The vortex flow of dilute polymer solutions


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


Subjects / Keywords:
Polymers   ( lcsh )
Polymerization   ( lcsh )
Vortex-motion   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


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

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University of Florida
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Full Text








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.







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 . .


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 .



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 . .


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 . .


v .

. 1

. 1
S 2

S. 5
S 9
. 17

S. 43







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


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


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






BIBLIOGRAPHY . . ... .. 292


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



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-


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-


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


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.



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.

Surface of Liquid

Figure 1-1

Illustration of vortex formation in water.


Figure 1-2

Surface of Liquid

Illustration of vortex inhibition with polymer

1 1


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)


(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)


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.



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)

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


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


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

S= rv + C (2-6)


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.


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


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-



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.

o 32
o o
>1 C

00) 0 C
OU 0 0 0
O O L O Ll

O 000 O


U 0

c2 >r- 4M r


0 ) 0 ) o E 0) H o ) 0)

S 0 0 ) -
T,-I L,-i I ) L
U O Lo o o L) m w 4J
, U 4 (g .,
E H -i ()
UC 0 0 00

0 6'

,-- t L O-
0--) Q C) L C) C C CD C)-
m rY in d ri m Cd n 0)

H H +J 4J 4J + +J to 4
-l 4-
r -1 f- 1

> 0

S 0 ) 0 .-
H -p H 0- )
0) (1

0 0) -i ;D4
04 4-) 0

wj U) 1 1 .
i0 1 0 to 4-

E-i 0

4-) H-0 U)
(n 0-i (0 0 Q) 0' *-4 ri
0) U r U) |o r Cd, (I
> H U) f Cd co

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


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

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

a = (2-9)
7Tr h

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


v ~ l eA 2 (2-10)


a T
A (2-11)

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

Sketch of geometry for a rotating porous cylinder.

Figure 2-6

first appear. Moreover, it can be seen that the axial
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


v + -


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

function, T, in the parameter e = (-Hn); here Q is a volume
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.



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


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


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





on y

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



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
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


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



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)





Poly(acrylic acid)







WSR 301

AP 273

AP 30




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).


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)

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 = -2 (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


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
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

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 = (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 = 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
mm R2 2
1 2











Figure 3-7





Rotating Magnetic Flux Eddy Current
(a) (b)

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


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

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


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



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


To Dye Reservoir


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


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.




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


0 ia

w a

c Q)L
.fl pC

Scl) U)
r-I bci



o W

c) .4 CLn

p o -W

QW ca

4J C04J
0d ) Crl

01. 4J 0)

4jt 0)

0- 4-J
c C

Q, .0 C:)

0 ),-




r-q u

C bfl

O n)
:4-) CO
p a)

-H C4J C
4 4- )

4-J WP
CO 0( a)

P. 1- 1 ni


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

C om

r4 0)Q.C'
I- 0)0



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.


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)


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


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
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.

O Water

SCMC 7H, 100 ppm


103 104
Reynolds Number, Re

Figure 4-7

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

number for CMC 7H









m-lWN4 mo 4r44'' r
* *c .'

i-) HH rH.HH

- miiaLn r oo r-
r( r % om r-00 -

mrO Cr4iooo

0 -H

O 0

0\o 0\0
0 0
C14 c

mN MLn mmCN r
NhlrmP me) mmmV
* **~I~

C)O m -mZoo Ln
SO\nr-. 00


0 N
4 -r-I


o o




























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-

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

do 'r KITSOOSTA euaqS





0 1 O


o 0
(I I)

o 4

d I


/. -I I

0 o