of Drag Reduction

At present, there is evidence which suggests that the

reduction of drag on a solid surface is associated with

changes in the turbulence structure in regions very close

to the surface. For Newtonian fluids, detailed flow visual-

ization experiments have been made on turbulent boundary

layer structure (Corino and Brodkey, 1969; Kim, Kline, and

Reynolds, 1971). The results of these studies have established

a relationship between the production of turbulence and the

behavior of low-speed "streaks" in the region near the wall.

Alternating regions of high and low velocity developed near

the wall were very much elongated in their streamwise extent,

thus appearing "streaky" in structure. These low-speed streaks

were seen (i) slowly to lift away from the wall, (ii) often to










begin a growing oscillation and (iii) finally, to break up

into a more chaotic motion. The whole cycle is termed the

bursting phenomenon. From estimates made from the hydrogen-

bubble data, Kim et al. (1971) showed that essentially all

the turbulence production occurs during bursting. As a re-

sult, it is now generally believed that bursting is the

mechanism for generation of turbulence.

Offen and Kline (1973, 1974), Donohue, Tiederman,and

Reischman (1972) carried out flow-visualization experiments

in dilute solutions of drag-reducing polymer and found that

the spatially averaged bursting rate is greatly decreased

by the addition of polymer. This then suggests that tur-

bulence production is also decreased, consistent with the

lowered wall stress. To explain the mechanism of drag re-

duction, therefore, it is necessary to understand how elas-

ticity and non-Newtonian effect of polymer solutions can

alter the bursting process.

Currently, two viscoelastic mechanisms which are con-

sistent with the bursting studies have been proposed in the

literature to account for the drag reduction effect:

(i) Transient Shear Flow:

One theory that has been proposed to explain drag reduc-

tion involves the viscoelastic effect of polymers on transient

shear flows (Ruckenstein,1971; Hansen, 1972a). It has been

suggested that the propagation velocity for the transient

shear disturbances in very dilute polymer solutions is sub-

stantially less than that in the solvent. Since a certain










stage of the bursting phenomenon may involve a deformation

of this type, a large decrease in the propagation velocity

may result in less bursting and thus less turbulent drag.

Since the propagations of a transient shear disturbance and

a simple sinusoidal shear wave are related, by evaluating the

phase velocity and amplitude attenuation coefficient for shear

wave propagation as a function of frequency, the proposed

mechanism can be examined (Little et al., 1975). A 100

parts per million by weight solution of a commercial sample

(Polyox WSR 301) was used as an example. The results clearly

indicated that no significant differences between solvent and

solution behavior were predicted, even at relatively high con-

centration levels. It appears-that this mechanism makes an

insignificant contribution to the observed drag reduction.


(ii) Elongational Flow:

A recent phenomenological explanation of drag reduction,

advanced by Metzner and Seyer (1969) involves the unusually

high resistance offered by polymer solutions to pure stretching

motions. According to these authors, the turbulent eddy struc-

ture near the pipe wall may be considered roughly as such a

stretching motion.

For viscoelastic liquids, it has been shown analytically

(Astarita, 1967; Everage and Gordon, 1971; Denn and Marruci,

1971) and partially confirmed experimentally (Ballman, 1965;

Metzner and Metzner, 1970; Balakrishnan and Gordon, 1975)

that the resistance to stretching may be much greater than

that of Newtonian fluids. The general concept is that this










resistance to stretching interferes with the production of

bursting, thus decreasing the turbulent dissipation and sub-

sequently the wall shear stress. This proposed mechanism

seems to have a high probability for success, since the

effects of elongational flow are predicted to be very large

and with the extremely large effects it is easy to understand

how small quantities of polymer additive can produce large

changes in flow.



1.3 Elongational Flows of Dilute Polymer Solutions

Among the proposed theories, the elongational viscosity

has been considered as the most promising explanation for

drag reduction. However, the large values of elongational

viscosity for very dilute polymer solution cannot be demon-

strated experimentally. While various estimates of the pro-

perties of dilute polymer solutions in elongational flow have

been made, no direct measurements appear to be available in

the literature. So far, only three methods of generating

reasonable approximations to "pure" elongational flow ap-

peared suitable:


(i) Converging Flow:

This method is based on the observation that for some

viscoelastic fluids the flow through an abrupt contraction

is restricted to a narrow conical region upstream. Metzner

and co-workers (1970, 1969) have described the kinematics of

such a flow utilizing an approximated velocity field and shown

that the deformation rate tensor is diagonal. The material










is thus expected to be subjected to simple Lagrangian un-

steady, extensional deformation. It is to be noted that

not all polymer solutions show this behavior. In many cases,

the flow enters the contraction from all directions upstream

just like a Newtonian fluid does. The deformation rate ten-

sor is no longer diagonal and the analysis of Metzner and co-

workers based on the assumption of simple elongational flow

is not applicable. It is not possible to obtain meaningful

elongational flow data when the flow does not exhibit the

conical pattern.


(ii) Fano Flow or Tubeless Siphon:

This experiment utilizes the ability of some visco-

elastic fluids to be drawn out of a reservoir into a tube

even when the liquid level in the reservoir falls below the

end of the tube. (The other tube end is connected to a

vacuum pump.) The deformation rate tensor for this flow has

also been shown to be diagonal and relevant stress-strain

rate information can be caclulated from the measurement of

the force exerted by the fluid column on the tube, the volu-

metric flow rate and the diameter-distance relationship.

Kanel (1972) showed, however, that no column can be obtained

with aqueous solutions having polymer (Separan AP 30 or

Polyox WSR 301) concentrations of the order of 100 ppm even

with tube diameters as small as 0.5 mm. By using a New-

tonian solvent of high viscosity (e.g., 50-50 glycerol-water,

viscosity -6 C.P.), columns about 1 mm in diameter can only

be realized for the same range of polymer concentrations.










Furthermore, the unavoidable errors in force measurement

limited the application of this technique in obtaining

accurate elongational data.


(iii) Fiber Spinning Flow:

In this method, the viscoelastic fluid is forced through

a die or a nozzle and the thread collected on a rotating

wheel downstream. The stress-strain rate information can

be obtained from the force measurements made at the wind-up

device, the volumetric flow rate and the diameter-distance

relationship obtained from photographs. This method has

been extensively used for elongational flow studies of polymer

melts (Spearot, 1972; Ziabicki and Kedzierska, 1960) and

concentrated solutions of polymers (Weinberger, 1970; Zidan,

1969). For dilute polymer solutions in water, it was found

that the solutions lacked tackiness and did not attach to the

rotating wheel (Baid, 1973). This situation has to be im-

proved by increasing the Newtonian solvent viscosity. It was

found that satisfactory data could be obtained by using a 95%

glycerol-5% water solvent for Separan AP 30 solutions having

concentrations of about 100 ppm (Baid, 1973) or using a 1%

solution of the same polymer in equal parts by weight of water

and glycerol (Moore and Pearson, 1975).

Based on the above discussions, it appears that no direct

measurement of elongational flow properties of certair dilute

polymer solutions are available, and the correlation between

the drag-reducing ability and elongational viscosity has,

therefore, never been fully established. For example, in the










case of Polyox WSR 301 with concentration varying from a

few ppm to about a thousand ppm, a significant drag reduc-

tion still can be detected while the elongational viscosity

of the solutions is still not measurable.

Recently, in the converging flow experiments, Bala-

krishnan (1976) found that Polyox solutions were unable to

show the wine-glass-stem shape flow so that no elongational

viscosity could be measured. Besides, no elongational in-

formation has ever been obtained on these effective, dilute

drag-reducing polymer solutions by the Fano flow or fiber

spinning techniques without highly increasing the Newtonian

solvent viscosity. It is therefore still not clear if the

actual mechanism would involve elongational flow.

An alternative approach to determine the mechanism of

drag reduction might be to investigate a new viscoelastic

effect recently discovered by Gordon and Balakrishnan (1972).

This effect, referred to as "vortex inhibition" shows that

the tendency of water (as solvent) to form a vortex, as it

drains from a large tank, is inhibited by the presence of

only minute quantities of various high molecular weight poly-

mers. This effect correlates extremely well with drag-

reducing ability. In other words, those polymers which show

vortex inhibition at the lowest concentration are also the

best drag-reducers. Both vortex inhibition and drag reduc-

tion are highly sensitive to polymer degradation, and both

also vary in the same fashion with fundamental change in

polymer conformation in solution (Balakrishnan and Gordon,









1975, Gordon and Balakrishnan, 1972). Studies of vortex

inhibition are therefore of interest both in their own right,

and also because of the possibility that may yield information

on the mechanism of drag reduction. It is the objective of

this study to characterize the velocity field through flow

visualization experiments and seek an explanation for the ef-

fect of vortex inhibition. These results should indirectly

contribute to our knowledge of drag reduction. Once the pro-

perty of dilute polymer solutions which is responsible for

vortex inhibition is identified, its dependence on macro-

molecular variables such as molecular weight, molecular weight

distribution, or molecular conformation would become immedi-

ately apparent. The effect of vortex inhibition may then be

useful, in conjunction with intrinsic viscosity measurement,

in providing both mean (viscosity-mean) molecular weight and

breadth of the molecular weight distribution. Since many use-

ful physical properties, as for example, the coating proper-

ties of acrylic paint (Billmeyer, 1971), depend on the "high

molecular weight tail" of the polymer, a simple means of

characterizing this "tail" would have immense significance.



1.4 Vortex Inhibition Phenomenon

A Plexiglas tank was filled with water and stirred

vigorously with a paddle in order to introduce some initial

circulation into the field. The bottom plug was then re-

moved, and after a few moments, an air core formed (Figure

1-1), extending from the free surface of the liquid to the

small drain hole at the bottom of the tank.














Vortex
Surface of Liquid


Figure 1-1


Illustration of vortex formation in water.


Vortex


Figure 1-2


Surface of Liquid


Illustration of vortex inhibition with polymer
solution.


1 1






i










The narrow air core, which was essentially cylindrical

over the greater part of its length, was extremely stable,

and once formed, it remained intact until the tank had drained.

The same experiment was then repeated with a dilute drag-

reducing polymer solution. This time, as long as the polymer

concentration was above some minimum value (denoted by CVI),

the vortex could not complete (Figure 1-2). The completeness

of the vortex was accompanied by a "slurping" sound as air

was drawn out of the drain hole. CVI was defined as the

lowest concentration at which the vortex would not complete.

Perhaps surprising, the results of this somewhat arbitrary

procedure were remarkably reproducible. The most striking

aspect of the vortex inhibition effect is the low concentra-

tion required and its extremely close correlation with the

drag-reducing polymers. Some important findings on the

effect of vortex inhibition that have been presented in the

earlier publications (Gordon and Balakrishnan, 1972, 1975b)

are:


(i) Correlation of Drag Reduction and Vortex Inhibition:

According to these authors, the drag-reducing ability

of various polymers was simply determined by measuring the

efflux time for the liquid level to drop between two

specific levels. If the efflux times for water and polymer

solution are denoted by tw and t the percentage of drag

reduction is defined as


% DR = 100(t -t )/t
w p w










The results of all the polymers used are summarized in

Table 1-1. As already mentioned, CVI represents the minimum

concentration for vortex inhibition, while CDR is the con-

centration at which maximum drag reduction was observed.

Inspection of Table 1-1 reveals that CVI is an excellent

ordering parameter for the drag-reducing effectiveness of

the different polymers. These results strongly suggest

that vortex inhibition may be used to predict a priori

whether or not a given polymer sample will be an effective

drag reducer. To the author's knowledge, no other rheo-

logical test presently available is capable of characterizing

drag-reducing ability for the polymer concentrations of

interest, varying from a few ppm on upward.


(ii) Influence of Polymer Degradation on Vortex Inhibition:

Theory and experiment both suggest that in dilute solu-

tion, mechanical degradation affects the higher molecular

weight components more than the lower molecular weight com-

ponents (Casale, et al., 1971). Paterson and Abernathy (1970)

found that the drag reduction depended predominantly on the

high molecular weight "tail." The results of the influence

of different degrees of mechanical degradation on CVI are

given in Table 1-2. This table illustrates that CVI is pri-

marily dependent on the high molecular weight species of a

given polymer, just as drag reduction is expected to be.


(iii) Influence of Molecular Conformation on Vortex Inhibition:

In contrast to an uncharged, random coiling polymer, a

polyelectrolyte is always composed of a macroion in which the













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charged groups are interconnected by chemical bonds, together

with an equivalent number of small oppositely charged counter-

ions. Thus polyelectrolytes undergo dramatic changes in con-

formation with change in their ionic environment. Polyelec-

trolytes are known to be highly coiled in acidic environment,

perhaps due to intramolecular hydrogen bonding, and has ex-

tended conformation at high pH due to ionization (Hand and

Williams, 1970; Mathieson and MacLaren, 1965). This tendency

of polyelectrolytes to change in conformation with change in

their ionic environment makes them suitable for studying

the effect of polymer conformation.

Parker and Hedley (1972) measured the drag-reducing

ability of an aqueous solution-of poly(acrylic acid) (PAA)

at different pH levels. They found that PAA exhibited re-

markable drag reduction at a pH of 6 to 8, while a lowering

of the pH to 2.2 greatly reduced the drag reduction. These

findings show the important effect of pH in controlling the

molecular extension and thus the drag reduction. Similar

results were also reported by Kim, Little, and Ting (1973)

and White and Gordon (1975). In order to see if the in-

fluence of molecular conformation would affect vortex in-

hibition, CVI has been measured for PAA (Versicol S25, Allied

Colloids) and PAM (polyacrylamide, Separan AP 273, Dow Chemical)

in different pH values. The results are listed in Table 1-3

where it is evident that the polymer conformation has strong

influence on the effect of vortex inhibition. This table

further indicates that CVI correlates closely with the drag-










Table 1-3

Vortex Inhibition Concentration C versus pH
for Polyacrylamide and Poly(acry ic acid)


pH CVI pH CVI



PAM 2.1 40 PAA 3.9 200


3.75 5 5.6 10


5 3 7.1 3


7 3 9.7 3


12 1.5 11.7 5






Source: C. Balakrishnan and R. J. Gordon, J. Appl. Poly.
Sci., 19, 909 (1975).










reducing ability of the polymers under the influence of

molecular conformation variation.



1.5 Scope of Problem

Based on the preceding discussion, it is strongly

suggested that vortex inhibition and drag reduction are due

to the same viscoelastic mechanism. The purposes of the

work are:


(i) To measure the velocity profiles experimentally for

both the solvent and the dilute polymer solutions in vortex

flows, especially in the core region where it is expected

to change significantly when polymer additive is present.


(ii) To provide a theoretical analysis of this problem,

using the momentum equations and realistic constitutive

assumptions, from which the velocity profiles may be predicted.
















CHAPTER II

BACKGROUND



2.1 General

This chapter will survey pertinent previous work on

vortex flows. Before discussing the published literature

in detail, two explanatory sections are presented to ac-

quaint the readers with some of the concepts and terminology

to be used in the main body of this thesis. Section 2.2

gives a brief description of the specific vortex flow en-

countered in the vortex inhibition experiment. Analytical

treatments of the classical vortex motion, in which various

definitions used to characterize rotating flows are intro-

duced, are discussed in Section 2.3. In Section 2.4,

published literature dealing with vortex flows of Newtonian

fluids is reviewed. Here mathematical treatment of the

problem to predict the velocity distribution within the vor-

tex is presented and the validity of each of the simplifying

assumptions is examined by comparison to available experi-

mental data. Although Newtonian vortices (in most cases, a

water vortex) have been studied extensively, both experi-

mentally and theoretically, there is very little published

information available on the behavior of non-Newtonian or

viscoelastic vortices, which is the point of emphasis of

this study. In Section 2.5, a description of some known










viscoelastic effects of dilute polymn- ...u ltions is pre-

sented which serves as a background for the present study

of the vortex flows of viscoelastic fluids.



2.2 A Description of the Vortex Flow with an Air Core Present

When a circular motion is induced in water within a

tank, either by tangentially directed jets or simply stirring

with a paddle, and shortly after a central exit hole has

been opened in the bottom of the tank, the resulting cen-

trifugal force field tends to form a depression in the water

surface. When the centrifugal force field is large enough,

the surface depression may reach the exit hole and thereby

form a hollow space in the core of the rotating fluid. In

general, these "air cores" are approximately of constant

diameter over the greater part of their length, and have

smooth, glassy surfaces as illustrated in Figure 2-1.

With smaller centrifugal force fields, the air core

will not complete, and only a small surface depression forms

(see Figure 2-2). In this case, it was observed that the

tip of the surface depression or "dimple" oscillated rather

rapidly and occasionally air bubbles appeared when the tip

broke. The air bubbles would fluctuate below the dimple for

a considerable time and then either rise to the free surface

or be discharged at the exit hole.



2.3 Classical Treatment of Vortex Flows

Before discussing the vortex motion in detail, some

quantities useful in describing vortex kinematics are intro-























































Figure 2-1


A typical air core in vortex flow.



















































Figure 2-2


A typical surface dimple in vortex flow.










duced: The circulation, which is customarily given the

symbol r, is defined as the line integral of the tangential

velocity component around a closed curve; thus


r = v*dr = v.d c (2-1)
c


where v is a velocity vector, c is some closed path, dc

an element of this path, and r a radius vector from an

arbitrarily located origin to a point on the path (Figure

2-3).

The circulation is related to the vorticity by Kelvin's

equation (Eqn. 2-2). The vorticity is defined as the curl

of the velocity vector: C = V x v. Consider a surface S

bounded by the contour C; if n is the component of vorticity

normal to the surface at any point, then from Kelvin's theorem



r = {f IC x dS| = f f r dS (2-2)



This relation follows from Stokes' theorem (Lamb, 1945) and

demonstrates that the circulation about any closed curve is

equal to the surface integral of the normal component of

the vorticity over any surface which is bounded by the given

curve.

While stream lines generally exist throughout all por-

tions of a fluid in motion, a single vortex line may exist

in an otherwise irrotational flow, so that only those infini-

tesimal fluid elements lying directly upon that line will

undergo "rotational" motion. If the vortex motion is viewed





















































Figure 2-3 Circulation and vorticity.










parallel to the vortex line on the cross-sectional plane,

it appears as a two-dimensional motion with the vortex line

appearing as a point. Given polar coordinates r and 0, with

r the radial distance from the axis of rotation, let v be

the circumferential velocity. Since the vortex proper is

considered as localized at a point and the motion is assumed

irrotational except at that point, one obtains


1 a
Sr (rv) = 0 (2-3)
r ar


and by integration of this equation


rv = constant (2-4)


The tangential velocity thus varies inversely with

the radius. This is the velocity relation for the so-called

potential or free vortex, since the motion is irrotational

except at the origin, where the vortex proper is located.

Another extreme case termed the forced vortex is formed

by assuming that the region occupied by the vortex is of

finite size with a uniform density of vorticity across it.

The cross section of the vortex tube is taken to be a circle.

The vorticity is now taken as constant rather than zero; thus,


1 3
r r -(rv) = constant (2-5)
r 2r


or integrating gives


2
gr
S= rv + C (2-6)
2


1










Since in such flows the tangential velocity is required to

go to zero on the axis of symmetry, one has C=0, i.e.,



v = r (2-7)


This indicates that the velocity now varies linearly with

the radius, as in solid (rigid) body rotation. This profile

is known as the forced vortex, since to produce it, a torque

must be applied to the fluid and different amounts of work

are done on different streamlines (i.e., at different radii).

Although both of the vortex motions (free and forced)

have a certain degree of artificiality, motions approaching

them are often encountered. A more logical and physically

rational vortex motion was introduced by W. J. M. Rankine

in 1858. This is the "combined vortex" in which the forced

vortex is considered limited to a core region outside of

which is a free vortex. This simple joining of the two

previous extreme cases of free and forced vortices is shown

in Figure 2-4.

The Rankine combined vortex is a closer approach to

real-fluid motion in that the forced vortex region--which

must have resulted from some viscous action--is assumed to

be of finite size. For the real vortex flows, however,

there are still some small but significant departures from

the ideal forms of these classical vortices. A literature

survey of the well represented experimental and theoretical

work on vortex flows of real fluids (water in most cases)

will be given in the next section.


1






























Vortex Core Diameter







S1i (Free Vortex)
V r
/ r


v = C2r (Forced Vortex)


Figure 2-4 Characteristics of Rankine's combined vortex.










2.4 Previous Investigations on Real Vortices

In addition to the theoretical analysis of the motion

of a vortex in an inviscid fluid, laboratory investigations

have been undertaken to study small but significant depar-

tures from the ideal forms. There have been a number of

experiments designed to generate concentrated vortices in

rotating tanks. The original aim of these works was to

obtain a laboratory model closely related to the meteoro-

logical vortices, tornadoes, dust whirls and water spouts.

Long (1958, 1961) has made a detailed investigation of

vortices produced by extracting fluid through a sink

situated just below the free surface of a rotating tank of

water. A very different mechanism for driving vortices in

rotating tanks has been described by Turner and Lilly (1963)

and used later by Turner (1966) to obtain quantitative

information about these vortices. The vortices of Turner

and Lilly are driven by drag force:exerted on the surrounding

fluid due to gas bubbles released near the axis of the tank.

These gas bubbles are caused by either nucleating carbonated

water or steadily injecting air through a fine tube. By

using a photographic method for velocity profile measurements

in these vortices, Turner found that the tangential and radial

velocities are independent of height, and the axial velocity

varies linearly with height. As will be seen later in Chap-

ter IV, these features of the velocity profiles are quite

similar to those observed in the present experiments.










Another way of generating a vortex is the well-known

"bathtub drain" vortex, in which initial circular motion

may be introduced by either stirring with a paddle or by

tangential injection of entering fluid at the near wall

region. Early studies on this type of vortex were made by

Binnie and coworkers (1948, 1955, 1957), Einstein and Li

(1951), Quick (1961), and Helmert (1963). All experimental

measurements made in these works have shown that the varia-

tion of tangential velocity is inversely proportional to

the radius. This will be true for a major region of the

vortex, but close to the center, the effect of fluid vis-

cosity cannot be neglected and the tangential velocity is

required to go to zero on the axis of symmetry.

Kelsall (1952) was probably the first to measure the

detailed velocity profiles close to the axis of a hydraulic

cyclone. He adopted a very successful optical method in-

volving suitable ultramicroscopic illumination and a micro-

scope fitted with rotating objectives. Kelsall found that

the tangential velocity appears to vary linearly with the

radius from the axis, even if an air core is present. If

an air core rotates as a solid body, within a large core

of liquid that also rotates as a solid body, then the

effects of an air core on the surrounding flow should be

negligible, except for axial effects. Axial effects may

be important if the presence of an air core serves as a

restriction within the exit hole, hence a resistance to

flow.










Roschke (1966), using a graphic differentiation of

the static pressure distributions measured on the closed-

end wall, calculated the tangential velocity profiles in

a jet-driven circular cylinder. These indirectly ob-

tained tangential velocity profiles were presented to give

some indication of the effect of exit-hole size and aspect

ratio (ratio of the length to diameter of the vortex cham-

ber). It was also noted by Rosche that use of probes in

vortex flows of this type should be avoided because even

very small probes have been found to produce significant

changes in the original vortex flow. Some of the problems

which arise in probing vortex flows have been discussed

elsewhere (Holman, 1961; Eckert and Hartnett, 1955).

The studies of strong jet-driven vortices were ex-

tended to the case of weak vortices in which the air core

was narrow or only a small surface depression occurred

(Anwar, 1966). In his experiments,Anwar found that there

was no downward movement of the air bubbles that appeared

when the tip of the surface depression or "dimple" broke

occasionally. This caused him to conclude that the axial

velocity was zero at the axis of symmetry. Following the

downward movements of drops of dye placed on the free sur-

face, Anwar estimated the position of maximum of the axial

velocity profile to be at a distance between 0.75 and 0.77

of the outlet radius. This finding contradicts the results

reported elsewhere (Granger, 1966; Dergarabedian, 1960) and

will be examined more carefully in this work.










The above experimental investigations on the vortex

flows are summarized in Table 2-1.

Perhaps the most striking single impression on pur-

suing these references is the large number of complicated

"secondary flow" patterns observed for a wide variety of

experimental conditions and apparatus. An example of such

secondary flows occurring in a simple bathtub drain vortex

is shown in Figure 2-5. As we know, the fluid elements

near the bottom move more slowly than those within the main

body of the vortex because of friction. The slower-moving

fluid in the boundary layer is then forced inward by the

radial pressure gradient established by the more rapidly

moving fluid in the main flow outside the boundary layer.

In Figure 2-5, it seems that most of the flow out of the

drain hole comes from the boundary layer and the fluid

above does indeed travel in circles.

The boundary layers that are formed on the end walls

of the vortex chamber have been studied previously without

regard to their effect on the primary or main vortex flow

that derived them (Taylor, 1950; Mack, 1962; Rott, 1962).

Rosenzweig, Lewellen and Ross (1964) investigated this

problem in further detail and attempted to compare theo-

retical results and experiments by matching the measured

circulation distribution with the analytical predictions.

In all cases, it was found that there was still large dis-

crepancy between experiments and theory: the measured

laminar Reynolds number was much higher than its theoret-














^Ii














(a)




















(b)
Figure 2-5 Flow visualization with dye injection.
(a) Dye marks the radially inward secondary flow near
the bottom of the vortex tank; (b) Dye marks the circular
streamlines of the primary flow high above the bottom.








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T,-I L,-i I ) L
U O Lo o o L) m w 4J
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U) IH
wj U) 1 1 .
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4-) H-0 U)
(n 0-i (0 0 Q) 0' *-4 ri
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d 0 H 0 0* l





H > m 9 0 p H ( o )
0 cr u3 q, L (rE









ical counterpart. According to these authors, the largest

discrepancy is probably due to the ambiguity concerning

the exhaust hole. Near the exhaust region, it was observed

that only a fraction of the boundary-layer flow passes

directly out of the exit hole while the remainder is ejected

axially to form the upward reverse flow (Rosenzweig, Ross

and Lewellen, 1962; Roschke, 1966; Kendall, 1962). As will

be seen later, the presence of boundary layers makes the

theoretical analysis of the vortex motion rather complex.

Recently, Dagget and Keulegan (1974) reported the ef-

fects of fluid properties on the vortex nature. The results

show that the surface tension of the fluid does not appear

to affect the vortex flow significantly, and fluid viscosity,

on the other hand, plays a very important role in vortex

flows. It was found that as viscosity increases, the cir-

culation decreases from inlet to outlet due to the increase

in viscous shear. Consequently, for the same initial circula-

tion, a vortex with working fluid of high viscosity forms no

air core, whereas the prototype using a low viscosity fluid

would form an air core. Anwar (1967) studied the effects of

fluid viscosity in more detail. The experimental results

showed that the relationship between tangential velocity, v,

and radius, r, appears to be v rn = constant, with the power,

n, increasing with increase in viscosity. It is worth noting

that the relationship, v r = constant transfers to V~r as r

becomes small, where a solid body rotation occurs.










Although d'Alembert and Euler in 1750 considered

vortex kinematics to some extent, it may be properly stated

that analytical treatment of vortex motion started with

Helmholtz' classical paper (1850) "On Integrals of the

Hydrodynamic Equation Corresponding to Vortex Motions."

The proofs of some of Helmholtz' theorems were given later

by Kelvin (1867). And since the 1951 paper of Einstein and

Li, steady progress has been made in the analytical descrip-

tion of laminar, incompressible vortex flows.

In an approximate treatment of vortices, originated

by Einstein and Li (1951), generalized later by Rott (1958),

and Deissler and Perlmutter (1958), the axial velocity, w,

is arbitrarily taken as a discontinuous function of the

radius and has a jump at the radius of the exhaust. That

is


w = 0 for r > r
e
w = a*z for r > r > 0 (2-8)
e -


where a is constant. From the total flow rate, QT' one gets


QT
a = (2-9)
7Tr h
e


where h is the depth of the flow, and r is the radius of

exhaust. The assumption is made that the axial flow out of

the orifice is uniform. Continuity is next used to deter-

mine a radial velocity which is independent of the axial

coordinate. The tangential velocity can then be determined










directly from the tangential momentum equation by simple

quadrature and is shown to be independent of the axial co-

ordinate. The calculated tangential velocity is of the

form



v ~ l eA 2 (2-10)

where

a T
A (2-11)
4Tirhvr
e


According to this solution, a definite magnitude of

the total volume flow, Q T is associated with a given tan-

gential velocity distribution. Experiments, however, show

that the measured velocity distribution corresponds to a

much lower value of A than the one computed from Eqn. (2-11).

This approach ignores the presence of the boundary layers on

the end walls of the chamber. A large radial flow occurring

in the boundary layers on the end walls offers a ready ex-

planation of the reason why the measured tangential velocity

distribution corresponds to a total volume flow much lower

than the one computed under the assumption that the entire

volume flow is distributed uniformly along the length of the

vortex chamber.

The volume flow which is available for the creation of

a vortex-like tangential velocity distribution is not the

total volume flow, rather it is the total volume flow minus

the secondary flow in the end-wall boundary layer. There-









fore, the assumed axial velocity (Eqn. 2-8) is not true and

a more realistic axial velocity distribution, obtained either

experimentally or theoretically, is definitely necessary in

predicting the tangential velocity profile.

Donaldson and Sullivan (1960) have shown that the most

general condition under which a solution of the Navier-Stokes

equations such that v = v(r) exists, is that the axial

velocity must be of the form


w = z f (r) + f2(r) (2-12)


Here functions of fl and f2 can be determined from the axial

momentum equation. As it turns out, the solutions of Ein-

stein and Li, etc., are special cases of this solution.

Unfortunately, the flow, so determined, cannot be made

to satisfy the boundary conditions corresponding to the

desired geometry of the actual vortex tank (with stationary

end walls and exhaust-hole geometry). Rather, the class of

physical flows which Donaldson and Sullivan investigated by

means of Eqn. (2-10) are the flows produced within a

rotating porous cylinder by uniformly sucking fluid out of

such a cylinder (Figure 2-6). The flow in Figure 2-6 is

bounded at radius r=ro by the rotating porous cylinder where

the axial component of velocity vanishes at the cylinder

surface. For the cases of jet-driven vortex flows, the axial

velocity vanishes at some distance between the axis and the

confined wall of the cylinder and hence the boundary condi-

tion for Eqn. (2-10) is not as straightforward as it might


































Rotating Porous
Cylinder


Sketch of geometry for a rotating porous cylinder.


Figure 2-6










first appear. Moreover, it can be seen that the axial
2
and radial momentum equations and Eqn. (2-10) lead to = 0.

It is thus impossible for a solution of this type to satisfy

any problem in which the axial boundary condition forces a

radial variation in the axial pressure gradient, which does

occur in most vortices.

Long (1961), using a dimensional analysis, gave another

mode of axial velocity which satisfies the Navier-Stokes

equations and certain radial boundary conditions. His mode

of the axial velocity distribution is of the form



w 1 f( ) (2-11)
r z


This equation is substituted into the Navier-Stokes equa-

tions and after some direct integration and rearrangement,

the original partial differential equations become ordinary

differential equations with only one independent variable,

r. Long actually solved the equation by using a numerical

method and showed that the behavior of the velocity compo-

nent at large distances from the axis of rotation is



r


V
v + -
r



WP
vr









where v is kinematic viscosity and F is the circulation

at a large enough distance from the axis. Clearly, it is

seen that a particular feature of Long's solution is that

the axial velocities are of the same order as tangential

velocities even at large radii. Long's flow problem can

perhaps be best described as a swirling jet exhausting into

an unbounded fluid which has constant circulation. Actually,

in the limit of zero circulation it reduces the Schlichting's

jet problem (Schlichting, 1968). The case of interest here,

that of radial sink flow with strong circulation and exhausts

axially near the center line, is thus not included in this

class of solutions.

In contrast to the above approaches, Lewellen's (1962,

1964) asymptotic expansion method provides a general

solution capable of satisfying the boundary conditions in

real flows. His method can, in principle, take into account

any variation of the axial pressure gradient which is over-

looked in Donaldson and Sullivan's solution. Lewellen's

analysis is in terms of a series expansion of the nondimen-

sional circulation function, F, and the nondimensional stream

QH
function, T, in the parameter e = (-Hn); here Q is a volume
F R
inflow per unit axial length, F is a constant circulation,

and H and R are characteristic axial and radial lengths re-

spectively. Physically, this parameter is the ratio of rela-

tive inertial to Coriolis force in a rotating fluid and is

known as the Rossby number which has been used to characterize

the effects of rotation on flows. Lewellen's argument is










that the parameter e is small in many flows of interest

where the radial velocity is small compared with the cir-

cumferential velocity. This method is interesting in that

it allows one to calculate also the higher-order correction

terms, and assess the accuracy of zeroth-order approxima-

tion (from equating the coefficient of se).

Although this expansion method is considered to be the

most relevant work on this subject, the accuracy of the solu-

tion is still principally limited by the knowledge of the

axial boundary conditions on the stream function. This is

caused by the fact that this method is to solve a boundary-

value problem for the stream function, and the answer to what

boundary conditions are physically imposed in any particular

problem is not explicit. Let us consider the jet-driven

vortex for instance. It is seen that the stream function can

be readily specified only if the viscous boundary layer on

the end wall(s) is ignored. However, the assumption of

neglecting the boundary layer(s) on the end wall(s) of the

container is very poor. This is exactly the regime in which

the so-called secondary flow induced by the boundary layer

can dramatically change the boundary conditions and hence

produce a major effect on the outside or main flow. This

boundary-layer interaction problem has been studied in greater

detail by Rosensweig, Lewellen and Ross (1964) and by Rott and

Lewellen (1965). However, the lack of agreement between ex-

perimental results and theory shows that the boundary condi-

tions must be determined experimentally. If the true stream










functions occurring at the boundaries were known, then an

accurate distribution for the circulation in the container

could be obtained by using Lewellen's method. The inherent

difficulties presented by the boundary layers would probably

be the most difficult points in the study of the flow in a

vortex tank.



2.5 Viscoelasticity Considerations

High molecular weight polymers dissolved in solutions

may exhibit some viscoelastic effects which cannot be found

in Newtonian fluids. These solutions differ from purely

viscous fluids in that they display elastic properties such

as recoil and stress relaxation. Such responses are related

to the ability of the macromolecular chains to assume dif-

ferent spatial configurations under deformation. As men-

tioned earlier, many viscoelastic effects of dilute polymer

solutions have been discovered and examined in detail, but

few workers have investigated the viscoelastic effects on

vortex motion. A summary of some characteristic viscoelastic

phenomena in particular flow fields is given here to serve

as background material which will be of help in understanding

the influence of viscoelastic effects on vortex inhibition.


(i) Turbulent Drag Reduction:

This effect has been discussed in detail in Chapter I.

A close correlation between drag reduction and vortex in-

hibition strongly suggests that both of these effects must be

due to similar, if not the same,viscoelastic mechanism.










(ii) Elongational Viscosity:

Viscoelastic theories predict that, under tension, a

viscoelastic fluid exhibits a very large elongational vis-

cosity which increases with stretch rate (Lodge, 1964; Dean

and Marucci, 1971; Everage and Gordon, 1971). This is in

contrast to the low shear viscosity which is a decreasing

function of shear rate. The experimental evidence for this

type of behavior is quite limited, but studies such as

Metzner and Metzner (1970), Balakrishnan and Gordon (1975a)

seem to indicate that large values of elongational viscosity

(two to three orders of magnitude larger than that for New-

tonian fluids) can be obtained at low concentration.


(iii) Recoil:

If the shearing stresses imposed upon a flowing visco-

elastic fluid are rapidly removed, the fluid will undergo a

partial recovery of strain. This has been observed exper-

imentally by following the movement of suspended air bubbles

in solutions (Lodge, 1964) or the deflection of an uncon-

strained rotor in a cone-and-plate system (Benhow and Howells,

1961). In a recent note, Balakrishnan and Gordon (1975a)

reported that recoil occurred, following sudden cessation in

flows through an orifice, even at concentration levels as low

as 10 ppm. It appears that the highly dilute polymer solu-

tion may still possess a "partial memory" of its rest state.
















CHAPTER III

EXPERIMENTAL



3.1 General Description of Experiments

Although the original vortex inhibition experiment

is easy to operate, it is practically impossible to ob-

tain any detailed knowledge of the velocity field kinematics

in such an experiment. The reasons are: (i) Since the

liquid in the tank is stirred arbitrarily with a paddle in

order to introduce some initial circulation motion, the

amount of vorticity introduced is hard to be controlled or

estimated accurately, and (ii) The system is unsteady, and

no measurements can be made during the short period in which

the vortex develops.

In this work, therefore, a steady, continuous, jet-

driven vortex will be used -as an approximation to the actual

vortex inhibition experiment. The vortex chamber comprises

a transparent cylindrical tank and the vortex flow is driven

by tangential injection near the cylindrical wall of the

chamber. Tracer particles were inserted into the fluids and

the primary velocity data were obtained from a series of

streak photographs of the steady vortex flow in the chamber.

The flow field was illuminated by an intense sheet of light.

Some such sheets of light were parallel to and included the










axis of the tank. Some other sheets of light were per-

pendicular to the axis. The entire vortex chamber was

mounted on a movable support so that it might move ver-

tically to be illuminated at any desired horizontal cross

section. A description of the photographic technique

along with the necessary optical equipment is given. The

physical properties of the test solutions were determined

using conventional or specialized techniques described

below.



3.2 Vortex Chamber and Flow System

A schematic diagram of the flow system including the

vortex tank along with its principal dimensions is shown

in Figure 3-1. The tank was 66 cm high and 45 cm in dia-

meter. The liquid level was maintained at 54.5 cm relative

to the tank bottom by manually controlling the rate of

fluid entering. The fluid circulating within the tank en-

tered through two vertical pipes placed opposite each other

on the circumference. The walls of both pipes were perfo-

rated at equal spaces, then welded with small injection tubes

so as to distribute the entering fluid uniformly in a ver-

tical direction. The fluid thus entered tangentially through

an exit hole centrally located in the bottom, and discharged

into air. For moderate circulation, injection tubes of 1/4

inch bore were used, and were replaced by tubes of 1/16 inch

bore when high circulations were required. The two vertical

pipes were fed from a 50-gallon head tank through two flex-

ible tubes. The feed rate was controlled by a gate valve.































Liquid Supply Pipe





Jet Orifice
















Randolph Pump


Gate Valve


Vortex chamber and flow system.


Head Tank


Tubing


Figure 3-1










The bottom of the vortex chamber was provided with a

central opening to which various sizes of orifices could be

fitted. Vortices with either an open air core or a small

dimple could be obtained by varying the size of orifice.

The circulated fluid was collected in a collecting tank,

pumped through the external closed circuit, and finally to

the head tank back into the supply tubings. The fluid was

originally pumped with a Randolph pump which is supposed

to impart gentle shearing action on the recirculating fluid.

Preliminary tests showed, however, that even this gentle

shearing action could cause polymer degradation after several

runs. It was found convenient to shear the solutions until

a stable state was reached which,however, in the present

study caused the solutions to become too hazy for good photo-

graphs to be obtained. It is, therefore, decided only the

fresh polymer solutions were used and would not be pumped

back to the head tank for repeated use. On the other hand,

for tap water and other Newtonian solutions we still operated

with the recirculation loop.



3.3 Optical Assembly

The optical assembly used for obtaining the streak photo-

graphs is illustrated schematically in Figure 3-2.

The flow field was illuminated in thin cross section by

an intense sheet of light. The light beam was interrupted

by a rotating pie-shaped disk, from which alternate pieces

of pies were cut. This was simply a mechanical strobing






47





























S0



LC-)



U
U)H




on y
CJ
Un
+,J







0
cr U)


O D E5
U)n z
oad m u < az
_J 0 -)



-< m m -- J 0-
SWW C C) 'Z X5

--mo m -j <[L un o Lw m -r


4 r_ J J..: 2cr

-J


UV










apparatus. The strobe disk was driven by a 1/4 HP, 1725

r.p.m. motor. The period of the successive illumination

intervals was measured with an ll91-Type, electric timer/

counter made by General Radio Company. The periods can
-12
be measured from 100 ns to 1 Gs. (10 12s) with a precision

of up to 1 fs (10 15s). Various illumination intervals

were obtained by changing the number of slits on the strobe

disk. In this experiment, all runs were measured with

either a one-slit of a two-slit disk.

All the photographs were taken with a Bell & Howell,

f-1.8 camera. To obtain closer photographs, three close-

up attachements were used with the camera. All photographs

were obtained using Kodak Tri-X films of ASA 400 rating.

The films were developed for 11 seconds at 650F in order to

increase the degree of contrast. Exposure times and depths

of fields were determined by trial and error. All the photo-

graphs were taken in complete darkness except for the light

plane. The entire system was covered with black plastic

cloth to stop all stray light.

The remainder of the optical assembly included a light

source, two condensing lenses, a heat-absorbing glass and a

mechanical slit. It turned out that getting a sufficient

amount of light to pass through the large vortex chamber of

liquid was one of the most difficult problems to be solved.

Various kinds of lamps including a carbon arc lamp were tried
*
without success. An ELH or ENG 300-Watt bulb with suitable


The 300-Watt ELH Quartzline bulb produces less heat than a
500-Watt CBH lamp, but offers equivalent light output. The
ELH or ENG bulb is made by General Electric Company.










optical assembly finally proved adequate. The ELH bulb

was placed at the focus point of lens L1. A thin slit

passed only the desired planar beam and the remaining light

was stopped. Lens L2 was placed in front of the slit and

served to form an image of approximately double magnifica-

tion of the slit inside the vortex. Under these circum-

stances, the image formed by lens L2 had a proper depth of

focus thus giving a clear slit of light across the tank.



3.4 Tracer Particles

Various types of tracer particles were tried without

success. Microglass beads of about 50 microns in diameter

were used first, but these could not be photographed with

the equipment available. Aluminum powder was also tried

without success due to the gravity effect. Finally the

particles called Pliolite (Goodyear Rubber Co., Akron, Ohio)

were found to be suitable to act as targets for velocity

measurement. Since the density of Pliolite is quite close

to that of water, the gravity effect is negligible. The optimum

size of particles was 150p ~ l00p. In this range, the par-

ticles are small enough for inertial effects to be negligible

but large enough to provide necessary reflected light. The

concentration of particles which gave the best photographs

was determined by trial and error.



3.5 Experimental Fluids

The Newtonian fluids used for comparison purposes were tap

water and Karo-brand corn syrup solutions of various viscosities.










The polymers used in this study are listed in Table

3-1. Separan AP 273 (S-273) and SP 30 (S-30) are partially

hydrolyzed polyacrylamides (approximately 25-35% hydrolysis).

Both S-273 and S-30 have the structure of


CH2 CH CH2 -CH

C=0 C=0

NH2 0
x y


while the S-273 is of higher molecular weight than S-30

(S-30 having a M.W. of about 3 x 106 in an undegraded state).

Versicol S25 (V-25) is a partially neutrallized poly(acrylic

acid) having the structure


CH2 CH

C=0
I-
-n


Due to the anionic character of these partially hydrolyzed

or neutralized polymers, their solution viscosities are

very sensitive to change in ambient ionic concentration.

Thus, molecular conformation studies could be carried out

with these polymer solutions by varying the solution pH or

by addition of salt.

Unlike the above polymers, Polyox WSR 301 (P-301) is

not a polyelectrolyte. It is a poly(ethylene oxide)

4 CH2-CH2-0 -)-n The average molecular weight of this

polymer has been estimated from the intrinsic viscosity

measurement to be about 4.0 x 10 P-301 is known to be









Table 3-1

Polymers Used


Manufacturers MW -
Chemical Manufacturer Trade Nane of Polyrers (x 10 )


Poly(ethylene oxide)



Polyacrylamide



Polyacrylamide


Sodium
Carboxymetyl
Cellulose


Carboxy-
Polymethylene


Poly(acrylic acid)


Union
Carbide


Dow



Dow



Hercules



B.F.
Goodrich


Allied
Colloids


Polyox
WSR 301


Separan
AP 273


Separan
AP 30


CMC 7H


Carbopol
934


Versicol
S25








very susceptible to shear degradation. Another polymer, CMC 711,

a high viscosity grade of sodium carboxymethylcellulose was

also used in this research. This polymer exhibits rather

weak drag-reducing properties in low concentration levels.

The range of polymer concentrations used was 1 to 100 ppm,

and all the solutions were made in deionized water.

The viscous properties of the Newtonian solutions were

determined using a capillary viscometer described elsewhere

(Van Wazer, et al., 1963). For the polymer solutions, the

viscosities at moderate shear rates were measured using a

Brookfield cone-and-plate viscometer while a specialized ap-

paratus was designed to obtain viscosity data at very low

shear rates. A description of this low shear viscometer will

be given later in this chapter.



3.6 Procedures

In order to prevent speed drift, the strobe disk was

allowed to run for several minutes to reach a steady operating

condition. The power for the light source was then turned on

and the frequency or period of the light pulses was readily

read out in 8 digits from the Type-1190 electric timer.

The preliminary experiments showed that the tracer par-

ticles previously added to the vortex might become less and

less and finally disappeared during the waiting period for

getting a steady state. This is caused by the radial influx

of the vortex flow in the present apparatus. Therefore, an

alternative was to insert the tracer particles a few minutes








before the photographs were taken. Tracer particles were

first dispersed in a beaker containing some working fluid

of the system. This particle solution was then introduced

into the system using a long, fine pipet. This procedure,

although suffering from introducing small disturbance into

the vortex flow, appeared to be adequate after some ex-

perience had been gained through trial and error manipula-

tion of the mixing conditions. Preliminary tests showed

that the introduced disturbance decayed very quickly and

the flow became steady in a few minutes.

With all of the operating condition of the system at

steady state a minimum of approximately 30 photographs were

taken at each given axial position.


3.6.1 Preparing polymer solutions

The solutions were prepared by making a master batch

of about 0.1% by weight in deionized water. The required

amount of polymer was weighed and added to about 50 ml of

isopropanol mixture and was then poured into a well-agitated

vessel containing deionized water. As soon as the mixture

was added the agitation was stopped. This procedure mini-

mized polymer degradation during agitation. The master

batch was allowed to sit for about 24 hours before used. The

master batch was then diluted with deionized water to yield

solution of the desired concentration and poured into both

the head tank and the vortex chamber. The level of the

solution in the vortex chamber was set at the desired position.








3.6.2 Making a run

At the beginning of a run the position of the light slit

was checked and the camera was brought to focus at the mid-

dle of the light slit by focusing on a horizontal or vertical

calibration rule. The magnification factor of the system

was determined from the photograph of the rule.

When making a run, the valve connecting the head tank

and the supply hoses was partially opened so that the liquid

entered the vortex tank through the tangentially positioned

injection tubes. The plug in the bottom of the tank was

then removed and the valve was readjusted until the level

of the liquid was kept at the desired position. The liquid

discharged into the air was collected and then pumped back to

the head tank for repeated use. For the cases where poly-

mer solutions were used as working fluids, the pump was

turned off and the system was left open. Approximately 40

minutes were required for the flow to reach a steady state

after the flow started. Measurements were taken only after

the steady conditions were established.



3.7 Velocity Data Analysis

The primary motion of a fluid element in the vortex flow

is motion in a circle about the cylinder axis. It has been

found that the axial flow occurs only in the core region of

radius of about one centimeter or less. Excluding this core

region, the paths of the tracer particles appear as concentric

circles when viewed parallel to the axis of the cylinder.








These facts make it convenient to photograph the paths of

particles in a thin illumination planar region of about

0.5 cm thick and 10 cm wide, centered about a particular

axial location. A typical top view photograph showing

particle traces in vortex flow is reproduced in Figure 3.3.

In principle, one should be able to obtain radial and

tangential velocities from such a photograph. In practice,

the radial velocity is too low to be measured satisfactorily,

and only tangential velocity data are obtained. The photo-

graphs were analyzed using a Beseler model 23C-Series II

enlarger in the darkroom. The enlarger projected the photo-

graph on a polar coordinate paper. The appropriate magnifica-

tion was obtained by adjusting the height of the enlarger

lamp house from the paper. The magnification factor was

determined using the photograph of the calibration rule.

The center point of the particle traces on the photograph

was located by slightly moving the polar coordinate paper

until the circles of the streaks coincided with the polar

coordinates. For a given particle travelling around the

center point, the following quantities were measured and

recorded: (a) The radius of the particular particle path

was recorded as r (b) The magnification factor was re-

corded as M, (c) If we define 01 and n as the angles of

the leading edge of one streak and the leading edge of the

next nth streak of the same trace in the polar coordinate,

respectively, the difference between 01 and n was recorded

as AO. The number of streaks was recorded as n. In this






























































Figure 3-3 Typical particle-trace photograph (Top view).





57


way the actual size of the individual tracer particle is

eliminated. The number of streaks was selected so as to

allow measurement of A6 to be large as compared to any

error in measurement, and (d) The frequency of the light

pulse was recorded as f.

The local tangential velocity v can be calculated by


[An/(n-l) ] -f- -r
v = 180.M P (3-1)
180*M


where i and 180 are the conversion factors for changing

the unit of AB from degree to radian.

The position coordinate was given by


r
r = -2 (3-2)
M


In the above manner,' tangential velocities were ob-

tained only in the region quite far away from the center

point. In the region near the center point the streaks of

the tracer particles became ambiguous due to the axial flow.

For this region where the axial flow is important, the flow

field must be illuminated in different ways and a side view

photograph is taken to obtain the velocity data.

The flow field in the core region was illuminated by a

slit of light of about 3 cm wide and 10 cm in height. This

sheet of light was parallel to and included the axis of the

cylinder. The camera was then placed at the right angle to

the slit of light and brought to focus at the axis by

focusing on a calibration rule placed in the middle of the


I









cylinder. A photograph of the rule was taken to determine

the magnification factor of the film.

In Figure 3-4 a schematic representation of the essen-

tial features of a streak photograph is given; samples of

the actual streak photographs are reproduced in Figures

3-5 and 3-6. The streaks appeared as a series of dashes-

helix due to the chopped light beam. For each of the

helixes, the following information was recorded: (a) The

radius of the helix was recorded as r (b) The magnifica-

tion factor M, (c) The frequency of the light pulse f,

(d) The apparent axial coordinate for the leading edge of

the streak on the end of a turn of the helix was recorded

as Zl, (e) The apparent axial coordinate for the leading

edge of the streak on the end of the "next" turn was re-

corded as Z (f) The axial distance between the leading

edges of the two streaks was recorded as 1 and (g) The

number of streaks in this turn was recorded as n The
p
following formula was then used to calculate local veloc-

ities and position coordinates. The local axial velocity

w and tangential velocity V are given by


1 f
w = (3-3)
n M




n M
p


The position cnordinates were given by






















































Figure 3-4 Illustration of the side view photograph.






























































Figure 3-5 Typical particle-trace photograph (Side view 1).































































Figure 3-6 Typical particle-trace photograph (Side view 2).









z + z
z = 2 (3-5)


r
r = (3-6)


The axial velocity along the centerline was measured

similarly by timing the motion of a tracer particle

judiciously placed on the tip of the dimple so as to flow

downward vertically along the axis of the cylinder. The

streaks appeared as a series of vertical dashes-line and

are shown in Figures 3-5 and 3-6.

The radial velocities in the body of the vortex were

too small to be measured satisfactorily. An alternative

for obtaining the radial velocity data was by applying con-

tinuity. Remembering that circular symmetry exists, the

vertical volume flow through any horizontal annulus is

given by
r
6 = 2rw rdr (3-7)



Consequently, if w-r is plotted against r for posi-

tions along a radius at a given horizontal level, the area

limited by the curve obtained, the r axis, and the limiting

values of w*r at rl and r2 are proportional to the volume

flow through the annulus. From graphs of w-r against r for

a series horizontal levels, by using a method of graphical

differentiation, average radial velocities were calculated

at selected positions in the vortex flow.








3.8 Concentric-Cylinder Viscometer for Very Low Shear Rates

There exist excellent concentric-cylinder or Couetter

viscometers which have been used to obtain viscosity data

at low shear rates, but because of the expense involved in

their construction, they have not found widespread use in

the laboratories. These viscometers generally consist of

an outer cylinder driven at a constant speed. The torque

transmitted through the fluid to a static inner cylinder

is measured by some appropriate device which is probably

the most expensive part in constructing the viscometers.

A very different mechanism for driving the rotating cylinder

has been described by Zimm and Crothers (1962). They used

a freely floating inner tube, supported by its own buoyancy

and held in place by surface forces such that the inner tube

or "rotor" floats concentrically with the outer tube or

"stator." A steel pellet is glued in the bottom of the rotor,

to which a constant torque is applied by the interaction of

the steel pellet with a rotating applied magnetic field.

This design utilized the magnetization properties of the ferro-

magnetic core in the rotor to generate a torque on the rotor

by an external rotating magnet. However, we found that the

torque so produced was not sufficiently stable for our pur-

poses, and moreover, the speed of the rotor drifted for sev-

eral hours after changing from a strong to a weak magnetic

field. In addition, a serious wobble of the rotor axis,

probably due to imperfectly placing the steel pellet in the

rotor, lead to erratic speeds and unreproducible data.








The present viscometer, shown schematically in Figure

3-7, is a modification of the original design, in which we

used a nonferromagnetic aluminum sheet instead of the steel

pellet in the rotor. The torque produced in the new instru-

ment comes from the interaction between the original applied

magnetic field and an induced magnetic field resulting from

the generated "eddy currents" throughout the surface of the

aluminum. Consider a cylindrical sheet in a rotating mag-

netic field perpendicular to the surface of the sheet but

confined to a limited portion of its area, as in Figure

3-8(a). The magnetic field is moving across element 0 in

which an emf is induced. Elements A and B are not in the

field and hence are not seats of emf. However, in common

with all the other elements located outside the field, ele-

ments A and B do provide return conducting paths along which

positive charges displaced along 00' can return from 0' to

0. A general eddy circulation is therefore set up in the

cylindrical sheet somewhat as sketched in Figure 3-8(b).

We therefore see that a torque is generated which attempts

to allign the induced and applied magnetic field. The period

of revolution of the rotor P (seconds per revolution) can be

represented in terms of apparatus constants and the liquid

viscosity p by (Appendix A)



m 8rr 2(h+Ah) (3-8)
P P (3-8) ;
m 1 1
K P
mm R2 2
R R
1 2









CORK
THERMOSTAT JACKET


CIRCULATING

TYGON INLET
MENISCUS


FLUID

TUBE


ROTOR
STATOR
ALUMINUM SHEET





IRON POLE PIECE




MAGNET




MOTOR SHAFT


SYNCHRONOUS MOTOR


Figure 3-7
SCHEMATIC


DRAWING OF A


LOW-SHEAR-RATE


CONCENTRIC- CYLINDER


VISCOMETER












































Rotating Magnetic Flux Eddy Current
(a) (b)





Figure 3-8 Eddy currents in the aluminum sheet located inside
a rotating magnetic field.





67


in which


R1 = rotor radius

R2 = stator radius

Pm = period of revolution of the magnet

K = Torque constant

h = rotor height

Ah = end correction


If h is held constant in order to obtain constant

rotor height, the relative viscosity of test solution

Orel is simply


P P
m
rel P -P (3-9)
o m


where P and Po refer to solution and solvent, respec-

tively. The average shear rate can be calculated when

some assumptions are made (Appendix A)


2 2
8T R R In R
S 1 2 (3-10)
ave P(R 2-R ) 1
2 2


For our viscometer, it was calculated that the apparatus

constant


2 2
87T R2 R2 R
1 2 In = 33.69358 (3-11)
(R -R2)2 2 R
2 1














CHAPTER IV

EXPERIMENTAL RESULTS



4.1 General Flow Pattern

The flow visualization results presented here give an

overall picture of the flow pattern in the vortex tank.

Visualization was accomplished by observing the motion of

a water-soluble dye, which is actively fluorescent under

illumination. The vortex apparatus was described earlier

in Chapter III. Injection was made using existing pres-

sure taps through which a probe (0.0005m. ID) was passed

(see Figure 4-la). The dye was injected continuously over

a short interval with the injection pressure controlled at

any desired level by controlling the dye-reservoir pressure

with a valve. The tank was illuminated using a Kodak slide

projector, and a "slit-shaped" slide allowed the light beam

to be focused on a narrow vertical section of the fluid.

With this arrangement, one observes primarily the axial but

also the radial flow components.

Initially, the dye was injected at different depths in

the axial direction along the vortex chamber. However, it

was discovered that if the probe tip is positioned near the

bottom boundary layer, the dye is convected upward and at a

later point in time the dye completely fills the main stream





69

To Dye Reservoir

t


Pressure Tapping


Injection P
(.05 cm ID)


Vortex-tank Wall




Silicone Rubber Seal


- Vortex Tank


Injection Probe
Probe Support
do CJ ==^/ L

Pressure Tapping Exit Orifice

To Dye Reservoir

(b)


Figure 4-. Dye injection probe arrangement.








from bottom to top and a picture of the overall flow struc-

ture is obtained. This occurs as a result of the large

"secondary" flow in the bottom boundary layer (Rosenzweig,

Ross, and Lewellen, 1962; Kendall, 1962). The injected dye

is carried along the tank bottom, a fraction passing directly

out of the exit orifice, while the remainder is "ejected"

upward (see Figure 4-3). Figure 4-lb illustrates the posi-

tion of dye injection, near the tank bottom.

All the photographs of the dye patterns were taken under

conditions of "steady" flow, with all controllable experi-

mental parameters held constant. The experimental conditions

for the quantitative flow field measurements are the same as

those for the present visualization study. Here, the term

"steady" flow is used rather loosely; the vortex flow was not

precisely steady and often was decidedly nonsteady.

The flow pattern was studied first in water. After the

steady state had been reached, a small amount of dye was in-

jected continuously for approximately 300 sec. A series of

photographs showing the dye pattern vs. time after injection

is given in Figure 4-2. Dye was released within the bottom

boundary layer and spiraled radially inward. Figure 4-2a

shows that a fraction of the dye passes directly out of the

exit orifice, while the remainder is abruptly ejected up-

ward near the sharp edge of the orifice. These "eruptions"

or "bursts" occur suddenly and are the origin of the tran-

sient, fast-moving counterflows. These counterflows in the

outer annular region, i.e., outside of the region of strong





















































Figure 4-2


Development of the dye pattern with water resulting
from bottom boundary-layer injection. The experimental
conditions are: Depth of the surface dimple = 1.2 cm,
diameter of the exit orifice = 0.516 cm, total volume
rate = 49 cm /sec, entering circulation = 25 cm /sec.
A small amount of dye is introduced continuously for a
period of 300 sec. The pictures show the dye pattern
in time-elapsing sequence: (a) 35 sec; (b) 100 sec;
(c) 400 sec; (d) 650 sec; (e) 1,200 sec.







72












































a
C-,

C"











axial downward flow, appear turbulent. At a later point

in time (Figure 4-2b), the outer annular region extends

further upward and becomes a typical conically shaped counter-

flow region within which is a center jet. Close observation

of the counterflow near the orifice revealed that the

bursting process was generally not steady. The unsteady

nature of the bursting process is illustrated by Figures 4-2c

and 4-2b. The dye front of the counterflow (see Figure 4-2b)

is observed to go up and down occasionally. This is probably

caused by the unsteadiness of the bursting process which is

the source of the counterflow. It is illustrated in Figure

4-2c (300 sec. after Figure 4-2b) that the dye front becomes

lower than that in Figure 4-2b. Of course, the dye front

will finally reach the top surface (see Figure 4-2e). The

outer annular region near the bottom is dark since it is

supplied with fresh fluid from near the wall. Some of the

injected dye is subsequently drawn into thin sheets wrapped

around the axis of rotation as shown in Figure 4-2d. The

formation of these sheets must be attributed to the axial

shear flow. The dye pattern in Figure 4-2e is seen to con-

tain a cell of recirculating flow between the axis and the

wall of the cylinder. The dye is found to recirculate in

the cell which dissipates very gradually and in some cases

remains detectable up to 1/2 hour after injection of dye.

Similar observations have been reported by Turner (1966),

and Travers and Johnson (1964).








From these dye studies, certain general features of

the flow could be deduced. As illustrated in Figure 4-3,

the actual flow consists of a strong vortex-type flow

superimposed on the indicated flow pattern. Along the

axis of the vortex, a strong center jet exists, designated

as region (I). This region is fed mainly from radial con-

vection from the outer annular region (II) throughout the

length of the vortex, and partly from the free surface

region (III) near the center of the surface. The radial

flow occurring in the boundary layer on the bottom of the

vortex chamber was found to be quite large using hot wire

probes or pitot tubes. The experimental results reported

by Kendall (1962) and Owen et al. (1961) indicate that the

boundary-layer flow (i.e., the so-called "secondary flow")

carried radially inward along the rigid end wall can be of

the same order of magnitude as the total mass flow through

the vortex chamber. The significant radial flow occurring

in the bottom boundary layer (IV) splits into two portions;

some is discharged directly out of the exit orifice, and

some is ejected outward from the edge of the exit orifice

into the tank. The ejected mass flow may be caused for

two reasons. First,boundary layer theory itself predicts

the sudden eruption of the secondary flow near the center

of the end wall (Moore, 1956; Burggraf et al., 1971).

Second, certain discotinuities in the end wall geometry,

such as the sharp edge of the exit orifice, can also induce

mass ejection (Rosenzweig, et al., 1962; Kendall, 1962;
































































Figure 4-3 Sketch of general flow structure in the vortex.








Roschke, 1966). The eruption region (VII) invariably

appeared to be turbulent. Outside of region (II) are two

annular zones (V) and (VI) in which the fluid possesses

only a downward and upward drifting velocity, respectively.

(These regions explain the stratified flow pattern as shown

in Figure 4-2d). In general, the annular structure

(regions (II), (V), and (VI)) remains qualitatively similar

when the circulation is varied. Outside of these regions,

the vertical velocity is very small. Rosenzweigh, Ross, and

Lewellen (1962) also gave a similar composite sketch of their

observed flow pattern in a closed jet-driven vortex.

Flow patterns for two typical polymer solutions known

to be effective drag reducers and to exhibit vortex inhibi-

tion at very low concentration levels were studied next:

Polyox WSR 301 (P-301), C = 3 ppm; and Separan AP 273

(S-273), CVI = 2 ppm. A slightly higher concentration above

the respective CVI for each polymer was used in the visualiza-

tion study in order to obtain greater contrast between the

flow patterns for water and the polymer solutions. Figure

4-4 is the time-lapse sequence of photographs showing the

dye pattern development for P-301 at a concentration of 10

ppm. Figure 4-5 is the similar time-lapse sequence for

S-273 at 3 ppm. The controllable conditions for both cases

were the same as those for the water vortices.

Comparing the sequence of the dye pattern between those

of water and P-301 solution, it appears that except for dif-

ferences in the shape of dye front the apparent difference






78


















C44
0 ia









w a
brh


c Q)L
.fl pC



Scl) U)
r-I bci

0Cc

XcuI


o W





c) .4 CLn
44-
Xd

p o -W




QW ca

4J C04J
0d ) Crl








01. 4J 0)

4jt 0)

0- 4-J
c C



Q, .0 C:)




0 ),-
>rUC










80











IU










0oC.)


r-q u


C bfl


O n)
:4-) CO
p a)













-H C4J C
4 4- )
caa










4-J WP
CO 0( a)











P. 1- 1 ni
coa

C4.J















0 -H C
Q) 0)
r Ir I
4
0J 4J Q


C om














::IP WU
FX4
0)l
r4 0)Q.C'
0)~
I- 0)0

If.)I


0)I
b13r











is the relative mixing or diffuseness of the dye (compare

Figures 4-2d and 4-4d). The dye streak in Figure 4-2d con-

sisting of well-defined concentric regions gives a good

demonstration of the laminar character of the flow. In

Figure 4-4d, there occurs a quasi-cyclic "eruption" or

"burst" at the intermediate distance from the vortex axis,

leading to a rapid mixing of the dye. Such an unsteady

bursting process is unable to be illustrated in Figure 4-4d

because it occurs instantaneously and quickly interacts with

the high-speed surrounding fluid and appears as a gray area

located near the vortex axis (see Figure 4-4d). The details

of the bursting process can be illustrated more clearly using

special dye injection technique which will be discussed in

Section 5-6.

The series of photographs shown in Figure 4-5 is for

S-273 solution. It is observed that the major difference

in the dye patterns from those of water is the remarkable

enlargement of the central axial flow region. (The bright

area in Figure 4-5d located near the vortex axis represents

this axial-flow region.) The axial velocity is seen to be much

smaller than that in water vortex, as qualitatively pre-

dicted by Rott (1958) as a consequence of the viscous effect.


A quasi-cyclic process means that a sequence of events
repeats in space and time, but not periodically at one place
of time nor at one time in space.

The "bursts" here are different from those described before
which originated near the sharp edge of the exit orifice.








More detailed pictures of the eruption region near the

exit orifice are given in Figure 4-6. Figure 4-6b shows that

the presence of S-273 greatly smoothes the fluctuations of

the ejected fluids and the counterflow originating at the

bottom of the chamber apparently spreads to a larger radius.

In Figures 4-6a and 4-6c, the stronger fluctuation of the

erupted fluids leads to more rapid mixing of dye; the finer

details of individual dye filaments become smeared out.

This makes the upward flowing regions for both water and

P-301 become vague and indistinguishable from each other.

In general, the photographs show that the difference

in flow pattern between water and S-273 is the "size" of

the central axial-flow and the annular counterflow regions.

For S-273, both these flows are much weaker than those of

water, which are actually observed by the movement of the

dye fronts during the course of experiments. However, ex-

cept for differences in the shape of stratified structures

of the dye interfaces, there is no apparent difference

between water and P-301. Therefore, it is rather difficult

to distinguish between water and P-301 from the dye pattern.



4.2 Physical Properties of Test Fluids

The test fluids used in this research may be conveniently

classified as follows:

*
Whether a flow is strong or weak is seen through the move-
ment of dye fronts and cannot be demonstrated in the present
still photographs.

















































Figure 4-6


Close observation of the dye pattern near the exit
orifice. The experimental conditions are the same
as those described in figure 4-2. (a) Water;
(b) Separan AP 273 solution, 3 ppm; (c) Polyox WSR
301 solution, 10 ppm.


























































(c)


Figure 4-6 (Continued)








1. Newtonian: Water, Corn syrup-water

2. Viscoinelastic: Carbopol 934

3. Viscoelastic: Polyox WSR 301 (P-301)

Separan AP 273 (S-273)

Separan AP 30 (S-30)

Versicol S 25 (V-25)

CMC 7H


The vortex inhibition ability of these test fluids is

listed in Table 4-1. It is worth noting that vortex for-

mation can also be suppressed in a Newtonian fluid if the

shear viscosity is high enough.

In the present experimental setup, a large amount of

liquid (about 300 liters) is required for each run to obtain

a complete flow field measurement. It is thus rather im-

practical to use corn syrup to raise the fluid viscosity up

to, say, 4 centipoise, as this can require as much as 150

liters of corn syrup per run. The corn syrup solutions also

promote bacterial growth within one day of preparation, thus

making the solution hazy and not usable. The addition of a

small amount of sodium benzonate fails to retard bacterial

growth. For these reasons, Carbopol 934 and CMC 7H were

used to increase viscosity in view of the fact these polymers

display negligible or very slight elasticity. Several inves-

tigators, including Dodge and Metzner (1959) and Kapoor (1963),

have demonstrated that Carbopol solutions are not drag re-

ducing nor do they demonstrate any recoil upon removal of

shear stress. However, a serious drawback of Carbopol 934





87


Table 4-1

Summary of Vortex Inhibition Data


Test Fluid CVI, ppm




S-273 2



S-30 7.5



P-301 3



V-25 3



CMC 7H 75



Carbopol 934a 400



Corn Syrup Water 450,000b




aNeutralized with 0.4 gm. of sodium hydroxide/gm. of
Carbopol 934.

bThe viscosity of this 45% corn syrup 55% deionized water
solution is about 5 centipoise.








in the present study is that above 200 ppm in water, the

solutions are so hazy that no streak photographs of the

tracer particles may be taken. For this reason, the test

fluid used to obtain higher viscosity levels was CMC 7H, a

high viscosity grade sodium carboxymethylcellulose. Even

at concentrations as high as 2,500 ppm in water, CMC 7H

was shown to be only very slightly viscoelastic (Goldin,

1970). Ernst (1966) and Pruitt (1965) have demonstrated

that CMC 7H is much less efficient in drag reduction than

the high molecular weight poly(ethylene oxides) and poly-

acrylamides. The measured friction factor Reynolds num-

ber data for a 100 ppm solution of CMC 7H are presented

in Figure 4-7. At low Reynolds number, the data of CMC 7H

lie above the line f 16 because of the increased solution
Re
viscosity (the Reynolds number is calculated by using the

solvent viscosity). At high Reynolds number, the friction

factor for CMC 7H is indistinguishable from that of the sol-

vent (water), implying the nondrag-reducing ability of CMC 7H

for a concentration of 100 ppm. This finding further con-

firms that CMC 7H does not display elasticity, at least at

low concentration levels.

For the corn syrup solution, viscosities were measured

with the Cannon-Fenske capillary viscometer. For the polymer

solutions, the specialized low-shear, coaxial-cylindrical

viscometer described in Chapter III was used to provide vis-

cosity data at shear rates ranging from 0.05 to 5 sec-i

The data of Table 4-2 indicate the precision of this instru-

ment as currently used.






























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SCMC 7H, 100 ppm


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Figure 4-7


Friction factor vs. Reynolds
100 ppm in 1.09 cm tube.


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Figure 4-8 shows the shear viscosity vs. shear rate

data for CMC 7H solutions at 75, 25, and 10 ppm. It is seen

that this polymer exhibits only slight shear-thinning ef-
-l
fects over the shear rate range 0.01 to 1 sec Figure 4-9

shows the viscosity measurements of the Carbopol 934 solu-

tions. CMC 7H solutions are more viscous than Carbopol 934

at equal concentration. Figures 4-10 and 4-11 show the vis-

cosity data of S-273 and S-30. Both of these polymers show

strong shear thinning behavior even at a concentration as

low as 1 ppm. To the best of our knowledge, this surprising

behavior has not been previously reported in the literature.

Figure 4-12 illustrates the shear viscosity data of the

P-301 solutions. A comparison can be made between the P-301

and S-273. The molecular weight of the two polymers is

nearly the same (according to the manufactures, the weight-

average molecular weight for P-301 and S-273 is 4x106 and

7.5x106 respectively). The distributions are unknown, but

it would be expected that their viscosities would also be

nearly the same. However, comparing Figures 4-10 and 4-12,

one finds that the viscosity of S-273 is much higher than

that of P-301, at low shear rates, and from this we deduce

that S-273 chains are considerably more expanded in solution.

This may be a result of the electrostatic repulsion between

the ionic groups on the chains of Separan polymers. Since

S-273 is a polyelectrolyte, the electrostatic repulsion be-

tween the ionic groups will cause significant expansion.

The P-301 chain, on the other hand, is known to be a random





























































































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81,9(56,7< 2) )/25,'$


THE VORTEX FLOW OF DILUTE POLYMER SOLUTIONS
By
CHII-SHYOUNG CHIOU
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1976

-77
ACKNOWLEDGMENTS
I wish to express my gratitude and appreciation to
my advisor, Professor R. J. Gordon, for his guidance and
instruction throughout the course of this work. I would
also like to thank the other members of my committee,
Professor R. W. Fahien, Professor R. D. Walter, Professor
D. W. Kirmse, and Professor U. H. Kurzweg for their par¬
ticipation. I am especially grateful for the numerous
hours of consultation Professor Kurzweg has provided on
the work covered here and related problems.
Many thanks are extended to my colleagues, Dr. C.
Balakrishnan, M. C. Johnson, D. White, and F. J. Consoli,
for their help and suggestions. The help given by the
Chemical Engineering Shop personnel, particularly by Mr.
M. R. Jones and Mr. R. L. Baxley, in the design and fabrica¬
tion of the experimental equipment, is greatly appreciated.
I am also grateful to the Chemical Engineering Depart¬
ment for financial support and the National Science Founda¬
tion (Grant GK-31590) for partial support of this research.
Finally, my wife, Mei, has provided the love and under¬
standing that have made the completion of this work possible.
ii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
ABSTRACT V
CHAPTER
I INTRODUCTION 1
1.1 Preliminary Remarks 1
1.2 Mechanisms of Drag Reduction 2
1.3 Elongational Flows of Dilute Polymer
Solutions 5
1.4 Vortex Inhibition Phenomenon 9
1.5 Scope of Problem 17
II BACKGROUND 18
2.1 General 18
2.2 A Description of the Vortex Flow with an
Air Core Present 19
2.3 Classical Treatment of Vortex Flows ... 19
2.4 Previous Investigations on Real Vortices 27
2.5 Viscoelasticity Considerations 41
III EXPERIMENTAL 43
3.1 General Description of Experiments ... 43
3.2 Vortex Chamber and Flow System 44
3.3 Optical Assembly 46
3.4 Tracer Particles 49
3.5 Experimental Fluids 49
3.6 Procedures 52
3.7 Velocity Data Analysis 54
3.8 Concentric-Cylinder Viscometer for Very
Low Shear Rates 63
IV EXPERIMENTAL RESULTS 68
4.1 General Flow Pattern 68
4.2 Physical Properties of Test Fluids ... 83
4.3 Flow Field Measurements--Water Vortices . 99
4.4 Influence of Newtonian Viscosity on Velo¬
city Distributions 107
iii

CHAPTER
Page
IV4.5 Flow Field Measurements of Polymer Solu¬
tions 114
4.6 Comparison of Experimental Results be¬
tween Those of Elastic and Inelastic
Fluids at Equivalent Shear Viscosities . 132
4.7 Conformational Studies 141
V THEORETICAL ANALYSIS 164
5.1 Fundamental Equations 165
5.2 Nonsteady-state Solution of Equations of
Motion (Following Dergarbedian) 168
5.3 Steady-state Solutions for Viscous Vortex
Flows 180
5.4 The Application of Lewellen's Asymptotic
Solution to our Laboratory Vortices . . . 199
5.5 Theoretical Analysis for Viscoelastic
Fluids 215
5.6 A Proposed Model for the Vortex Inhibi¬
tion Process 233
VI DISCUSSION OF RESULTS 249
6.1 Air Core Formation 249
6.2 Significance of Secondary Flow in Bottom
Boundary Layer 253
6.3 Zero-shear Viscosities of the Polymer
Solutions 260
6.4 Theoretical Analysis on Vortex Flows with
Polymer Additives 262
6.5 Dependence of Drag Reduction and Vortex
Inhibition on Polymer Conformation . . . 271
6.6 Conclusions and Recommendations 273
APPENDIX
A DERIVATION OF EQUATIONS (3-8) THROUGH (3-10) . 277
B ORDER-OF-MAGNITUDE ANALYSIS OF VORTEX FLOWS . 281
C PREDICTION OF MAXWELL MODEL—RECONSIDERATION
IN A TRANSIENT VORTEX FLOW 283
D STABILITY OF VORTEX FLOWS (NEWTONIAN) .... 286
BIBLIOGRAPHY 292
BIOGRAPHICAL SKETCH 298
IV

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
THE VORTEX FLOW OF DILUTE POLYMER SOLUTIONS
By
Chii-Shyoung Chiou
December, 1976
Chairman: Ronald J. Gordon
Major Department: Chemical Engineering
When minute quantities of various high molecular
weight polymers are added to a solvent, the tendency of
the solvent to form an air core or vortex as it drains
from a large tank, is severely inhibited. This effect,
which is referred to as "vortex inhibition," correlates
extremely well with the drag-reducing ability of the
polymers. It would appear that the two phenomena are
closely related, and a thorough understanding of vortex
inhibition may shed light on the mechanism of drag reduc¬
tion.
In this work, a steady three-dimensional vortex flow
was used as an approximation to the actual vortex inhibition
experiment. The fluid enters tangentially, spirals radially
inward and exits axially at the bottom of a tank. The
three-dimensional velocity distribution within the vortex
was determined using a photographic technique. It was
found that the shearing nature of the flow field is much
more important than its stretching nature which only exists
in the very near vicinity of the vortex axis.
v

Flow field measurements show that the presence of the
polymer causes a dramatic change in the originally steady
flow field, leading to a strong flow fluctuation near the
vortex axis. It is the flow fluctuation which breaks up
the concentrated vortex core and results in the vortex in¬
hibition.
A theoretical analysis of the problem using the equa¬
tions of motion and constitutive theories of dilute polymer
solutions in terms of experimentally supported kinematics
shows that the normal stress difference, (Tn.-T )--the
0 0 rr
tension along the lines of flow in the circumferential
direction--is of significance in suppressing the vortex
formation.
The roles of polymeric additives in the mechanism of
vortex inhibition and drag reduction were closely examined.
The results of this work offer considerable support to
theories that explain these phenomena as a consequence of
the viscoelastic behavior of polymer solutions in time-
varying shear fields.
vi

CHAPTER I
INTRODUCTION
1.1 Preliminary Remarks
Since the drag reduction phenomenon was first reported
by Toms in 1948, an intensive interest in this topic has
developed. Especially in recent years, extensive research
has been conducted by a large number of investigators.
Drag reduction is defined by Savins (1964) as the reduction
in pressure drop in turbulent pipe flow following addition
of various substances such as some high molecular weight
polymers, soaps (Agoston, 1954) or suspended particles
(Bobkowicz, 1965; Zandi, 1967). Drag reduction occurs in
turbulent flow and is, therefore, of great potential value
to industry, where most flows are turbulent. Promising ap¬
plications of drag reduction are in long crude petroleum
pipelines, in brine disposal pipelines, in sewer disposal
systems, in fire fighting, and in reducing drag in the flow
of fluids past submerged vessels.
The recent development of drag reduction has provided
a fairly comprehensive picture of the drag reduction phe¬
nomenon. Many important aspects of drag reduction have been
identified, such as its dependence on the average molecular
weight of the polymer, polymer concentration, and the wall
1

2
shear stress. In the last few years, the study of new
polymer-solvent pairs has either added new insight to our
knowledge or has been used in solving practical problems; the
development of new flow-visualization technique has aided
in understanding the influence of drag-reducing polymers
on turbulence in near-wall regions; the study of polymer
degradation has shown that some highly effective drag-
reducing polymers appear to break up with mechanical shear
and thus lose their effectiveness. However, in spite of
the considerable accomplishments, the actual mechanisms
which govern the magnitude and character of the drag reduc¬
tion phenomenon are still unresolved.
1.2 Mechanisms of Drag Reduction
At present, there is evidence which suggests that the
reduction of drag on a solid surface is associated with
changes in the turbulence structure in regions very close
to the surface. For Newtonian fluids, detailed flow visual¬
ization experiments have been made on turbulent boundary
layer structure (Corino and Brodkey, 1969; Kim, Kline, and
Reynolds, 1971). The results of these studies have established
a relationship between the production of turbulence and the
behavior of low-speed "streaks" in the region near the wall.
Alternating regions of high and low velocity developed near
the wall were very much elongated in their streamwise extent,
thus appearing "streaky" in structure. These low-speed streaks
were seen (i) slowly to lift away from the wall, (ii) often to

3
begin a growing oscillation and (iii) finally, to break up
into a more chaotic motion. The whole cycle is termed the
bursting phenomenon. From estimates made from the hydrogen-
bubble data, Kim et al. (1971) showed that essentially all
the turbulence production occurs during bursting. As a re¬
sult, it is now generally believed that bursting is the
mechanism for generation of turbulence.
Offen and Kline (1973, 1974), Donohue, Tiederman, and
Reischman (1972) carried out flow-visualization experiments
in dilute solutions of drag-reducing polymer and found that
the spatially averaged bursting rate is greatly decreased
by the addition of polymer. This then suggests that tur¬
bulence production is also decreased, consistent with the
lowered wall stress. To explain the mechanism of drag re¬
duction, therefore, it is necessary to understand how elas¬
ticity and non-Newtonian effect of polymer solutions can
alter the bursting process.
Currently, two viscoelastic mechanisms which are con¬
sistent with the bursting studies have been proposed in the
literature to account for the drag reduction effect:
(i) Transient Shear Flow:
One theory that has been proposed to explain drag reduc¬
tion involves the viscoelastic effect of polymers on transient
shear flows (Ruckenstein,1971; Hansen, 1972a). It has been
suggested that the propagation velocity for the transient
shear disturbances in very dilute polymer solutions is sub¬
stantially less than that in the solvent. Since a certain

4
stage of the bursting phenomenon may involve a deformation
of this type, a large decrease in the propagation velocity
may result in less bursting and thus less turbulent drag.
Since the propagations of a transient shear disturbance and
a simple sinusoidal shear wave are related, by evaluating the
phase velocity and amplitude attenuation coefficient for shear
wave propagation as a function of frequency, the proposed
mechanism can be examined (Little et al., 1975). A 100
parts per million by weight solution of a commercial sample
(Polyox WSR 301) was used as an example. The results clearly
indicated that no significant differences between solvent and
solution behavior were predicted, even at relatively high con¬
centration levels. It appears that this mechanism makes an
insignificant contribution to the observed drag reduction.
(ii) Elongational Flow:
A recent phenomenological explanation of drag reduction,
advanced by Metzner and Seyer (1969) involves the unusually
high resistance offered by polymer solutions to pure stretching
motions. According to these authors, the turbulent eddy struc¬
ture near the pipe wall may be considered roughly as such a
stretching motion.
For viscoelastic liquids, it has been shown analytically
(Astarita, 1967; Everage and Gordon, 1971; Denn and Marruci,
1971) and partially confirmed experimentally (Ballman, 1965 ;
Metzner and Metzner, 1970; Balakrishnan and Gordon, 1975)
that the resistance to stretching may be much greater than
that of Newtonian fluids. The general concept is that this

5
resistance to stretching interferes with the production of
bursting, thus decreasing the turbulent dissipation and sub¬
sequently the wall shear stress. This proposed mechanism
seems to have a high probability for success, since the
effects of elongational flow are predicted to be very large
and with the extremely large effects it is easy to understand
how small quantities of polymer additive can produce large
changes in flow.
1.3 Elongational Flows of Dilute Polymer Solutions
Among the proposed theories, the elongational viscosity
has been considered as the most promising explanation for
drag reduction. However, the large values of elongational
viscosity for very dilute polymer solution cannot be demon¬
strated experimentally. While various estimates of the pro¬
perties of dilute polymer solutions in elongational flow have
been made, no direct measurements appear to be available in
the literature. So far, only three methods of generating
reasonable approximations to "pure" elongational flow ap¬
peared suitable:
(i) Converging Flow:
This method is based on the observation that for some
viscoelastic fluids the flow through an abrupt contraction
is restricted to a narrow conical region upstream. Metzner
and co-workers (1970, 1969) have described the kinematics of
such a flow utilizing an approximated velocity field and shown
that the deformation rate tensor is diagonal. The material

6
is thus expected to be subjected to simple Lagrangian un¬
steady, extensional deformation. It is to be noted that
not all polymer solutions show this behavior. In many cases,
the flow enters the contraction from all directions upstream
just like a Newtonian fluid does. The deformation rate ten¬
sor is no longer diagonal and the analysis of Metzner and co¬
workers based on the assumption of simple elongational flow
is not applicable. It is not possible to obtain meaningful
elongational flow data when the flow does not exhibit the
conical pattern.
(ii) Fano Flow or Tubeless Siphon:
This experiment utilizes the ability of some visco¬
elastic fluids to be drawn out of a reservoir into a tube
even when the liquid level in the reservoir falls below the
end of the tube. (The other tube end is connected to a
vacuum pump.) The deformation rate tensor for this flow has
also been shown to be diagonal and relevant stress-strain
rate information can be caclulated from the measurement of
the force exerted by the fluid column on the tube, the volu¬
metric flow rate and the diameter-distance relationship.
Kanel (1972) showed, however, that no column can be obtained
with aqueous solutions having polymer (Separan AP 30 or
Polyox WSR 301) concentrations of the order of 100 ppm even
with tube diameters as small as 0.5 mm. By using a New¬
tonian solvent of high viscosity (e.g., 50-50 glycerol-water,
viscosity ~6 C.P.), columns about 1 mm in diameter can only
be realized for the same range of polymer concentrations.

7
Furthermore, the unavoidable errors in force measurement
limited the application of this technique in obtaining
accurate elongational data.
(iii) Fiber Spinning Flow:
In this method, the viscoelastic fluid is forced through
a die or a nozzle and the thread collected on a rotating
wheel downstream. The stress-strain rate information can
be obtained from the force measurements made at the wind-up
device, the volumetric flow rate and the diameter-distance
relationship obtained from photographs. This method has
been extensively used for elongational flow studies of polymer
melts (Spearot, 1972; Ziabicki and Kedzierska, 1960) and
concentrated solutions of polymers (Weinberger, 1970; Zidan,
1969). For dilute polymer solutions in water, it was found
that the solutions lacked tackiness and did not attach to the
rotating wheel (Baid, 1973). This situation has to be im¬
proved by increasing the Newtonian solvent viscosity. It was
found that satisfactory data could be obtained by using a 95%
glycerol-5% water solvent for Separan AP 30 solutions having
concentrations of about 100 ppm (Baid, 1973) or using a 1%
solution of the same polymer in equal parts by weight of water
and glycerol (Moore and Pearson, 1975).
Based on the above discussions, it appears that no direct
measurement of elongational flow properties of certair dilute
polymer solutions are available, and the correlation between
the drag-reducing ability and elongational viscosity has,
therefore, never been fully established. For example, in the

8
case of Polyox WSR 301 with concentration varying from a
few ppm to about a thousand ppm, a significant drag reduc¬
tion still can be detected while the elongational viscosity
of the solutions is still not measurable.
Recently, in the converging flow experiments, Bala-
krishnan (1976) found that Polyox solutions were unable to
show the wine-glass-stem shape flow so that no elongational
viscosity could be measured. Besides, no elongational in¬
formation has ever been obtained on these effective, dilute
drag-reducing polymer solutions by the Fano flow or fiber
spinning techniques without highly increasing the Newtonian
solvent viscosity. It is therefore still not clear if the
actual mechanism would involveâ– elongational flow.
An alternative approach to determine the mechanism of
drag reduction might be to investigate a new viscoelastic
effect recently discovered by Gordon and Balakrishnan (1972)
This effect, referred to as "vortex inhibition" shows that
the tendency of water (as solvent) to form a vortex, as it
drains from a large tank, is inhibited by the presence of
only minute quantities of various high molecular weight poly
mers. This effect correlates extremely well with drag-
reducing ability. In other words, those polymers which show
vortex inhibition at the lowest concentration are also the
best drag-reducers. Both vortex inhibition and drag reduc¬
tion are highly sensitive to polymer degradation, and both
also vary in the same fashion with fundamental change in
polymer conformation in solution (Balakrishnan and Gordon,

9
1975, Gordon and Balakrishnan, 1972). Studies of vortex
inhibition are therefore of interest both in their own right,
and also because of the possibility that may yield information
on the mechanism of drag reduction. It is the objective of
this study to characterize the velocity field through flow
visualization experiments and seek an explanation for the ef¬
fect of vortex inhibition. These results should indirectly
contribute to our knowledge of drag reduction. Once the pro¬
perty of dilute polymer solutions which is responsible for
vortex inhibition is identified, its dependence on macro-
molecular variables such as molecular weight, molecular weight
distribution, or molecular conformation would become immedi¬
ately apparent. The effect of- vortex inhibition may then be
useful, in conjunction with intrinsic viscosity measurement,
in providing both mean (viscosity-mean) molecular weight and
breadth of the molecular weight distribution. Since many use¬
ful physical properties, as for example, the coating proper¬
ties of acrylic paint (Billmeyer, 1971), depend on the "high
molecular weight tail" of the polymer, a simple means of
characterizing this "tail" would have immense significance.
1.4 Vortex Inhibition Phenomenon
A Plexiglas tank was filled with water and stirred
vigorously with a paddle in order to introduce some initial
circulation into the field. The bottom plug was then re¬
moved, and after a few moments, an air core formed (Figure
1-1), extending from the free surface of the liquid to the
small drain hole at the bottom of the tank.

10
Figure 1-2 Illustration of vortex inhibition with polymer
solution.

11
The narrow air core, which was essentially cylindrical
over the greater part of its length, was extremely stable,
and once formed, it remained intact unt il the tank had drained.
The same experiment was then repeated with a dilute drag-
reducing polymer solution. This time, as long as the polymer
concentration was above some minimum value (denoted by Cv^),
the vortex could not complete (Figure 1-2). The completeness
of the vortex was accompanied by a "slurping" sound as air
was drawn out of the drain hole. C was defined as the
lowest concentration at which the vortex would not complete.
Perhaps surprising, the results of this somewhat arbitrary
procedure were remarkably reproducible. The most striking
aspect of the vortex inhibition effect is the low concentra¬
tion required and its extremely close correlation with the
drag-reducing polymers. Some important findings on the
effect of vortex inhibition that have been presented in the
earlier publications (Gordon and Balakrishnan, 1972, 1975b)
are:
(i) Correlation of Drag Reduction and Vortex Inhibition:
According to these authors, the drag-reducing ability
of various polymers was simply determined by measuring the
efflux time for the liquid level to drop between two
specific levels. If the efflux times for water and polymer
solution are denoted by t^ and t^, the percentage of drag
reduction is defined as
% DR = 100(t -t )/t
w p w

12
The results of all the polymers used are summarized in
Table 1-1. As already mentioned, C ^ represents the minimum
concentration for vortex inhibition, while c n is the con-
centration at which maximum drag reduction was observed.
Inspection of Table 1-1 reveals that C is an excellent
ordering parameter for the drag-reducing effectiveness of
the different polymers. These results strongly suggest
that vortex inhibition may be used to predict a priori
whether or not a given polymer sample will be an effective
drag reducer. To the author's knowledge, no other rheo¬
logical test presently available is capable of characterizing
drag-reducing ability for the polymer concentrations of
interest, varying from a few ppm on upward.
(ii) Influence of Polymer Degradation on Vortex Inhibition:
Theory and experiment both suggest that in dilute solu¬
tion, mechanical degradation affects the higher molecular
weight components more than the lower molecular weight com¬
ponents (Casale, et alâ– , 1971). Paterson and Abernathy (1970)
found that the drag reduction depended predominantly on the
high molecular weight "tail." The results of the influence
of different degrees of mechanical degradation on C ^ are
given in Table 1-2. This table illustrates that is pri¬
marily dependent on the high molecular weight species of a
given polymer, just as drag reduction is expected to be.
(iii) Influence of Molecular Conformation on Vortex Inhibition
In contrast to an uncharged, random coiling polymer, a
polyelectrolyte is always composed of a macroion in which the

Table 1-1
Summary of Data
Polymer designation
Polymer type
Manufacturer
CVI,
wppm
CDR'
wppm
[o ] ,
dl/g
Polyox FRA
poly(ethylene oxide)
Union Carbide
7
.5 9
23.5
Polyox WSR 301
poly (ethylene oxide)
Union Carbide
30
20
15.1
Polyox WSR 205
poly(ethylene oxide)
Union Carbide
600
150
4.1
Polyox WSR N750
poly(ethylene oxide)
Union Carbide
3000
550
3.3
Separan AP 273
polyacrylamide
Dow
3
5
50a
Separan AP 30
polyacrylamide
Dow
40
35
31a
Polvhall 295
polyacrylamide
Stein Hall
10
9
25a
Cellulose gum 7H
carboxymethylcellulose
Hercules
400
600
25a
These values are for
â–  hSp/c at 20 wppm. The n
/c-vs.-c curves
turned
up sharply at
lowe
concentrations, and
precise extrapolation to
c - 0 was not possible.
All data are
for
tap water.
Source: R. J. Gordon and C. Balakrishnan, J. Appl. Poly. Sci., 1_6 , 1629 (1972).

Table 1-2
Results of Degradation Study
Polymer
Degree of
degradation
0
1
2
3 4
5
Polyox FRA
HR @
200
wppm
1.57
1.46
1.29
1.26
CVI
15
40
175
>200
Polyox WSR 301
nR @
400
wppm
1.75
1.68
1.57
cvi
4 0a
75
>400
Separan AP 273
r|R @
100
wppm
1.61
1.61
1.56
1.53
1.49
cT7T
3
3
7.5
40
75
VI
aCVI varied from 30 to 40 for this polymer.
Source: R. J. Gordon and C. Balakrishnan, J. Appl. Poly. Sci., 16, 1629 (1972).

15
charged groups are interconnected by chemical bonds, together
with an equivalent number of small oppositely charged counter¬
ions. Thus polyelectrolytes undergo dramatic changes in con¬
formation with change in their ionic environment. Polyelec¬
trolytes are known to be highly coiled in acidic environment,
perhaps due to intramolecular hydrogen bonding, and has ex¬
tended conformation at high pH due to ionization (Hand and
Williams, 1970; Mathieson and MacLaren, 1965). This tendency
of polyelectrolytes to change in conformation with change in
their ionic environment makes them suitable for studying
the effect of polymer conformation.
Parker and Hedley (1972) measured the drag-reducing
ability of an aqueous solution of poly(acrylic acid) (PAA)
at different pH levels. They found that PAA exhibited re¬
markable drag reduction at a pH of 6 to 8, while a lowering
of the pH to 2.2 greatly reduced the drag reduction. These
findings show the important effect of pH in controlling the
molecular extension and thus the drag reduction. Similar
results were also reported by Kim, Little, and Ting (1973)
and White and Gordon (1975). In order to see if the in¬
fluence of molecular conformation would affect vortex in¬
hibition, CVI has been measured for PAA (Versicol S25, Allied
Colloids) and PAM (polyacrylamide, Separan AP 273, Dow Chemical)
in different pH values. The results are listed in Table 1-3
where it is evident that the polymer conformation has strong
influence on the effect of vortex inhibition. This table
further indicates that CVI correlates closely with the drag-

16
Table 1-3
Vortex Inhibition Concentration C versus pH
for Polyacrylamide and Poly(acrylic acid)
pH
CVI
pH
CVI
PAM
2.1
40
PAA
3.9
200
3.75
5
5.6
10
5
3
7.1
3
7
3
9.7
3
12
1.5
11.7
5
Source: C. Balakrishnan and R. J. Gordon, J. Appl. Poly.
Sci., 19, 909 (1975).

17
reducting ability of the polymers under the influence of
molecular conformation variation.
1.5 Scope of Problem
Based on the preceding discussion, it is strongly
suggested that vortex inhibition and drag reduction are due
to the same viscoelastic mechanism. The purposes of the
work are:
(i) To measure the velocity profiles experimentally for
both the solvent and the dilute polymer solutions in vortex
flows, especially in the core region where it is expected
to change significantly when polymer additive is present.
(ii) To provide a theoretical analysis of this problem,
using the momentum equations and realistic constitutive
assumptions, from which the velocity profiles may be predicted.

CHAPTER II
BACKGROUND
2.1 General
This chapter will survey pertinent previous work on
vortex flows. Before discussing the published literature
in detail, two explanatory sections are presented to ac¬
quaint the readers with some of the concepts and terminology
to be used in the main body of this thesis. Section 2.2
gives a brief description of the specific vortex flow en¬
countered in the vortex inhibition experiment. Analytical
treatments of the classical vortex motion, in which various
definitions used to characterize rotating flows are intro¬
duced, are discussed in Section 2.3. In Section 2.4,
published literature dealing with vortex flows of Newtonian
fluids is reviewed. Here mathematical treatment of the
problem to predict the velocity distribution within the vor¬
tex is presented and the validity of each of the simplifying
assumptions is examined by comparison to available experi¬
mental data. Although Newtonian vortices (in most cases, a
water vortex) have been studied extensively, both experi¬
mentally and theoretically, there is very little published
information available on the behavior of non-Newtonian or
viscoelastic vortices, which is the point of emphasis of
this study. In Section 2.5, a description of some known
18

19
viscoelastic effects of dilute polymei. .. jiutions is pre¬
sented which serves as a background for the present study
of the vortex flows of viscoelastic fluids.
2.2 A Description of the Vortex Flow with an Air Core Present
When a circular motion is induced in water within a
tank, either by tangentially directed jets or simply stirring
with a paddle, and shortly after a central exit hole has
been opened in the bottom of the tank, the resulting cen¬
trifugal force field tends to form a depression in the water
surface. When the centrifugal force field is large enough,
the surface depression may reach the exit hole and thereby
form a hollow space in the core of the rotating fluid. In
general, these "air cores" are approximately of constant
diameter over the greater part of their length, and have
smooth, glassy surfaces as illustrated in Figure 2-1.
With smaller centrifugal force fields, the air core
will not complete, and only a small surface depression forms
(see Figure 2-2). In this case, it was observed that the
tip of the surface depression or "dimple" oscillated rather
rapidly and occasionally air bubbles appeared when the tip
broke. The air bubbles would fluctuate below the dimple for
a considerable time and then either rise to the free surface
or be discharged at the exit hole.
2.3 Classical Treatment of Vortex Flows
Before discussing the vortex motion in detail, some
quantities useful in describing vortex kinematics are intro-

20
Figure 2-1
A typical air core in vortex flow.

21
Figure 2-2 A typical surface dimple in vortex flow.

22
duced: The circulation, which is customarily given the
symbol T, is defined as the line integral of the tangential
velocity component around a closed curve; thus
r
o vdr
v d c
c
(2-1)
where v is a velocity vector, c is some closed path, dc
an element of this path, and r a radius vector from an
arbitrarily located origin to a point on the path (Figure
2-3) .
The circulation is related to the vorticity by Kelvin's
equation (Eqn. 2-2). The vorticity is defined as the curl
of the velocity vector: £ = V x v. Consider a surface S
bounded by the contour C; if is the component of vorticity
normal to the surface at any point, then from Kelvin's theorem
r =
£ x dS| =
r, ds
n
(2-2)
This relation follows from Stokes' theorem (Lamb, 1945) and
demonstrates that the circulation about any closed curve is
equal to the surface integral of the normal component of
the vorticity over any surface which is bounded by the given
curve.
While stream lines generally exist throughout all por¬
tions of a fluid in motion, a single vortex line may exist
in an otherwise irrotational flow, so that only those infini¬
tesimal fluid elements lying directly upon that line will
undergo "rotational" motion. If the vortex motion is viewed

23
v
Figure 2-3 Circulation and vorticity.

24
parallel to the vortex line on the cross-sectional plane,
it appears as a two-dimensional motion with the vortex line
appearing as a point. Given polar coordinates r and 0, with
r the radial distance from the axis of rotation, let v be
the circumferential velocity. Since the vortex proper is
considered as localized at a point and the motion is assumed
irrotational except at that point, one obtains
(2-3)
and by integration of this equation
(2-4)
rv = constant
The tangential velocity thus varies inversely with
the radius. This is the velocity relation for the so-called
potential or free vortex, since the motion is irrotational
except at the origin, where the vortex proper is located.
Another extreme case termed the forced vortex is formed
by assuming that the region occupied by the vortex is of
finite size with a uniform density of vorticity across it.
The cross section of the vortex tube is taken to be a circle.
The vorticity is now taken as constant rather than zero; thus,
1 8
C = — (rv) = constant
(2-5)
or integrating gives
2
2
rv + C
(2-6)

25
Since in such flows the tangential velocity is required to
go to zero on the axis of symmetry, one has C=0, i.e.,
This indicates that the velocity now varies linearly with
the radius, as in solid (rigid) body rotation. This profile
is known as the forced vortex, since to produce it, a torque
must be applied to the fluid and different amounts of work
are done on different streamlines (i.e., at different radii).
Although both of the vortex motions (free and forced)
have a certain degree of artificiality, motions approaching
them are often encountered. A more logical and physically
rational vortex motion was introduced by W. J. M. Rankine
in 1858. This is the "combined vortex" in which the forced
vortex is considered limited to a core region outside of
which is a free vortex. This simple joining of the two
previous extreme cases of free and forced vortices is shown
in Figure 2-4.
The Rankine combined vortex is a closer approach to
real-fluid motion in that the forced vortex region--which
must have resulted from some viscous action—is assumed to
be of finite size. For the real vortex flows, however,
there are still some small but significant departures from
the ideal forms of these classical vortices. A literature
survey of the well represented experimental and theoretical
work on vortex flows of real fluids (water in most cases)
will be given in the next section.

Figure 2-4 Characteristics of Rankine's combined vortex.

27
2.4 Previous Investigations on Real Vortices
In addition to the theoretical analysis of the motion
of a vortex in an inviscid fluid, laboratory investigations
have been undertaken to study small but significant depar¬
tures from the ideal forms. There have been a number of
experiments designed to generate concentrated vortices in
rotating tanks. The original aim of these works was to
obtain a laboratory model closely related to the meteoro¬
logical vortices, tornadoes, dust whirls and water spouts.
Long (1958, 1961) has made a detailed investigation of
vortices produced by extracting fluid through a sink
situated just below the free surface of a rotating tank of
water. A very different mechanism for driving vortices in
rotating tanks has been described by Turner and Lilly (1963)
and used later by Turner (1966) to obtain quantitative
information about these vortices. The vortices of Turner
and Lilly are driven by drag force exerted on the surrounding
fluid due to gas bubbles released near the axis of the tank.
These gas bubbles are caused by either nucleating carbonated
water or steadily injecting air through a fine tube. By
using a photographic method for velocity profile measurements
in these vortices, Turner found that the tangential and radial
velocities are independent of height, and the axial velocity
varies linearly with height. As will be seen later in Chap¬
ter IV, these features of the velocity profiles are quite
similar to those observed in the present experiments.

28
Another way of generating a vortex is the well-known
"bathtub drain" vortex, in which initial circular motion
may be introduced by either stirring with a paddle or by
tangential injection of entering fluid at the near wall
region. Early studies on this type of vortex were made by
Binnie and coworkers (1948, 1955, 1957), Einstein and Li
(1951), Quick (1961), and Helmert (1963). All experimental
measurements made in these works have shown that the varia¬
tion of tangential velocity is inversely proportional to
the radius. This will be true for a major region of the
vortex, but close to the center, the effect of fluid vis¬
cosity cannot be neglected and the tangential velocity is
required to go to zero on the axis of symmetry.
Kelsall (1952) was probably the first to measure the
detailed velocity profiles close to the axis of a hydraulic
cyclone. He adopted a very successful optical method in¬
volving suitable ultramicroscopio illumination and a micro¬
scope fitted with rotating objectives. Kelsall found that
the tangential velocity appears to vary linearly with the
radius from the axis, even if an air core is present. If
an air core rotates as a solid body, within a large core
of liquid that also rotates as a solid body, then the
effects of an air core on the surrounding flow should be
negligible, except for axial effects. Axial effects may
be important if the presence of an air core serves as a
restriction within the exit hole, hence a resistance to
flow.

29
Roschke (1966), using a graphic differentiation of
the static pressure distributions measured on the closed-
end wall, calculated the tangential velocity profiles in
a jet-driven circular cylinder. These indirectly ob¬
tained tangential velocity profiles were presented to give
some indication of the effect of exit-hole size and aspect
ratio (ratio of the length to diameter of the vortex cham¬
ber) . It was also noted by Rosche that use of probes in
vortex flows of this type should be avoided because even
very small probes have been found to produce significant
changes in the original vortex flow. Some of the problems
which arise in probing vortex flows have been discussed
elsewhere (Holman, 1961; Eckert and Hartnett, 1955).
The studies of strong jet-driven vortices were ex¬
tended to the case of weak vortices in which the air core
was narrow or only a small surface depression occurred
(Anwar, 1966). In his experiments, Anwar found that there
was no downward movement of the air bubbles that appeared
when the tip of the surface depression or "dimple" broke
occasionally. This caused him to conclude that the axial
velocity was zero at the axis of symmetry. Following the
downward movements of drops of dye placed on the free sur¬
face, Anwar estimated the position of maximum of the axial
velocity profile to be at a distance between 0.75 and 0.77
of the outlet radius. This finding contradicts the results
reported elsewhere (Granger, 1966; Dergarabedian, 1960) and
will be examined more carefully in this work.

30
The above experimental investigations on the vortex
flows are summarized in Table 2-1.
Perhaps the most striking single impression on pur¬
suing these references is the large number of complicated
"secondary flow" patterns observed for a wide variety of
experimental conditions and apparatus. An example of such
secondary flows occurring in a simple bathtub drain vortex
is shown in Figure 2-5. As we know, the fluid elements
near the bottom move more slowly than those within the main
body of the vortex because of friction. The slower-moving
fluid in the boundary layer is then forced inward by the
radial pressure gradient established by the more rapidly
moving fluid in the main flow Outside the boundary layer.
In Figure 2-5, it seems that most of the flow out of the
drain hole comes from the boundary layer and the fluid
above does indeed travel in circles.
The boundary layers that are formed on the end walls
of the vortex chamber have been studied previously without
regard to their effect on the primary or main vortex flow
that derived them (Taylor, 1950; Mack, 1962; Rott, 1962).
Rosenzweig, Lewellen and Ross (1964) investigated this
problem in further detail and attempted to compare theo¬
retical results and experiments by matching the measured
circulation distribution with the analytical predictions.
In all cases, it was found that there was still large dis¬
crepancy between experiments and theory: the measured
laminar Reynolds number was much higher than its theoret-

31
(b)
Figure 2-5 Flow visualization with dye injection.
(a) Dye marks the radially inward secondary flow near
the bottom of the vortex tank; (b) Dve marks the circular
streamlines of the primary flow high above the bottom.

Table 2-1
Summary of Experimental Investigations on Water Vortices
Particles Used in
Vortex Chamber
Range of Velocity
Investigator
Velocity Measurements
D(cm) x L(cm)
Vortex Driven Force
v(cm/sec)
w (cm/sec)
o
Kelsall (1952)
Aluminium powder
Hydraulic cylone
Sink/Jets
200-1000
0-200
(ID: 9)
Binne (1956)
Lycopodium powder
150 x
90
Sink/Jets
—
—
Long (1956)
Aluminium tristerate
6 x
125
Sink/Rotating tank
-
u>
ro
Anwar (1965)
Suspended fine particles
90 x
150
Sink/Jets
10-200
-
Roschke (1966)
Static pressure data
10 x
10 to 120
Sink/Jets
100-1500
-
Turner (1966)
Neutrally boyoant particles
15 x
30
Release of air bubbles/
0-30
0-3
Rotating tank
Granger (1966)
Globule of dye
60 x
120
Sink/Jets
—
0-700
Anwar (1966)
Suspended fine particles
90 x
150
Sink/Jets
0-350
0
the axial velocity along the centerline of the vortex chamber.

33
ical counterpart. According to these authors, the largest
discrepancy is probably due to the ambiguity concerning
the exhaust hole. Near the exhaust region, it was observed
that only a fraction of the boundary-layer flow passes
directly out of the exit hole while the remainder is ejected
axially to form the upward reverse flow (Rosenzweig, Ross
and Lewellen, 1962; Roschke, 1966; Kendall, 1962). As will
be seen later, the presence of boundary layers makes the
theoretical analysis of the vortex motion rather complex.
Recently, Dagget and Keulegan (1974) reported the ef¬
fects of fluid properties on the vortex nature. The results
show that the surface tension of the fluid does not appear
to affect the vortex flow significantly, and fluid viscosity,
on the other hand, plays a very important role in vortex
flows. It was found that as viscosity increases, the cir¬
culation decreases from inlet to outlet due to the increase
in viscous shear. Consequently, for the same initial circula¬
tion, a vortex with working fluid of high viscosity forms no
air core, whereas the prototype using a low viscosity fluid
would form an air core. Anwar (1967) studied the effects of
fluid viscosity in more detail. The experimental results
showed that the relationship between tangential velocity, v,
and radius, r, appears to be v rn = constant, with the power,
n, increasing with increase in viscosity. It is worth noting
that the relationship, v r11 = constant transfers to V~r as r
becomes small, where a solid body rotation occurs.

34
Although d'Alembert and Euler in 1750 considered
vortex kinematics to some extent, it may be properly stated
that analytical treatment of vortex motion started with
Helmholtz' classical paper (1850) "On Integrals of the
Hydrodynamic Equation Corresponding to Vortex Motions."
The proofs of some of Helmholtz' theorems were given later
by Kelvin (1867). And since the 1951 paper of Einstein and
Li, steady progress has been made in the analytical descrip¬
tion of laminar, incompressible vortex flows.
In an approximate treatment of vortices, originated
by Einstein and Li (1951), generalized later by Rott (1958),
and Deissler and Perlmutter (1958) , the axial velocity, w,
is arbitrarily taken as a discontinuous function of the
radius and has a jump at the radius of the exhaust. That
is
o
II
S
for r > r
e
w = a* z
for r > r > 0
e -
(2-8)
where a is constant. From the total flow rate, QT, one gets
a
2.
it r h
e
(2-9)
where h is the depth of the flow, and rQ is the radius of
exhaust. The assumption is made that the axial flow out of
the orifice is uniform. Continuity is next used to deter¬
mine a radial velocity which is independent of the axial
coordinate. The tangential velocity can then be determined

35
directly from the tangential momentum equation by simple
quadrature and is shown to be independent of the axial co¬
ordinate. The calculated tangential velocity is of the
form
where
4iihvr^
e
(2-10)
(2-11)
According to this solution, a definite magnitude of
the total volume flow, Q , is associated with a given tan¬
gential velocity distribution. Experiments, however, show
that the measured velocity distribution corresponds to a
much lower value of A than the one computed from Eqn. (2-11).
This approach ignores the presence of the boundary layers on
the end walls of the chamber. A large radial flow occurring
in the boundary layers on the end walls offers a ready ex¬
planation of the reason why the measured tangential velocity
distribution corresponds to a total volume flow much lower
than the one computed under the assumption that the entire
volume flow is distributed uniformly along the length of the
vortex chamber.
The volume flow which is available for the creation of
a vortex-like tangential velocity distribution is not the
total volume flow, rather it is the total volume flow minus
the secondary flow in the end-wall boundary layer. There-

36
fore, the assumed axial velocity (Eqn. 2-8) is not true and
a more realistic axial velocity distribution, obtained either
experimentally or theoretically, is definitely necessary in
predicting the tangential velocity profile.
Donaldson and Sullivan (1960) have shown that the most
general condition under which a solution of the Navier-Stokes
equations such that v = v(r) exists, is that the axial
velocity must be of the form
w = z f^(r) + Í2(r) (2-12)
Here functions of f^ and can be determined from the axial
momentum equation. As it turns out, the solutions of Ein¬
stein and Li, etc., are special cases of this solution.
Unfortunately, the flow, so determined, cannot be made
to satisfy the boundary conditions corresponding to the
desired geometry of the actual vortex tank (with stationary
end walls and exhaust-hole geometry). Rather, the class of
physical flows which Donaldson and Sullivan investigated by
means of Eqn. (2-10) are the flows produced within a
rotating porous cylinder by uniformly sucking fluid out of
such a cylinder (Figure 2-6). The flow in Figure 2-6 is
bounded at radius r=rQ by the rotating porous cylinder where
the axial component of velocity vanishes at the cylinder
surface. For the cases of jet-driven vortex flows, the axial
velocity vanishes at some distance between the axis and the
confined wall of the cylinder and hence the boundary condi¬
tion for Eqn. (2-10) is not as straightforward as it might

37
Q
out
Rotating Porous
Cylinder
Figure 2-6 Sketch of geometry for a rotating porous cylinder.

38
first appear. Moreover, it can be seen that the axial
32
and radial momentum equations and Eqn. (2-10) lead to „ ^ = 0.
3r3z
It is thus impossible for a solution of this type to satisfy
any problem in which the axial boundary condition forces a
radial variation in the axial pressure gradient, which does
occur in most vortices.
Long (1961), using a dimensional analysis, gave another
mode of axial velocity which satisfies the Navier-Stokes
equations and certain radial boundary conditions. His mode
of the axial velocity distribution is of the form
w = p f(f) (2-11)
This equation is substituted into the Navier-Stokes equa¬
tions and after some direct integration and rearrangement,
the original partial differential equations become ordinary
differential equations with only one independent variable,
r
—• Long actually solved the equation by using a numerical
method and showed that the behavior of the velocity compo¬
nent at large distances from the axis of rotation is
u â– +
-v
r
r
oo
V -* —
r
r
oo
w ->■ —
v/2

39
where v is kinematic viscosity and r is the circulation
at a large enough distance from the axis. Clearly, it is
seen that a particular feature of Long's solution is that
the axial velocities are of the same order as tangential
velocities even at large radii. Long's flow problem can
perhaps be best described as a swirling jet exhausting into
an unbounded fluid which has constant circulation. Actually,
in the limit of zero circulation it reduces the Schlichting's
jet problem (Schlichting, 1968). The case of interest here,
that of radial sink flow with strong circulation and exhausts
axially near the center line, is thus not included in this
class of solutions.
In contrast to the above approaches, Lewellen's (1962,
1964) asymptotic expansion method provides a general
solution capable of satisfying the boundary conditions in
real flows. His method can, in principle, take into account
any variation of the axial pressure gradient which is over¬
looked in Donaldson and Sullivan's solution. Lewellen's
analysis is in terms of a series expansion of the nondimen-
sional circulation function, T, and the nondimensional stream
?
OH
function, 1, in the parameter e = (^—=) ; here Q is a volume
1 K
00
inflow per unit axial length, is a constant circulation,
and H and R are characteristic axial and radial lengths re¬
spectively. Physically, this parameter is the ratio of rela¬
tive inertial to Coriolis force in a rotating fluid and is
known as the Rossby number which has been used to characterize
the effects of rotation on flows. Lewellen's argument is

40
that the parameter e is small in many flows of interest
where the radial velocity is small compared with the cir¬
cumferential velocity. This method is interesting in that
it allows one to calculate also the higher-order correction
terms, and assess the accuracy of zeroth-order approxima¬
tion (from equating the coefficient of e°) .
Although this expansion method is considered to be the
most relevant work on this subject, the accuracy of the solu¬
tion is still principally limited by the knowledge of the
axial boundary conditions on the stream function. This is
caused by the fact that this method is to solve a boundary-
value problem for the stream function, and the answer to what
boundary conditions are physically imposed in any particular
problem is not explicit. Let us consider the jet-driven
vortex for instance. It is seen that the stream function can
be readily specified only if the viscous boundary layer on
the end wall(s) is ignored. However, the assumption of
neglecting the boundary layer(s) on the end wall(s) of the
container is very poor. This is exactly the regime in which
the so-called secondary flow induced by the boundary layer
can dramatically change the boundary conditions and hence
produce a major effect on the outside or main flow. This
boundary-layer interaction problem has been studied in greater
detail by Rosensweig, Lewellen and Ross (1964) and by Rott and
Lewellen (1965). However, the lack of agreement between ex¬
perimental results and theory shows that the boundary condi¬
tions must be determined experimentally. If the true stream

41
functions occurring at the boundaries were known, then an
accurate distribution for the circulation in the container
could be obtained by using Lewellen's method. The inherent
difficulties presented by the boundary layers would probably
be the most difficult points in the study of the flow in a
vortex tank.
2.5 Viscoelasticity Considerations
High molecular weight polymers dissolved in solutions
may exhibit some viscoelastic effects which cannot be found
in Newtonian fluids. These solutions differ from purely
viscous fluids in that they display elastic properties such
as recoil and stress relaxation. Such responses are related
to the ability of the macromolecular chains to assume dif¬
ferent spatial configurations under deformation. As men¬
tioned earlier, many viscoelastic effects of dilute polymer
solutions have been discovered and examined in detail, but
few workers have investigated the viscoelastic effects on
vortex motion. A summary of some characteristic viscoelastic
phenomena in particular flow fields is given here to serve
as background material which will be of help in understanding
the influence of viscoelastic effects on vortex inhibition.
(i) Turbulent Drag Reduction:
This effect has been discussed in detail in Chapter I.
A close correlation between drag reduction and vortex in¬
hibition strongly suggests that both of these effects must be
due to similar, if not the same, viscoelastic mechanism.

42
(ii) Elongational Viscosity:
Viscoelastic theories predict that, under tension, a
viscoelastic fluid exhibits a very large elongational vis¬
cosity which increases with stretch rate (Lodge, 1964; Dean
and Marucci, 1971; Everage and Gordon, 1971). This is in
contrast to the low shear viscosity which is a decreasing
function of shear rate. The experimental evidence for this
type of behavior is quite limited, but studies such as
Metzner and Metzner (1970), Balakrishnan and Gordon (1975a)
seem to indicate that large values of elongational viscosity
(two to three orders of magnitude larger than that for New¬
tonian fluids) can be obtained at low concentration.
(iii) Recoil:
If the shearing stresses imposed upon a flowing visco¬
elastic fluid are rapidly removed, the fluid will undergo a
partial recovery of strain. This has been observed exper¬
imentally by following the movement of suspended air bubbles
in solutions (Lodge, 1964) or the deflection of an uncon¬
strained rotor in a cone-and-plate system (Benhow and Howells,
1961). In a recent note, Balakrishnan and Gordon (1975a)
reported that recoil occurred, following sudden cessation in
flows through an orifice, even at concentration levels as low
as 10 ppm. It appears that the highly dilute polymer solu¬
tion may still possess a "partial memory" of its rest state.

CHAPTER III
EXPERIMENTAL
3.1 General Description of Experiments
Although the original vortex inhibition experiment
is easy to operate, it is practically impossible to ob¬
tain any detailed knowledge of the velocity field kinematics
in such an experiment. The reasons are: (i) Since the
liquid in the tank is stirred arbitrarily with a paddle in
order to introduce some initial circulation motion, the
amount of vorticity introduced is hard to be controlled or
estimated accurately, and (ii) The system is unsteady, and
no measurements can be made during the short period in which
the vortex develops.
In this work, therefore, a steady, continuous, jet-
driven vortex will be used -as an approximation to the actual
vortex inhibition experiment. The vortex chamber comprises
a transparent cylindrical tank and the vortex flow is driven
by tangential injection near the cylindrical wall of the
chamber. Tracer particles were inserted into the fluids and
the primary velocity data were obtained from a series of
streak photographs of the steady vortex flow in the chamber.
The flow field was illuminated by an intense sheet of light.
Some such sheets of light were parallel to and included the
43

44
axis of the tank. Some other sheets of light were per¬
pendicular to the axis. The entire vortex chamber was
mounted on a movable support so that it might move ver¬
tically to be illuminated at any desired horizontal cross
section. A description of the photographic technique
along with the necessary optical equipment is given. The
physical properties of the test solutions were determined
using conventional or specialized techniques described
below.
3.2 Vortex Chamber and Flow System
A schematic diagram of the flow system including the
vortex tank along with its principal dimensions is shown
in Figure 3-1. The tank was 66 cm high and 45 cm in dia¬
meter. The liquid level was maintained at 54.5 cm relative
to the tank bottom by manually controlling the rate of
fluid entering. The fluid circulating within the tank en¬
tered through two vertical pipes placed opposite each other
on the circumference. The walls of both pipes were perfo¬
rated at equal spaces, then welded with small injection tubes
so as to distribute the entering fluid uniformly in a ver¬
tical direction. The fluid thus entered tangentially through
an exit hole centrally located in the bottom, and discharged
into air. For moderate circulation, injection tubes of 1/4
inch bore were used, and were replaced by tubes of 1/16 inch
bore when high circulations were required. The two vertical
pipes were fed from a 50-gallon head tank through two flex¬
ible tubes. The feed rate was controlled by a gate valve.

45
Figure 3-1 Vortex chamber and flow system.

46
The bottom of the vortex chamber was provided with a
central opening to which various sizes of orifices could be
fitted. Vortices with either an open air core or a small
dimple could be obtained by varying the size of orifice.
The circulated fluid was collected in a collecting tank,
pumped through the external closed circuit, and finally to
the head tank back into the supply tubings. The fluid was
originally pumped with a Randolph pump which is supposed
to impart gentle shearing action on the recirculating fluid.
Preliminary tests showed, however, that even this gentle
shearing action could cause polymer degradation after several
runs. It was found convenient to shear the solutions until
a stable state was reached which, however, in the present
study caused the solutions to become too hazy for good photo¬
graphs to be obtained. It is, therefore, decided only the
fresh polymer solutions were used and would not be pumped
back to the head tank for repeated use. On the other hand,
for tap water and other Newtonian solutions we still operated
with the recirculation loop.
3.3 Optical Assembly
The optical assembly used for obtaining the streak photo¬
graphs is illustrated schematically in Figure 3-2.
The flow field was illuminated in thin cross section by
an intense sheet of light. The light beam was interrupted
by a rotating pie-shaped disk, from which alternate pieces
of pies were cut. This was simply a mechanical strobing

c
A: LIGHT SOURCE
B: HEAT-ABSORBING GLASS
C: SLIT
D: STROBE DISK
E: PULLY BELT
F; STROBE MOTOR
G: POWER SUPPLY
H; ELECTRIC TIMER/COUNTER
I : PHOTOCELL
J: SLIT
K: CAMERA
L,8 L2: CONDENSING LENSES
M: VORTEX CHAMBER
Figure 3-2 Schematic optical assembly
J
4^

48
apparatus. The strobe disk was driven by a 1/4 HP, 1725
r.p.m. motor. The period of the successive illumination
intervals was measured with an 1191-Type, electric timer/
counter made by General Radio Company. The periods can
-12
be measured from 100 ns to 1 Gs. (10 s) with a precision
-15
of up to 1 fs (10 s). Various illumination intervals
were obtained by changing the number of slits on the strobe
disk. In this experiment, all runs were measured with
either a one-slit of a two-slit disk.
All the photographs were taken with a Bell & Howell,
f-1.8 camera. To obtain closer photographs, three close-
up attachements were used with the camera. All photographs
were obtained using Kodak Tri-X films of ASA 400 rating.
The films were developed for 11 seconds at 65°F in order to
increase the degree of contrast. Exposure times and depths
of fields were determined by trial and error. All the photo¬
graphs were taken in complete darkness except for the light
plane. The entire system was covered with black plastic
cloth to stop all stray light.
The remainder of the optical assembly included a light
source, two condensing lenses, a heat-absorbing glass and a
mechanical slit. It turned out that getting a sufficient
amount of light to pass through the large vortex chamber of
liquid was one of the most difficult problems to be solved.
Various kinds of lamps including a carbon arc lamp were tried
* *
without success. An ELH or ENG 300-Watt bulb with suitable
*
The 300-Watt ELH Quartzline bulb produces less heat than a
500-Watt CBH lamp, but offers equivalent light output. The
ELH or ENG bulb is made by General Electric Company.

49
optical assembly finally proved adequate. The ELH bulb
was placed at the focus point of lens . A thin slit
passed only the desired planar beam and the remaining light
was stopped. Lens L2 was placed in front of the slit and
served to form an image of approximately double magnifica¬
tion of the slit inside the vortex. Under these circum¬
stances, the image formed by lens L2 had a proper depth of
focus thus giving a clear slit of light across the tank.
3.4 Tracer Particles
Various types of tracer particles were tried without
success. Microglass beads of about 50 microns in diameter
were used first, but these could not be photographed with
the equipment available. Aluminum powder was also tried
without success due to the gravity effect. Finally the
particles called Pliolite (Goodyear Rubber Co., Akron, Ohio)
were found to be suitable to act as targets for velocity
measurement. Since the density of Pliolite is quite close
to that of water, the gravity effect is negligible. The optimum
size of particles was 15Op ~ lOOp. In this range, the par¬
ticles are small enough for inertial effects to be negligible
but large enough to provide necessary reflected light. The
concentration of particles which gave the best photographs
was determined by trial and error.
3.5 Experimental Fluids
The Newtonian fluids used for comparison purposes were tap
water and Karo-brand corn syrup solutions of various viscosities.

50
The polymers used in this study are listed in Table
3-1. Separan AP 273 (S-273) and SP 30 (S-30) are partially
hydrolyzed polyacrylamides (approximately 25-35% hydrolysis)
Both S-273 and S-30 have the structure of
ch2
CH
00
nh2
CH2
CH
C= 0
0~
y
while the S-273 is of higher molecular weight than S-30
(S-30 having a M.W. of about 3 x 10^ in an undegraded state)
Versicol S25 (V-25) is a partially neutrallized poly(acrylic
acid) having the structure
C=0
0~
n
Due to the anionic character of these partially hydrolyzed
or neutralized polymers, their solution viscosities are
very sensitive to change in ambient ionic concentration.
Thus, molecular conformation studies could be carried out
with these polymer solutions by varying the solution pH or
by addition of salt.
Unlike the above polymers, Polyox WSR 301 (P-301) is
not a polyelectrolyte. It is a poly(ethylene oxide)
4 CH2-CH2-0 4n . The average molecular weight of this
polymer has been estimated from the intrinsic viscosity
measurement to be about 4.0 x 10^. P-301 is known to be

51
Table 3-1
Polymers Used
Chemical
Manufacturer Trade Name
Manufacturers _g
of Polymers (x 10 )
Poly(ethylene oxide) Union Polyox
Carbide WSR 301
4.0
Polyacrylamide Dow
Separan
AP 273
7.5
Polyacrylamide Dow
Separan
AP 30
Sodium
Carboxymetyl Hercules CMC 7H
Cellulose
3.0
0.2
Carboxy-
Polymethylene
B.F. Carbopol
Goodrich 934
Allied Versicol
Colloids S25
Poly(acrylic acid)
20

52
very susceptible to shear degradation. Another polymer, CMC 711,
a high viscosity grade of sodium carboxymethylcellulose was
also used in this research. This polymer exhibits rather
weak drag-reducing properties in low concentration levels.
The range of polymer concentrations used was 1 to 100 ppm,
and all the solutions were made in deionized water.
The viscous properties of the Newtonian solutions were
determined using a capillary viscometer described elsewhere
(Van Wazer, et al., 1963). For the polymer solutions, the
viscosities at moderate shear rates were measured using a
Brookfield cone-and-plate viscometer while a specialized ap¬
paratus was designed to obtain viscosity data at very low
shear rates. A description of this low shear viscometer will
be given later in this chapter.
3.6 Procedures
In order to prevent speed drift, the strobe disk was
allowed to run for several minutes to reach a steady operating
condition. The power for the light source was then turned on
and the frequency or period of the light pulses was readily
read out in 8 digits from the Type-1190 electric timer.
The preliminary experiments showed that the tracer par¬
ticles previously added to the vortex might become less and
less and finally disappeared during the waiting period for
getting a steady state. This is caused by the radial influx
of the vortex flow in the present apparatus. Therefore, an
alternative was to insert the tracer particles a few minutes

53
before the photographs were taken. Tracer particles were
first dispersed in a beaker containing some working fluid
of the system. This particle solution was then introduced
into the system using a long, fine pipet. This procedure,
although suffering from introducing small disturbance into
the vortex flow, appeared to be adequate after some ex¬
perience had been gained through trial and error manipula¬
tion of the mixing conditions. Preliminary tests showed
that the introduced disturbance decayed very quickly and
the flow became steady in a few minutes.
With all of the operating condition of the system at
steady state a minimum of approximately 30 photographs were
taken at each given axial position.
3.6.1 Preparing polymer solutions
The solutions were prepared by making a master batch
of about 0.1% by weight in deionized water. The required
amount of polymer was weighed and added to about 50 ml of
isopropanol mixture and was then poured into a well-agitated
vessel containing deionized water. As soon as the mixture
was added the agitation was stopped. This procedure mini¬
mized polymer degradation during agitation. The master
batch was allowed to sit for about 24 hours before used. The
master batch was then diluted with deionized water to yield
solution of the desired concentration and poured into both
the head tank and the vortex chamber. The level of the
solution in the vortex chamber was set at the desired position.

54
3.6.2 Making a run
At the beginning of a run the position of the light slit
was checked and the camera was brought to focus at the mid¬
dle of the light slit by focusing on a horizontal or vertical
calibration rule. The magnification factor of the system
was determined from the photograph of the rule.
When making a run, the valve connecting the head tank
and the supply hoses was partially opened so that the liquid
entered the vortex tank through the tangentially positioned
injection tubes. The plug in the bottom of the tank was
then removed and the valve was readjusted until the level
of the liquid was kept at the desired position. The liquid
discharged into the air was collected and then pumped back to
the head tank for repeated use. For the cases where poly¬
mer solutions were used as working fluids, the pump was
turned off and the system was left open. Approximately 40
minutes were required for the flow to reach a steady state
after the flow started. Measurements were taken only after
the steady conditions were established.
3.7 Velocity Data Analysis
The primary motion of a fluid element in the vortex flow
is motion in a circle about the cylinder axis. It has been
found that the axial flow occurs only in the core region of
radius of about one centimeter or less. Excluding this core
region, the paths of the tracer particles appear as concentric
circles when viewed parallel to the axis of the cylinder.

55
These facts make it convenient to photograph the paths of
particles .in a thin illumination planar region of about
0.5 cm thick and 10 cm wide, centered about a particular
axial location. A typical top view photograph showing
particle traces in vortex flow is reproduced in Figure 3.3.
In principle, one should be able to obtain radial and
tangential velocities from such a photograph. In practice,
the radial velocity is too low to be measured satisfactorily,
and only tangential velocity data are obtained. The photo¬
graphs were analyzed using a Beseler model 23C-Series II
enlarger in the darkroom. The enlarger projected the photo¬
graph on a polar coordinate paper. The appropriate magnifica¬
tion was obtained by adjusting the height of the enlarger
lamp house from the paper. The magnification factor was
determined using the photograph of the calibration rule.
The center point of the particle traces on the photograph
was located by slightly moving the polar coordinate paper
until the circles of the streaks coincided with the polar
coordinates. For a given particle travelling around the
center point, the following quantities were measured and
recorded: (a) The radius of the particular particle path
was recorded as r , (b) The magnification factor was re¬
corded as M, (c) If we define 0. and 0 as the angles of
the leading edge of one streak and the leading edge of the
next nth streak of the same trace in the polar coordinate,
respectively, the difference between 0^ and 9^ was recorded
as A9. The number of streaks was recorded as n. In this

56
Figure 3-3 Typical particle-trace photograph (Top view).

57
way the actual size of the individual tracer particle is
eliminated. The number of streaks was selected so as to
allow measurement of A0 to be large as compared to any
error in measurement, and (d) The frequency of the light
pulse was recorded as f.
The local tangential velocity v can be calculated by
[An/ (n-1) ] • f • tt • r
V = 18 0•M E (3_1
where r and 180 are the conversion factors for changing
the unit of A0 from degree to radian.
The position coordinate was given by
r
(3-2)
In the above manner,' tangential velocities were ob¬
tained only in the region quite far away from the center
point. In the region near the center point the streaks of
the tracer particles became ambiguous due to the axial flow.
For this region where the axial flow is important, the flow
field must be illuminated in different ways and a side view
photograph is taken to obtain the velocity data.
The flow field in the core region was illuminated by a
slit of light of about 3 cm wide and 10 cm in height. This
sheet of light was parallel to and included the axis of the
cylinder. The camera was then placed at the right angle to
the slit of light and brought to focus at the axis by
focusing on a calibration rule placed in the middle of the

58
cylinder. A photograph of the rule was taken to determine
the magnification factor of the film.
In Figure 3-4 a schematic representation of the essen¬
tial features of a streak photograph is given; samples of
the actual streak photographs are reproduced in Figures
3-5 and 3-6. The streaks appeared as a series of dashes-
helix due to the chopped light beam. For each of the
helixes, the following information was recorded: (a) The
radius of the helix was recorded as r , (b) The magnifica¬
tion factor M, (c) The frequency of the light pulse f,
(d) The apparent axial coordinate for the leading edge of
the streak on the end of a turn of the helix was recorded
as Z^, (e) The apparent axial coordinate for the leading
edge of the streak on the end of the "next" turn was re¬
corded as Z , (f) The axial distance between the leading
edges of the two streaks was recorded as 1 , and (g) The
number of streaks in this turn was recorded as n . The
P
following formula was then used to calculate local veloc¬
ities and position coordinates. The local axial velocity
w and tangential velocity V are given by
1
f
w
P
(3-3)
n
. m
P
v
2tt • r • f
P
n • M
P
The position coordinates were given by

59
Figure 3-4 Illustration of the side view photograph.

Figure 3-5 Typical particle-trace photograph (Side view 1).

Figure 3-6 Typical particle-trace photograph (Side view 2).

62
z
z, + z
1 n
(3-5)
r
(3-6)
The axial velocity along the centerline was measured
similarly by timing the motion of a tracer particle
judiciously placed on the tip of the dimple so as to flow
downward vertically along the axis of the cylinder. The
streaks appeared as a series of vertical dashes-line and
are shown in Figures 3-5 and 3-6.
The radial velocities in the body of the vortex were
too small to be measured satisfactorily. An alternative
for obtaining the radial velocity data was by applying con¬
tinuity. Remembering that circular symmetry exists, the
vertical volume flow through any horizontal annulus is
given by
6
Q
2itw rdr
(3-7)
Consequently, if w*r is plotted against r for posi¬
tions along a radius at a given horizontal level, the area
limited by the curve obtained, the r axis, and the limiting
values of w*r at r^ and are proportional to the volume
flow through the annulus. From graphs of w*r against r for
a series horizontal levels, by using a method of graphical
differentiation, average radial velocities were calculated
at selected positions in the vortex flow.

63
3.8 Concentric-Cylinder Viscometer for Very Low Shear Rates
There exist excellent concentric-cylinder or Couetter
viscometers which have been used to obtain viscosity data
at low shear rates, but because of the expense involved in
their construction, they have not found widespread use in
the laboratories. These viscometers generally consist of
an outer cylinder driven at a constant speed. The torque
transmitted through the fluid to a static inner cylinder
is measured by some appropriate device which is probably
the most expensive part in constructing the viscometers.
A very different mechanism for driving the rotating cylinder
has been described by Zimm and Crothers (1962). They used
a freely floating inner tube, supported by its own buoyancy
and held in place by surface forces such that the inner tube
or "rotor" floats concentrically with the outer tube or
"stator." A steel pellet is glued in the bottom of the rotor,
to which a constant torque is applied by the interaction of
the steel pellet with a rotating applied magnetic field.
This design utilized the magnetization properties of the ferro¬
magnetic core in the rotor to generate a torque on the rotor
by an external rotating magnet. However, we found that the
torque so produced was not sufficiently stable for our pur¬
poses, and moreover, the speed of the rotor drifted for sev¬
eral hours after changing from a strong to a weak magnetic
field. In addition, a serious wobble of the rotor axis,
probably due to imperfectly placing the steel pellet in the
rotor, lead to erratic speeds and unreproducible data.

64
The present viscometer, shown schematically in Figure
3-7, is a modification of the original design, in which we
used a nonferromagnetic aluminum sheet instead of the steel
pellet in the rotor. The torque produced in the new instru¬
ment comes from the interaction between the original applied
magnetic field and an induced magnetic field resulting from
the generated "eddy currents" throughout the surface of the
aluminum. Consider a cylindrical sheet in a rotating mag¬
netic field perpendicular to the surface of the sheet but
confined to a limited portion of its area, as in Figure
3-8(a). The magnetic field is moving across element 0 in
which an emf is induced. Elements A and B are not in the
field and hence are not seats of emf. However, in common
with all the other elements located outside the field, ele¬
ments A and B do provide return conducting paths along which
positive charges displaced along 00' can return from O' to
0. A general eddy circulation is therefore set up in the
cylindrical sheet somewhat as sketched in Figure 3-8(b).
We therefore see that a torque is generated which attempts
to allign the induced and applied magnetic field. The period
of revolution of the rotor P (seconds per revolution) can be
represented in terms of apparatus constants and the liquid
viscosity u by (Appendix A)
P - P
8tt y (h+Ah)
m
K P
m m
(3-8)

65
CORK
THERMOSTAT JACKET
CIRCULATING FLUID
TYGON INLET TUBE
MENISCUS
ROTOR
STATOR
ALUMINUM SHEET
IRON POLE PIECE
MAGNET
MOTOR SHAFT
SYNCHRONOUS MOTOR
Figure 3-7
SCHEMATIC DRAWING OF A LOW-SHEAR-RATE
CONCENTRIC-CYLINDER VISCOMETER

66
Figure 3-8 Eddy currents in the aluminum sheet located inside
a rotating magnetic field.

67
in which
R1=
rotor radius
R2 =
stator radius
P =
m
period of revolution of the magnet
3
ll
Torque constant
h =
rotor height
Ah =
end correction
If h
is held constant in order to obtain constant
rotor height, the relative viscosity of test solution
^rel
is simply
P - P
V)rel P - P (3 9)
o m
where P and Pq refer to solution and solvent, respec¬
tively. The average shear rate can be calculated when
some
assumptions are made (Appendix A)
8tt R2 R2 In fR N
p(R2_R2)2 (RJ
For our viscometer, it was calculated that the apparatus
constant
8it R2 R2 rR -v
5 In = 33.69358 (3-11)
(R^-rJ)2 IriJ

CHAPTER IV
EXPERIMENTAL RESULTS
4.1 General Flow Pattern
The flow visualization results presented here give an
overall picture of the flow pattern in the vortex tank.
Visualization was accomplished by observing the motion of
a water-soluble dye, which is actively fluorescent under
illumination. The vortex apparatus was described earlier
in Chapter III. Injection was made using existing pres¬
sure taps through which a probe (0.0005 m. ID) was passed
(see Figure 4-la). The dye was injected continuously over
a short interval with the injection pressure controlled at
any desired level by controlling the dye-reservoir pressure
with a valve. The tank was illuminated using a Kodak slide
projector, and a "slit-shaped" slide allowed the light beam
to be focused on a narrow vertical section of the fluid.
With this arrangement, one observes primarily the axial but
also the radial flow components.
Initially, the dye was injected at different depths in
the axial direction along the vortex chamber. However, it
was discovered that if the probe tip is positioned near the
bottom boundary layer, the dye is convected upward and at a
later point in time the dye completely fills the main stream
68

69
To Dye Reservoir
(a)
Support
V''Ns'^_ Exit Orifice
(b)
Figure 4-1 Dye injection probe arrangement.

70
from bottom to top and a picture of the overall flow struc¬
ture is obtained. This occurs as a result of the large
"secondary" flow in the bottom boundary layer (Rosenzweig,
Ross, and Lewellen, 1962; Kendall, 1962). The injected dye
is carried along the tank bottom, a fraction passing directly
out of the exit orifice, while the remainder is "ejected"
upward (see Figure 4-3). Figure 4-lb illustrates the posi¬
tion of dye injection, near the tank bottom.
All the photographs of the dye patterns were taken under
conditions of "steady" flow, with all controllable experi¬
mental parameters held constant. The experimental conditions
for the quantitative flow field measurements are the same as
those for the present visualization study. Here, the term
"steady" flow is used rather loosely; the vortex flow was not
precisely steady and often was decidedly nonsteady.
The flow pattern was studied first in water. After the
steady state had been reached, a small amount of dye was in¬
jected continuously for approximately 300 sec. A series of
photographs showing the dye pattern vs. time after injection
is given in Figure 4-2. Dye was released within the bottom
boundary layer and spiraled radially inward. Figure 4-2a
shows that a fraction of the dye passes directly out of the
exit orifice, while the remainder is abruptly ejected up¬
ward near the sharp edge of the orifice. These "eruptions"
or "bursts" occur suddenly and are the origin of the tran¬
sient, fast-moving counterflows. These counterflows in the
outer annular region, i.e., outside of the region of strong

71
(a)
Figure 4-2 Development of the dye pattern with water resulting
from bottom boundary-layer injection. The experimental
conditions are: Depth of the surface dimple =1.2 cm,
diameter of t^he exit orifice = 0.516 cm, total ^olume
rate = 49 cm /sec, entering circulation = 25 cm /sec.
A small amount of dye is introduced continuously for a
period of 300 sec. The pictures show the dye pattern
in time-elapsing sequence: (a) 35 sec: (b) 100 sec;
(c) 400 sec; (d) 650 sec; (e) 1,200 sec.

(b)
Figure 4-2 (Continued)
t\j
(c)


axial downward flow, appear turbulent. At a later point
in time (Figure 4-2b), the outer annular region extends
further upward and becomes a typical conically shaped counter¬
flow region within which is a center jet. Close observation
of the counterflow near the orifice revealed that the
bursting process was generally not steady. The unsteady
nature of the bursting process is illustrated by Figures 4-2c
and 4-2b. The dye front of the counterflow (see Figure 4-2b)
is observed to go up and down occasionally. This is probably
caused by the unsteadiness of the bursting process which is
the source of the counterflow. It is illustrated in Figure
4-2c (300 sec. after Figure 4-2b) that the dye front becomes
lower than that in Figure 4-2b. Of course, the dye front
will finally reach the top surface (see Figure 4-2e). The
outer annular region near the bottom is dark since it is
supplied with fresh fluid from near the wall. Some of the
injected dye is subsequently drawn into thin sheets wrapped
around the axis of rotation as shown in Figure 4-2d. The
formation of these sheets must be attributed to the axial
shear flow. The dye pattern in Figure 4-2e is seen to con¬
tain a cell of recirculating flow between the axis and the
wall of the cylinder. The dye is found to recirculate in
the cell which dissipates very gradually and in some cases
remains detectable up to 1/2 hour after injection of dye.
Similar observations have been reported by Turner (1966),
and Travers and Johnson (1964).

75
From these dye studies, certain general features of
the flow could be deduced. As illustrated in Figure 4-3,
the actual flow consists of a strong vortex-type flow
superimposed on the indicated flow pattern. Along the
axis of the vortex, a strong center jet exists, designated
as region (I). This region is fed mainly from radial con¬
vection from the outer annular region (II) throughout the
length of the vortex, and partly from the free surface
region (III) near the center of the surface. The radial
flow occurring in the boundary layer on the bottom of the
vortex chamber was found to be quite large using hot wire
probes or pitot tubes. The experimental results reported
by Kendall (1962) and Owen et al. (1961) indicate that the
boundary-layer flow (i.e., the so-called "secondary flow")
carried radially inward along the rigid end wall can be of
the same order of magnitude as the total mass flow through
the vortex chamber. The significant radial flow occurring
in the bottom boundary layer (IV) splits into two portions;
some is discharged directly out of the exit orifice, and
some is ejected outward from the edge of the exit orifice
into the tank. The ejected mass flow may be caused for
two reasons. First, boundary layer theory itself predicts
the sudden eruption of the secondary flow near the center
of the end wall (Moore, 1956; Burggraf et al., 1971).
Second, certain discotinuities in the end wall geometry,
such as the sharp edge of the exit orifice, can also induce
mass ejection (Rosenzweig, et al., 1962; Kendall, 1962;

76
Figure 4-3 Sketch of general flow structure in the vortex.

77
Roschke, 1966). The eruption region (VII) invariably
appeared to be turbulent. Outside of region (II) are two
annular zones (V) and (VI) in which the fluid possesses
only a downward and upward drifting velocity, respectively.
(These regions explain the stratified flow pattern as shown
in Figure 4-2d). In general, the annular structure
(regions (II), (V), and (VI)) remains qualitatively similar
when the circulation is varied. Outside of these regions,
the vertical velocity is very small. Rosenzweigh, Ross, and
Lewellen (1962) also gave a similar composite sketch of their
observed flow pattern in a closed jet-driven vortex.
Flow patterns for two typical polymer solutions known
to be effective drag reducers and to exhibit vortex inhibi¬
tion at very low concentration levels were studied next:
Polyox WSR 301 (P-301) , Qyj- = 3 ppm; and Separan AP 273
(S-273), CVI = 2 ppm. A slightly higher concentration above
the respective for each polymer was used in the visualiza¬
tion study in order to obtain greater contrast between the
flow patterns for water and the polymer solutions. Figure
4-4 is the time-lapse sequence of photographs showing the
dye pattern development for P-301 at a concentration of 10
ppm. Figure 4-5 is the similar time-lapse sequence for
5-273 at 3 ppm. The controllable conditions for both cases
were the same as those for the water vortices.
Comparing the sequence of the dye pattern between those
of water and P-301 solution, it appears that except for dif¬
ferences in the shape of dye front the apparent difference

(a)
(b)
Figure 4-4 Development of the dye pattern with Polyox WSR 301 solution, 10 ppm.
The experimental conditions are the same as those described in figure 4-2.
The pictures show the dye pattern in time-elapsing sequence: (a) 35 sec;
(b) 100 sec; (c) 400 sec; (d) 650 sec.

(a)
Figure 4-4 (Continued)
(b)

Figure 4-5 Development of the dye pattern with Separan AP 273 solution, 3 ppm.
The experimental conditions are the same as those described in figure 4-2.
The pictures show the dye pattern in time-elapsing sequence: (a) 100 sec;
(b) 200 sec; (c) 400 sec; (d) 650 sec.


82
is the relative mixing or diffuseness of the dye (compare
Figures 4-2d and 4-4d). The dye streak in Figure 4-2d con¬
sisting of well-defined concentric regions gives a good
demonstration of the laminar character of the flow. In
â– k
Figure 4-4d, there occurs a quasi-cyclic "eruption" or
"burst"^ at the intermediate distance from the vortex axis,
leading to a rapid mixing of the dye. Such an unsteady
bursting process is unable to be illustrated in Figure 4-4d
because it occurs instantaneously and quickly interacts with
the high-speed surrounding fluid and appears as a gray area
located near the vortex axis (see Figure 4-4d). The details
of the bursting process can be illustrated more clearly using
special dye injection technique which will be discussed in
Section 5-6.
The series of photographs shown in Figure 4-5 is for
S-273 solution. It is observed that the major difference
in the dye patterns from those of water is the remarkable
enlargement of the central axial flow region. (The bright
area in Figure 4-5d located near the vortex axis represents
this axial-flow region.) The axial velocity is seen to be much
smaller than that in water vortex, as qualitatively pre¬
dicted by Rott (1958) as a consequence of the viscous effect.
*
A quasi-cyclic process means that a sequence of events
repeats in space and time, but not periodically at one place
of time nor at one time in space.
rThe "bursts" here are different from those described before
which originated near the sharp edge of the exit orifice.

83
More detailed pictures of the eruption region near the
exit orifice are given in Figure 4-6. Figure 4-6b shows that
the presence of S-273 greatly smoothes the fluctuations of
the ejected fluids and the counterflow originating at the
bottom of the chamber apparently spreads to a larger radius.
In Figures 4-6a and 4-6c, the stronger fluctuation of the
erupted fluids leads to more rapid mixing of dye; the finer
details of individual dye filaments become smeared out.
This makes the upward flowing regions for both water and
P-301 become vague and indistinguishable from each other.
In general, the photographs show that the difference
in flow pattern between water and S-273 is the "size" of
the central axial-flow and the annular counterflow regions.
:k
For S-273, both these flows are much weaker than those of
water, which are actually observed by the movement of the
dye fronts during the course of experiments. However, ex¬
cept for differences in the shape of stratified structures
of the dye interfaces, there is no apparent difference
between water and P-301. Therefore, it is rather difficult
to distinguish between water and P-301 from the dye pattern.
4.2 Physical Properties of Test Fluids
The test fluids used in this research may be conveniently
classified as follows:
•k
Whether a flow is strong or weak is seen through the move¬
ment of dye fronts and cannot be demonstrated in the present
still photographs.

84
(a)
Figure 4-6 Close observation of the dye pattern near the exit
orifice. The experimental conditions are the same
as those described in figure 4-2. (a) Water;
(b) Separan AP 273 solution, 3 ppm; (c) Polyox WSR
301 solution, 10 ppm.

(c)
Figure 4-6 (Continued)

86
1. Newtonian: Water, Corn syrup-water
2. Viscoinelastic: Carbopol 934
3. Viscoelastic: Polyox WSR 301 (P-301)
Separan AP 273 (S-273)
Separan AP 30 (S-30)
Versicol S 25 (V-25)
CMC 7H
The vortex inhibition ability of these test fluids is
listed in Table 4-1. It is worth noting that vortex for¬
mation can also be suppressed in a Newtonian fluid if the
shear viscosity is high enough.
In the present experimental setup, a large amount of
liquid (about 300 liters) is required for each run to obtain
a complete flow field measurement. It is thus rather im¬
practical to use corn syrup to raise the fluid viscosity up
to, say, 4 centipoise, as this can require as much as 150
liters of corn syrup per run. The corn syrup solutions also
promote bacterial growth within one day of preparation, thus
making the solution hazy and not usable. The addition of a
small amount of sodium benzonate fails to retard bacterial
growth. For these reasons, Carbopol 934 and CMC 7H were
used to increase viscosity in view of the fact these polymers
display negligible or very slight elasticity. Several inves¬
tigators, including Dodge and Metzner (1959) and Kapoor (1963),
have demonstrated that Carbopol solutions are not drag re¬
ducing nor do they demonstrate any recoil upon removal of
shear stress. However, a serious drawback of Carbopol 934

87
Table 4-1
Summary of Vortex Inhibition Data
Test Fluid
CVI, ppm
S-273
2
S-30
7.5
P-301
3
V-2 5
3
CMC 7H
75
Carbopol 934a
400
Corn Syrup - Water
450,000b
Neutralized with 0.4 gm. of sodium hydroxide/gm. of
Carbopol 934.
bThe viscosity of this 45% corn syrup - 55% deionized water
solution is about 5 centipoise.

88
in the present study is that above 200 ppm in water, the
solutions are so hazy that no streak photographs of the
tracer particles may be taken. For this reason, the test
fluid used to obtain higher viscosity levels was CMC 7H, a
high viscosity grade sodium carboxymethylcellulose. Even
at concentrations as high as 2,500 ppm in water, CMC 7H
was shown to be only very slightly viscoelastic (Goldin,
1970). Ernst (1966) and Pruitt (1965) have demonstrated
that CMC 7H is much less efficient in drag reduction than
the high molecular weight poly(ethylene oxides) and poly¬
acrylamides. The measured friction factor - Reynolds num¬
ber data for a 100 ppm solution of CMC 7H are presented
in Figure 4-7. At low Reynolds number, the data of CMC 7H
16
lie above the line f = — because of the increased solution
Re
viscosity (the Reynolds number is calculated by using the
solvent viscosity). At high Reynolds number, the friction
factor for CMC 7H is indistinguishable from that of the sol¬
vent (water), implying the nondrag-reducing ability of CMC 7H
for a concentration of 100 ppm. This finding further con¬
firms that CMC 7H does not display elasticity, at least at
low concentration levels.
For the corn syrup solution, viscosities were measured
with the Cannon-Fenske capillary viscometer. For the polymer
solutions, the specialized low-shear, coaxial-cylindrical
viscometer described in Chapter III was used to provide vis¬
cosity data at shear rates ranging from 0.05 to 5 sec ^.
The data of Table 4-2 indicate the precision of this instru¬
ment as currently used.

Friction Factor
89
Figure 4-7 Friction factor vs. Reynolds number for CMC 7H
100 ppm in 1.09 cm tube.

Table 4-2
Comparison of Low-Shear Viscometer with Capillary Viscometer
Solution
rel
Shear Rate
, -1.
(sec )
Capillary / Viscometer
^rel^rel ^rel
20% gycerol- 1.071
80% deionized water
30% glycerol- 1.100
70% deionized water
3.749
1.962
2.763
1.914
1. 694
1.923
0.806
1.943
0.355
1.958
0.186
1.987
0.117
1.962
0.096
Ave.
1.943
1.949+0.0219
2.403
2.723
1.462
2.712
0.920
2.742
0.540
2.738
0. 253
2.785
0.187
2.763
0.133
2.786
0.084
2.752
0.045
Ave.
2.791
2.755± 0.0270
1.9704
\o
o
2.7606
Pre^ is the density relative to water at 30°C.
ky is the viscosity relative to water at 30°C.
rel J

91
Figure 4-8 shows the shear viscosity vs. shear rate
data for CMC 7H solutions at 75, 25, and 10 ppm. It is seen
that this polymer exhibits only slight shear-thinning ef¬
fects over the shear rate range 0.01 to 1 sec Figure 4-9
shows the viscosity measurements of the Carbopol 934 solu¬
tions. CMC 7H solutions are more viscous than Carbopol 934
at equal concentration. Figures 4-10 and 4-11 show the vis¬
cosity data of S-273 and S-30. Both of these polymers show
strong shear thinning behavior even at a concentration as
low as 1 ppm. To the best of our knowledge, this surprising
behavior has not been previously reported in the literature.
Figure 4-12 illustrates the shear viscosity data of the
P-301 solutions. A comparison can be made between the P-301
and S-273. The molecular weight of the two polymers is
nearly the same (according to the manufactures, the weight-
g
average molecular weight for P-301 and S-273 is 4x10 and
g
7.5x10 respectively). The distributions are unknown, but
it would be expected that their viscosities would also be
nearly the same. However, comparing Figures 4-10 and 4-12,
one finds that the viscosity of S-273 is much higher than
that of P-301, at low shear rates, and from this we deduce
that S-273 chains are considerably more expanded in solution.
This may be a result of the electrostatic repulsion between
the ionic groups on the chains of Separan polymers. Since
S-273 is a polyelectrolyte, the electrostatic repulsion be¬
tween the ionic groups will cause significant expansion.
The P-301 chain, on the other hand, is known to be a random

Shear Viscosity y, cp
Shear Rate
Y.
-1
sec
10
0
NJ
Figure 4-8 Shear viscosity of CMC 7H solutions.

Viscosity cp
u>
Figure 4-9 Shear viscosity of Carbopol 934 solutions.

Viscosity cp
\D
Figure 4-10 Shear viscosity of Separan AP 273 solutions.

Viscosity cp
Figure 4-11 Shear viscosity of Separan AP 30 solutions.

Viscosity cp
Figure 4-12 Shear Viscosity of Polyox WSR 301 solutions.

97
flexible coil. For the case of V-25, large gel-like particles
or lumps were noticed in the solution which did not disappear
even when the solution was kept for several weeks. These
particles are so large that at times they blocked the annular
gap between the inner and outer cylinders of the viscometer
and cause the inner cylinder or rotor to stop. However, in
the rare cases when the inner cylinder did rotate properly,
viscosity data were obtained as shown in Figure 4-13.
4.2.1 Discussion of experimental error
The viscosity data presented above are subjected to both
random and systematic errors:
(a) Systematic Errors
Variations in frequency will cause the synchronous
motor to rotate at speeds other than those stated on the
calibration chart. Since the frequency in this laboratory
is regulated this factor is immediately discounted. Voltage
variations do not matter. For example, a 110-volt motor
will operate accurately between 70 and 130 volts, as long as
the frequency is constant. Variations of surface tension of
the test fluids are unimportant since no observable change
has been found for a reduction of the surface tension by a
factor of one half. The only other source of systematic
error that has been observed is the geometrical irregularity
in the rotor. One rotor in which the top was not perpendi¬
cular with the axis showed anomalous behavior with rotor
speeds; the anomalies disappeared when the top was properly
squared off.

Viscosity cp
10
0
Shear
Rate y,
-1
sec
•DO
Figure 4-13 Shear viscosity of Versicol S 25 solutions.

(b) Random Errors
The timing error has its counterpart in capillary vis-
cometry; the average deviation of a set of measurements of
a 100-sec interval is about 0.1%. The other random error
is in the leveling of the rotor. It is easy to adjust the
bottom gap between the rotor and stator and the meniscus
above the top of the rotor by adding or removing liquid
with a fine-typped dropper to a repeatability of 0.1 mm.
In the present case, the height of the bottom gap between
the rotor and stator was adjusted to be the same as the
annular gap, i.e., 1 mm. The height of the meniscus was
adjusted to be equal to 1.9 mm. An error of 0.1 mm in mea¬
suring the bottom gap between the rotor and stator corre¬
sponds to about 0.3% error in the speed; this measurement
error in the height of the meniscus corresponds to 0.1%
error in the speed. A slight wobble of the rotor axis, of
perhaps 0.1 mm off the center line, due to improper balancing
causes an error of about 2% in measuring relative viscosities
between two and three. This error can be readily eliminated
by proper positioning and balancing of the viscometer support
For the polymer solution, the specialized low-shear, coaxial-
cylindrical viscometer (Chapter III) was designed to provide
viscosity data at shear rates ranging from 0.05 to 5 sec .
4.3 Flow Field Measurements -- Water Vortices
In the following sections, the velocity distributions
in the vortex flow measured by tracking the motion of single,

100
neutrally buoyant particles are presented. Experimental
conditions are listed in Table 4-3. To specify the three
velocity components we use a cylindrical coordinate sys¬
tem (r, 0, z) with the z-axis pointing downward and having
its origin at the center of the liquid surface in the ab¬
sence of any rotational motions. The radial, tangential,
and axial components of velocity are denoted by u, v, and
w, respectively.
"v" and "w" were measured at various levels and radii
within the vortex. Figure 4-14 shows the distribution of
tangential velocity along radii at different horizontal
levels. The tangential velocity increases with decreasing
radius, reaches a maximum at approximately the radius of
the exit orifice and decreases rapidly to zero at the center.
This figure also shows that except for the region at which
maximum tangential velocity is obtained, the relationship
between "v" and "r" is independent of horizontal position.
The maximum tangential velocity appears to decrease slightly
with increasing distance below the surface. Figure 4-15 shows
the circulation distribution replotted from the tangential
velocity profile. It is apparent that the use of driving jets
in the present experiments gives rise to a disturbance zone
in the near-wall region. The turbulence created in the mixing
zone of the tangential influx jets results in a large fluctua¬
tion of velocity for r/r > 60, where r is the radius of the
e — e
exit orifice. Moreover, the circulation decreases with de¬
creasing radius over a large portion of the tank, presumably

101
Table 4-3
Experimental Conditions for Velocity
Profile Measurements
Injection tube number =
10
Injection tube bore =
0.635 cm, 0.159 cm
Exit orifice diameter =
0.516 cm
Depth of fluid =
54.5 cm
Total discharge
48.9 cm^/sec
Temperature
27 °C

102
e
Figure 4-14 Tangential velocity distribution for water.

103
due to viscous losses of angular momentum. However, for
the region 10 < r/re £ 20, there exists a flat portion in
the curve of Figure 4-15. Evidently, an inviscid tangen¬
tial velocity profile is reached there. Anwar (1968) re¬
ported a similarly shaped curve of the circulation dis¬
tribution in his experiments. In the present studies, it
is convenient to define the "entering circulation," r ,
as the constant circulation at the radius r/r = 15.
e
As described earlier, profiles of axial velocity were
obtained by measuring the axial distance between the leading
edges of streaks in a "dashed helix" (see Figure 3-4). It
is estimated that the accuracy of this measurement is on
the order of +0.2 cm/sec. Accurate measurements of the
axial velocity were difficult because of the vertical os¬
cillations still present (see Section 4-1). Figure 4-16
shows the axial velocity as a function of radius measured
at different depths relative to the tank bottom. The following
observations are worthy of note:
(a) The maximum axial velocity occurs at the axis of the
vortex, and as the radius is increased the axial veloc¬
ity decreases to zero. In the range 2.5 axial velocities become negative (obviously not the ex¬
perimental error), implying the existence of reversed
flow. This counterflow in the annular region involves
the secondary boundary-layer flow originating at the
bottom of the tank. This finding confirms the previous
visual observation of the dye injection flow pattern.

e
Figure 4-15
Circulation distribution for water.
104

Axial Velocity w, cm/sec
105
Figure 4-16 Radial distribution of the axial velocity, water.

106
(b) The axial-counterflow appears to oscillate; its
radial thickness decreased slightly with increasing
axial distance from the free surface. The axial-
counterflow is extremely weak compared with the cen¬
tral axial flow.
(c) Photographic measurements of this kind are certainly
not good enough to define the radius at which the
axial velocity is zero. The reasons include velocity
fluctuations and the limitations of the photographic
technique.
(d) If the axial counterflow is neglected, a Gaussian dis¬
tribution fits the axial velocity profile very well;
the Gaussian distribution is of the form
-r2/2a2
f(r) = wQe / , where wq is the axial velocity along
the center line and a is the variance of the distribu¬
tion. In the present case, a = 1.022 r . Comparing
this characteristic parameter with Figure 4-14, it was
found that a, obtained by fitting a Gaussian distribu¬
tion on the axial velocity profile, agrees fairly well
with the core radius, rc (=1.1 r ), for which the
forced vortex field meets the free vortex field, or
stated alternatively, when the tangential velocity is
a maximum.
(e) The three curves of the axial velocity profile at dif¬
ferential horizontal planes can be reduced to a single
. w . .
one by plotting — vs. r. This is because a, the

107
variance of the distribution, is independent of z and
w is approximately linearly dependent on z.
Axial velocity distributions along the center line
from the bottom to the top surface are shown in Figure 4-17.
They are failrly well fitted by a straight line for z/H < 0.6
where H is the depth of the liquid. However, for the region
close to the exit orifice, the axial velocity increases
rapidly with z. The approximation of linear dependence of
axial velocity on the axial coordinates is, therefore, only
good for smaller values of z. From a consideration of the
equations of motion in the vortex flow, it can be shown that
this nonlinearity of axial velocity with respect to the
axial coordinate is related to the dependence of tangential
velocity on the axial coordinate. This will be discussed
in more detail in Chapter V.
4.4 Influence of Newtonian Viscosity on Velocity Distributions
In describing the influence of polymer additives on
vortex flow, the relative importance of viscous and elastic
effects must be obtained. In this section, the effect of
slight increase of viscosity on vortex flow is investigated
using corn syrup solutions of different viscosities. Corn
syrup is replaced by CMC 7H when high viscosity solutions
are required.
Figure 4-18 shows the tangential velocity profiles at
z/H = 0.394 for viscosity = 1.16, 1.29, and 1.46 relative to
that of deionized water at 27°C. It was found that the tan-

Axial Velocity w , cm/sec
108
Figure 4-17 Variation of axial velocity along the vortex axis, (water)

Tangential Velocity v, cm/sec
109
e
Figure 4-18 Tangential velocity distribution in com syrup solutions.

110
gential velocity rapidly decreases with a slight increase
of the radius at which maximum tangential velocity is ob¬
tained, which is related to the viscous core size. The
size of the viscous core generally plays an important part
in determining the vortex flow. Rott (1958) assumed a
stagnation-point flow field in which the axial gradient of
the axial velocity is independent of fluid viscosity and
showed that the viscous core increases with the square root
of the viscosity. In reality, however, the situation is
much more complex; the axial gradient also depends on the
viscosity. This is expected because the circulation dis¬
tribution is strongly affected by the change of the fluid
viscosity. In the present work, the viscous core was found
to increase with 1.656 power of the viscosity, much higher
than Rott's prediction of rc ~ /v.
The axial velocity profiles are shown in Figure 4-19.
The axial velocity distribution also becomes broader with
increasing viscosity. Axial velocity profiles for all four
fluids can be expressed by a single empirical equation,
w = ~"i'^42 5 exp(-r2/2a2) (4-1)
a
where C is a function of the axial coordinate. In Figure
4-19 where z/H = 0.394, C is equal to 2.981. Since a is
related to the core radius, the influence of fluid viscos¬
ity on the suppression of axial velocity is readily obtained
from Eqn. (4-1).

Axial Velocity w, cm/sec
111
Figure 4-19 Axial velocity distribution in corn syrup solutions.

112
The other reason for a smaller tangential velocity for
a fluid of high viscosity is the use of driving jets. Since
the vortex was created by driving jets, a higher viscosity
leads to a reduction of the jet intensity and hence a reduc¬
tion of the entering circulation due to the larger momentum
loss. The angular momentum or circulation graphed against
r/re on log/log paper at horizontal level z/H = 0.394 is
given in Figure 4-20. A study of this figure leads to the
following observations:
(a) For all four fluids, the relationship between circula¬
tion, T, and radius, r, appears similar; the circula¬
tion decreases from the near-wall region toward the
center of the vortex, reaches a stage of constant cir¬
culation and then decreases quickly as the center is
approached. The constant-circulation portions in these
curves become shorter as the fluid viscosity increases.
This is reasonable in view of the fact that the dis¬
tribution of circulation in a viscous vortex approaches
that of a solid body rotation for a fluid of very high
viscosity.
(b) The magnitude of circulation throughout the flow field
decreases with increasing viscosity. This implies that
the viscosity effect plays an important part in deter¬
mining the flow, not only in the vicinity of the axis,
but also in the main body of the vortex, including the
near-wall region.

e
Figure 4-20 Measured circulation distribution in Newtonian fluids.
113

114
(c) Empirically, the constant circulation, may be
related to the fluid viscosity, y, by the following
-1 29
experiments, f^-y
The influence of fluid viscosity on the depth of the
free surface depression was also measured. The 1/4-inch
bore of injection tubes in the vortex chamber was replaced
by tubes of 1/16-inch bore. The smaller injection tubes
provided stronger entering circulation and caused a deeper
surface depression which can be taken as a measure of the
vortex strength. Figure 4-21 is a plot of the surface de¬
pression versus the relative viscosity for liquid. The sur¬
face depression is very sensitive to the fluid viscosity,
especially when the surface depression is large. The strong
dependence of depth of surface depression on viscosity com¬
plicates the study of vortex inhibition, since most polymer
solutions have high viscosities in addition to strong
elastic effects. The problem of separating the relative in¬
fluence of viscous and elastic effects will be discussed in
more detail in the following chapters.
4.5 Flow Field Measurements of Polymer Solutions
Experimental velocity distributions for the various
polymer solutions are presented here. For each solution,
axial and tangential velocities were obtained as a function
of radius at different axial locations. Data are given for
polyacrylamide, polyacrylic acid, and poly(ethylene oxide)
solutions.

115
Figure 4-21 Dependence of surface depression on fluid viscosity.

116
4.5.1 PAM
As in the case of water, it was found that the axial
velocity profiles in PAM solutions can be reduced to a sin¬
gle curve by plotting w/z vs. r. A typical radial varia¬
tion of the axial velocity over a small range of depths
near the middle of the vortex is given in Figure 4-22.
This is for S-273 at 3 and 1 ppm as a function of r for
z/H~0.4. The negative axial velocity data (due to the
counterflow in the outer annular region surrounding the
strong axial flow) has been omitted for clarity. The data
in this region scatter considerably. This is due to the
fluctuating structure of the counterflow as observed in the
dye injection experiments. With the omission of the counter¬
flow, the axial velocity in the central axial-flow region
can again be well fitted.by Eqn. (4-1), which has been shown
to hold for Newtonian fluids. It was found that the charac¬
teristic parameter, a, was 2.003 rg and 4.853 rg for 1 ppm
and 3 ppm solutions of S-273, respectively. As described
earlier, this characteristic parameter is a measure of the
amount of "spreading" of the axial velocity distribution,
and is closely related to the radius of the viscous core.
Since the core radius for water is about the radius of the
exit orifice, rg (Section 4-3), the core radius of the 3 ppm
S-273 solution is thus 4.853 times that of water. The en¬
largement of the viscous core in S-273 solution has been ob¬
served in the dye injection studies (see Figures 4-2 and 4-5).
It is of interest to point out that the central axial flow

Axial Velocity w, cm/sec
117
e
Figure 4-22 Axial velocity distribution in Separan AP 273 solutions.

118
region (which is approximately equivalent to the viscous
core) is also about 5 times larger in the S-273 solution.
It is probable that the reduced axial velocities in S-273
solutions, as well as enlargement of the core radius, is
caused at least partially by viscous effects. This hypo¬
thesis will be examined more carefully later in this chap¬
ter.
In Figure 4-23, the tangential velocity profiles for
the S-273 solutions are plotted against dimensionless radius.
It is apparent that the viscous core, in which the tangen¬
tial velocity increases linearly with the radius, becomes
larger as the polymer concentration is increased. Further¬
more, it was found that in the polymer solution the tangen¬
tial velocity was suppressed over the entire flow field from
the axis of symmetry to the cylindrical wall. The velocity
reduction in the near-wall region signifies a higher viscous
loss due to wall shear and, perhaps, jet mixing. In the re¬
gion near the axis, the flow field is no longer "purely"
shearing; instead, it is rather complex. It is this region,
if any, in which the viscoelastic effects of polymer solu¬
tions may play an important part in determining the struc¬
ture of the flow field.
Figure 4-24 shows the axial variation of the axial
velocity along the centerline for S-273 solutions. The
axial velocity increased linearly with the increasing axial
distance from the free surface except in the region very
close to the tank bottom. Comparing these findings with

Tangential Velocity v, cm/sec
119
Figure 4-23 Tangential velocity distribution in Separan AP 273 solutions.

O ppm (water)
o Ti ppm (S-273)
0. 0.2 0.4 0.6 0.8 1.0
z/H
Figure 4-24 Axial velocity along the vortex axis.
120

121
the case of water, the linear dependence of w on z holds
true over a greater portion along the vortex length for
the S-273 solution than for the water. It is apparent
that the approximation of a linear dependence of w on z
is better when the axial velocity is suppressed.
The velocity profiles for the other polyacrylamide,
S-30 at 3 ppm, are shown in Figures 4-25 and 4-26. It is
important to note that S-30 is similar to S-273 in the way
that it causes the velocity suppression. The only differ¬
ence is that S-30 causes less reduction of both the axial
and tangential velocities than S-273 at equal concentra¬
tions. This is reasonable because the of S-30 is higher
than that of S-273 (Table 4-1).
4.5.2 PAA
The results obtained with PAA (V-25) were similar to
those obtained with the PAMs. The axial and tangential
velocity distributions of V-25 at 1.0, 1.5, and 3 ppm were
shown in Figures 4-27 and 4-28. The data points of the
negative values of the axial velocity were again omitted.
The velocity distributions exhibited by V-25 fell between
those exhibited by S-273 and S-30. It appears that the
vortex-inhibition properties of PAA are similar to those of
the PAMs and both PAA and PAM may be grouped in one category.
It is of interest to point out that the axial velocities at
the axis, measured at the same horizontal level z/H = 0.394,
are 1.32, 1.82, and 3.34 cm/sec for S-273, V-25 and S-30,
respectively (the polymer concentration was 3 ppm). These
values are consistent with their respective C : S-273:

Radius Ratio r/r
e
Figure 4-25 Axial velocity distribution in Separan AP 30, 3 ppm.
122

o
OJ
w
>
4J
O
o
a)
>
ft
•H
c
QJ
60
e
H
e
Figure 4-26 Tangential velocity distribution in Separan AP 30 solution.
123

Axial Velocity w, cm/sec
124
e
Figure 4-27 Axial velocity distribution in Versicol S 25 solutions.

125
25
CJ
U)
'e
o
>
4-1
•H
CJ
O
T—H
cu
>
•H
U
c
(U
bO
C
H
20
15
10
o
r I
T 1
o
O 0
ppm (water)
-
A 1
ppm
-
• 3
ppm
o
o
-
A
A o
1—
â–º
o
-
â–²
O
O
â–²
A
Aa O
-
o
o
o
o
â—„
• •
•
A * °
A
A
• •
• •
•
J 1
° O
A A A A
• • •
_! L_
° c
A A*
•
10 15
Radius Ratio r/r
20
25
Figure 4-28 Tangential velocity distribution in Versicol S 25
solutions.

126
2 ppm; V-25: 3 ppm; and S-30: 7.5 ppm. These figures sug¬
gest that in these polymer solutions vortex inhibition is
caused by the suppression of axial velocity gradient and
hence suppression of the axial velocity. This suppression
leads to the corresponding suppression of the tangential
velocity distribution and subsequently the vortex motion.
The coupling between the axial and tangential velocity and
hence the vortex strength will be analyzed later in Chapter V.
4.5.3 PEO
The typical poly(ethylene oxide) used in this work was
P-301. The viscosities of P-301 solutions were found to be
sensibly constant and not much different from that of water.
>
The measured zero shear viscosity for P-301 at 100 ppm is
1.5 relative to that of the solvent (Section 4-2). Therefore,
the viscous effect on the velocity profile rearrangements
resulting from polymer addition is negligible.
The vortex flow behavior of PEO is quite different from
those of the PAMs and PAA. Figure 4-29 shows that the tan¬
gential velocity distributions of P-301 solutions in the outer
radii (r/re>4) is completely identical to that of water. In
the inner region close to the vortex axis, an anomalous be¬
havior of the velocity distribution was found. Above its ,
P-301 solutions exhibit strong fluctuations in the velocity
distributions, and the amplitude and frequency of the fluc¬
tuations increase with increasing polymer concentration. The
increase in fluctuations was clearly visible during the course
of the velocity profile measurements, but was too random and

J 1 I I
10 20 30
Radius Ratio r/r
e
Figure 4-29 Tangential velocity distributions in Polyox WSR 301 solutions.
12 7

128
unstable to be expressed quantitatively. The characteristics
of the fluctuations were more obvious in the axial velocities.
The appearance of the flow field fluctuation is illustrated by
streak photographs of the tracer particles as shown in
Figures 4-30a and 4-30b. The particles in the vicinity of
the axis appeared to flow an "acceleration-deceleration"
process. These particles accelerated as they spiralled down
in the axial direction, but were decelerated "abruptly" and
sometime even reversed upward for a few mm, while the par¬
ticles in the outer annular region proceeded with no apparent
unusual behavior. Figure 4-31 shows a typical axial velocity
distribution along the centerline, at a particular instance,
for P-301 at 40 ppm. It is somewhat surprising that the axial
velocity fluctuation in P-301 solutions is quite extreme in
comparison to that of water. The fluctuation decays slowly
with increasing radius. Plots of the axial velocity against
the radius for P-301 solutions are given in Figure 4-32. The
data points in the near vicinity of the vortex axis were
scattered over the limits of the axial velocity. All of the
data points of water in Figure 4-31 have been omitted for
clarity. Due to the fluctuating nature of the axial velocity,
the tangential velocity also fluctuated, especially in the
vicinity of its maximum (Figure 4-29), and the upper limit
of the fluctuating tangential velocity is lower than the
steady value of the solvent.
Intuitively, it appears that the fluctuation process
here in the vortex flow may be analogous to the Uebler effect

Figure 4-30 Typical particle-trace photographs showing the flow fluctuation

Axial Velocity Ratio
130
z/H
Figure 4-31 Typical axial velocity fluctuation in Polyox WSR 301,
20 ppm.

131
Figure 4-32 Axial velocity distribution in Polyox WSR 301
solutions.

132
(Metzner, 1967). Uebler found that the finite normal stress
near a gas bubble in an accelerating flow field such as the
tubular entry flow will tend to impede its motion and may
stop it. Apparently, this effect must be associated with
the behavior of large particulate matter in the rapidly ac¬
celeration viscoelastic fluids. However, in the viscoelastic
vortex flow the flow fluctuation can still be observed with
dye injection. In a test conducted in a centrifuge, it was
found that the dye could not be separated from the solution.
This implies that the dye was tracing streamlines and the
fluctuating velocity field in the vortex flow is not caused
by the Uebler effect.
4.6 Comparison of Experimental Results Between Those of
Elastic and Inelastic Fluids at Equivalent Shear
Viscosity
Originally it was intended to compare the velocity
profile measurements of viscoelastic fluids with those of
Newtonian or near-Newtonian fluids at equal shear viscosity.
However, since most viscoelastic fluids used in this study
display strong shear thinning effects, such exact comparisons
are not possible. One must simply select a shear rate at
which to characterize viscosity in the non-Newtonian fluids.
The shear rate was calculated by the equation
i - r If (?) U-2)
From the measured tangential velocity data, by using a method
of graphical differentiation, the shear rate was obtained as

133
a function of the radius. Shear rates of water and S-273
solutions graphed against the radius on log-log paper are
given in Figure 4-33 for the experimental conditions shown
in Figure 4-23. For all three fluids (i.e., water, S-273
at 3 and 1 ppm), the relationship between y and r appears
to be yr = constant for all positions within the vortex
from the near-wall region to within a small distance from
the axis of the tank. The shear rates increase with de¬
creasing radius, reach a maximum, and decrease steeply down
to zero. The zero shear region corresponds to the solid-
body rotation core of the vortex in which the shearing mo¬
tion of the fluid elements is negligible. Under all con¬
ditions for the three fluids, the radius at which maximum
shear rate is obtained coincides with the radius at which
the tangential velocity is maximum. It is noted that the
shear rate of water varies over three orders of magnitude
and ranges from 0.1 to 100 sec At progressively higher
concentrations of S-273 solutions the tangential shear in
the main body of the vortex becomes smaller due to the vis¬
cous effect. The pertinent viscosity of S-273 at 3 ppm in
the vortex flow would be its zero-shear viscosity.
In order to better compare the velocity profiles at
reasonably equivalent viscosities, the average shear rate
was calculated to estimate the corresponding shear visosity.
The average shear rate is
Y =
R
yrdr/
R
o
rdr
(4-3)

Shear Rate Y, sec
134
q 0 ppm (water)
• 1 PPm
Figure 4-33 Shear rate in vortex flow with Separan AP 273.

135
where R is the radius of the vortex chamber. From Figure
4-33 the average shear rates for S-273 at 3 and 1 ppm are
0.25 and 0.58 sec \ respectively. From Figure 4-10, these
shear rates correspond to the shear viscosities of 1.4 and
2.8 centipoise for S-273 solutions at 1 and 3 ppm, respec¬
tively. Therefore, the experimental results of S-273 at
3 ppm will be compared with those of Newtonian or near-
Newtonian fluids having shear viscosities of about 2.8 centi¬
poise. Similarly, the solution of S-273 at 1 ppm will be
compared with fluids having viscosities of about 1.8 centi¬
poise. For viscosities above 1.5 centipoise, the solutions
of CMC 7H and Carbopol 934 were used as the "equivalently"
Newtonian fluids. From the viscosity data of CMC 7H and
Carbopol 934 as shown in Figures 4-8 and 4-9, the corre¬
sponding solutions with the equivalent shear viscosities
were listed in Table 4-4. In Figures 4-34 and 4-35, the
velocity distributions of the "viscosity equivalent" fluids
are shown along with the results of S-273 at 3 ppm. The
agreement between the three sets of data of different fluids
is quite good and confirms that indeed S-273 is indistinguish¬
able from the viscosity-equivalent inelastic fluids. In
Figure 4-34, no data points of Carbopol 934 were shown because
the fluid was so optically ambiguous that the tangential velo¬
city profile measurements were impossible. Comparison of
5-273 at 1 ppm with Newtonian and non-Newtonian inelastic
fluids are given in Figures 4-36 and 4-37. It is seen that
the Newtonian and non-Newtonian velocity profiles are matched

Table 4-4
Comparison of Shear Viscosities of PAM Solutions
with Shear Viscosities of Equivalent Fluids
Viscoelastic Fluid
Shear Viscosity
at y (Eqn. 4-3)
Equivalent Fluid
Equivalent Viscosity
at y
3 ppm
2.8 c.p.
25
ppm CMC 7Ha
2.95 c.p.
S-273
300
ppm Carbopol 934^
2.75
1 ppm
1.4 c.p.
100
ppm Carbopol 934^
1.40
S-273
10%
corn syrup
1.41
3 ppm
1.85 c.p.
10
ppm CMC 7Ha
1.80 c.p.
S-30
150
ppm Carbopol 934^
1.80
aViscosity data are shown in Figure 4-8.
Viscosity data are shown in Figure 4-9.
136

Tangential Velocity v, cm/sec
â–¡ Water
O Separan AP 273
• CMC 7H
0 10 20 30 40
Radius Ratio r/r
e
Figure 4-34 Comparison of tangential velocity profiles of fluids at equivalent viscosity, y = 2.8 cp
137

Axial Velocity w, cm/sec
O
â– 
Separan AP 273
Carbopol 934
CMC 7H
Radius Ratio r/r
e
Figure 4-35 Comparison of axial velocity profiles of fluids at equivalent
shear viscosity, y = 2.8 cp.
138

Axial Velocity w, cm/sec
1
1 r
, 1
1
r
â–¼
il
o
T^o
â–¼
O Separan AP 273
• Carbopol 934
-
â–¼
^ Water-corn syrup
—
â– y
o
â–¼
â– 
Qd
o
â– 
o
CP
o â– 
â–¼
O
â–¼
â–¼
â–¼
1
1 1
1 1 o
—©—■
0
2
4
6
Radius Ratio r/r
e
Figure 4-36 Comparison of axial velocity of fluids at equivalent shear viscosity,
U = 1.4 cp.
139

Tangential Velocity v, cra/sec
15 -
OT
O
10
o
o
â–¡
o
â–¡
°D
â–¡
â–¡
Water
O Separan AP 273
â–  Carbopol 934
â–¼ Water-corn syrup
â–¡
O,
'O
OT
O
D â–¡
OT ° °* TO* cFt
O
Cl
o *
10 15
Radius Ratio r/r
20
25
Figure 4-37 Comparison of tangential velocity of fluids at
equivalent shear viscosity, y = 1.4 cp
140

141
very well with those of S-273. The viscosity-equivalent
solutions of S-30 are also listed in Table 4-4, the results
of which are shown in Figures 4-38 and 4-39. Again a close
agreement between the different sets of data are obtained.
For the case of PAA, as described earlier, large gel¬
like particles existed in the solution which did not dis¬
appear even after several weeks. For this reason the vis¬
cosity measurements for V-25 were probably misleading. How¬
ever, if we assume the viscosity data in Figure 4-13 are
correct, the viscosity of a 3 ppm solution V-25 at its
average shear rate is about 2 centipoise. This is equivalent
to a 200 ppm Carbopol 934 solution which has a viscosity of
approximately 2 centipoise. As demonstrated in PAM solutions,
the results obtained with the PAA solution also agreed very
well with those obtained with viscoinelastic liquid (Figures
4-40 and 4-41).
The vortex flow behavior of PEO was quite different from
those of PAMs and PAA. In experiments conducted with PEO
solutions, the flow field in the near vicinity of the vortex
axis strongly fluctuated, which did not occur in its viscosity-
equivalent, Newtonian or viscoinelastic fluids.
4.7 Conformational Studies
As mentioned earlier in Chapter III, conformational
studies can be carried out with solutions of polyelectrolytes
in different ionic media. In this study aqueous solutions of
polyacrylamide (S-273) and polyacrylic acid (V-25) were used.

1
1—
i
1 1
<â–º
0
<)
o
©
€
0
©
©
O
Separan AP 30
(
0
0
©
€
0
CMC 7H
—
•
o •
0
O
•
Carbopol 934
o©
_ 0
O
•
_
o
o 0
-
0 8
cl
1
1
1
©o
! 1
O
0
2
Radius
Ratio r/r
4
6
e
Figure 4-38 Comparison of axial velocity of fluids at equivalent
shear viscosity, y = 1.85 cp.

CJ
cn
5
>
Ps
4-J
O
o
r—H
Q)
>
•H
U
C
(1)
W)
C
n?
H
15
-
l
,
â–¡
—1
Water
r
—
â–¡
o
Separan AP
30
â–¡
Q
CMC 7H
•
Carbonol 934
â–¡
10
©
â–¡
-
V
cS
DD
© o
â–¡
O»
o
â–¡
â–¡
5
-8»
o
O ©
o ©
D â–¡
-
o®
©
n
©
o
¿V
O
O
•
°o
u []
Sd© 0o«>
o
0
1
1
!
1
Ü
5
10
15
20
25
Radius Ratio r/r
e
Figure 4-39 Comparison of tangential velocity profiles of
fluids at equivalent shear viscosity, y = 1.85 cp.
143

Axial Velocity w, cm/sec
T
T
T
O Versicol S 25
• Carbopol 934
3
O
0
o
I 1 L
12 3
Radius Ratio r/r
e
1
4
5
Figure 4-40 Comparison of axial velocity profiles of fluids
at equivalent shear viscosity, y= 2.1 cp.
144

1
—r~
o
Versicol
1
S25
1
•
Carbopol
934
-
o
—I
o
t
•
o
•
1
o
o •
•
•
o
• •
•
1
1
O
o •
O "
0 5
10
15
20
Radius Ratio r/r
e
Figure 4-41 Comparison of tangential velocity profiles of
fluids at equivalent shear viscosity, y = 2.1 cp.

146
4.7.1 Viscosity measurements
Changes in polymer conformation were followed by visco¬
sity measurements as shown in Figures 4-42 to 4-44. In
Figure 4-42, viscosities of 200 ppm solutions of these poly¬
mers were plotted as a function of pH at a shear rate of 115
sec . The maximum viscosity is found in the pH range of
6-9 for PAA and 7-10 for PAM. For the high pH solution, the
viscosity is decreased significantly. The viscosity reduction
effect is even more pronounced at low pH. Figure 4-43 shows
the variations in viscosity of a 3 ppm and a 10 ppm solution of
S-273 with pH at a very low shear rate of approximately 0.1
sec . Figure 4-43 is very similar to Figure 4-42 implying
that changes in polymer conformation occur similarly at both
the high and low concentration levels in the proper shear
rate range. The data concerning the influence of pH on low-
shear viscosity for PAA at low concentration levels are un¬
available due to the presence of large gel-like particles in
the solution. The particles were so large that they inter¬
fered with the shearing flow and caused the viscosity measure¬
ments to be erroneous. Macromolecular conformation can also
be changed by the addition of salts to polyelectrolyte solu¬
tions. The effect of salt concentration on the viscosity of
the 10 ppm solution of S-273 is shown in Figure 4-44.
4.7.2 Drag reduction measurements
The effect of polymer conformation on drag reduction has
recently been determined by White and Gordon (1975) using the
same polymers as those used in this study. The friction fac-

Viscosity cp
147
O Separan AP 273, 200 ppm
• Versicol S 25, 200ppm
Figure 4-42 Viscosity vs. pH @ shear rate = 115 sec

Viscosity cp
148
Figure 4-43 Viscosity vs. pH @ shear rate = 0.1 sec

Viscosity cp
T
T
T
T
Salt Concentration, %
Figure 4-44 Viscosity vs. salt concentration @ shear rate = 0.1 sec
149

150
tor - Reynolds number data for S-273 and V-25 was measured
in four solutions: in deionized water, in high pH and low
pH solutions, and in salt solution. This data is summarized
in Figures 4-45 and 4-46. In each figure, the three solid
lines correspond to
(i)laminar flow
f = 16/Re
(ii)turbulent flow
— = 4.0 log..(Re/f) - 0.4
/f 10
(iii)Virk's maximum drag reduction asymptote (Virk, 1971)
— = 19.0 log. n (Re/f) - 32.4
/f iU
In all cases, the Reynolds number Re is calculated by using
the solvent viscosity in order to portray the true drag-
reducing ability of the various solutions.
4.7.3 C measurement
The vortex-inhibiting ability of S-273 and V-25 was
determined by measuring at different pH levels and in
0.5% salt solution. The results are summarized in Table 4-5,
in which the low-shear viscosity data at the respective C^
of each polymer solution was also included. It is clear from
this table that the vortex-inhibiting ability is not changed
significantly by the addition of salt or an increase in pH,

151
Table 4-5
Vortex Inhibition versus pH for PAM and PAA
c
Viscosity (@shear rate
S/i
0 . 1 sec--*-)
PH
(ppm)
(relative to solvent)
PAM
(S-273)
2.5
400a
1. 38
3.75
20
1.29
5.0
5
3.80
7.0
3
3.50
12.0
7.5
1. 17
PAA
(V-2 5)
2.5
400
1. 50
3.9
200
_b
7.1
3
_b
9.7
5
__b
11.7
15
1.80
0.5% salt
20
2.06
a
The solution becomes very hazy. This is probably due to
the "poor solvent" of the high pH solution in which a lot
of polymer was observed to precipitate. In about 2 weeks
a thin layer of polymer slurry or sediment was present
over the bottom of a container.
The large gel-like particles or lamps usually block the
annular gap of the rotating cylinder viscometer and cause
the inner cylinder or "rotor" to stop. The viscosity
measurements for such solutions are impossible to be made.

Friction Factor
152
Reynolds Number Re
Figure 4-45 Friction factor vs. Reynolds number—Separan AP 273.
Comparison of all solutions in the 1.09 cm tube,
ODeionized water solution; • High pH solution;
©Salt solution; (1 Low pH solution. Polymer con¬
centration, 10 ppm.

Reynolds Number Re
Figure 4-46 Friction factor vs. Reynolds number—Versicol S 25.
Comparison of all solutions in the 1.09 cm tube,
ODeionized water solution; • High pH solution;
©Salt solution; 3 Low pH solution. Polymer con¬
centration, 20 ppm.

154
while it is considerably reduced by a decrease in pH. Further¬
more, the low shear viscosity data alone cannot explain the
effect of vortex inhibition. The PAM solution exhibits
vortex inhibition at low and high pH or in salt solution
while its viscosity is not much different from that of water.
It is therefore very important to study the velocity field
kinematics for each solution at its respective C, from
which an explanation for vortex inhibition may be obtained.
This will be done in the following flow field measurements.
4.7.4 Vortex flow field measurements
Initially, the measurements were made in the vortex
chamber used in the previous experiments (that is, with
moderate entering circulation). However, the axial flows
were found to be somewhat anomalous; at C , the axial
velocity near the vortex axis fluctuated "occasionally."
The fluctuations of the axial flow appeared quite similar
to the "acceleration-deceleration" process describer earlier
in the case of P-301 solution (Section 4-5). Since the
fluctuation is not very clear, originally we had intended
to increase the polymer concentrations in order to amplify
this fluctuation effect as was the case for P-301 solution.
However, as the polymer concentration increased, the visco¬
sity increased and the viscous effect tended to "stabilize"
the flow thereby weakening the fluctuation of the flow field.
We therefore need another way to exhibit the fluctuation
effect, if it indeed exists.

155
Since the viscosity difference between solvent and solu¬
tions would not cause the flow fluctuation, it should be
related to elastic forces, which are present in the defor¬
mation process of the polymer solutions. It is expected
that the flow fluctuation may be more evident in a deforma¬
tion process with higher stress levels, which is the case.
The way to elevate the stress level in a vortex flow
field is to increase the strength of the tangential influx
jets. The injection tube of 1/4-inch bore which was used
previously was thus replaced by tubes of 1/16-inch bore.
A higher entering circulation and hence, higher deforming
process, is thus obtained in the new system.
Using the new injection tubes, velocity profiles for
the polymers in the different solutions (i.e., deionized
water, high pH, low pH, and salt solution) were compared in
two ways. First, the velocity profiles were measured for
the polymers at a constant concentration, say 10 ppm.
Water and a viscoinelastic Carbopol solution (100 ppm) were
also studied for comparison. Second, the measured velocity
profiles were compared at the respective CVI in the dif¬
ferent solution. The velocity profiles for S-273 and V-25
are similar in most features and only the results of S-273
will be presented here.
In 10 ppm, the elevated viscosity in the deionized
water solution causes the vortex to exhibit near rigid-body
rotation in which the axial velocity component is negligibly
small. In the case of high pH and salt solutions, we see

156
completely different results. During the course of taking
the axial velocity data, the tracer particles were observed
to move very unstably. The streak photographs of the tracer
particles appear very similar to those of P-301 as shown in
Figure 4-30. In Figure 4-47, the axial velocity data ob¬
tained from such photographs showed the strongly fluctuating
nature with respect to time. The axial velocities also fluc¬
tuated with respect to the axial positions. The low pH solu¬
tion (below its CVI at 20 ppm) shows indistinguishable axial
velocity profiles from those of water. Both water and low
pH solution profiles are quite stable compared with the dra¬
matic fluctuation found in the high pH and salt solutions.
The viscoinelastic Carbopol solution shows even more stable
axial velocities than those of water. This is because of
the fact that the viscous process tends to stabilize the
flow. It was noted that as the radius increases, the velo¬
city fluctuations faded rapidly and died off within 1 or 2
cm from the tank axis.
The corresponding tangential velocity profiles for the
different solutions are shown in Figure 4-48. All data points
have been omitted for clarity because of considerable scatter
among the data of the high pH and salt solutions, particularly
in the vicinity of the vortex axis, where a shaded area re¬
presents the scattering data. The tangential velocity pro¬
file for the low pH solution is indistinguishable from that
of water, while the high pH and salt solutions exhibit some
fluctuation. In the overall flow region, except very close

Axial Velocity along the Vortex Axis w , cm/sec
157
Figure 4-47 Axial velocity fluctuation for Separan AP 273 in
the vortex flow. (1) = water; (2) = low pH solution;
(3) = Carbopol 934, 100 ppm; (4) = salt solution
(0.5%); (5) = high pH solution. Separan AP 273
concentration, 10 ppm.

158
Radius Ratio r/r
e
Figure 4-48 Tangential velocity distribution for Separan AP 273
in different ionic solvents.

159
to the vortex axis (r the high pH and salt solutions are also close to that of
water, while in the vicinity of the vortex axis, the tangen¬
tial velocities fluctuate to some extent, although their in¬
tensity is not so high as in the axial velocities.
The velocity measurements were then made at the respec¬
tive CVI of the polymer in different solutions. This time,
except in the dionized water solution, all the three other
solutions (that is high pH, low pH and salt solutions) ex¬
hibit the similar fluctuating behavior as those found in the
preceeding experiments. Figure 4-49 shows a typical axial
velocity along the centerlines as a function of the axial
coordinate. Velocity in the deionized water solution in¬
creases linearly with increasing axial distance, while the
other solutions display strong fluctuations. Figure 4-50
shows the corresponding tangential velocity profiles. The
tangential velocities in the deionized water solution are
substantially suppressed, very similar to the case of a
highly viscous fluid. The other three solutions again ex¬
hibit the similar behavior as those found before, that is,
indistinguishable from water for r>l cm and show scattering
for r As mentioned earlier, the dimple of the surface depres¬
sion oscillates associated with the fluctuating velocity
field. Table 4-6 summarizes the depth of dimple data with
S-273 in high pH, low pH, and salt solutions. It is apparent
that oscillation of the surface dimple prevents the develop-

Axial Velocity along the Vortex Axis w , cm/sec
20
10
0
0.3
0.4
z/H
0.5
Figure 4-49 Axial velocity along the vortex axis for Separan AP 273
in different ionic solvents. ■ = deionized water solu¬
tion (C = 3 ppm); â–² = low pH solution (pH = 3.75, C
= 20 ppm); O = high pH solution (pH = 12.0, C = 7.5V~
ppm); •= salt solution (0.5% sodium chloride, C = 7.5
ppm) . 1
160

161
Figure 4-50 Tangential velocity distribution for Separan Ap 273
in different ionic solvents (at C ).

162
ment of a complete air core and hence must be responsible
for the suppression of the vortex formation.

Table 4-6
Dependence of Depth of Dimple on Flow Fluctuation
Polymer
Concentration
(ppm)
c
„ VI
pH
(ppm)
Relative Viscosity
@shear rate = 0.1 s
Depth of
Dimple
(cm)
Flow
Fluctuation?
S-27 3
10
12.0 7.5
1.55
1.5-2.4
yes
7.5
12.0 7.5
1.23
3. ~ 7 .
yes
3
7.0 3
3.50
0.5
no
20
3.75 20
1.29
2.5-6.
yes
10
3.75 20
1.24
23.
no
400
2.50 400
1.38
2. -3.5
yes
10
0.5% salt 7.5
1.22
2 . -4 .
yes
7.5
0.5% salt 7.5
1.17
3. ~7.
yes
Carbopol
100
400
1.40
2. 5
no
Water
1. 00
25.
no
163

CHAPTER V
THEORETICAL ANALYSIS
In this chapter, various theoretical analyses of
the results observed in the previous experiments are pre¬
sented. The analysis of the formation of a vortex core
during the emptying process of a tank falls into the class
of nonsteady free-boundary problem, and appears to be too
difficult to handle by direct analytical means. A simple
analysis similar to that presented by Dergarabedian (1960)
will be used first to examine the case of an inviscid
fluid. The result may yield some idea as to the nature of
the flow during the vortex developing process. Previous
work which relates to the jet-driven vortex is mentioned
briefly and limits of application of the solution discussed.
The account given here begins with the solutions of Long
(1958, 1961), and Einstein and Li (1951). A theoretical
solution (Anderson, 1961) for the secondary flow on the
bottom surface of the vortex chamber is closely examined.
The predicted radial mass flow in the bottom boundary layer
is compared with values of present experimental results.
Finally, an account is given of the very general asymptotic
expansion techniques of Lewellen (1962).
164

165
Analyses of the vortex flow are then extended to in¬
clude models of viscoelastic liquids. An analysis of the
problem using the equations of motions and constitutive
theories of dilute polymer solutions in terms of experimentally
supported kinematics shows that the normal stress difference,
(x00-l^rr)—the tension along the lines of flow in the cir¬
cumferential direction--is of significance in causing flow
fluctuation which seems to serve as the origin of the vortex
inhibition phenomenon.
5.1 Fundamental Equations
Let (r,0,z) be cylindrical coordinates with the z-axis
pointing downward along with the vortex axis. The origin of
the coordinate system is chosen at the center of the free
surface with the liquid at rest. The radial tangential and
axial components of velocity are denoted by u, v, and w res¬
pectively .
In the usual notation, the equations of continuity and
motion for an incompressible fluid, assuming axial symmetry,
are
1 9(ru) 3 (w)
r 3 r 3 z
(5-1)
3u
31
+
3u
3
w
3u
3 z
1 3P + 1
p 3r p
r-i 3 T 3x t
£ rr , rz rr 00
r 3r 3 z t
3v
31
+
3v
3
3 v uv
3 z r
1 ] _3_
p 2 3 r
Lr
(r‘
Tr0}
(5-2)
(5-3)
/

166
u
1 8P 1 fl _9_ f . atzz
p 3z p r 3r rTrz 3z
+ g
(5-4)
where p is the density, P the isotropic pressure, the
extra-stress, and g the gravity. In a general fluid, the
stress is taken as depending on the entire past history of
the deformation gradient. This formulation is too general
for comparison with experiment, however. If we take stress
to be proportional to the velocity gradient, the shearing
stress terms in Eqns. (5-2) to (5-4) become
= y [ r
_3_
3r
(p) 1 ,
rz
= y (
3u
3 z
3w,
3r
0z
,9v\
= y(37}
(5-5)
•k
The viscosity y is, in general, a function of shear rates.
The normal stress terms in Eqns. (5-2) through (5-4) are
left as unknown functions.
It is convenient to introduce the axisymmetric stream
A A
function 4; and circulation T such that
u
1 3j¿
r 3 z '
w
1 3ijj_
r 3r
(5-6)
The tilde (~) on i|i and r denotes a dimensional quantity.
By eliminating the pressure by cross differentiation of Eqns.
(5-2) and (5-4) and substituting Eqs. (5-5) and (5-6) into
the resulting equations, the equations of motion reduce to
1 3¿ 3¿ 3T _ 3¿ 3T _ I" 2 1 3J1 , _ 32?
2 31 3z3r 3r3z 3r'r3r; .2
L 3 z -
(5-7)
The essential shear in the vortex flow is
on the cylindrical surface in the tangential
tangential
momentum
direction.

167
3_r
3z
- r
3 _1_
3t
r32í ^ 3 .13$'
u2 + r 3?
*-3 z
. , li M . . 31 O.
3r 3z 3z _ 2
3r
2 3¿ 32j 2 3¿ 3^i _ 2 3Í 33j 2r ¿Í- + r2 ¿Í i2í
3r 3r3z 3z . 3 3r . 2. 3z 3z _ - 2
3r 3r 3z 3r3z
A ^ A O
_ 2 3i 3fi 4 3Z r
3r . 3 3r3z
3 z
, .33, .
(t -t ) - r 77— (t -tqq)
zz rr 3z rr 90
(5-8)
axial +
radial
momentum
Equations (5-7) and (5-8) can be somewhat simplified by
2
writing them in terms of r . For convenience, the following
dimensionless parameters are introduced
2 , 2
n = r /ro ,
5 - —
r = JL
r '
CO
, \p
* = Ql
2
F - — T
s - SI
li Qy li
S 2
r
(5-9)
where r , 1, rm, and Q are characteristic dimensional quan¬
tities, the meaning of which will be made more explicit later.
Equations (5-7) and (5-8) become
1 II + 11 11
2 3s 3? 35
ü II = 2H <¿T o_ 3^T
3g 3 5 Re . 2 Re ._2
3h 3 5
(5-10)
3_T
35
r _ 3 3 >
'
in2 3.1 + 3h 3>
9 1 9 9
2 2
Ro = Ro
. 2
4n
1 3s3n 3 s3 5Z
.
3ip 3 ^ip
3T 7M
- ^ 3 n
dip 3 ip
9n 353n2
2_
Re
2 ÍJÜ
3
3n
+ n
,4,r
0 l¡J.
, 4
3 n
+ a
_ 11 Ml + n 11 3 ip
L ^ 352 3* 3n352
- n
3j¿ 3 3i^
3n g ^ 3
2
+ 3-
!— (p _p ) _ ,_Q_ _!_ (p _p \
Re 3n 35 ' ZZ r rr 2Re 35 1 rr *ee;
(5-11)

168
This form is similar to that presented by Lewellen (1962)
except for adding the transient and non-Newtonian stress
terms. It is noted that the flow is governed by three para¬
meters
Re
the Reynolds number,
r 2
a = (-£-) the square of the ratio of characteristic
lengths,
QÍ,
Ro = = the ratio of volume flow to circulation
1 IT
00 o times the radius, commonly called the
Rossby number in meterological literature
(Morton, 1966),
and perhaps, the normal stress functions, t , xAn and x
rr 66 zz
5.2 Nonsteady-State Solution of Equations of Motion
(Following Dergarabedian)
It is useful to examine first the formation of a vortex
core during the emptying process of a tank. Initially, the
case of inviscid fluid will be considered. Setting Re = °°
in Eqns. (5-10) and (5-11)
2 9s 9E, 9n 9n 9£ U
(5-12)
r
9?
- RO
2n
93^ + _n_
9s9r)2 2a 9s9^2
9ip 92^ 9t|i 93t|>
" ^ ^ n ^ 91 i I 2
Ro2
4n2
.
~9 ip 33\¡j
9iJ> 93^ “
9£ a 3
L 9n
9g _ 2
9£9n J
ki
3 n
kjkll
(5-13)
+ a

169
Following Degarabedian, the simplest form of ¥ which,
(a) satisfies the boundary conditions that the radial velo¬
city must go to zero on the axis of symmetry, and the axial
velocity must become negligible at large radii, and (b) re¬
presents a central axial flow, is
ip = -U1 - e n)
(5-14)
The corresponding radial and axial velocities are
u = - a
r
1 - e
-(r/ro)
(5-15)
w =
2QZ -(r/ro>
—=r- e
(5-16)
o
It is worth noting that this assumed velocity profile agrees
well with the experimental results of the present work (see
Figure 4-15). It can be seen that
Q Q _
u — as r °o ; u -* r as r ->■ 0
r 2
r
o
Explicit meanings can now be given to rQ and Q: rQ is the
radius at which the axial velocity is e ^ times its value
2
ro wo
at the centerline and Q is equal to —= , where w is the
2z o
centerline axial velocity. Q is thus a measure of the cen¬
tral axial flow rate and is a constant if the axial velocity
can be assumed to be linearly dependent on z.
With iJj of the form given by Eqn. (5-14), the equation
of F (Eqn. 5-12) becomes

170
1 dT_
2 9s
- (1
e"n)
9_F
9n
+ e
-n
9_r
= o
(5-17)
The characteristic equation of the above first-order
differential equation is expressed as
ds
1/2
from which one obtains
(5-18)
(e^-lje23 = constant
(5-19)
£(l-e 1-1) = constant
Thus the general solution of Eqn. (5-17) for F can be
expressed as a function of the two arguments given by Eqn.
(5-19), or
r = F[en-l)e2s , £(l-e“n)] (5-20)
Now for time s = 0, it is assumed that r is independent of E,,
hence
F = F[(en-l)e2s] (5-21)
The actual function chosen to represent T is dependent upon
the choice of the initial distribution of F which is arbi-
*
trary. Degarabedian used the known solution for diffusion
of a line vortex for the initial form of F. The line vortex
of strength T = is initially concentrated along the z-axis.
:k
In the original vortex inhibition experiment, the circulational
motion in the liquid is introduced by stirring with a paddle.
This procedure is, of course, arbitrary.

171
Defining z-component vorticity £ as
' - (,xV>z - ?+ H
(5-22)
the vorticity equation is of the form (Lamb, 1945),
3t = V
file + 1 ie ]
Ur2 r 9r
(5-23)
Equation (5-23) is identical with the equation of radial flow
of heat in two dimensions. The thermal analogy is the dif¬
fusion of heat from an instantaneous line source in infinite
medium.
The solution of Eqn. (5-23) is (Carslaw and Jaeger, p. 258)
r
2
5 2vt
exp(- 4vt>
from which the expression for the
circulation 1
1
r r
r 2
r = —
r
CO
£r dr =
o
L1 ' exp<- 4vt>J
(5-24)
(5-25)
or, in dimensionless form
r =
1 - exp (-
Q
4vs
(5-26)
The initial distribution for T could be selected by
choosing a time, s = s-^, for which Eqn. (5-26) would be
1 - exp(
Q
4vs.
(5-27)

172
or
In order to determine if Eqn. (5-27) can properly ex¬
press the initial condition of the circulation in the vortex
developing process, some simple experiments were done. The
cylindrical tank was filled with water and stirred gently
with a paddle. The bottom plug was then removed and the
valve for the influx jets was adjusted to keep the liquid
level constant. The tangential velocities at a specific
horizontal level were measured a few moments (about 15 sec.)
after the flow started (using the trace-particle technique
described in Chapter III). Perhaps surprisingly, the results
of this somewhat arbitrary procedure were reproducible.
Figure 5-1 shows the measured circulation, r, as a function
of the square of the radius. They are compared with Eqn.
(5-27) ; the parameters of T^ and vt^ have been adjusted to
give the best fit.
2
vt^ = 4.5 cm
2
F = 53.6 cm /sec.
The agreement is only fair, but it is clearly impossible
with the limited knowledge about the initial condition to
provide a better result. Assuming a low kinematic viscosity
for most fluids, the decay period is much longer than the
time involved in the formation of the vortex core, if it can

Circulation f, cm /sec
r , cm
Figure 5-1 Initial distribution of circulation.
173

174
2
occur. For example, if the viscosity is 0.001 Ns/m , the
decay period would be 450 sec., compared with a few seconds
for the core formation. Because of the short time needed
for core formation, a feasible assumption may be made that
during the emptying process the viscous effects may be
neglected in the analysis. Thus the problem is formulated
in such a way that the viscous effects are neglected during
the air-core-forming process; however, the viscosity does
play a role in determining the initial r distribution in
the waiting period.
From Eqn. (5-21), the general solution,
T = F[en-l)e2s]
the following solution is considered,
F = 1 - exp
Q (en-l)e2s
4vs
(5-28)
For s=0, Eqn. (5-28) becomes
f = 1 - exp
Q
4vs-
â– (en-l)
where
f ~ 1 - exp
Q
4vs.
for n << 1
It can be seen that the asymptotic form of Eqn. (5-28) is
identical to Eqn. (5-27).
To find the shape of the free surface, consider Eqns.
(5-2) and (5-4) and neglect all the inertial forces except

175
the centrifugal force, p — . Thus Eqns. (5-2) and (5-4)
become
1 3P = v_
p yr r
1 3P
p 3z
= g
or, integrating these two equations one gets
P - pgz +
r r 2
pv
o
dr -
2
pv
o
dr
(5-29)
Since the free surface is a constant pressure surface, the
locus of the free surface is then
1
z = —
s g
p- r00 2
v
o
dr -
T 2
— dr
r
o
(5-30)
or, in dimensionless form
H 0 2„ ,
2gr H L
23 o
nr00 2
r A
~2 dn ~
o n
n r 2
o n
dp
where H is the height of the liquid.
Substituting Eqn. (5-28) into Eqn. (5-30) gives
z r2
r i
r
S _ 00
l-exp
H „ 2
2
2gr H
o
_
o n
*
-Q-â– 
4 vs
(e — 1) e
2s
dn
o n
l-exp -
Q (en-l)e2s
L4 v s i
2
dn
(5-31)

176
Now the term in brackets { } on the right hand side
of the equation can be approximated as
{ }
Qne
2s
4vs.
for n << 1
~ 1 for g >> 1
In order to simplify the integration of Eqn. (5-31)
the asymptotic functions of { } may be taken by considering
{ } =
Qne
2s
4vs-
for o < n < n
for
n > n
Here n is the position at which the two asymptotic func¬
tions meet. Hence n is calculated by equating — and 1,
2s
4vs-
or
* 4vs1
" =
Qe
Using these approximations in Eqn. (5-31) , one obtains
H
2 2
8tt r gH
o^
0
Í Q
e2s
Í Q 1
4s
[ 4 v s x J
[ 4 v s ! J
e n
for o < n < n
r
2 2
8tt rQgH
1
n
for
(5-32)
*
n > n
and for n = 0, the depth of dimple can be expressed as

177
*s,0
QF2e2s
2 2
16tt r gvs,
o 1
(5-33)
In terms of the original variables, Eqn. (5-33) can be written
as
.2
's, 0 2
16tt gvt^
exp
'2Qt'
(5-34)
which means the depth of dimple increases exponentially with
respect to time. Since the coefficient,
, on the
167T gvtqJ
right hand side of Eqn. (5-34) is determined by the initial
distribution of the circulation r and the gravity g, it
appears that the growth rate of the surface dimple depends
2
upon the parameter (r /2Q), if the initial circulation is
fixed. Figure 5-2 shows a plot of the developing depths of
dimples calculated by Eqn. (5-34) using rQ/2Q as a parameter.
They are compared with experimental results obtained by
measuring the variation of dimple depths with time following
the initial circulation distribution given in Figure 5-1.
2
It was found that the values for the parameter (r /2Q) appear
to range between 3 and 7 seconds. The question is how to
determine the magnitude of this parameter in experiments.
2
From Eqn. (5-16) the physical meaning of (r /2Q) is made
explicit, i.e.,
2Q
(5-35)

Depth of Surface Dimple, cm
178
Figure 5-2 Dependence of surface depression on time.

179
Therefore, this parameter is nothing but the inverse
of the axial stretch rate or velocity gradient at the vortex
axis. As seen in Figure 5-2, the higher the axial velocity
gradient, the faster the depth of dimple increases exponen¬
tially with respect to time. Since the axial velocity is
indeed dependent on time, the assumed value of w in Eqn.
(5-16) is not strictly correct. Because of the time-dependent
nature of the axial velocity gradient during the core devel¬
oping process, the actual central depression does not quite
follow a single predicted curve with a specific constant
value of the axial velocity gradient. The dependence of the
axial velocity gradient on time is impossible to measure in
experiments because of the short times involved for the core
formation.
As mentioned earlier, the time involved for the core
formation should be "short." This concept can be made clearer
here. It can be seen in Figure 5-2 that if the axial velocity
gradient is suppressed; for instance, due to the high visco¬
sity, the time involved for the core formation will increase.
However, the longer the time involved, the more important
the viscous effects become, causing the liquid motion to decay
more gradually with time. Therefore, when the axial velocity
gradient is suppressed low enough, the viscous effects cannot
be neglected in the analysis anymore and the central depres¬
sion no longer increases exponentially with respect to time.
Consequently, the surface depression would not develop to
form a complete central air core even if the time approached

180
infinity as the present analysis predicted. It is concluded
that during the emptying process of a tank, either an air
core forms in a very short time, or no air core forms at
all (i.e., only a shallow dimple forms at the free surface).
In order to find out whether or not the assumed stream
function, ip, represents a possible flow field and how the
suppression of the axial velocity gradient causes a corre¬
sponding suppression of the tangential velocity, the next
section will investigate the problem of finding solutions
to the steady equations of motion of a viscous fluid.
5.3 Steady-State Solutions for Viscous Vortex Flows
5.3.1 Similarity solution (Long)
The similarity transformation is a common method of
obtaining exact solutions to the equation of motion in which
the variables are transformed to functions of a single ele¬
mentary function of the coordinate. Long (1961) found that
the original equations of motion (steady state) could be re¬
duced to ordinary differential equations in terms of the
similarity variable
when T = T (y) and ip = £f(y). The definition of the variables
has been given in Eqn. (5-9). These quantities are substituted
into Eqns. (5-10) and (5-11); the resulting steady equation
can be written after some direct integration and rearrangement

181
2xr"
r r •
RefF' + a (3T' + 2X2r") = 0
-2Ro"
ff,,, + 3f,f’1 - ^ (Xf"*' + 2f")
+ | [2Xff"' + 6f' f" + ff" - 3 f' 2 ]
- ^ [(4X+2aX2)f'''' + (14X+12aX2)f'''
+ (4 + Q aX) f" - 3af ]
(5-36)
(5-37)
a, Re and Ro have been defined in Eqns. (5-10) and (5-11).
Long actually solves the equations obtained by regarding
a as small compared with unity. This approximation means
that variations in the axial direction are small compared
with variations in the radial direction. Numerical solutions
for different combinations of conditions are obtained by an
iterative method in which a parameter is varied until the
outer boundary condition r-*l is satisfied. The radius of
the viscous core, r , is an important feature of the solutions,
i. e. ,
r
c
zy
r
(5-38)
The series of solutions shows that the axial velocities are
of the same order as the tangential velocities even at larger
radii. However, in our laboratory vortex, the viscous core
is independent of the axial position (see Figure 4-14) and
the central axial flow drops off rather quickly as the radius
increases (see Figure 4-15). Thus our laboratory vortex is
not included in Long's solutions. Long himself concluded

182
that the jet-driven vortex is not characterized by his
solutions. Actually, Long's flow problem can perhaps be
best described as a swirling jet exhausting into unbounded
fluid which has constant circulation. In the limit of zero
circulation, it reduces to Schlichting1s jet problem (1968).
5.3.2 Approximate solution (Einstein and Li)
Einstein and Li (1955) divided the flow into two re¬
gions, an inner region or central core in which the axial
motion toward the exit port takes place and an outer region
in which the axial velocity is zero. Their solution in the
outer region is a generalization of the well-known two-
dimensional viscous solution for an axisymmetric rotating
flow. In the inner region, their solution is the same as
â– k
Rott's (1958) and Burgers (1940) vortex solution assuming
stagnation-point flow. This stagnation-point flow is
assumed to extend to infinite and be superimposed by a
rotating motion in which
u = -ar, v = v(r) and w = 2az (5-39)
where a {- constant) is the "inflow gradient" (Rott, 1958).
With such a velocity profile, the continuity equation is
satisfied and the steady Navier-Stokes equations are reduced
to
★
Their solutions were obtained independently, but the results
are identical.

183
2
a r
2
v
r
1 9JP
p 3r
radial momentum
v
J_ 1 _3_
3r r 3r
vr
tangential momentum (5-40)
2
4azz
1
p 3 z
axial momentum
With the boundary condition, vr = at r=co, the tan¬
gential momentum equation is readily integrable and the solu¬
tion is
v =
r t 2 /0
1 _ e-ar /2v
Rott defined the "viscous core," r as
c
r
c
(5-41)
(5-42)
whose physical meaning is that if r >> r , the solution re¬
duces to potential flow and only for r comparable to rc is
the viscosity effect felt. Considering the case of the bath¬
tub vortex, Rott assumed that a is a constant and can be found
from the total discharge, QT> and the vortex height, H,
a
Qn
2ttH'
(5-43)
Experimentally, the actual value of a (estimated from the
axial gradient of the measured centerline axial velocity) is
a few orders of magnitude higher than the above prediction.

184
This implies that the inflow gradient a should be predicted
in some other way.
Einstein and Li made their assumption that inside the
inner region (the radius of which is simply taken as that
of the exit hole, r ), the axial velocity at the exit hole
2
is reduced to a value (Q^/irrg) with the total flow dis¬
charge. This is equivalent to assume the inflow gradient
a as
a
2„
tt r H
e
(5-44)
Again, solutions based on this arbitrarily assumed axial velo¬
city profile have not been successful at all in predicting
the measured tangential velocity distribution. In comparing
the experimental values with those assumed by Eqn. (5-44) , it
was found that the predicted values were about an order of
magnitude lower, the discrepancy being even larger for more
viscous fluid (see Figure 5-7).
Equation (5-44) implies that the total volume flow con¬
centrates in the central core to form a central axial flow.
This aspect ignores the presence of the boundary layer on
the end walls. However, at the end walls, the well-known
"tea cup" effect exists, and a secondary flow is produced
in the boundary layer. A large secondary flow in the end-
wall boundary layers offers a ready explanation of the rea¬
son why the measured axial velocity is much lower than the
one computed with the assumption of Eqn. (5-44). The volume

185
flow rate in Eqn. (5-44) is not the total discharge rate
but the total discharge rate minus the secondary flow in
the boundary layers.
Two analyses presented independently by Rosenzweig,
Lewellen and Ross, (1964) and by Anderson (1961) give solu¬
tions for the secondary flow on the end wall of a vortex
chamber for the turbulent and laminar cases, respectively.
The Anderson analysis is discussed in the next section in
order to determine if the boundary layer analysis can in¬
deed explain the observed axial velocity distributions.
The Anderson analysis is chosen because, at the tangential
Reynolds number ordinarily encountered in the present work
(Re^: 500-5000), the boundary layer is most likely laminar.
(The transition tangential Reynolds number observed in tests
of a rotating disk is on the order of 10^ (Bilgen, 1971).)
5.3.3 Theoretical solution for the secondary flow (Anderson)
The flow was analyzed under the assumption that the
flow may be divided into three regions, as indicated in
Figure 5-3. In the primary flow region, the tangential velo¬
city is assumed to be a function of radius only. This assump¬
tion has been confirmed by the present experiments. In the
secondary flow region in the boundary layer on the end wall,
the axial velocity gradients are considered to be an order of
magnitude larger than the radial velocity gradients due to
the large axial viscous stresses. Inside the viscous core
region, the downward axial velocity profile is assumed to be
flat. The radius of the viscous core is simply assumed to be

Secondary Flow Region
Viscous Core Region
Plane of Symmetry
(Imaginary Free Surface)
Primary Flow Region
Secondary Flow Region
Q
T
Figure 5-3
Flow model in the vortex chamber. The
is considered as the symmetry plane in
free surface
the model.

187
equal to that of the exhaust hole. This assumption, which
has been generally made in vortex investigations will be
shown later in this analysis to be a very bad one. In an
actual vortex, the viscous core must be dependent on the
flow conditions and fluid properties.
5.3.3.1 Derivation of interaction equations
The radial flow occurring in the boundary layer on the
end wall is coupled through the continuity equation to the
axial flow in the primary flow region. As a result, the
secondary flow may have a strong interaction with the pri¬
mary flow. In the three flow regions, respective governing
equations of motion are employed.
1. Primary Flow Region:
The governing equations for this region can be derived
from equations of motion (5-1) through (5-4) by an order of
magnitude analysis. The equations of motion reduce to
TT- (ru) + ~ (rw) = 0 Continuity (5-45)
o T o Z
UP _ v
p 9r r
2 3 19
r 9r r 3r
rv
vr _9_
v 9 r
rv
9P
9 z
Radial Momentum (5-46)
Tangential Momentum (5-47)
0
Axial Momentum
(5-48)

188
The tilde (~) denotes a quantity in the primary flow region,
and the letter without a tilde will be subsequently used for
a corresponding variable in the boundary layer. The following
dimensionless variables are now introduced:
^ *
u
u
vr
vrr
*
r
r
R
Re =
r
urr
Ret =
vrr
where R denotes the quantities evaluated at the outer radius
of the vortex chamber. Equation (5-47) in dimensionless
form is
dzr
^ *
Re u
r
(5-49)
2. Secondary Flow Region:
The governing equations of motion (5-1) through (5-4)
are transformed into boundary layer equations in the usual
manner. The derivatives in the z direction are considered
to be an order of magnitude larger than those in the r
direction. The resultant boundary layer equations are
d 3
(ru) + tñ (rw) = 0 Continuity (5-50)
dr a Z
2*2?
9u 3u v v 9 u „ , . . .. . /c
u -s— + w 7T— - — = - — + v —Radial Momentum (5-51)
dr dz r r ^2

189
9 3 3
-x-rv + w^-=v —- Tangential Momentum (5-52)
1C O 1C U Z r\ Z
3z
0 = -
3P
3 z
Axial Momentum (5-53)
where Eqn. (5-46) has been used to replace the pressure
/\
gradient by the primary tangential velocity v. Equations
(5-51) and (5-52) are now integrated from z=0 to z=6 where 6,
a function of r to be determined, is the boundary layer
thickness. The integrated equations are
d_
dr
ru dz +
2 2, , ,3u,
(v -v )dz - -vr (H
o Z
o z=0
Radial Momentum
(5-54)
d_
dr
r6
(ru)(rv)dz - rv
dr
, v , 2 , 3v.
(ru) dz = -vr (-5—)
o Z «
o z = 0
Tangential Momentum (5-55)
The velocity components u and v in these two equations
are functions of both r and z. In order to proceed further,
Anderson assumed u and v could be represented by simple poly¬
nomials and F2 in which the variables r and z are separated.
~ 2
vS z * * * 2 * 3
u = F2 (p F2(z ) = 12 [ z -2 (z P+(z )J] (5-56)
v = VFX(f)
* * * o
Fx(z ) = 2z - (z )
(5-57)
where F^ and F2 were chosen based on the viscous flow adjacent
to a rotating disk. Since Ó is the only unknown function, the

190
radial momentum equation (5-54) can be eliminated. This
is a result of the assumed radial velocity profile (Eqn.
5-56). By substituting Eqns. (5-56) and (5-57) into Eqn.
(5-55) and nondimensionalizing the terms, the following
equation is obtained.
d
*2-,
J { ¿
Í ^ * d
r- * 3-
2 5
*
dr
r
1 *2
l. r J
i -^dz i *
o dr
1 *2
l r J
*
- ^ TF'(0) - 0
6
*
where 6
Re
1/2 6
t R '
(5-58)
3. Viscous Core Region:
In the viscous core, the end-wall boundary layer does
not exist and the governing equations are identical with
those for the primary flow. Inside this region, the sim¬
plest possible assumption will be made, namely, the radial
velocity falls to zero at r=0 linearly with r. When the
result is matched with the primary flow solution at the outer
edge of the viscous core, the following relationship is ob¬
tained
A A V"
u = ue — (5-59)
e
where e denotes the quantities evaluated at the exit orifice.
Substituting Eqn. (5-59) into Eqn. (5-49) and introducing
/\ /\
= Re u r one obtains
r e e
the constant A

191
r
A
r
e
1
- e
1 -
(A/2)r*2
7^
(5-60)
Thus solution of the viscous core is given by Eqns. (5-59)
and (5-60), provided ug and are known from the primary
and secondary flow.
4. Method of Solution:
Since it is assumed that the boundary layer flow grows
from zero near r, any increase (or decrease) in the radial
flow in the boundary layer must result in a corresponding
decrease (or increase) in the radial flow through the pri¬
mary flow region. Thus
2ttHu R
K
2rHur + 2 tt
[8
ru dz
o
(5-61)
Substituting Eqn. (5-56) into Eqn. (5-61) and nondimension-
alizing the term yields
u
JL_
*
r
+ BT 2 (—-£■)
r
3
(5-62)
2 R R
where B = —=—-pr — — is called the secondary flow parameter
Re/2 “r
which is a measure of the fraction of the total flow which
passes through the boundary layer. Equations (5-62) , (5-58) ,
* *
and (5-49) determine u , T, and 6 as functions of the radius
and the parameters Re^_ and B. These equations were solved
iteratively using a numerical method (Anderson, 1961).

192
5.3.3.2 Results and comparisons with experiments
A large number of solutions were obtained for some fixed
values of Re^ over a range of B. A typical boundary layer
flow is shown as a function of radius in Figure 5-4. It can
be seen that the radial flow occurring in the boundary layer
decreases with decreasing radius near the vortex axis. From
the continuity consideration, this implies the passage of
flow in the boundary layer to the mainstream. This is also
predicted by the theory of Rosenzweig, Lewellen and Ross
(1964). Since only a fraction of the fluid's angular momentum
is lost in the boundary layer (due to viscous shear), mass
return from the boundary layer tends to support the angular
momentum distribution, retarding its further decline. The
elimination of the secondary flow from re-entering the primary
flow utilizing wall suction has been investigated experimen¬
tally (Travers and Johnson, 1964), and found to be effective
in decreasing the vortex strength and thus confirm the
boundary layer analysis. The effect of an increase in B on
the variation of tangential velocity with radius is shown in
Figure 5-5. It can be seen that for a fixed total flow
through the vortex chamber an increase in B and the accom¬
panying increase in the secondary flow (see Eqn. 5-62) results
in a decrease in the vortex strength.
In order to determine whether the boundary layer analysis
can indeed explain the experimentally observed axial velocity
distribution, the prediction of the analysis is compared with
data for a typical run with water. The experimental condi-

Secondary Flow on Boundary Layer
Total Volume Flow Rate
193
Dimensionless Radius r/R
Figure 5-4 Effect of radius on fraction of total flow
within boundary layers on the end walls,
Re^ = 10, r^/R =0,1.

Tangential Velocity
Tangential Velocity at
Figure 5-5 Effect of radius on tangential velocity in primary flow.
194

195
tions are listed in Table 4-1 and the measured velocity-
profiles are shown in Figures 4-13 through 4-16. The
variation in the circulation (Figure 4-14) indicated for
r > 15 cm is believed to be due to the influence of the
turbulent mixing region between the jet and vortex flows.
Therefore, the outer radius of the vortex chamber, R, is
set at r = 15 cm. From the mass-flow measurement
3
(Qt = 49 cm /sec) and tangential velocity measured at
r=R, the calculated Re^, Ret and B are found to be 14.3,
2800 and 1.0, respectively. The predicted radial mass
flow in the boundary layer is shown in Figure 5-6. Once
this radial mass flow is known, the axial velocity in the
primary flow can be obtained by integrating the continuity
equation. If the axial velocity is assumed to be linearly
dependent on z, then from Eqn. (5-45) one obtains
A
w(r,z)
zQT dG
2ttH rdr
for r > r
e
(5-63)
where G is the ratio of the boundary-layer mass flow to the
total discharge (Figure 5-6).
Inside the viscous core, the axial velocity profile is
assumed to be flat and can be calculated from Eqn. (5-59)
through continuity equation,
Qtz
1 - G
l
r=r,
for r < r
e
w (z)
2-itH
(5-64)

Secondary Plow within Boundary Lay
Total Volume Flow Rate
e
Figure 5-6 Predicted secondary flow within the boundary layer.
196

197
In Figure 5-7 the variation of axial velocity with
radius calculated by this method is compared with that in
Figure 4-15 for ^ = 0.4. It was found that the axial velo-
n
city profiles are poorly predicted as compared with the
experimental results. The solution of Einstein and Li is
also shown in Figure 5-7 for comparison. It is apparent
that the axial velocity profile, especially near the vortex
axis, cannot be correctly predicted by either the oversim¬
plified solution of Einstein and Li or the boundary layer
theory. However, Owen et al. (1961) reported that the radial
mass flows in the boundary layer (calculated directly from
the measured variation of velocity and flow angle for each
of the boundary layer transverse) are in good agreement
with values predicted by Anderson's theory. Examination of
the work of Owen et al. reveals that this theory can ade¬
quately predict the secondary flow only in the outer region
of the boundary layer (r >> rg) for the strong-vortex case,
i.e., Rer^-<». In the present experiment, Rer is an order of
magnitude lower than that of Owen et al. More important,
this theory (and also the theory of Rosenzweig, Lewellen and
Ross) overlooks the presence of the boundary-layer discontinuity
at the exit orifice. There, abruptly, the boundary layer
theory is not applicable because the no-slip condition at
the wall is invalid. Besides, near the exit orifice, the
boundary-layer flow may have such a strong interaction with
the central axial flow that the actual flow is entirely dif¬
ferent from the one predicted in the theory. Another weak

198
Radius Ratio r/r
e
Figure 5-7 Comparison of axial velocity profiles between
theory and experiment.

199
point of the boundary layer theory is the arbitrary specifica¬
tion of the radius of the viscous core. As the present study
shows, the core radius is nearly proportional to the square
root of the ratio of the fluid viscosity and the axial gradient
of the centerline axial velocity (Section 5-4).
In summary, the size of the viscous core and the inter¬
action between the boundary-layer flow and central axial flow
which have been overlooked in the boundary layer theory make
the prediction of the axial velocity distribution in the pre¬
sent experiments invalid!
5.4 The Application of Lewellen's Asymptotic Solution to Our
Laboratory Vortices
5.4.1 Lewellen's solution
The most relevant work on the subject of vortex flow
is that of Lewellen (1962). He has developed a series ex¬
pansion method for studying such flows when the ratio of
stream function to circulation is small, which assures that
the Rossby number, Ro (defined earlier in Eqn. 5-11), is
small. A summary of his work follows, modified such that it
may be applicable to our case.
The Navier-Stokes equations in the dimensionless forms
equivalent to Eqns. (5-10) and (5-11) are written as
ü 1L _ M il = 2jx ¿r _o_ a2r
3 5 On 3n 9C Re „ 2 Re „r2
an a^
(5-65)

200
r iL
3C
Ro2-! 4n2
~di¡j d3ip
3n3
+ 9
- Re
3ip 3_ip
3E, g^2
+ n
4n‘
~4 ,
9 ip
2 2
an 3?
3 ip
9 3ip
2 Í 2
3 3ip
3n
3C3n2
Re [2
30 3
3 ip
3 3ip
3 ip
n ^x
3n
3 3ip
3?
3n3?2
3?3
1
3 ^ip
ll
2
an J
srJ
+ 0
.4 .
3 ip
~ 4
3n
(5-66)
Lewellen considered asymptotic expansion of ip and T in
2
terms of Ro (which, since it is small, is denoted by e) with
the leading term of f independent of C. He assumed
r = r (n) + r,(n,?)e + r (n,?)e + ...
^ = ^0(o,C) + ip1(n,?)e + ip2(n,C)e + ...
and showed that the Navier-Stokes equations became
(5-67)
3^o
zeroth-order 2 nf" - ReT ' -â–  â– â–  â–  = 0
o o 3£¡
(5-68)
ror!2 = 212lf01fd' - fÓlf01 - le >“01
first-order â– 
(2fñi' + nfJJ)
(5-69)
r r = 4n2Tf f • ' ' - f f1' - — + nf^l
o 11 Lr0l 00 01 00 Re ^00 nI00J
(5-70)
2r|r 12 Re f01rÍ2 3Re f13r0 + 2Re f01F 12 ~ 0
2nríl f0iríl 2Re f12r 0 + 2Re f00ri2
+ Re f¿irn = 0
2nro Re f01110 Re fliro + Re f00ril
(5-71)
(5-72)
+ ar12 - 0
(5-73)

201
*0 =
f00 (n)
+
C£0i(D
with r1 =
rio ^
+
eqpn)
+ £2r12(n
*1=
fio(n)
+
€fi:L (n)
+ £2f12 (n
where primes denote differentiation with respect to g.
The system of differential equations (5-68 through 5-73)
provides a set of 6 equations with 9 knowns.
The extra degrees of freedom can in principle take into
account any axial boundary conditions which most theoretical
investigators drop as a result of the restrictive require¬
ments imposed upon their solutions (see Section 5-3). The
major contribution of Lewellen was that he identifies the
vortex problem to be of the boundary-value problem for the
stream function, i[i. This means that the set of equations
governing the flow can be made complete only by supplying in¬
formation at one or more axial positions in the form of boun¬
dary conditions. This point is very important because it
demonstrates the important role that the conditions at the
end walls of the chamber may play in the set of solutions.
In order to apply Lewellen's method (or any general method
without restrictive requirements imposed upon the solution)
to the real vortices, the exact boundary axial boundary con¬
ditions at two suitable axial positions must be given. For
the case of flow in a vortex chamber, with tangential injec¬
tion and axial withdrawal of fluid, the boundary conditions
for the stream function may be

202

(5-75)
(5-76)
In our laboratory vortex model (see Figure 5-3), one can
set 5=0 and 5=1 at the free surface and the bottom end wall
respectively. Considering the free surface to be a plane of
symmetry at E, = 0, one immediately obtains that g(n) = 0 since
the strem function is zero at the symmetry. Also from Eqn.
(5-74) one obtains the three additional equations
(5-77)
f0i(n) = f(n)
(5-78)
f 11 (n) + fi2^ + fi3^n^ = 0
(5-79)
This forms a complete set of 9 equations and 9 unknowns.
Lewellen assumed a special form of f(n) which he expected to
be appropriate to the flow through the exit hole near the
end walls and proceeded further to solve the set of equations
by simple differentiation and integration. Lewellen showed
that the behavior of solutions is determined from the form
of f. It is therefore apparent that Lewellen's method only
determines the solution uniquely to the same degree to which
the boundary condition f(n) is uniquely specified. If f (n)
is specified incorrectly, the solutions obtained may probably
be erroneous.

203
As described earlier in Section 5-3, however, the situa¬
tion in the near vicinity of the end wall is much more com¬
plex, since the flow there is accelerated radially inward
producing a very large radial mass flow (that is the secon¬
dary flow). The boundary layer flow imposes different
boundary conditions on the flow to the ones assumed neglecting
the influence of such a flow on the outer flow. It is hence
necessary to have the boundary layer solution before pre¬
scribing a stream function as a boundary condition at the
end wall. In the boundary layer analyses as summarized in
the previous section it is shown that the predicted axial
velocity profile at the edge of the bottom boundary layer
(which is related to the radial flux in the boundary layer
through continuity of mass) is still not in agreement with
experiments. This is directly caused by the ambiguous in¬
teraction between the boundary-layer mass flow and axial
downward flow near the exit orifice. Such an interaction
is as yet undetermined. The difficulty of specifying f(n)
near the exit orifice is therefore quite serious in Lewellen's
and other existing methods in the theoretical study of a
vortex flow, since the flow is of the nature of the boundary-
value problem.
5.4.2 Comparisons of theoretical solution with experiments
Rather than predicting the function of f(n) as a boun¬
dary condition, it would seem appropriate here to specify f
according to the axial velocity profiles actually observed
in our laboratory vortices. It has been indicated in Chapter

204
IV that the axial velocity gradient is approximately in¬
dependent of z and its distribution can be well fitted by
a Gaussian distribution function,
w(r,z)
az
(5-80)
where a is the axial gradient of the axial velocity at the
axis and a is the variance of the Gaussian distribution.
The characteristic dimensional quantities of Eqn. -(5-80)
are now defined (refer to Section 5-1),
r = /2 a
o
Q
2
= aa
(5-81)
Substitution of Eqn. (5-81) into Eqn. (5-80) and using equa¬
tions (5-6) and (5-9) yield the stream function
iMn/£) = Ue"n - 1) (5-82)
From Eqn. (5-71) it follows that
f(n) = e n - 1 (5-83)
which is the stream function at the edge of the boundary layer.
This function is chosen to represent the required features
of the flow (that is Eqn. 5-80). Equations (5-68) through
(5-73) can be readily solved to give the complete set of solu¬
tions. Experimental velocity profiles for water and some vis¬
cous fluids with different viscosities (Sections 4-3 and 4-4)

205
are compared with the thoeretical solution in order to
determine whether Lewellen's method can indeed predict all
or part of the experimentally observed velocity distribu¬
tions. The characteristic dimensional quantities, Q, r ,
and 1 for the various viscous liguids are listed in Table
CO A
5-1. It was found that r increases with increasing visco-
o
sity, which implies the central axial flow tapers off more
slowly with respect to the radius in the more viscous fluid.
This is reasonable in view of the fact that the viscous core
becomes thicker in the radial direction with more viscous
fluids.
From Eqns. (5-68) and (5-78), the zeroth-order solutions
are
in
Re
exp
ft -i
n x(e n-i)dn
o
o
r =
o
foi = e_n - 1
dt
o L-
exp
Re
o
n 1 (e n-l)dn
(5-84)
dt
The systematic procedure devised by Lewellen allows us to
calculate the higher-order solutions, and assess the accuracy
of the zeroth-order approximation. As listed in Table 5-1,
the value of the expansion parameter, e, is of order 10
In this case, the zeroth-order solution can well represent
the essential part of the asymptotic solution. Comparisons
between the experimental results and the zeroth-order approx¬
imate solutions are shown in Figures 5-8 through 5-11. Close
agreement was obtained for all the viscous fluids with dif¬
ferent experimental conditions, thus confirming the validity

Table 5-1
Values for Characteristic Parameters
Fluid
Relative Viscosity3
2
F , cm /sec Q
2 /
cm /sec
r , cm
o
Re t
£
Water
1.00
19.0
0. 035
0.37
2030
0.074
Corn syrup +
water 1.16
16.1
0.040
0.45
1483
0.091
Corn syrup +
water 1.29
13 . 6
0.044
0.54
1127
0.187
Corn syrup +
water 1.46
11.8
0. 050
0.66
864
0.122
CMC 7H
300 ppm or
S-273 3 ppm
2. 80
5. 9
0.093
1.89
225
0.207
aThe relative
: viscosity is based on the
viscosity of water
at 23°C,
that is
0.9358
cp.
206

207
Figure 5-8 Comparison of observed tangential velocity profiles
with the prediction, water.

208
Figure 5-9 Comparison of observed tangential velocity profiles
with the prediction, corn syrup-water.

209
Figure 5-10 Comparison of observed tangential velocity profiles
with the prediction, corn syrup-water.

Tangential Velocity v, cm/sec
210
O CMC 7H, 300 ppm
• S-273, 3 ppm (in deionized water)
Radius Ratio r/re
Figure 5-11 Comparison of observed tangential velocity profiles
with the prediction, 300 ppm CMC 7H solution or
3 ppm Separan AP 273 solution.

211
of the zeroth-order approximation to our problem. It is
of interest to point out that the present analysis, which
is derived from the Navier-Stokes equations, also correctly
predicts the velocity distribution for a viscoelastic S-273
solution. This is possibly caused by its high zero shear
viscosity which might offset any viscoelastic effect in the
flow field. This will be discussed in more detail in the
next section.
The boundary condition of the stream function is pre¬
scribed in a general form as given by Eqn. (5-83). The
dimensionless form of Eqn. (5-83) implies that the boundary
condition for a particular flow field depends only on the
particular characteristic dimensional quantities rQ and Q.
Thus the problem of predicting the velocity distributions
in that particular flow field is to determine r and 0.
O '
From Eqn. (5-81) rQ is related to the variance of the axial
velocity distribution, a, while Q is related to both o and
the axial flow gradient, a. It has been found theoretically
and confirmed experimentally (see Figures 5-8 through 5-11)
that rQ is almost identical to the "viscous core" of the vor¬
tex, in which the tangential velocity decreases linearly with
decreasing radius and the flow reduces to that of rigid-body
k
rotation. Since the flows are laminar throughout the core,
it is natural to assume that, in the absence of an exact
theoretical solution for the present problem, the expressions
k
The dye streak study of Section 4-1 gives a good demonstra¬
tion of the laminar character of the flow in the core region.

212
for the variation of the viscous core radius with Reynolds
number are similar to the corresponding expressions for
flow over a flat plate (Schlichting, 1968). That is
vortex flow r ~ ^~TTT = ^(core radius)
° (Ret)V¿
flow over a flat plate 6 ~ —-—^--rió = /rp— (boundary layer
(Re ) ' °°X thickness)
In Figure 5-12 the observed value of rQ is plotted against
Re^ on a log/log scale. A straight line was obtained, with
a slope of -0.7 rather than -0.5 as we expected. Figure 5-12
may be used to predict rQ in the flows of our laboratory vor¬
tices. If r is known, Q can then be determined from the es-
o
timation of the axial flow gradient, a. Rott (Section 5-3)
has discussed an exact solution of the Navier-Stokes equations
and shown that the radius at which the tangential velocity is
maximum is determined by the Kinematic viscosity r and the
axial velocity gradient a,
r
max
o/i
2 4^ —
a
(5-85)
Equation
in Table
(5-85)
5-2.
gives excellent
Experimentally,
prediction of
it was observed
r
max
as
that
shown
r
max
0.85 r
o
(5-86)
Substitution of Eqn. (5-86) into Eqn. (5-85) yields
v
~2
r
o
a
7.0
(5-87)

Table 5-2
Comparison of Rott's Prediction of r to Experimental Results
^ max
a(axial flow r (Experimental), r Rott s solution,
max ^ ' max' , c oc.,.
Fluid gradient), sec x cm Eqn. (5-85)), cm
Water
0.558
0.314
0. 300
Corn syrup + water
0. 396
0.382
0. 383
Corn syrup + water
0.302
0.460
0.463
Corn syrup + water
0.228
0.561
0.567
CMC 7H 300 ppm
or
S-273 3 ppm
0. 052
1. 607
1.644
213

Viscous Core Radius r , cm
214

215
Also from Eqn. (5-81), a simple relationship between the
characteristic parameter, Q, and the fluid viscosity v is
obtained,
Q = 3.5 v (5-88)
In summary, in the present vortex flow field, if the
entering circulation, r , and the kinematic viscosity, v,
00
of the working fluid are specified, the three components of
the velocity can be predicted. This is very important in
view of the fact that there are still large discrepancies
between the experimental results and the available theory
(Anderson, 1961; Rosenzweig, Lewellen and Ross, 1964) which
has studied in greater detail the boundary-layer interaction
problem.
5.5 Theoretical Analysis for Viscoelastic Fluids
Considering the equations of motion in the form of Eqns.
(5-10) and (5-11), some useful features of the solutions can
be elucidated. It is seen from Eqn. (5-10) that the relation¬
ship between the circulation and stream function is governed
by a single characteristic parameter - Reynolds number, Re.
As described earlier, Re is based upon the volume flow through
the main body of the vortex chamber and the fluid viscosity.
Therefore, the circulation-stream function relation is in¬
dependent of the other material parameters of the fluid, such
as the relaxation time. If a precise value of the stream
function could be obtained, then Eqn. (5-10) would yield an
accurate description of the circulation without any informa-

216
tion about the fluid properties, except the shearing visco¬
sity. In Eqn. (5-11), it should be noted that the normal
stress terms enter into the determination of the stream func¬
tion and circulation. Since a viscoelastic fluid is usually
characterized by its large normal forces which in general
depend on the entire past history of the deformation process,
the structure of the problem is quite complex. For simplicity,
Section 5.5.1 begins by considering a simple model of the flow
such as Rott's and introduces the model into a proper con¬
stitutive equation such as the convected Maxwell model, and
then the values of the different components of the stress ten¬
sor are backed out.
5.5.1 Kinematics of vortex flow
It has been shown in Section 5.4 that the experimental
velocity field within the vortex matches very well with Eqn.
(5-84). However, Eqn. (5-84) is still too complicated to be
introduced into the constitutive equation to calculate the
stress field. For simplicity, by applying a Taylor series
expansion of Eqn. (5-84) with respect to r=0, the whole equa¬
tion reduces to the earlier solution of Burgers (1940) and
Rott (1958) (see Section 5.3). In terms of a cylindrical co¬
ordinate system (r,0,z) with the radial, tangential, and
axial velocity components designated by u, v, and w, respec¬
tively, the velocity field can be given by

217
u = -
a
2
r
v
(5-89)
w = az
where a is a constant related to the axial gradient of the
axial velocity profile which Rott calls the "inflow gradient,"
and is the constant circulation at infinity. Figure 5-13
shows some typical tangential velocity profiles for water
vortices in various flow conditions. With proper values of
the parameter a and 1^, Eqn. (5-89) is in fair agreement with
the experimental data.
For the above velocity field, it is necessary to deter¬
mine whether such flows are extensional in nature. For flows
which are materially steady, several criteria for extensional
character are available or can be developed from existing
definitions of extensional flow (Wang, 1965; Noll, 1962).
The velocity gradient tensor is found to be
r
r
r 2 ii
a
CO
ar
2
2
r
1-exp
l"4 JJ
r i
r 2\
7 al r
2
vy =
CO
J-exp
ar
4v
CO f
+ 2v eXT
ar
4v
r
0
(5-90)
0
0
a
It is clear that the flow considered here is not materially
steady. For materially unsteady flows, there are no known
tests for their extensional character. Intuitively, Kanel
|IIul
(1972) suggested that used I ~| as a simple measure of the
9

218
extensional or shearing character of a flow, where d and oj
are the deformation rate and vorticity tensors and II's are
the second invariant of these tensors. That is
d = j (VV + vvT)
w = \ (VV - VVT) (5-91)
11A = \ {tr2(A) ~ tr (A2) >
n0)
The group 1=—^1 goes to one for shearing and to zero
IJd
for steady extensional flow. However, this is not a frame-
indifferent (Astarita and Marrucci, 1974) quantity and the
flow character must certainly be independent of the reference
frame. Until some frame-indifferent measure of the exten¬
sional or shearing character of a flow is found, it is im¬
possible to determine the relative importance between shear
and extension. Here, we simply compare the magnitude of the
shearing and stretching (diagonal) terms in the kinematic
tensor in order to shed some light on the relative shearing
and extensional characters of the vortex flow. For the typ¬
ical runs shown in Figure 5-13, the shearing component, drg,
and the stretching component, d , are calculated according to
z z
Eqn. (5-90). They are plotted as functions of radius in
Figure 5-14.
It is obvious that the flow we are considering is not
an extensional type flow. The shearing terms are much more

Tangential Velocity v, cm/sec
219
Figure 5-13 Comparison of observed tangential velocity profiles with
predictions, water.
V

Deformation Rate d, sec
220
Figure 5-14 Typical deformation rate tensor components.

221
important in comparison to the stretching terms over most
of the flow field. It can be seen clearly that the deforma¬
tion is shearing in nature in the main body of the vortex.
At the axis, the deformation becomes extensional but this
is of very small magnitude in comparison to the shearing
deformation at outer radius, r>0.
5.5.2 Prediction of Maxwell model
The simplest of the convected Maxwell constitutive
models may be written
t + A ^ t = 2yd (5-92)
where t is the extra stress tensor, and A,y are the relaxa¬
tion time and the viscosity of the fluid, respectively, and
the Oldroyd convective derivative is given by
6I m
^7 = + Vx • y - vv • T - t • vy1 (5-93)
In indicial notation, Eqn. (5-93) is written as
6x
61
13
ij
at
k 13
+ v t •
;k -
kj i , ik j .
t v ; k - i v ,• k
(5-94)
Substitution of Eqns. (5-89) and (5-94) into Eqn. (5-92)
yields the following equations for the physical components
of the extra stress tensor,
T
rr
+
3t
rr
3 r
+
az
a t
rr
a z
+
-ya
(5-95)

222
T 0 0 + A
a - 3TO0 , „„ 3l00 o-
r + aZ ~TT ' 2YTr0 + aT00
2 9r
= - ya
(5-96)
T + X
zz
_ 9x 9x
a zz , zz „
- r —— + az — - 2ax
2 9r 9z zz
= 2ya
(5-97)
Tr0 =
(5-98;
x = xQ = 0
rz 0z
(5-99)
r
where y = -
1 - exp
ar
2>n
4 v
+
r a
oo
2v
exp
ar
4v
is the tan¬
gential shear rate which is a function of r only.
The first-order partial differential Eqns. (5-95) through
(5-97) are decoupled and can therefore be solved one at a
time. The equations are solved in the usual manner using the
characteristic equations for the differential equations.
The characteristic equations for Eqn. (5-95) are ex¬
pressed as
dr
_r
2
dz
z
dx
rr
- (5-100)
from which is obtained
zr = constant
(5-101)
ya
rr 1 + Aa
1 + aA
aX
constant
(5-102)
Then the general solution for x^ can be expressed as

223
i + ya
rr 1+aX
= z"(1 + 1/a^ F(r2z:
(5-103)
Now, for z=0 (at the free surface), it is assumed that the
stresses there are all finite, if not negligibly small.
Hence, the solution to be considered must be of the form
rr
, r 2(l+l/aX)
1+aX 1
(5-104)
where is an arbitrary constant.
and
Similarly, the general solutions for rn„ and t are
0 0 zz
00
H (a-2Xy ) 2 (1+1/aX;
1+aX u2
(5-105)
zz
2ya 2(l/aX-2)
1-2aX ^3r
(5-106)
respectively.
The choice of the arbitrary function F in Eqn. (5-103)
is guided by the assumption that the stresses at the free
surface are not infinite. Since the normal stresses all
vanish at r->0, the constants C^, C2 , and must be zero.
Hence the normal stress difference can be expressed as
T - T,
rr (
2Xpy
1+a X
(5-107)
3ya
Tzz Trr (1+aX)(l-2aX)
(5-108)

224
In Eqn. (5-108), the well-known limiting behavior for
Tzz ~ Trr a Maxwe^--*- fluid in homogeneous steady extension
is predicted in the present analysis. Denn and Marrucci
(1971), and Everage and Gordon (1971), using the limiting
case of constant stretching in a transient process, have,
however, shown that 2Aa = 1 represents only a critical con¬
dition in the sense that for stretch rates lying below that
value, the rate of increase in relative tensile stress with
time is bounded and a finite asymptote is approached. At
the limiting condition 2Aa = 1, Denn and Marrucci showed
that a time t = 15A was required for the stress (in an ini¬
tially unstressed fluid) to increase an order of magnitude
above the Newtonian value. In the present study, the fluid
at the free surface is also in an unstressed state and the
total time for the fluid element to be stretched within the
vortex before passing out of the exhaust hole is limited.
This implies that even at the limiting condition (t -t )
of the fluid element in the vortex flow would not possibly
reach a high level before it passes out of the exit orifice
(because the vortex depth is finite). Considering further
that the vortex flow field is dominated by shearing instead
of stretching deformation, we conclude that (t -t ) would
be negligibly small comparing with the tangential shearing
stress. The effect of (t -t ) > if any, on the velocity
field rearrangement resulting in the polymer addition is
negligible.

225
In order to see the effect of t - Ton in inhibiting
vortex formation, it is necessary to determine the material
parameter, A, the relaxation time of the polymer solution.
Due to the form of (x -t„-) we would expect this normal
stress difference to increase with A. A is considered to
be an extremely important viscoelastic parameter which
characterizes the viscoelastic nature of the material as
the ratio of the viscous portion to the elastic portion.
With the aid of the concept of bead-segment model Zimm
(1956) derived the following eguation for the relaxation
time, A,
0.42(y-y ) M
A “ CRT (5-109)
where y is the solution viscosity and yg the solvent visco¬
sity. In principle, Eqn. (5-109) is valid only for a mono-
disperse solution in a theta solvent, but it is expected to
also give a reasonable estimate for polydisperse solution
in good solvents. In this case M must be some suitable mole¬
cular weight average. The manufacturer's estimates for P-301
and S-273 are 4x10^ and 7.5x10^ respectively. The zero shear
viscosities at the respective CVI's for the two typical poly¬
mers have been measured using the specialized low shear visco¬
meter (Chapter III),
S-273
ll
M
>
u
2ppm)
y : 1.35 c.p.
P-301
II
H
>
u
3ppm)
y: 1.09 c.p.

226
Using the given data, calculated values of X for S-273 and
P-301 solutions are 1.4 and 0.056 sec. respectively. Con¬
sidering uncertainty in M and the validity of Eqn. (5-106),
these values must be considered no more than order of mag¬
nitude approximations. In Figure 5-15 the radial distribu¬
tion of (t -tqq) in the vortex field is plotted against
l l a U
the dimensionless radius r/r (r =2/— , Rott's core radius).
c c a
The values of (t -t__) have been dimensionalized with
rr 60 pr2a2x
K oo
respect to the reference stress, ~—p¡——rr • It appears that
2v(1+aX) ^
the "hoop" stresses, (t„ -t ), the tensions along the line
0 0 rr ^
of flow in the 0-direction, are built up when the vortex flow
is "drawn in" toward the core. But inside the core, where
the flow is rapidly decelerated, the hoop stresses or the
tensile forces are released, much like a spring under tension
being suddenly removed of tension. Such consideration might
lead to an explanation for the appearance of the flow fluc¬
tuation. (The details of the fluctuating velocity field will
be presented and discussed in the next section.) Recently,
Rama Murthy (1976) reported a fluctuating surface velocity
in the die exit region in the extrusion of polymer solutions.
He observed that the fluid elements on the surface of the
emerging jet accelerated and then decelerated. This accelera¬
tion-deceleration process leads to a major fluctuation of the
surface velocity as a consequence of the propagation if in¬
stability. The fluctuating surface velocity of the jet looks
remarkably like the fluctuating axial velocity in our vortex
core described earlier in Chapter IV. The analogue between

Dimensionless Normal Stress Difference
227
u
u
H
I
CD
CD
H
r<
cO
+
CN
r<
CNJ
cO
CN 8
2 3
Dimensionless Radius r/r
Figure 5-15 Dimensionless normal stress differences.

228
the two phenomena might lead us to the speculation that the
fluid, as it flows through the tube and recoils as the
constrains of the tube wall are removed act similarly as it
is drawn in toward the vortex core and recoils inside before
passing out of the orifice. The situation for both cases of
flow instability can be imaged as being similar to damped
oscillation of a spring after removal of tension. It is
argued that the distortion of the emerging polymer stream
and the flow fluctuation in polymer vortex core are analogous
effects.
In order to see if the presence of (x -x„_) can affect
rr 0 0
the isotropic pressure distribution, we write down the r-com-
ponent of the equation of motion for the vortex flow,
2 (T — T )
9P = p/ _ v rr 189
Or r r
(5-110)
where we have set 0/9r)x equal to 0. Introducing v from
Eqn. (5-89) and (x -x„„) from Eqn. (5-107), and by use of
r r t) o
the dimensionless variable
, r , 2
x = <—)
c
2
ar
4v
P is given by the following expression after some rearrange¬
ment ,
(5-111)
8r

229
where is the dimensionless pressure due to the
culatory" motion, P =
' OO
1-e
X
l x
dx and P_ is the
E
less pressure due to the "elastic" normal stress,
"cir-
dimension-
p* = 2Xa
E (1+Xa)
r 1
i -x -x
1-e - xe
3
X X
-
dx . Numerical values
of the integrations can be found with the help of the
•k
tabulated "exponential integral." Curves showing Pr and P
are plotted in Figure 5-16. Arbitrarily assuming that X*a
is of value 1 which is believed to be a critical value to
exhibit viscoelastic effect, it is found that the elastic
pressure effect does not cause a reduction in the total
pressure above 20%. Thus the above steady-state analysis
indicates that the normal stress difference, ), is
not able to cause the liquid to "bulge up" at r=0. Arm¬
strong (1972) did show that (xnn-T ) became infinite at r=0
00 rr
while using a different constitutive model. His result is
caused by an unrealistic assumption of the potential vortex
flow field that v becomes infinite at r=0.
Assuming the connected Maxwell model can adequately
describe the deformation behavior of the polymer solutions
of interest, one may question if the relatively small (Tgg“Trr
plays the role in inhibiting the vortex formation. To answer
it the vortex flow is examined more carefully. It is found
that in an actual vortex inhibition experiment the flow is
indeed not steady; instead, the flow is a "quasi-cyclic"
process (see Section 4-1). Experimental results of the depth
of surface depression show that the surface dimple fluctuates

230

231
violently (Chapter IV). Therefore we should not expect
(T00Trr^ to cause a "stationary" or "equilibrium" reduction
of the depth of the surface depression. As will be shown in
the next section the flow field is in transient state. It is
possible that the growth rate of (TQQ-Trr) can t>e much larger
2
than that of pv (see Eqn. 5-110). In order to test this
idea an analysis similar to the steady-state case is applied
to a transient vortex during the period of the air core for¬
mation. The transient flow field similar to Degarabedian's
solution (Section 5-2) is considered. The details of the
analysis are reported in Appendix C. A typical result is
shown in Figure 5-17 in which P = -
/<» 2
pa
dr +
yoo T — T
' 00 rr
o
dr
is plotted against the duration time t. The depth of the
surface depression can be calculated by dividing P by the
gravity g (see Eqn. 5-30). The relaxation time is used as a
parameter. When A = 0 it vanishes and the result is identical
to that presented in Section 5-2 for a Newtonian fluid. The
2
centrifugal force term grows very fast with time, re¬
sulting in the core formation in a short period, as observed
experimentally. When A is large enough (in the present typ¬
ical case, A~1 sec) the centrifugal force initially dominates
T — T
over the elastic force term — , but as time proceeds,
the latter may be comparable with the former and even domi¬
nate over the former. This result, theoretically, indicates
that the normal force grows much faster than the centrifugal
force and may finally overcome the growing centrifugal force
field in the transient vortex motion. Therefore, in the for-

Depth of Surface Dimple, cm
232
0 10 20 30
Time, sec
Figure 5-17 Theoretical prediction of variation of the
surface depression with time.

233
mation of a vortex core during the emptying process of a
container with some polymer solutions, the normal stress
difference is primarily responsible for the induction of
flow fluctuations as is discussed in the following section.
5.6 A Proposed Model for the Vortex Inhibition Process
The main purpose of this section is to justify the
theoretical results presented in the preceeding section which
indicate that the normal stress difference causes vortex in¬
hibition. As described earlier in Chapter IV, when a few
drops of liquid containing small particles were added to the
vortex center at its open end, the particles spread along the
vortex inner region and exhibited a very strong fluctuation.
This fluctuating velocity field has been regarded as being
important in the vortex inhibition process. These fluctuating
velocity measurements, however, cannot reveal all the details
of the flow field when investigating the structure of the vor¬
tex itself.
In an attempt to view the structure more clearly than
the particles had allowed, dye was injected into the flow.
After injecting the dye, the three-dimensional details were
more clearly observed directly by the dye pattern in the in¬
ner region. For this reason, visual studies of the flow
characteristics were hereafter made in detail with the dye
injection technique. The flow field is illuminated through
a thin slit to allow the observation of the fluorescent dye
in the vertical cross-sectional area of the vortex. In the

234
following subsections, results of the direct visual observa¬
tions are presented. In Section 5.6.1, the detailed descrip¬
tions of the nonbursting or quiescent periods are given as the
first stage in a repeating sequence. Characteristics asso-
. .
ciated with the second stage, i.e., sudden eruption or burst
and the following interaction with the existing vortex-core
flow are described in Section 5.6.2. The characteristics asso¬
ciated with the third stage, i.e., the "break-up" of the vor¬
tex core into turbulent fluctuations, are described in Section
5.6.3. Finally, the clarification of the model hypothesis is
given in Section 5.6.4. It must be strongly emphasized that
the entire process Stage I-* Stage 2-* Stage 3^ Stage 1 ... is
continuous in nature and appears to be a repeating sequence;
we speak of these three stages only for clarity in description.
5.6.1 Nonbursting period--first stage of the vortex inhibition
process
This is the time during which no burst is observed to
occur. The flow structure is essentially laminar in nature.
The dye streak as shown in Figure 5-18a consists of concentric
regions and gives a good demonstration of the laminar charac¬
ter of the flow. The flow pattern in this stage closely re¬
sembles those patterns known from dye studies in stable water
vortices (Section 4.1). The line-up of the dye into cylin¬
drical formations must be attributed to the axial shear flows
(Basinger, 1968 ) .
* ...
The term "burst" does not relate with the mass ejection in¬
duced at the sharp edge of the exit orifice resulting from
the boundary layer flow over the bottom surface.

235
Tank Bottom
Figure 5-18
Surface Dimple
1
(a)
Y_
Exhaust
Development of the dye pattern in the vortex inhibition
process: Polyox WSR 301, 3ppm. View field 20x30 cm.
(a) Quiescent of nonbursting flow; (b) Onset of the
bursting process. Notice the localized "bursts" occur
at the intermediate distance from the vortex axis;
(c) The bursting fluid interacts with the vortex-core
flow and causes local velocity fluctuations; (d) Breakup
following amplification of the bursting interaction.
Notice the vortex core becomes less coherent and more
random; (e) Restoration of the quiescent state.

Figure 5-18 (Continued)
236

(d)
Figure 5-18 (Continued)

Figure 5-18 (Continued)

239
5.6.2 Interaction of instantaneous eruption with vortex-core
flow -- second stage of the vortex inhibition process
This stage is the first step in the bursting process.
This and the next stage are quite difficult to follow in detail
because chaotic motion of the localized "bursts" occurring
near the vortex axis appears not to be a single isolated pheno¬
menon pertaining to the bursting fluid alone, but rather a re¬
sult of complicated interaction of the bursting fluid with sur¬
rounding high-speed elements in the vortex core. However, use
of the present dye injection technique made it possible to ob¬
serve typical motions of the localized bursting which ulti¬
mately break up into turbulent fluctuations. In this second
stage, sudden eruption occurs at some point in the near vici¬
nity of the axis of the vortex chamber. These eruptions
carry fluid upward to form a local, temporary reverse flow.
Careful observation indicates that the eruptions in general
occur under the dimple of the surface depression immediately
after the vortex oscillates downward to become deeper, but
sometimes it just occurs abruptly. Interaction of the bursting
mass with the central axial flow exhibits instability that
causes some kind of flow fluctuations in the longitudinal
direction. These fluctuating velocity patterns are more
clearly demonstrated by injecting neutrally buoyant particles
into the vortex center. A typical photograph showing several
of these fluctuating tracks has been shown in Figure 4-30.
Since the eruptions are strongly three dimensional and
they occur at random, they also interact with the surrounding
high-speed rotating fluid adjacent to the bursting mass. The

240
interaction appears to cause localized turbulent fluctuation
in the otherwise laminar vortex. Figures 5-18b and 5-18c
illustrate the growth of the localized turbulence in still
pictures; these photographs were taken successively in very
short time intervals.
5.6.3 Break-up--third stage of the vortex inhibition process
In the third stage, the eruption interacts even more
strongly with the vortex flow in the core region, and the
localized turbulence becomes much more chaotic. It is this
more chaotic, more "random" motion that we call "break-up."
Figure 5-18d illustrates the ultimate break-up of the con-
★
centrated vortex core into a turbulent fluctuation. After
the break-up, the flow will go back to the first stage or
quiescent state (Figure 5-18e) and begin another burst cycle.
The results of direct visual studies of the vortex in¬
hibition process are recapitulated as follows:
Although the details of the vortex inhibition process
after the eruption or burst are variable and highly compli¬
cated, there are certain common features that need to be em¬
phasized. First is the nonbursting period during which the
flow structure resembles that known in a water vortex.
Second is the formation of the localized burst followed by
its interaction with the vortex flow. And third, the am¬
plification of the interaction that causes ultimate break-up
into turbulent fluctuations in the chaotic state. The whole
★
Concentrated vortices here mean those vortices which have
concentrated axial flow restricted to a small radius.

241
process suggests not only instability of the instantaneous
velocity profiles but also the sequential formation of the
secondary and tertiary instability following from the un¬
stable mass eruption.
The process described above leads to the suppression
of the air core formation in vortices or vortex inhibition.
The stable central downward flow which has been shown theo¬
retically and confirmed experimentally to be required to
support a strong open vortex is always interfered with
the sudden eruption near the vortex axis. It is impossible
to maintain a high level of central downward flow with the
presence of a random sequence of eruptions or bursts.
Furthermore, these eruptions strongly interact with the
surrounding vortex flow which causes the coherent concen¬
trated vortex to break up into a chaotic, localized tur¬
bulence and ultimately leads to the apparent local tempo¬
rary destruction of the vortex core. It is virtually im¬
possible to form a complete air core when the vortex inhibi¬
tion process is strong enough because once the surface de¬
pression begins to "dip down" to reach the exhaust hole, the
bursts always interrupt its movement.
The vortex inhibiting process described above bears a
striking resemblance to all the flow patterns observed in
various vortex inhibition experiments using polymers of dif¬
ferent structures or conformations. In the vortex inhibi¬
tion experiments of S-273 in tap water, the whole process is
shown in time sequence still pictures in Figures 5-19a through

242
(a)
Figure 5-19 Development of the dye pattern in the vortex inhibition
process: Separan AP 273, 3 ppm in tap water. View
field 20x30 cm. (a) Quiescent of nonbursting stage;
(b) and (c) Interaction of the bursting fluid with the
vortex-core flow; (d) and (e) Breakup of the vortex core.

Figure 5-19 (Continued)
243

Figure 5-19 (Continued)

245
5-19e. A typical bursting process is also shown in Figures
5-20a and 5-20b for S-273 in salt solution.
5.6.4 Clarification of the model hypothesis
The model will be clarified by discussing the origins
of the bursts and the vortical interactions between flow
structures which explain the quasi-cyclic vortex inhibition
process.
According to the model, localized "bursts" occur below
the dimple of the surface depression. This can be adequately
explained by a local temporary adverse pressure gradient near
the vortex dimple. As predicted in the preceeding section,
a large normal stress difference, t -t__, developed in the
vortex flow field increases rapidly with decreasing radius in
a strong vortex.
0 . *2
t - t
rr 00 1+aA
where y = tangential shear rate
y = fluid viscosity
A = relaxation time of the viscoelastic fluid
a = axial velocity gradient
The pressure distribution in the vortex is readily obtained
from the integration of the r-component of the equations of
motion neglecting the less important terms,
2
P = -
p — dr +
r
r Trr T00
dr + constant
centrifugal force
elastic force

246

247
It is obvious that both the centrifugal and elastic forces
increase with increasing strength of the rotation motion.
Owing to the opposite sign of the two forces, they contri¬
bute to the pressure distribution in the adverse way, with
the elastic force increasing more rapidly with decreasing
radius than the centrifugal force does. During the emptying
process of a tank, the depression of the liquid surface dips
down as a result of the development of a centrifugal force
field due to the rotating motion. During this same time,
the elastic force also develops and tries to prevent the
surface depression from dipping further down. Since the
elastic force increases faster than the centrifugal force,
the latter may be suddenly overcome. As this occurs, an
abrupt increase of the pressure below the surface dimple may
cause the liquid to "bulge out" rather than "dip down" and
the apparent localized "bursts" or "eruptions" result.
Since, in the absence of momentum influx, fluid can only move
from a region of higher to one of lower pressure, the bursts
lead to a reverse flow directed to the liquid surface which
is simply under atmospheric pressure. The upward-moving
bursting fluid will further interact with the existing vortex
core and cause a local turbulence. As mass eruption progresses,
the interaction becomes even more violent and the whole flow
field adjacent to the bursting fluid becomes more random and
chaotic and leads to the ultimate break-up of the concentric
vortex. The break-up of the concentrated vortex subsequently
results in the local temporary collapse of the established
rotating flow field.

248
Because the vortex collapses locally, the developed
elastic force also collapses. The turbulent fluctuations
then die off quickly and the flow field goes back to the
original nonbursting stage. A "burst cycle" is thus complete.
Evidence to support this explanation of the burst cycle
comes from combining visual observations of the dye patterns
with the fact that all effective vortex inhibiting fluids
exhibit somewhat similar nonbursting and bursting processes.
It is reasonable to believe that the bursting process, which
is closely related to the rapidly increasing elastic force,
^rr ^00
— , with respect to increasing rotation motion, is
responsible for the phenomenon of vortex inhibition.

CHAPTER VI
DISCUSSIONS OF RESULTS
In this chapter, the various experimental and theo¬
retical considerations presented in Chapters IV and V are
discussed. The dependence of air core formation on the
velocity field is analyzed in Section 6.1. This is fol¬
lowed by a discussion of the significance of the boundary
layer in the tank bottom and the possible influence of
polymer additives (Section 6.2). In Section 6.3, the zero-
shear viscosity data of the polymers used in this study are
discussed. The velocity field rearrangements resulting
from the polymer addition are discussed in Section 6.4 In
Section 6.5, the dependence of both drag reduction and vor¬
tex inhibition on polymer conformation is examined. The
conclusions of this work are summarized in Section 6.6.
Recommendations are also presented on possible extensions
of the studies.
6.1 Air Core Formation
That air core formation is in large part a consequence
of the positive radial pressure gradient is well known (see
Chapter V). The pressure is least at the center of the vor¬
tex and a surface depression forms in order to compensate.
249

250
What is important in this work is how the radial distribu¬
tion is influenced by the velocity and stress fields in the
vortex flow. The general characteristics of air core forma¬
tion may be predicted from an order of magnitude analysis of
the terms in the equations of motion, similar to that used
in obtaining the boundary layer equations, in which the ratio
of tangential velocity to radial velocity is large. This
analysis (see Appendix B) leads to the following relations,
P
(6-1)
u
9v
9r
+ w
9v
9 z
+
uv
r
_9_ 1 _9
9r r 9
(6-2)
pu
pw
9w
9 z
3t
z z
9 z
+ pg
(6-3)
Here, the shearing stresses are assumed to be given by the
corresponding expressions for a Newtonian fluid. Examination
of Eqn. (6-2) indicates that if the axial velocity is known
(and hence the radial velocity from the continuity equation),
the tangential velocity can be determined without any infor¬
mation about the normal stress field. In this equation, the
only material parameter involved is the fluid viscosity.
The axial velocity measurements presented in Chapter IV jus¬
tify rejection of the inertial terms in Eqn. (6-3) as they
are small compared to the gravity force. With these terms
eliminated, integration of Eqns. (6-1) and (6-3) yields

251
P(r,z) = pgz +
r 2
pv
dr +
o
r Trr T00
o
dr + t (r,z)
zz
+ constant
(6-4)
The constant may be evaluated by requiring that P(r,z) be
equal to zero (i.e., the atmospheric pressure) on the free
surface z = 0 as r+oo. in addition, the normal stresses must
vanish here. Equation (6-4) becomes
P(r,z) = pgz -
ar
(6-5)
For a Newtonian fluid, the last two terms in Eqn. (6-5)
are negligible and the pressure distribution is completely
determined by the centrifugal force associated with the
existence of curved streamlines in rotating flow. The situa¬
tion for the viscoelastic fluid becomes more complex. Both
the axial normal stress, x , and the normal stress differ-
zz
ence, TQ0_Trr would be expected to be nonzero, and may lead
to a modification of the overall pressure field.
It v/as originally intended to use probes placed tangen¬
tially to the flow to measure stresses along the centerline
of the tank. Since the shear rate in the r direction is zero
on the centerline, the pressure reading would be
T
rr
r=0
-P
r-0
(6-6)
Since the tangential velocity distribution is known, the
pressure data may be used to see whether the normal stress

252
terms in Eqn. (6-5) are comparable to the centrifugal term.
However, flow visualization studies indicated that the
presence of even a probe of minimum size, as determined by
structural requirements, produces pronounced disruption in
core flow, greatly retarding axial motion in that region.
Other observed effects of the probe were an increase in
the core diameter, generation of the counterflows not
previously present in the vortex and greatly retarded or
eliminated the axial shear flows. The last effect is prob¬
ably due to the fact that the radial inflow must occur in
the wake of the probe. The overall effect produced by a
probe should be similar to that produced by an end-wall,
except that the latter will be more pronounced.
With the results of visualization studies in hand, it
was felt that the probes could not be used meaningfully
(not to mention the possible "hole errors" of the probe
readings in viscoelastic fluids). Attempts to probe the
flow within the vortex were, therefore, abandoned. Dif¬
ficulties in probing vortex flows, and some of the problems
which arise in that connection, have been discussed by Keyes
and Dial (1960), Roschke (1966), and Holman and Moore (1961).
The normal stress x in Eqn. (6-5) can be estimated
as follows. We consider the central axial flow region as a
cylinder with some small radius. Take the control surface
which contains the cylinder and apply the momentum balance
in the z direction. If the x term in the momentum balance
rz
can be neglected, we obtain

253
T
zz
(6-7)
2
As the term pw is, under most conditions, less than 5 per¬
cent of the centrifugal term, t in Eqn. (6-5)' may be jus-
z z
tifiably neglected. We then have
P = pgz -
£V
dr t
00
rr
dr
(6-8)
This result is in agreement with the conclusion reached
in Section 5.5, that it is the normal stress difference
(Tgg-xrr) which is primarily responsible for modifying the
tendency toward air core formation.
6.2 Significance of Secondary Flow in Bottom Boundary Layer
It was noted briefly in Chapter IV that a rather compli¬
cated "secondary" flow occurs in the bottom boundary layer of
the vortex chamber. The well-known "tea-cup" effect is a
good demonstration of the existence of such secondary flows.
Tea leaves in the bottom of a cup move to the center during
the settling of the tea. This accumulation of tea leaves at
the center is an indication of the presence of a secondary
flow in the boundary layer on the cup bottom. The effect of
friction on the bottom is to decrease the tangential velocity
of the fluid in the vicinity of the rigid end wall and there¬
by to cause a local imbalance between the centrifugal force
acting on the fluid element and the force due to the pressure
gradient imposed by the outer primary rotating flow. As a

254
result, flow near the bottom is accelerated radially inward
producing a secondary flow characterized by a local increase
in the radial mass flow per unit length. Measurements of
the radial flow in the end-wall boundary layers of a vortex
chamber reported by Kendall (1962) and Owen et al. (1961)
indicate that more than three-fourths of total discharge
came from the boundary layers in some cases. Secondary flow
may therefore play an important role in air core formation.
It was reported by Anwar (1965, 1968) that excessive rough¬
ness in the tank bottom can prevent the full development of
an air core.
Anderson, in theoretical investigations of vortex flows
(Section 5.3), found very strong coupling between the primary-
and secondary-flow regions with a large amount of flow inter¬
change for some flow conditions, which have been confirmed
experimentally (Owen et al., 1961). The secondary flow
in the present study leaves the boundary layer through a sud¬
den eruption near the sharp edge of the exit orifice (see
Figures 4-2 through 4-6). Consequently, it is not apparent
whether the polymer additive has affected the primary flow,
the secondary flow, or both.
Literature is available on the problem of a rotating
disk in dilute polymer solutions. Experimental studies on
the effects of polymer additives on the friction torque on
the rotating disk have been reported by Hoyt (1972), Kale
et al. (1973), Kato et al. (1972), and Bilgen (1971). The
situation of a rotating disk is related to the present case

255
in which the fluid rotates over a stationary disk (i.e.,
the tank bottom). The difference in the secondary flow be¬
tween the two cases is simply that with fluid at rest and
the disk rotating, the direction of the secondary flow in the
boundary layer is radially outward rather than inward as in
the case of the present vortex flow.
The investigations in the rotating disk reported that
drag reduction only appears in turbulent regime at a tangen¬
tial Reynolds number higher than 3xl05 based upon the disk
diameter. Below this value of Re^_ the flow appears laminar
and the experimental results of the torque data with our
without polymer additives agree approximately with the theo¬
retical calculation using von Kárman's laminar flow formula
(Kato et al., 1972). It was concluded that the onset of
drag reduction begins at approximately Re^ = 3x10 . Before
the onset, the secondary flow in the boundary layer is not
affected by the polymer additive and no drag reduction can
be observed. In the present study, Re^ is between 500 and
5000, implying that the boundary-layer flow in the vortex
chamber is laminar. Therefore, we expect that the boundary-
layer flow and thereby the amount of flow interchange be¬
tween the primary- and secondary-flow regions is not affected
by the polymer additive.
We have argued so far that the secondary flow in the
boundary layer of polymeric vortices appears to be of roughly
the same magnitude as that in water vortices. However, the
question remains as to whether the fraction of the boundary-

256
layer mass flow (which is ejected near the edge of the exit
orifice and ultimately finds its way back into the primary
flow) can be affected by polymer additives. If this is
the case, does the polymer suppress the mass eruption (and
in essence act similarly to its role in drag reduction) to
reduce the vorticity of the core flow? In order to clarify
this problem, some new experiments were conducted as follows.
Experiment (A): Polymer Injection Throught Radially Directed
Probes
The vortex chamber was installed with a series of injec¬
tion ports similar to those used in Chapter IV. These ports
were distributed at approximately 3-inch intervals in the
axial direction along the cylindrical wall of the tank. Each
of these ports was provided with a sliding hollow probe that
could be retracted flush with the cylindrical wall or inserted
to any desired radial depth into the vortex flow. A dyed
P-301 solution at 30 ppm (10 times C ^) was used as the
typical injection fluid.
The vortex chamber was filled with fresh water and driven
in a steady vortex motion through the tangential influx jets,
•k
such that a complete air core was present. With the probe
bore at an intermediate distance from the air/liquid inter¬
face, the injected solution moved axially in the annulus
surrounding the air core (due to the axial shear flow). In
a few moments the air core broke and started to oscillate
★
An air core is complete means that it extends from the top
surface of the liquid to the exit hole at the tank bottom.

257
rather rapidly. Except for the injection port very close
to the tank bottom all other port-injection similarly cause
vortex inhibition in a short time. For the near-bottom
probe, the injected solution moved along with the boundary-
layer flow, passing partially out of the exit hole with the
remainder being ejected upward. The air core was not
readily inhibited as before. It took a much longer time
for the injected solution to affect the air core. This
result suggests that the effect of polymer addition on vor¬
tex inhibiton occurs through its influence on the primary
flow, especially in the core region.
In this experiment the polymer solution was injected
through a single probe. When the probe was placed near the
tank bottom, the amount of the injected solution was not
enough to cover the entire bottom surface. This is the result
of the strong radial mass flow in the boundary layer. In
order to supply enough polymer solution to the boundary-layer
flow and thereby further test the conclusion derived from
Experiment (A), another experiment was undertaken with poly¬
mer uniformly injected around the cylindrical wall at the
bottom of the tank.
Experiment (B): Polymer Injection in the Boundary Layer
The vortex chamber was modified as illustrated in Figure
6-1. A copper tube was manipulated to form a coil circling
against the cylindrical wall over the tank bottom. The wall
of the coil near the bottom surface was perforated at equal
spacing (about 2 in.) with 1/16-inch holes so as to distribute

258
Figure 6-1 Schematic illustration of the additional experimental
apparatus.

259
the injected solution uniformly over the bottom without
causing too much disturbance in the boundary layer. The
injected fluid thus followed the boundary-layer flow and
formed a thin layer near the tank bottom. It was observed
that the dyed solution erupted in the near vicinity the
exhaust-hole radius, resulting in the familiar upward
flowing annular region that surrounds the air core. No
apparent suppression of this counterflow due to the poly¬
mer additive was observed. In the early stage of the in¬
jection, the air core remained intact. As time proceeded,
the injected polymer ejected more and more into the counter¬
flow annulus. When this polymer reached the primary central
flow, vortex inhibition occurred. Even after the polymer
injection was shut off, vortex inhibition was still pre¬
served for a considerable length of time. It appears that
the polymer does not affect the "bursting process."
The same experiment was then repeated with salt added
to the polymer solution to raise the specific gravity of
the injected fluid to about 1.2 (about 25% salt). With such
a density gradient, the familiar mass ejection was suppressed
significantly. The buoyancy acted to inhibit the formation
of the axial-counterflow and most of the polymer solution
passed directly out of the exit orifice. The heavier poly¬
mer solution thus formed a thin layer about 0.2-inch thick
over the injection polymer solution while the fluid in the
primary-flow region was still only fresh water. This time,
the air core remainded complete as long as the experiment
proceeded.

260
It is concluded from the above experiments that the
polymer additive does not affect the bursting process and
the effect of vortex inhibition resulting from the polymer
addition is through its influence on the primary core flow.
Although the bottom boundary layer is of significance in
determining the overall vortex flow, the polymer additive
has no influence on it.
6.3 Zero-Shear Viscosities of the Polymer Solutions
It is commonly known that viscosity of dilute polymer
solutions of ppm order in weight concentrations is only a
few percents higher than that of solvent and does not vary
with shear rates significantly (Hayes and Hutton, 1972).
It should be pointed out here that Ostwald-Ubellode visco¬
meters usually used for measuring the viscosity of low-
2 3
viscosity systems operate at shear rates of about 10 to 10
sec ^ (Van Wazer et al., 1963).
The viscosity measurements presented in Section 4.2
-2 1 -1
were made within a relatively low shear range (10 -10 sec ).
Both PAM and PAA are markedly shear thinning when dissolved
in deionized water, even at a shear rate as low as 0.1 sec .
Under zero shear conditions (at a shear rate of about 0.05
sec â– *") with the specialized concentric-cylindrical viscometer,
viscosities of these polymers at concentrations of a few ppm
were found to be several factors higher than that of solvent.
This surprising behavior has not been previously reported.
It will be important to investigate whether the viscous
abnormality of polyelectrolytes in deionized water at low

261
shear rates is a real fact or an experimental error. As
the instrument currently used in this study can measure vis¬
cosities of standard solutions precisely and repeatedly
(see Table 4-2), the error may be small.
One may question if some solid surface films form at
the liquid-air interface. The surface might prevent or
grossly delay the rotor movement, leading to erratic vis¬
cosity readings. To answer it the rotational viscometer
is operated with a thin lighter organic solvent (e.g.,
anhydrous ether) over the surface of the fluid such that
the interface is eliminated. The data taken are almost
the same as those with liquid-air interface, and therefore,
solid suface films do not exist, at least with the polymer
solutions used in this study. It is suggested that the
high zero-shear viscosity of dilute polyelectrolyte solutions
(in deionized water) is a result of their polyelectrolytic
nature in nonionic solvents--polymer molecules expand as the
polymer concentration decreases.
The viscosities of the polyelectrolytes are very sensi¬
tive to the ionic strength of the solvent. The polymer dif¬
fered very noticeably in the ionic strength necessary to
cause a collapse of the molecules. It can be seen that 0.1%
salt is sufficient to almost completely collapse the mole¬
cules of S-273 (see Figure 4-44).
Poly(ethylene oxide), on the other hand, does not ex¬
hibit significant shear thinning or viscous abnormalities
within a relatively wide range of shear rates. Caraher

262
(1965), who has experimented with aqueous solutions of P-301
• - 2 1 -1
within the shear rates from 10 to 10 sec , has arrived
at the same conclusion.
6.4 Theoretical Analysis on Vortex Flows with Polymer
Additives
6.4.1 PAM and PAA in deionized water
It was shown in Chapter IV that the velocity rearrange¬
ments exhibited by PAM and PAA in deionized water are sim¬
ilar to those exhibited by high viscosity Newtonian fluids.
Inspection of Figures 4-22, 4-25, and 4-27 shows that the
axial velocity becomes smaller in magnitude while the size
of the central axial flow region increases ("thickened
core"). This same effect occurs following elevation of the
solution viscosity. The tangential velocity distributions
shown in Figures 4-23, 4-26, and 4-28 indicate the consistent
increase of the radius at which maximum tangential velocity
is obtained. Since this radius of maximum tangential velo¬
city corresponds to the position where the forced vortex
field meets the free vortex field, the "viscous core" in
the above polymer solution has expanded effectively, as qual¬
itatively predicted by theory (Rott, 1958). Comparisons of
the velocity profiles of the polymer solutions with those
of Newtonian or non-Newtonian inelastic fluid at roughly
equivalent shear viscosity (Section 4.6) confirm that the
fluids viscosity is the most significant factor in deter¬
mining the velocity field rearrangements in these solutions.

263
The close agreement between different sets of data with
the same solution viscosities further suggests that the
elastic properties possessed by those polymer solutions
are probably offset by the high viscosity which will result
in large loss of the angular momentum due to the viscous
shear. Furthermore, the high viscosity tends to stabilize
the vortex flow if at some instant an instability occurs.
6.4.2 PEO
As described in Chapter IV, the flow kinematics of PEO
are completely different from those of PAM or PAA in de¬
ionized water. A drastic oscillation in the near vicinity
of the vortex axis in the longitudinal direction was clearly
observed for polymer concentrations above a few parts per
million. As the polymer concentration was increased further,
the oscillation of the axial stream became more violent.
In the presence of these flow fluctuations or instabilities,
the air core would not complete; instead, it dipped below
the top surface of the liquid, and oscillated back and forth
in a very random manner. The viscosity difference between
the solvent and solution is too small to induce any viscous
effects in the flow field. As confirmed in the preceeding
section, the secondary flow is not affected by the polymer
additive, and the effect of polymer must be through the
primary flow in the main body of the vortex.
In Section 5.5, the components of the deformation rate
tensor are plotted against radius as shown in Figure 5-14.
This figure shows that the shearing term is orders of magni-

264
tude greater than the stretching term, which implies the
predominant shearing nature of the primary flow field. In
the same section we have examined the development of the
normal stress function, (t-.-t ), for such a flow field
0 9 rr
using a constitutive equation of the convected Maxwell type
It was shown that (Tg0-Trr) increases radially inward from
the cylindrical wall a high value before rapidly dropping
to zero on the vortex axis.
It is this normal stress function which is responsible
for the well-known "climbing effect" in Couette flow of
elastic fluids, in which the free surface climbs around the
rotating cylinder (Lodge, 1964; Coleman, Markovitz, and
Noll, 1966; Joseph and coworkers, 1973, 1975). In vortex
flow of elastic fluids, however, the free surface does not
"bulge up" as in the climbing effect. As described in
Chapters IV and V, the dimple of the surface depression in
the PEO solution flaps up and down. The dimple might dip
steadily down with no apparent unusual behavior, but it
stops abruptly and retracts back to the top surface rather
quickly. It flaps there for a few moments before starting
to dip down again. It is obvious that the inertial centri¬
fugal force and elastic normal force cannot come to a "com¬
promise" so that the surface depression may finally reach
an equilibrium position.
In Section 5.6, the irregular and unsteady flow was
described in detail and a model proposed to explain the spa
tial and temporal structures in the fluctuating velocity
field. The flow was only investigated qualitatively with

265
flow visualization technique. One of the reasons for this
was the very unsteady flow characteristics involved; another
was the considerable difficulty encountered in the stability
analysis of the viscoelastic vortex flows. Indeed, there
are some investigations with respect to stability in flows
of viscoelastic fluids published during the past few years,
but involving only the simple rectilinear flows such as the
•k
plane Couette or Poiseville flows (Denn, 1975). In the
vortex flow, however, all three velocity components cannot
be neglected and the flow is certainly not rectilinear.
Therefore, even assuming a simple constitutive equation,
the kinematics of the vortex flow leads to complicated ex¬
pressions for the additional disturbance stresses
(Chandrasekhar, 1961). As a result, the problem of solving
the linearized disturbance equations becomes immensely com¬
plicated. Therefore, it is quite difficult, if not impos¬
sible, to apply the conventional technique of stability
analysis to the present problem and make quantitative com¬
parison of the experimental and theoretical results for the
vortex inhibition flow field. A simple stability criterion
for the onset of flow fluctuation to the steady vortex flow
is unlikely and we seek here to discuss in general possible
mechanisms for the instability that appears to be relevant.
The normal forces which arise from fluid elasticity
could play a large part in an instability in the core flow.
As predicted from the convected Maxwell model (Section 5.5),
k
The main result of those analyses is that viscoelasticity
of the fluid is found to have a destabilizing effect.

266
the normal stress difference, (t -t ), is positive and
increases rapidly with shearing rate (and time in transient
state). Suppose there is an upward counterflow occurring
locally at the axis of the vortex (caused by a small dis¬
turbance most probably due to the unstable bursting process).
This local axial counterflow is coupled to a local radial
outflow through the continuity equation. For a Newtonian
fluid, (T -T ) = 0 (here T indicates the "total" normal
Ob rr
stress). From the r-component equation of motion, the total
normal force in the radial direction decreases with decreasing
radius due to the centrifugal force in the rotational flow, or
dT
rr
dr
(6-9)
Therefore, there is a net force, and so a flow from the
outer radius to the centerline. As a result, the radial out¬
flow, the counterflow, and hence the upward disturbance, will
be damped out rather quickly. The above argument can be
further tested by more rigorous hydrodynamic stability
analyses (Chandrasekhar, 1961). An appendix is included
which studies the stability of the vortex core region by im¬
posing a small perturbation on the equation governing the
flow (Appendix D). The result confirms that the vortex flow
in Newtonian fluids is stable, that is, if the system is
disturbed, the disturbance gradually damps out.
For a viscoelastic fluid such as a solution of PEO, how¬
ever, the total normal force in the radial direction becomes

267
dT
rr
pv
T -T
00 rr
dr
(6-10)
In Section 5-5, the value of (T0Q-Trr) in a typical
vortex flow was predicted from the convected Maxwell model
assuming that the relaxation time is 1 sec. It is recognized
that the relaxation time for the PEO solution of interest
calculated from Zimm theory was much less than 1 sec. The
result shows that during the transient process for the forma¬
tion of a vortex core, (Tno-T ) grows much faster than the
t) b rr
2
centrifugal force, pv , and the former may dominate over the
latter, if the solution is "elastic" enough. According to
this result, at some instant during the vortex-developing
process, the total normal force in the radial direction (T ,
in Eqn. 6-10) may have a negative radial gradient. Under
this condition, the fluid on the centerline will flow radially
outward to the outer radius. As a result, the perturbed
radial outflow caused by the axial counterflow disturbance
will be strengthened, and the disturbed counterflow increases.
Consequently, the disturbance will grow without bound resulting
in the flow instability. However, since (Tnn-T ) increases
with the transient shear gradient, this disturbance is "self-
limiting." As the counterflow grows, and so the radial out¬
flow, the concentrated vortex core collapses since the angular
momentum is not able to be carried inward to the center to
main a near-constant circulation distribution in the vortex.
★
In the practical situation, the flow instability is observed
clearly from the dye pattern that is triggered by the localized
radially outward "bursts," followed by a temporary axial-flow
reversal (refer to Figure 5-18).

268
As the vortex core collapses, the developed normal stress
disappears, and the disturbance will be damped out.
This mechanism offers an explanation for the periodic
fluctuations in the axial velocity measurements and as to
why the instability is less intense when the vortex flow
is weaker. This mechanism is also consistent with the ob¬
servation that increasing the polymer concentration am¬
plifies the instability due the elasticity.
In terms of the above ideas, an explanation for the
vortex inhibition phenomenon can be given as follows. As
a concentrated vortex core develops, the dimple of the sur¬
face depression dips down rather quickly to form a complete
air core, as qualitatively predicted in Chapter V. However,
the amplitude of the flow disturbance increases abruptly,
and the vortex core collapses. At the time the vortex core
collapses, the already dipped-down surface depression re¬
tracts back to the top surface because the centrifugal force
to maintain a deep dimple also diminishes with the collapsing
vortex core. Just at the same moment, the disturbance starts
to damp out because the normal stress function which is the
source of instability decays with tangential shear rate.
Then the surface depression becomes a small dimple and the
vortex flow starts to develop and another cycle of vortex
inhibition process begins. The proposed mechanism would ex¬
plain most of the observations observed in the course of the
vortex inhibition experiment.

269
In order to clarify the ideas presented here, a pic¬
torial representation is given below. A sketch of the
various stages during the vortex inhibition process is
illustrated in Figure 6-2. According to the dye pattern
described in Section 5.6, the vortex develops from an ini¬
tial circulation distribution and the surface dimple ex¬
tends downward with no apparent unusual behavior (Figure
6-2a). But prior to reaching the exit hole the dip of
the dimple stops abruptly and simultaneously a localized
axial-flow reversal occurs near the vortex axis at an in¬
termediate distance from the surface depression (see Figure
6-2b). Since, in the absence of momentum influx fluid can
only move from a region of higher to one of lower pressure,
the reverse flow near the axis implies that the pressure
field there is temporarily characterized by a local adverse
pressure gradient. This adverse pressure gradient
seems to serve as the origin of the bursting process (Figure
6-2c). When the bursting fluid interacts violently with
the vortex-core flow the concentrated vortex core collapses
or "breaks up" (Figure 6-2c). As the vortex core breaks up
locally the surface dimple retracts quickly backward to the
top surface (see Figure 6-2d). In this stage the localized
disturbance caused by the bursting damps out gradually and
the flow field becomes laminar again and starts to develop
another cycle of the vortex inhibition process.

270
(a)
(b)
YT
l
J
^7777\
¡I
i
v
777777
(d)
Figure 6-2 Instantaneous view of the vortex inhibition process.
(a) Development stage and the surface dimple extending
downward steadily. (b) Axial-flow reversal occurs and
the downward moving surface dimple stops abruptly. (c)
Localized "bursts" occur near the vortex axis and interact
with the axial shear flow and the surrounding rotating flow
leading to local breakup of the vortex core. (d) Following
the locally collapsed vortex core, the surface dimple
retracts backward and the localized disturbance damps out.

271
6.5 Dependence of Drag Reduction and Vortex Inhibition on
Polymer Conformation
Changes in polymer conformation were followed by visco¬
sity measurements as shown in Figures 4-41 through 4-43.
The polyelectrolytes exhibit maximum viscosities at inter¬
mediate pH levels, where carboxyl groups are ionized, leading
to chain expansion. At high pH, the excess electrolyte is
believed to shield the carboxyl groups from one another,
ultimately reducing coil expansion. Viscosity reduction at
low pH occurs through suppression of carboxyl group ioniza¬
tion. The effect of polymer conformation on drag reduction
may be deduced from Figures 4-44 and 4-45. The salt and low
pH solutions show low levels of drag reduction at low Reynolds
number due to their collapsed conformations. At high Reynolds
number, however, coil expansion should occur, with solution
behavior approaching that of the deionized water or high pH
solutions. This clearly is the case for the salt solution,
and the fact that it does not occur with the low pH solution
implies the presence of strong intramolecular bonding, most
probably in the form of hydrogen bond (White and Gordon, 1975).
The results of CVI measurement given in Table 4-3 show
that the vortex inhibition ability is not changed significantly
by the addition of salt or an increase in pH while it is con¬
siderably reduced by a decrease in pH. Thus, changes in the
effectiveness of drag reduction resulting from polymer con¬
formation variation appear to be the same as changes in vortex
inhibition ability. The vortex flow field measurements of
polymer solutions at different pHs and salt concentrations

272
show some important findings. The tangential and axial
velocity profiles in different medium are shown in Figures
4-47 and 4-48. The low pH solutions show indistinguishable
velocity profiles from those of water due to the collapsed
conformation of the macromoles. However, the velocity
profiles of the salt and high pH solutions are quite dif¬
ferent. Both of these show a very large degree of flow
fluctuation, much like the PEO solution (Section 4.5).
Comparison of Figure 4-47 with Figure 4-30 clearly shows
that the fluctuations in the axial velocity near the vortex
axis of the three solutions, PEO, PAA, or PAM in high pH
and PAA or PAM in salt solution, are quite similar in
nature. The corresponding tangential velocity profiles of
these polymer solutions also exhibit the same degree of
fluctuation.
The similarity of the fluctuating velocity fields among
PEO and PAM or PAA in ionic solvents strongly suggests that
the polymer coils in these solutions are similar, most likely
flexible and extendible. It is very important to note that
these polymer solutions also exhibit similar effectiveness
in drag reduction and vortex inhibition. Since no correla¬
tion can be made between steady-state elongational viscisity
and drag raduction at the present time (Balakrishnan, 1976),
it is not yet clear that the reduced drag turbulent flow
involves the elongational properties of the polymeric material.
On the other hand, since the vortex inhibition and drag reduc¬
tion are closely correlated and since it has been shown that

273
vortex inhibition results from the viscoelastic behavior of
polymer solutions in the large, transient shear gradient in
the vortex-core region, it appears that examination of
transient shear flow in the wall region of turbulent flows
(Hansen, 1971, 1972; Ruckenstein, 1971) may be more promising
than elongational flow for investigating and understanding
the polymer solution properties that are important in poly¬
meric drag reduction. The present work gives an indirect
support for the proposed mechanism that drag reduction in¬
volves the viscoelastic effect of polymers on transient
shear flow in the near wall region (Hansen, 1973).
6.6 Conclusions and Recommendations
The conclusions of this work are summarized as follows:
(1) The flow field measurements of the vortex flow indicate
that the shearing nature of the flow field is much more im¬
portant than the stretching nature which only exists in the
very near vicinity of the vortex axis. An analysis of the
equations of motion for the flow of this type indicates that
the normal stress difference xn -t --the tension along the
lines of flow in the circumferential direction--is of sig¬
nificance in suppressing the vortex formation.
(2) The axial and tangential velocities in the vortex flow
are coupled through the tangential momentum equation in which
the viscoelastic properties of the polymer solutions have no
effect. The axial and tangential velocities are solely
determined by each other depending only on the fluid viscosity.

274
Given the axial velocity profiles and the fluid viscosity,
the tangential velocity profiles can be determined.
(3) The boundary-layer flow on the tank bottom is laminar.
The range of the tangential Reynolds numbers in the present
study was from 500 to 5000, well below the transition point
at a Reynolds number of 3xl05. The secondary flow in the
boundary layer would not be affected by the polymer addi¬
tives and the effect of vortex inhibition resulting from
polymer addition is through the primary flow in the main
body of the vortex.
(4) By using a Maxwell model to predict the normal stress
difference x_ -x in terms of experimentally supported
on r r
kinematics, it is clearly shown that t.„-t increases from
1 00 rr
the cylindrical wall, approaching a very high value in the
vicinity of the vortex.
(5) Flow field measurement for P-301 or some other polymers
demonstrates a strong flow fluctuation near the vortex axis.
A flow model was proposed for the vortex inhibition process.
The fluid motions which cause the localized turbulence and
which seem to serve as the origin of the fluctuating velocity
field, were explained qualitatively.
(6) PAM and PAA in dilute solution (say, 10 ppm) in high
pH or salt solution exhibit similar flow fluctuation as ob¬
served in PEO solution. All these solutions are effective
in drag reduction and vortex inhibition. It was suggested

275
that these polymers with similar flexible, random coil con¬
formation will be aligned in the deformation processes in
reducing the drag in turbulent pipe flow or suppressing
the air core formation in vortex flow.
(7) The high viscosities of PAM or PAA solutions may offset
the elastic effects in the deformation processes of vortex
flows. Since both viscous and elastic effects of dilute
polymer solution lead to the suppression of the air core for¬
mation, it is very difficult, if not impossible, to examine
the two effects separately on their influence on the vortex
flow field.
The possible extension of this work would be in the
following areas:
(1) A detailed and careful study of instability in the
viscoelastic vortex flows. A laser Doppler Anemometer might
be the best instrument for this investigation because of
its precise velocity measurement without having to disturb
the flow field. An instability analysis similar to the
viscoelastic stability theory for fluid motion between con¬
centric cylinders (Taylor vortices) might be used as a re¬
ference. An understanding of the mechanism of the unstable
flows in viscoelastic vortices might also help explain the
instability in the extrusion of polymer melts or some poly¬
mer solutions through capillary dies, the available explana¬
tion of which is still qualitative.

276
(2) Assessment of the utility of the convected Maxwell
constitutive equation at concentrations of ppm levels.
If this relationship is adequate, values of relaxation
time are required to make quantitative comparisons of the
experimental and theoretical results of the fluctuating
velocity field. This work will presumably require the
development of new theories and techniques to estimate
correctly the relevant relaxation time of a material.

APPENDIX A
DERIVATION OF EQUATIONS (3-8) THROUGH (3-10)
The low shear viscometer is illustrated in Figure
A-l, in which h is the height of the inner cylinder or
"rotor." Let the radius of the rotor be R^ and that of
the outer cylinder or "stator" be let the tangential
shear stress on the cylindrical surface in the liquid at
radius r be x, then the resultant moment arising from the
tangential traction on the cylindrical surface becomes
2
2iTr hx. Assume the motion of the rotor is stationary,
this moment must be equal to the torque M applied to the
rotor. Hence we have
M = 2iTr2hx = 2irR2hx1 = 2iTR2hx2 (A-l)
where x^ and i^ are the shear stresses at r = R^ and r = R2
respectively. If we denote the angular velocity of the
liquid element around the axis by w(r) then the shear rate
is given by
Y
(A- 2)
Then we have from Eqn. (A-l)
277

278
Figure A-l
Coaxial cylinder viscometer

279
; - 2T ^ ^ f(T)
(A-3)
T
Substituting - for f(T) in Eqn. (A-3), then integrating
2 7T
Eqn. (A-3) under the boundary conditions co(R^) = — and
oj (R2) = 0, we obtain
P
8TT2yh
(A-4)
where P is the period of revolution of the rotor (seconds
per revolution) and y is the apparent shear viscosity of
the liquid.
The preceding treatments are valid only for the case
of infinite long inner and outer cylinders. In an actual
rotating coaxial cylinder viscometer, for example, as shown
in Figure A-l, there is always a viscous drag due to the
stress on the bottom surface of the inner cylinder, and
besides, the distribution of the stress on the cylindrical
surface differs from that for infinite long cylinders, because
the state of flow is affected by the existence of the ends.
Consequently, there is in general an end-effect. The end-
effect may be considered as equivalent to an increase in
the effective depth of immersion from h to h + Ah. Thus
Eqn. (A-4) may be modified as
P
8tt y (h+Ah)
M
R
R?
)
(A-5)

280
The torque, M, produced on the rotor comes from the inter¬
action between the original magnetic field and the induced
field resulting from the induced "eddy currents." This
torque is generated to align the two magnetic fields.
Hence M should be proportional to the relative angular
velocity between these two fields, or
M
(A-6)
where K , the proportional constant, is in general a func¬
tion of the magnet properties, the size and properties of
the aluminum sheet inside the rotor, and the geometry of the
viscometer. Substituting Eqn. (A-6) into Eqn. (A-5) one
obtains Eqn. (A-7)
P
P
m
8TT M (h+Ah)
K P (^x- - -^x-)
m m r2 r2
(A-7)
Since
test
P is held constant,
m
solution (in general,
the relative viscosity of the
relative to that of water) is
rel
_ u _
o
p - p
m
P - P
o m
(A- 8)
where P and Pq refer to solution and solvent, respectively.
The average shear rate can be obtained by integrating Eqn.
(A-4) from r = R^ to r = R2
ave
R.
R,
rdr 8ttr^R2 r2
' p(r2 r2,2 in g.
rdr 2 1
(A- 9)

APPENDIX B
ORDER-OF-MAGNITUDE ANALYSIS OF VORTEX FLOWS
The important properties and the governing equations
of motion for the primary flow can be derived from Eqns.
(5-2) through (5-4) by an order of magnitude analysis.
Assume, as is the case for most vortex flows, that the
radial velocity is much smaller than the tangential velo¬
city, i. e. ,
v ~ 0(1) , u - 0(e) (B-l)
and that variations in the axial direction are small com¬
pared with variations in the radial direction, i.e.,
~ 0(1) , |p ~ 0(6) (B-2)
Introducing these orders into Eqn. (5-1) one finds
w ~ 0(|) (B-3)
and into Eqn. (5-2) one finds
P ~ 0(1)
so that the order of magnitude in each term in Eqns. (5-2)
through (5-4) is readily known,
281

282
— + w — = - — ZZ + I iTrr T68 ^Trz~
Or 3z r p 3r p r dz
0 (e2) - 0 (e2) << 0 (1) 0 (1)
(B— 4)
w
uv
r
1
P
_3_
3r
2
r
3 T
Tr 0 +
9z
3 z
0(e) ~ 0(e) ~ 0(e)
(B-5)
3w
U -r— +
3r
w
3w
3 z
1 3P + 1
p 3z p
'1 3
r 3r rTrz
+
3t
zz
3 z
+ g
p2 2
0(V) ~ o(V] 0(6)
(B-6)
If we take the shearing stress to be proportional to the
velocity gradient, Trz and x0z become negligible and
Tr0 = (~) 1 • Equations (B-4) through (B-6) can be
simplified by dropping the smaller terms,
P
3P Trr x69
3r r
(B-7)
u
w
uv
r
v
_3_ 1 _3_
3r r rr
rv
(B-8)
pu
pw
3w
rr
3t
zz
3 z
+ pg
(B- 9)
The terms neglected in these simplified equations are
indeed much less important than those retained.

APPENDIX C
PREDICTION OF MAXWELL MODEL—RECONSIDERATION IN A
TRANSIENT VORTEX FLOW
For the sake of simplicity, only the tangential velo¬
city component will be taken to be dependent on time.
The following type of flow field will be considered (see
Section 5-2),
u
(5-15)
v =
1 - exp
4 vt -
2 , 2
r /r
' o
2Qt n
\
r2
ro
e
w
2
r
(5-28)
(5-16)
Substituting Eqn. (5-28) into Eqn. (5-107) and assumed aA = 1
one obtains
T00 Trr
4F2uM
1 - exp -
4 v t
2Qt
2 2,2 2 i
r /r r
o o . o
(e - 1) e
4 vt
2Qt
2 2,2
r r /r
o o ' o
e e exp
4 v t
2 2.2
o , r /ro ,. „ rg '
(e - 1) e
(6-1)
283

284
In order to see if the presence of (t.a-t ) can
09 rr
affect the isotropic pressure and thereby the surface
depression, Eqn. (5-112) is integrated from r=0 to r=°°.
Substituting Eqns. (5-28) and (C-l) into the integrand
and dimensionalizing the terms yields
a = A
{1-exp[-D(en-l)eS]} dn
o n
- 4AB
o
^ {- ^ [1-exp(D(en-l)eS)
+ DeSen (-D(en-l)eS) ] } dn
(C-2)
where
a
A =
pj- the dimensionless dimple depth
r2
2gr2H
^ o
B =
S =
vA
2
r
o
2Qt
(C-3)
D =
o
4vtn
The result of Eqn. (C-2) for a typical case given in
Section 5.2 that

285
T =53.6
CO
v tx = 4.5
2
r
cm /sec
2
cm
sec
(C — 4 )
is illustrated in Figure 5-17. This is the time history
of the formation of the vortex core. The relaxation time X
is used as a parameter.

APPENDIX D
STABILITY OF VORTEX FLOWS (NEWTONIAN)
D.1 Hydrodynamic Stability
The problem of the stability of fluid flow is well
known in classical mechanics. The basic concept of the
hydrodynamic stability may be stated in general terms as
follows: given a well-defined boundary value problem,
the governing equation is
x = f(x) (D-l)
where x(t) is a set of parameters which define the system
in an n-dimensional Euclidean space. Let xg be the steady-
state solution satisfying all the boundary conditions, i.e.,
0 - f(xg) (D-2)
Such a solution may be unstable, in the sense that, if the
system (D-l) is disturbed, the disturbance grows in ampli¬
tude in such a way that the system progressively departs
from the steady state and never reverts to it. On the other
hand, we may say that the system is stable if the disturbance
gradually damps out.
The analysis of stability is typically carried out as
follows. Let a small perturbation E, be imposed on the steady
flow such that
286

287
x = x + r
~ s ~
(D-3)
If the assumption is made that the perturbation is infini¬
tesimal, the governing equation (D-l) can be linearized
with respect to the perturbation variable, £; the linearized
perturbation equation becomes
§ - f(x) = f(xs) + Vf(xs) • £ + 0(?)
(D-4)
where Vf (xg) is the gradient of f evaluated at the steady
state. The dependence on time can then be eliminated by
seeking solution of the form
(D-5)
where E, is the local amplitude describing the disturbance
and a is constant (in general, a complex) to be determined.
When the imaginary part of a (I (a)) is zero, the perturbation's
amplitude does not change in time, and the flow is called
"neutral stable." When I (a) > 0, the disturbance grows
in time and hence the flow is unstable; when I (a) < 0, the
m
disturbance decays in time and the flow is stable. In light
of the above stability theory, we may study the rotational
stability in a vortex flow.
D.2 Stability of Vortex Flows
It has been found that the vortex core region generates
the instability when some polymer additives are present. It
is of interest therefore to study the instability of this

288
core region by imposing a small perturbation on the equa¬
tions governing the flow. This appendix is an analysis- of
the flow stability in Newtonian fluids.
Under a very special assumption on the flow geometry,
namely, a stagnation-point flow extending to infinite, the
steady-state solution for the governing equation is (Rott,
1958) ,
2
r -ar
a 00 4 \)
us = - 2 r' vg = — d - e ) , wg = az (D-6)
For an incompressible Newtonian fluid, the governing
equations are the Navier-Stokes equations and the continuity
equations,
p + PV ’ VV = - VP + yV2V (D-7)
VV = 0
/\
An axisymmetric stream function ip is introduced and
Eqn. (D-7) are made dimensionless by introducing
CO
/
e
Ql
r r
2
The resulting equations are

289
2 3s 9£ 9n 3h 3£ n ,2
dri
r
iL
3 £
9 2
2 n c
3 3ip
3 s 9 n
eí
4n‘
~dü) d3\p 3 ib
w 9n3 " 9n
d3il>
3£3n2
(D-8)
3 4 s-,
3 ^ . 3
3 + n 4
3n 3n
in which the terms involved (-g—) are neglected for the cases
of narrow vortices. The steady-state solution (Eqn. D-6)
can be rewritten as
s
-n£ ,
r =
s
1 - e
-n
(D-9)
If one assumes that the flow gradient "a" is negligibly
small, the dependence on z in the steady-state solution may
be eliminated. Equation (D-9) becomes
ip = 0
s
r =
s
1 - e
-n
(D-10)
To simplify the problem, it is further assumed that the
parameter e (square of the Rossby number) is very small.
In that case the terms multiplying e in Eqn. (D-8) drop and
we obtain
1 3r 3ijj 9r dip 91 9 92r
2 3s 3£ 3n 3n 3? n a 2
d n
(D-ll)
0

290
If we define the perturbation variables as
<í» = 'J'g + $
(D-12)
r = r + f
s
and substitute Eqns. (D-10) and (D-12) into Eqn. (D-ll)
we obtain
1 9£ -n
2 3s 6 3C
2n
0
(1
e'n)
3T
3 K
0
(D-13)
We therefore seek solution of the form (Chandrasekhar, 1961),
7 * . . i(k£-at)
ip = ip (n)e
r = r*(n)ei(kC_at)
(D-14)
here we assume the disturbance is axisymmetric and hence the
0-dependence is eliminated. Equation (D-13) then becomes
10
2
*
ikr
The prime indicates
Boundary conditions
* -n * *"
F - ike 1 ijj - 2nF
- 0
differentiation with
are
= 0 (D-15)
(D-16)
respect to n.
r*(0) = r* (1)
0
0

291
Combining Eqns. (D-15) and (D-16) leads to a single second-
order equation
* »
r
o
(D-17)
★
Equation (D-17) is an eigenvalue problem. T (n) = 0
is always a solution, but nontrivial solution will exist
under certain conditions. By appealing to the standard
theorems in boundary-value problem, it follows (Denn, 1975)
that a nontrivial solution of Eqn. (D-17) exists only if
sup
a i
4n
> TT
(D-18)
where sup[ ] denotes the least upper bound (supremum) of
^ in 0 < n < 1. It is readily shown that 1^(0) < 0 such
that the inequality is satisfied. Since a negative 1^(0)
means stability, it is clear that this type of flow is
always stable. The foregoing analysis gives an explanation
for the observation of a draining vortex: an air core forms
in the vortex, extending from the surface of the liquid to
the drain hole at the bottom of the tank. The vortex is
extremely stable, and once it forms, it remains intact until
the tank has drained (Gordon and Balakrishnan, 1972).

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BIOGRAPHICAL SKETCH
Chii-Shyoung Chiou was born on March 7, 1948, in
Tainan, Taiwan. In the summer of 1970, he graduated
from the National Cheng Rung University with a B.S. in
Chemical Engineering. Shortly after his graduation,
he entered the National Taiwan University for his
graduate studies. In June, 1972, he attained his M.S.
degree in Chemical Engineering. Following graduation,
he was drafted for one year of ROTC service in the
Chinese Army before enrolling in the Graduate School of
the University of Florida. From September, 1973, until
the present time, he has been a graduate assistant in
the Chemical Engineering Department, pursuing his work
towards the degree of Ph.D. in Chemical Engineering.
298

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
â– " ; .4,
Ronald J. Gordon, Chairman
Associate Professor of
Chemical Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Raymond W. Fahien
Professor of Chemical Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Robert D. Walker, Jr.
Professor of Chemical
\
Engineering
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Dale W. Kirmse
Assistant Professor of
Chemical Engineering

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
o / ’ 1 L
¿ / / f - A h y ¿ ^
Ulrich H. Kurzweg
Professor of Engineering Sciences
This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate Council, and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
December, 1976
Dean, Graduate School

UNIVERSITY OF FLORIDA
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