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Rheology and streaming birefringence of an anisotropic fluid

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Title:
Rheology and streaming birefringence of an anisotropic fluid
Creator:
Schonblom, James Eric, 1934-
Publication Date:
Language:
English
Physical Description:
xxiv, 230 leaves. : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Birefringence ( jstor )
Calibration ( jstor )
Cylinders ( jstor )
Flow velocity ( jstor )
Fringe ( jstor )
Shear stress ( jstor )
Sine function ( jstor )
Velocity ( jstor )
Viscometers ( jstor )
Viscosity ( jstor )
Liquid crystals ( lcsh )
Rheology ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 221-228.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by J. Eric Schonblom.

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Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
000580716 ( ALEPH )
ADA8821 ( NOTIS )
14076297 ( OCLC )

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.AND STREAMING BIREFRINGENCE


OF AN

ANISOTROPIC FLUID



By
J. ERiC SCHOONBLOM







A DISoSR .TI0 PR.3EETED F0 T:- GRADJi.3 COUNCIL OF :C'
ULIVTERSIL'T OF IIDA IN PARTIAL FULFILLChITT OF 7:-i
REOUI'::Z:'3 ?2 "FO? -THE DEGREE OF DOCTOR OF PHILOSO?0i'








UnIVRSITY OF FLORIDA


1974













ACi"TOWLEDGMENTS


To Dr. E. Rune Lindgren, for stimulating my interest

in anisotropic liquids, for suggesting Milling Yellow as

an experimental medium, for unstinting support and encour-

agement, and for unwillingness to accept facile explan-

ations or unnecessary assumptions.

To Richard R. Johnson, for a hundred instances in

which he interrupted work upon his own dissertation to

provide practical suggestions or physical assistance.

To Dr. Ulrich H. Kurzweg, teacher, for assistance

and advice in the mathematical formulation of the problem,

for constant interest in day by day developments, and for

exemplary performance as a lecturer.

To Dr. Martin A. Eisenberg, for serving on my advisory

committee and for providing references concerning the

torsion analogue of cylindrical fluid floor.

To Dr. RoI>rb L. Sierakovski, for serving on my

advisory committee, for useful c orm-nts concerning th2

organization of ry dissertation, and for excellence as a

lecturer.

To Dr. C. Michael Levy, for supervision of my Minor

Program.

To Dr. Craig Hartley, for the loan of optical ite,'s.







To Dr. Gene Hecp, for the allocation of computer time.

To Frank Hearne, for microscopic e;:amnration of a

Milling Yellow solution.

To John Tang, for the tedium of checking some trigo-

nometric relationships.

To Max Suarez, for making the viscosity measurements

which were necessary during my absence in July, 1972.

To Bill Wilson, for assistance in the preparation of

the flow apparatus.

To Jerry Hornbuckle, for encouragement, for a useful

discussion of variational solutions for the determination

of shear-rate distributions, and for assistance in prepar-

ation for the qualifying examination.

To Carl Langner, for suggestions concerning numerical

integration.

To Edward Tess:a and William Loc.,_urst, for help in

the shop.

To Dr. Donald E. Swarts, then President of the

University of Pittsburgh Bradford Campus, for a grant of

extended leave front my Assistant Profoezjr2-1 to s.-.-

doctoral studies and for acceptance, without predjudice,

cf ry resignati-. n ,hen it was n reces ...- for me t become

a "permanent Flo-:rda resi- -t."

To David Alford, for giving me something else to

think about.






The research reported herein was supported in part by

the National Science Foundation. At other times the author

was supported by a National Science Foundation Traineeship.

A portion of the computer cost was borne by the Department

of Engineering Science, Mechanics, and Aerospace Engineering

at the University of Florida














TABLE OF CC.:'TrS


LIST OF TABLES

LIST OF FIGURES

T ': TO SYI30LS

ABSTRACT

CHAPTERS:


I1TR ODTCT IO


Scooe of Dissertation


RESULTS OF PREVIOUS I:NVEST IGAT S ATND
THEIR TILPLICATIO ,S

3irefr:in ent Flow -Fields

Physical Propertires of _-illi-ng


G e nera Fro pr te s
i Y- 3


Theory of b'irof ce

_ f-! 1 s 2^ C
r'


P-do cst, L -


V o -0 D 1 i .t... p-_.in Ree an-
" '. '.' -


x

xii

xv


0O-E


-.,o







CHAPTERS:

,TWO (Continued)

Newtonian Flow in
Cylinders 29

Non-Neewtonian Fluids in
Rectangular Conduits 31

THREE OPTICAL ATI:AL-, IS 38

Two-Dimensional Flow 39

Three-Di ensional Flow 43

Assumn-otions 44

Definition of Effective
Optical Properties 47

Analysis 48

Integration to Obtain Fringe
Pattern

FOUR DETERMAT0TION OF OPTICAL PROPERTIES 62

Birefringence 62

Aonaratus 63

Procedure 72

Prediction of Results 73

.roerinTental Data 7

Discussion of Results 83

Prelm in: r' tests 88


data 9

Computation of optical
coefficients 94



Ass.ption of For 100






C IAPTERS:

FIVE DZ' i ..R..:.IT OF 0 E GEOLOGICAL PR r I?.'iS 103
Enpir-" :.'.."'L i \L R:L)!o,:!L ic.T.!L
For:..ul r.s 103
--.-ss .- Sliding Ball
v i. ,'t :- 110

Relate ,'1. In'.:' eti at ions 111

Ana lys is 110
Integration of visco-
meter equation 124

Exnerimental Results 130

Discussion of Results 140

Curve Fitting 140

Determirnation of g(j) from
L .. --1 .. -. 3-.3 146

Range of Aolication 151

SIX DISTRI3DTIOK OF SI-IEAR RAES I7 TL REC'TA TGULAR
CC DU IS 1i6

Pow'er- Ljw Fluids 156

,.-na_- _t! ,"ajti v,-s 161
s -,'z _,-"S :: Y :D ,oITC,. -" 01 ~s 163




Otical Pro-norties o:f i:ei_'u. 164

Rheo?]o.,ic31 Pro .-'ties of_ ..i..r 1.


vii~







APPENDICES:

A THE EFFECTIVE BIREFRI:GENCE AND 167
ORi: STATION ANGLE OF THE OPTICAL ELLIPSE
FORiKED BY THE INTERSECTION OF THE OPTICAL
ELLIPSOID WITH THE PLANE ORTHOGONAL TO
THE PATH OF LIGHT

B PREPARATION OF EILLIG YELLOW SOLUTIONS 171

Original Stock Solution 171

Fresh Stock Solution 173

C VARIATION IN FLOW RATE AS AMOUNT OF
LIQUID IN O-VER:HE TANK DECREASES 175

D ALIIG:.ET OF POLAR~IING ARRAYS 177

E C.TAI:P.'TION OF HOPPLE-, RHEO-VISCOMETER 179

F BILINTEAR ,MATERIALS 187

G SOME0 CiHAITEL CCiSIANT4S 191

H VARIATIONS IN MILLING YELLOW! SOLUTIONS 192
WITH TIME

I VARIATIONS IN .ILLi:'G YELLOW APPARElT 195
VISCOSITY WITH CONCENTRATING AND
TEMPERATURE

Apparatus 19-

Preparation of Samples 196

Experirmenta.l Data 197

Comnutation of Temnerature 197
Co0 icients

J RESTRICTION OF VISCOMETER TO LTIUIDS 20-
lHAVING VISCOSITY ABOVE 4 CENTIPOISE

K VISCOIET : RESPCO.'E AT VERY SLOW 212
FALL TIMES

L DETE:MI:IATION OF SAMPLE CONCE3:TRATIONS 218


vili








3ILIO GRAPHY 221

BIOGRAPHI-AL SKETCH 229












LIST OF TABLES


INTERNAL DIMENSICIS OF CTAIELS

TVALLS OF y AID y' FOR SUCCESSIVE ELEMEIITS
OF POL-LIiZ-'G ARRAYS


ORDER

ORDER

ORDER

OF RED

ORDER

ORDER

ORDER

ORDER
ORDER
ORDER
BORDER


CRDER
*^,~3r\T-- -
^~^:~\ *'7

^-iU~i->


AT WALL RUN 123

AT WALL RUN 130

AT WALL RUN 131

FRINGE AT WALL RUN' 218

AT WALL RUN 42

AT WALL RU: 44

AT WALL RUN 44A

AT WALL RUN 45

AT WALL RUN 4-18

AT WALL RUN L 1i9

AT W.!ALL RUNi 420A

AT WTALL RUT 420D

-rT W-ALL RUN 424

AT ALL RUTN 429


98




132


.O C OR S:, 42 /', 44A
AIZ 4 (OIGIITAL STOCK -;0 LUTIN)

XVIII. BIREFITiGENT CONISTANTS FOR RUNS 418, 419,
420A, 420D, 424, AND 425 (FRESH STOCTK
SOLUTION)

XI,. VISCOILER EASU'E S RU 42

XX. VISC T.. ERP IEAS'L:EIp : :3 RUT 4-4


I.

II.


III.

IV.

V.

VI.

VIIT.

VIII.

IX.

Xi.

XI.







- .

+.


FR INTGE

FRINGE
FRI: IG-


ORDER



FR IT:,GE





FRINGE


FRINGE



FR I: :

FRI -E






LIST OF TABLES (Continued)

XXI. VISCC:.:TR --.ASrEJ ~ 1i3 S- RUI: 418 133

XXII. VISCOICTER MEASUTREI!.1 ;T .UM- 419 134

XI II. VISCO'ETER 1EASE .:.T -- RU7N 420 135

XXIV. VISCOMETER IzASUII -::3 RUN 420B 136

.:v. VISCOI~TER 1.ASUT',E! EITS R"T 424 137

XXVI. VISCOIMETER IEASUREIBNTS RUN 4243 138

XXVII. VISCO LTER I EAS E: ;TS RUNT 425 139

CXVIII. THEOLOGICAL COI:STAI-S S 145

XXIX. ES-INATION OF APPARENT VISCOSITY 154

C-I C;:.dI3G IN FLOW RATE AS OVERHEAD TAI1T 176

E-I CALIRATIOM: DATA 181

E-II C.. Z.1- : iON CHECK (WATER) 184

I-I TEI.PERATURE COEFFICIE TS 203

I:-I FALL TIL.-S FOR 1-1: INCEEI7TS 214

I:-II CORRECTED FALL TIZS FOR FIRST 10 .F:
OF FAL'L 3Y 1-I TI: ..:..:S3 217

C-I CC OCEI.. T_7 O:S OF SCIYLS 219












LIST OF FIGURES


1. Dispersion in concentration data of Peebles, 18
Prados, and Honeycutt (1965).

2. Unit vectors and angles relating to the flow. 40

3. Optical ellipsoid showing parameters 8, 4, 45
and An.

4. Poincare sphere showing P, the condition of 49
the polarized light bean, R, the principal
(fast) optic axis of the medium, and AP, an
arc on the surface representing the change
in polarization which occurs.

5. Elliptically polarized light in the yz-plane. 52

6. Projection of OP on OR and definition of r. 52

7. Definition of ar. 52
8. Schematic representation of successive deter- 59
mination of the variables.

9. Renlot of data of Peebles, Prados, and 64
Honeycutt (1965) to show linear relatio3nsi o
between square of fringe order and shear rate.

1. Schematic of rexperimental apparatus. 6

1 Cut -a0way i;: of rectangular conduit sno.:g 7
roughered L.:--ls, gaskets, and spacing wiros.

12. OrieltitLon of elements in polarizer cn-d 70

1-. Fringe order at -all as function of ass 4
flow rate. Runs: 123; 131; 130; 213.

14. Fringe order at wall as function of mass 85
flow rate. Runs: 42; 44; 44A; 4$.

15. Fringe order at wall as function of mass 86
flow rate. Runs: 412, 419; 420A.


xii







LIST OF FIGU3BES (Continued)

16. Fringe ordar at wall as function of mass 87
flow rate. Runs: 420D; 424; 425.

17. Birefringence of original stock solution 95
c:r.-.:red with an e::tra-olation of the data
of Peebles, Prados, and Honeycutt (1965).

18. Birefringence of fresh stock solution. 97

19. Extinction angles measured by Peebles, 102
Prados, and Honeycutt (1965) replotted to
obtain straight lines.

20. Data of Peebles, Prados, and Honeycutt 108
(1965) replotted to obtain a linear
relationship.

21. Hoppler Rheo-Viscometer. 112

22. Geometry in annulus. 116

23. Response of viscometer, runs: 42; 420; 425. 141

24. Test of functional form: Pt = k + kg. 143
P
25. Test of functional form: P[Pt-(Pt)oo = k + 144

26. Constructed relationship between shear stress 149
and shear rate for Milling Yellow at 250 C.

27. Comparison of equations (.) and (.7), run 150
4243, at 25= 5 .

2-. I-....::.,.eter readings cor:-pared with those 15'
exT ected for ewtonian fluid or a po-er-


29. Schechter's (1961) co-fficients plotted as
function f power-law exponent.

or. ball.

-1. 3-i-functional material. 138

F-2. Flow field within viscometer showing bo --.-y 18
between regions obeying se-.r :te constit;'ive
equations.
r-o. Vr"tn I- n --' ar i 194


ziii






LIST OF FIGU? D (Continued)

I-i. Temperature variation of samples 103-30 198
and 1012-30.

1-2. Temperature variation of samples 107-25' 198
and 1011-25.

1-3. Temperature variation of samples 103-20 199
and 109-20.

1-4. Temperature variation of samples 104-17 199
and 1011-17.

I-$. Temperature variation of sample 104-14. 200

1-6. Temperature variation of original stock 200
solution.

J-l. Transition in measlirement of apparent 206
viscosity when P is too large.

J-2. Transition in measurement of apparent 206
viscosity when temperature is too high.


Xiv










TABLE OF SYMBOLS


a

ao, al, ..an

A

Ap
As

Al, A2, ..A5


bo, bi, ..bn


bst

Bs

B1,
C
c
es


Co,

do,
d


B2, B3








C1, C2
dl, ..dn


Radius of viscometer ball

Coefficients of polynomial fit of
P-'d(P2/t)/dP as function of P

Cross-sectional area

Constant in Powell-Eyring equation

Amplitude of sinusoidal

Geometric factors relating AN and i
to an, J, and 6 (see Appendix A)

Coefficients of polynomial fit of
1/t as function of P

Coefficients of variational solution

Amplitude of sinusoidal

Geometric factors (see Appendix A)

Speed of light in vacuum

Speed of light when E is parallel
to ni

Speed of light when E is parallel
to n2

Experimental constants

Coefficients in curve-fitting

Director a unit vector character-
izing directional property of
anisotropic fluid

Diameter

Jaumann derivative -.-gij /t = agij/t

+ uk ij/6xk 'ikgkj wkijk










E



Ea
Ee

Emax' E-i
Eo


Eo

F(x,y,z)


g



gij
G

h


i








-
:< kA k2, k1 K,.



n


Unit vector characterizing principal
direction of elliptically polarized
light

Electric field vector

Unit vector, E/Eo

Component of E parallel with na

Electric field on emergence

Axes of elliptically polarized light

Amplitude of electric field entering
flow

Rms value of Eo

F(x,y,z) = 0 is equation for surface
of optical ellipsoid

Shear rate: in conduit, V(au/ox)2
+ (8u/6y)2; in viscometer, bu/bo;
in capillary, 8u/ar

Rate-of-deformation tensor

.ass flow rate

Manometer reading, difference in
fluid levels

Unit vector, x-direction (usually
direction of the light path)

Rms value, light intensity

Initial intensity of light

Unit vector, y-direction (usually
direction alo:- which fringe pattern


Empirical constant, defined at point
of use

Unit vector, z-direction (flow
direction is usually -k)

Fall distance: distance bali moves in
visco,-eter during timed interval


_ 1-






L Effective length of eccentric annulus
in viscometer

L Length of capillary

m Power law exponent
m' (l-m)/2m

M (l+2m)/m

n Exponent in power series
SAverage refractive index

n Analyzer direction: direction of E
a for maximum transmission through
analyzer

p Polarizer direction: direction of E
n for maximum transmission through
polarizer

ni Index of refraction, c/ci (c> ci > c2)

na Index of refraction, c/c2 (c> Ci > 2)

n, Direction of major optic axis

n2 Direction of minor optic axis

Ln Birefringence, n, n2
Integer, often the fringe order

';,unber of data pairs

,s Ns' Integers tabulated by Sc:c~- (1961)
Fringe order at the wall

N2 Characteristic coordinatess, optical
ell-_:e (see -'+uendi": 2)

L.. IEffective birefringence

AN Effective birefri-.gnce at wall

p Static press'lre
P Force on viscometer ball/area of ball,
"average sheari.-._. stress" or
"load on ball"


:C :v.i1







P(n, )

A
P

PI P, P3

q


Q

r

r


Air

R

Requiv.


R(f, I)


R

Re

s

s2(Xs)

So


t (as variable)


t (as integer)

t
m

to

t'

T


Point on Poincare sphere representing
polarization of light

Position vector OP, Poincare sphere

Components of P

Concentration of Milling Yellow,
weight percent

Volumetric flow rate

Radial coordinate

Radius for movement on Poincare
sphere

Change in r, chord of AP

Radius of capillary

Equivalent radius of rectangular
pipe, 2b6j2/(b1+b2)
Point on Poincar6 sphere representing
principal axis of medium

Position vector, OR, Poincare sphere

Reynolds number

Summation index

Estimated variance of variable Xs

Constant: see Appendix G and the
definition on page 74

Time, especially fall time in visco-
meter

Summation index

Experimental fall time, expressed in
terms of Tm

Initial time

Time much greater than \i/c

Temperature


xviii






u

u


Ul


wij

x

X'

X1


U2, U3








X2, X3


A A A
X1, X2, X3


Xs

y'

y'

z
' lI

oc


Velocity of fluid

Mean velocity, A1'Su dA

Components of velocity

Speed of viscometer ball, Y/t

Vorticity tensor

Spatial coordinate

See Appendix A

Characteristic coordinates of Poin-
care sphere; generalized coord-
inates

Unit vectors associated with xl, x2,
and x3

Experimental variable

Spatial coordinate

See Appendix A

Spatial coordinate

See Appendix A

Half-angle of divergent channel

Coefficients of power law expansion
of g(T)

Secondary normal stress function

m/4L

Polarizer angl3, measured from prin-
cin~~ flow axis

Analyzer -l--, me.-.:. _'d from prin-
cipal flow axis

Average field angle, (y+y')/2

Field difference, y y'

Gamnma function of M


o0 ..OCn


AT

[(M)







6 Width of viscometer annulus

T Thickness of optical field

bij Kronecker delta

5m Maximum width of annulus
E Angle between e and principal flow
axis

SSpatial coordinate in viscometer,
radial distance from surface of
ball into fluid in plane of minimum
clearance

Intrinsic viscosity, r/g



0 Angular coordinate in viscometer,
zero where ball contacts wall

On Primary normal stress function
0 Plane, 9 = constant, in which major
axis of optical ellipsoid is
inclined

X Wave length of incident light in
vacuum

p Viscosity

Pa Apparent or average viscosity

u Newtonian viscosity
.uo Limiting value of viscosity as s2:-r
rate approaches zero

oo Limiting value of viscosity as shear
rate increases without limit

p Density
a Phase angle, elliptically polarized
light

ao (Fo Pon)/oPoo
c2(Xs) Variance of variable Xs






Shearing stress


rc Critical value of T at which g(T)
changes its characteristics in a
bi-functional constitutive rela-
tionship
Tij .Stress tensor

T Maximum shear stress in viscometer,
m&m/2L

Tw Shear stress at the wall

Y(,m), T'<'m), Y(O) Experimentally derived functions
defined by equation (5.4)

9 0/2
$ Coordinate of "latitude" on Poincare
sphere

X Extinction angle: angle between prin-
cipal optic axis and principal
direction of polarizer (or analyzer)
when polarizer and analyzer are
crossed. Of the four angles thus
defined, the extinction angle is
the only one which is less than i,/4
and positive.

Orientation angle .rhich 1 makes with
principal flow axis
-0 Limiting value of orientation angle
as shear rate increases without
limit
Effective orientation angle

-E Effective orientation angle at :':ll

Circular fre-e cy of light

I, I, II Invariants of gij


xx i













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

RIEOLOGY AND STREAMING BIREFRINGENCE
OF AN ANISOTROPIC FLUID

By

J. Eric Schonblom

:.rch, 1974


Chairman: Dr. E. Rune Lindgren
-.'ior Department: Engineering Science, ilechanics, and
Aerospace Engineering

The intrinsic viscosity and birefrTigenoc of an

aqueous solution of killing Yellow NGS, a co:m-:ercial

organic dye, are obtained experimentally. Each property

is measured in a flow where the velocity is depen:cnt upon

to spatial cc.rinties It is shown. tat the rhe.olocl

r. optical properties thus obtained may be used to coimpara

-:pothetical v-, e ity distributions i- stea three-

lne1nsional flo;s.

The rheol ical invo stigation employs a Hpp?..er n-Rheo-

isco--eter in whi c a ball slide_ s _ithot rotting through

the fluid within a closely fitted cylinder. This instrument

has previously been considered unsuited for the deter-

nination of basic rheological constints. B/y odelg the


:;xx.i







flow past the ball on steady flow in an eccentric anrnulus,

it is shown that the distribution of shear rates can be

integrated to obtain a unique relationship between the

shear rate and the shear stress for the fluid. The

analysis is valid for all fluids and can be extended

without difficulty to viscometers in which the tightly

fitted ball is replaced by a cylinder.

Values for the birefringence (maximum difference in

refractive index between the principal optic axes) of

killing Yellow have been previously reported. The present

study shows that the previous data exhibit a linear rela-

tionship between the square of the birefringe nce and the

shear rate. An analysis demonstrates that as a result, in

a square pipe, the fringe order at the wall, squared, should

vary linearly with the mass flow rate through the pipe.

This expectation is confirmed experimentally, and the bire-

fringence is calculated from the data.

When birefringent fluids are observed in flow oet wen

t'o polarizers a fringe pattern is seen, Such pattern

x-ve been usel -2 obtain pressure ad velo ity~ distri-

jions in tr-dimensional flows and to estimate lift and

coeff'ici2r
such studies have been limited to the extre':ely low flow

rates at which Milling Yellow's birefringence and shear

stress vary linearly with the shear rate.


xxi iai







The results of the present study extend these methods

to include steady three-dimensional flows in which velocity

variations along the light paths are permissible. Further,

share rates for which the birefringence and shear stress

vary non-linearly are no longer excluded. Although the

direct determination of velocity distributions from fringe

patterns remains impractical, the pattern which corresponds

to any assumed velocity distribution may be computed and

compared with the fringe pattern obtained experimentally.

The method by which fringe patterns may be calculated once

the velocity distribution has been assumed is outlined

schematically.

A hypothetical distribution of shear rates for Hilling

Yellow flowing in a rectangular conduit has not been

attempted for the theological relationship obtained with

the H6ppler Rheo-Viscometer; however, the application of

other constitutive relationships, notably that for a pce'r-

_.w fluid, is considered briefly.












CHAPTER ONE

INTRODUCTION


The velocity distribution of anisotropic liquids

flowing steadily in rectangular pipes can be constructed in

certain cases from a knowledge of the optical and rheo-

logical properties of the fluid. Specifically, if the

material is birefringent, so that the refractive index

bears a directional dependence upon the shear rate, th-i

any hypothetical velocity distribution may be confirmed

or denied by observing the fringe pattern which results

when the flow is observed between crossed polarizers. The

successive steps in such an evaluation are as follows:

Determination of constitutive relaticshizs. Con-

stitutive relationships must be provided :'..'h. describe

the optical and theological properties of the material.

De'rrrin of sr. r rte distrbut:. Based

upon the rheological properties of the medium, a compat-

ible distribution of shear rates for st.,dy flow in rectan-

gular pipes must be calculated. De =--:.ic upon the complex-

ity of the rheological relationship, the mathematical

solution of this boundary value problem may be exact or

appro.:.r-.*ate.








Integration to obtain fringe patt--rns. Once the

distribution of shear rates is known, integration of the

dependent optical properties along each light path will

determine the relative intensity of the emergent light

beam. Fringe patterns thus obtained may be compared with

experimental data to evaluate the relationships derived

in the previous steps.

Scope of Dissertation

This dissertation is concerned primarily with the

first of the three steps just listed and with the

properties of a single birefringent medium: an aqueous

solution of a commercial organic dye, Milling Yellow G3S.

In previous investigations the optical and theological

properties of Milling Yellow solutions (referred to here-

after as s '.ply "Hilling Yellow") have been measured in

viscometric* flows using a concentric cylinder polariscope

and a capillary viscoreter respectively. The present

study, utilizes non-viscometric flows to measure the optical

properties at the wall of a nearly square conduit and the

rheological -propertis within the eccentric oanulus of a

sliding bll visco-eter. Since neither of these meas-

urements seem to have been employed previously, analyses

are provided to support the presen-t applications.


*A flow is viscometric for the purpose of this dis -t-' tion
if the velocity field hos the form u, = 0, u? = 0, u3 = u(x)
where : is a single spatial coordinate. For a more general
defin itio soe Colen, ar. kovitz, and ioll (1966).








The determination of the distribution of shear rates

for Milling Yellow flowing in a rectangular conduit has

not been atteo.ipted for the r.e ;sur'l theological rela-

tionships; however, the application of other constitutive

relationships, notably that of a so-c.-1,lle-1 power-law fluid,

is considered briefly.

An optical analysis is performed to demonstrate the

means by which the resultant fringe pattern may be obtained

once the preceding steps have been accomplished. It is

shown that if the optical properties do not change along

a given light path through the flowing medium the optical

relationshiD simplifies to a familiar result from two-

dimensional optical stress analysis.











CHAPTER TWO

RESULTS OF PREVIOUS IT ;? TIGATIONS
AND THEIR IMPLICATIONS


This assessment of the present state-of-the-art

is in three parts. The first is devoted to studies in

which birefringent liquids have been used to obtain

information concerning velocity fields. Emphasis is

laid upon those studies in which Milling Yellow was the

birefringent medium. The second part is concerned with

the physical properties of Milling Yellow and includes

a discussion of continuum mechanics and model construction

as they relate to Killing Yellow's rheology. The final

part describes previous investigations of velocity

distributions in rectangular pipes.

In Chapter Five, preceding the analysis of the

c~opler Rheo-Viszometer, is a review of the rolling ball

.-iscometer, the falling cylinder viscometer, and the ball

-.d. tube flow meter, subjects too specific for inclusion

i- this more general chapter.








Birefringent Flow Fields


The first reports of streaming birefringence are

those of Mach (1873) and Maxwell (1873) a century ago.

Said Maxwell (1873, p. 46):
I am not aware that this method of rendering
visible the state of strain of a viscous liquid has
been hitherto employed.

Although many theories have arisen from this humble

beginning, the employment of birefringence for the quanti-

tative investigation of flow fields has remained scant to

the present day.

The most popular media for these studies have been

suspensions of colloidal bentonite and solutions of organic

dyes, notably Milling Yellow. Dewey (1941) observed two-

dimensional flow patterns with bentonite and concluded tat

quantitative velocity gradients could be obtained from such

data. Similar studies by Weller (1947) rere hampered by

the high viscosity of the polyme&ric medium which he used.
Tinogradov's (1950) work with colloids provided pictures

of two-dimensional flows around circular o'itacles and

suggested applications for lubrication theory. Rosernber

(1992), another user of bentonite, described the optical
properties of his medium, recommended suitable concen-

trations and colloidal dimensions, and sii. ;ested .eth-ods

of using two-dimensional models to calculate pressure

disitribu;ions, lift and drag coefficients, velocity distri-

butions and streamlines. He concluded that applications








to turbulence would remain qualitative reflecting Binnie's

(1945) experience with dilute solutions of benzopurpurin.
Later Lindgren (1953 et sea.) and Wayland (1955) used
bentonite to visualize turbulence but not for the purpose
of computing the velocity field.
All of the early quantitative studies were hampered
either by high viscosities (as in Weller's case) or by

marginal birefringence (with bentonite). These consid-
erations prompted Jury in 1950 to suggest to Fields the
investigation of various organic dyes for their feasibility

as birefringent media. Fields (1952) concluded that the

most likely candidate for such use was an aqueous solution
of commercial Milling Yellow. A preliminary study of its
usage by Peebles, Garber, and Jury (1953) ratified this

conclusion and sparked some independent studies by other
investigators. Although the 1953 report did not attempt
a quantitative evaluation of the flow fields which it
investigated, it did include photographs of the two-

dimensional flow patterns, details for the preparation of
the dye solution, determination of the density (1.005 gm/c.f)
sad a plot of apparent viscosity versus temperature for four
different dye concentrations. The latter information is

discussed in more detail in the next section.

The first of the independent studies using Milling

Yellow was completed by Hargrove and Thurstone (1957) who

observed flow through an orifice. Although this study was








not quantitative, the usefulness of the medium prompted
its further use by Thurstone for the investigation of

wave pro3paation: Thurstone (1961); Thurstone and Schrag

(1962, 1964); Cerf and Thurstone (1964). From these

studies emerged numerical values for the viscoelastic

properties of Milling Yellow which correlated well with

the measured wavelengths and propagation velocities of

small amplitude waves. Thurstone (1961) also replicated

the earlier density measurement.
Other independent studies were conducted by Swanson,

Scheuner, and Ousterhout (1965) and Swanson and Ousterhout

(1965). They assiusd a linear relationship between bire-

fringence and shear rate and demonstrated the means by

which a two-dimensional flow field could be calculated from

such data. They also described a flow tunnel built for

this purpose.
;1hile these independent studies -.rre under way,

Peebles and his students, particularly Prados, continued

the original work at the University of Te "nnessee. Prados

(1957) and Prados and Peebles (1959) obtained velocity
profiles for t:-.c-dimensional flow in straight channels,

in convergir. and divcrg'-.S channels, and in a stra.i-'.t

channel around a cylinder. Bogue and Peebles (1962)

suggested a technique for obtaining velocity profiles from

isochromatic* fringes only. Their technique was applied

*Isochromatic: optical response (colored in white light)
which depends only upon the birefrirnrnce, An.








to data obtained in a converging channel by Liu. Liu and

Peebles (1963) indicated that Milling Yellow can be used

to describe two-dimensional flows in converging and

diverging channels, in free jets, and in wall jets.

Building upon work by Bogue and Peebles (1962) and Eirsch

(1964), Peebles and Liu (1965) describe in detail the

numerical technique by which two-dimensional velocity

profiles may be obtained from an isochromatic pattern using

a lam inar expanding jet as the experimental configuration.

An important restriction upon this research was that

quantitative evaluation of flow fields using killing Yellow

appeared possible only at extremel- low, flow rates where

the optical and rheological properties vary linearly with

the shear rate as predicted by the theories described 3nter

in this chapter. Further, only two-dim.ensional config-

-u-atiocns, in which the variation in fluid velocity along a

given light path can be neglected, were considered

-rCctable for analysis. When these restrictions were

served, velocity fields could be calculated with an

. ;e.ge error f about 13 percent according to Peebles

-et Liu (1965).

It should be mentionedd that o.st of the flow field

investigation were accon.anied by concurrent examinations

of the optical and theological properties of the medium.1.

ITo fi. "'".- which relate to i:illing Yellow are reviewed

in the section which follows.

Excluding poly.oeric media in which elastic propuertios








przdo.ir."'te extrudedd po lye'. lene :ws o'.-:.!rved by .I.als,

.169, through windows set in the long walls of a capllr:-

slit, for e::a .nple) one three-dimensional flow field has

been examined quantitatively. Durelli and iiorgard (1972)

h:ot..;:r.hed flow around a cylinder in a rectangular

channel with an aspect ratio of 0.75; that is, the li:;-

path .was actually shorter than the channel width. This

arr-Iangernt violated the requirement set by the Tennessee

studies for two-dimensional flows based on Purday's (1949)

estimate that the light path must be 5 to 10 times the

width of the channel. Durelli and Iorgard chose to treat

the flow as two-diz.'.nsional and calculated average veloc-

ities along each light path. This assu:cm.;.tion yielded good

agreement m with local velocities obtained by averaging

speeds measured tfro streak photographs of h:Srogen bubbles

at three locations in the same channel.

From this review of f.c; analyse using birefrirnrent

redia it is evident that there is a rned for a technique by

".'ich three-dicme nsional_ flows can be considered. It would

o helpful if the restriction to e:tre.ely _lo flow ratas

7:.;I be relaxed or eli-inatedc








Physical Properties of Milling Yellow


General Pronerties

The medium for the present investigation was obtained

by dissolving in water a commercial dye designated by the

Society of Dyers and Colourists (1971) as Colour Index

Acid Yellow 44. The common name is Milling Yellow. The

trade name for the commercial product supplied by the

Keystone Aniline and Chemical Company, Incorporated,

Chicago, is Milling Yellow NGS. It is this product which

was used in the current investigations and unless otherwise

indicated, the term Milling Yellow in this dissertation

will refer to solutions of this commercial product rather

than the pure dyestuff.

Swanson and Green (1969) provide a number of details

concerning the physical chemistry of Milling Yellow. They

give the structural formula as
CH3 CH3

COE CH3 CH3 COH

:TIi; \ \ C h
0 ;aS03 NaS03 0

and state that the birefringence is due to a solid phase

precipitated frco- solution. They describe this solid

phase as consisting of transparent, rhombic crystals with

an aspect ratio of 5.7 and "strong inherent polarization."








Swanson and Green state, and the supplier confirms, that

the presence of impurities, notably NaCI and :a2SO4. with

some sodium acetate, may constitute more than 30 percent

of the commercial product.

In dilute solution (less than 1 percent) Milling

Yellow is lemon yellow and highly transparent. Swanson

and Green obtained birefringence with pure dye solutions

having concentrations as low as 0.1 percent by salting

the solution with electrolytes, but such solutions were

highly unstable.

At higher concentrations Milling Yellow is orange

and deeply colored. The preparation of the medium used

in this dissertation was basically that described by

Peebles, Garber, and Jury (1953). The dye was mixed with

water at a weight concentration below that desired and

heated to just under 1000 C. At this ter.parature water

was evaporated until the desired concentration was

obtained. Further details are provided in Appendix B.

Although earlier investigators follow Peebles, Garber,

and Jury in suggesting dilution of the concentrated medium

to obtain the desired level of birefringence for a given

=":peri..ent, this dissertation concurs with ZL .:'on ';ho

cautioned against dilution since the equilibrium of the

medium is disturbed when distilled watere r is added, and,

on occasion, siimnjntation may result.

It was observed that when Millin: Yellow is suddenly








diluted to about 1 percent there is a short period during

which significant birefringence remains in the dispersed

mixture, but the viscosity approaches that of water.

Investigators who are willing to tolerate rapid changes

in the optical properties of the medium may find this

unstable dispersion a useful medium for the observation

of qualitative phenomena. Although the birefringence

soon disappears, the color remains the deep orange which

characterizes solutions concentrated by heating.

There is disagreement concerning the stability of

Milling Yellow preparations. Peebles, Garber, and Jury

(1953) detected no qualitative differences in the observed

optical properties of their 1.5 to 1.8 percent medium

over a period of 10 months, nor was there a perceptible

darkening after more than a week's continuous contact with

iron pipe, steel, copper, brass, or rubber. Prados and

Peebles (1959) did report darkening of their 1.3 percent

solution within two weeks of preparation except for small

samples stored in glass bottles which remained unchanged

after three monr-hs Peebles, Prados, and Honeycutt (1965)

emphasized that small changes in concentration due to evap-

oration have a marked influence upon both the optical and

theological properties of the medium.

During the current investigation it was found that a

significant concentration gradient may develop between the

surface and the bottom of the storage container due to

evaporation within the container followed by draining of








condensate from the lid. The refluxing action, unless

controlled by floating plastic sheeting on the fluid

surface, seems to lead to sedimentation.

From the preceding paragraphs it is clear that the

preparation of standardized birefringent media having

specified properties is not practical due to variability

in the commercial dyestuff, apparent instability, and

marked variation in properties arising from evaporation.

The recommended procedure is to measure all significant

properties at the time of each use. This has been done

in the present study and in every previous quantitative

investigation using Milling Yellow.

Optical Properties

In non-steady shearing flow the optical properties

of Milling Yellow have both in-phase and out-of-phase

components which have been studied by Thurstone and Schrag

(1962, 1964) and Cerf and Thurstone (1964). They found

that the birefringence is highly dependent upon strain as

well as strain rate particularly below room temperature

at oscillatory rates less than 1 Hz. Thus, as Harris

(1970) concludes, the analysis of unsteady flows is not
possible in the general case. In the present study only

steady state coalitions are considered in the measurement

of the optical properties and the out-of-phase components

are neglected.

Two in-phase optical characteristics, the birefringence







and the extinction angle, serve to define streaming bire-
fringence in steady flow. Besides Harris (1970), Jerrard

(1939) and Peterlin (1956) have described these character-
istics in useful review articles.
Theory of birefringence
Based upon earlier work by Jeffery (1922), Boeder

(1932), Peterlin and Stuart (1939), and Snellman and
Bjornstahl (1941), the birefringence of a suspension of
rigid non-interacting ellipsoids has been calculated by

Scheraga, Edsall, and Gadd (1951). This body of theory,
which predicts a linear dependence of birefringence upon
shear rate at low flow rates, has been applied to Milling
Yellow by Cerf and Thurstone (1964) for the assessment

of small amplitude oscillations and by Peebles, Prados,

and Honeycutt (1965) and Swanson and Green (1969) to
estimate particle size. The theory fails when there are

interactions between particles or when the particle
dimensions exceed the upper limit of 106 meters set by
Peterlin and Stuart (1939) and Snellman and Bjornstahl
(1941). Little is known about the microstructure of bire-
fringent solutions of Milling Yellow. Cerf and Thurstone

1964) observed crystals between 1 and 2 microns in length
under the electron microscope but do not report how the

solution was prepared for viewing in a vacuum.* Recent

*At my request Mr. Frank Hearne made a microscopic obser-
vation of a 2.8 percent Milling Yellow solution under an
oil-immrersion magnification of X1000. He observed no
crystals but did obtain stress birefringence in the clear
medium by pressing upon the cover glass.







descriptions of lyotropic mesophases, such as ;.riose given
by Hartshorne and Stuart (1970) suggest a viable alter-
native to the usual assumption that the proper h;L of
Milling Yellow are due to a crystalline precipitate of the

type described by Swanson and Green (1969). Thl h-dro-
dynamics of such mesophases requires further investi,,ation
before the applicability of suspension theory can be
assessed.
Experimental measurement of birefringence
The introductory studies of Fields (1952) azd Peebles,

Garber, and Jury (1953) provided qualitative information

about the optical properties of Milling Yellow, The first
quantitative evaluation was reported by Prados (1957) and
Peebles and Prados (1959) who calibrated their solution

(roughly 1.3 percent dye) in simple shearing flo;j using a
concentric cylinder polariscope. They also verified that
the measurement of the distance between fringes in parallel
channel flow could be used as an alternative method of
calibration. The latter method is still in use (e.g.._ Dur-
elli and Norgard, 1972). Peebles and Prados obtained a
nearly linear relationship between birefringence and rate
of deformation for rates up to 19 see-'. A marked temper-
ature dependence was observed. The birefringence at

24.750 C was 11 percent higher than at 24.950 C and 36
percent higher than at 25.200 C. The extinction angle
.was measured only at 250 C and dropped monotonically from








450 at negligible rates of shear to about 280 at 20 sec-1.

The data of Thurstone and Schrag (1962) are of

limited usefulness to the present study since they consid-

ered oscillatory, rather than steady shear flows. Their

1.72 percent solution showed a progressive, 20-fold
reduction in the optical coefficient (loosely, the bire-

fringence) as the temperature was raised from 120 C to

420 C. When the temperature was held constant at 230 C,

the coefficient showed little change as oscillatory

frequencies increased from 10-2 to 1 Hz, but dropped

rapidly with further increases.

The range of dependence of birefringence upon shear

rates was extended by Hirsch (1964) in his study of

diverging ducts, but the most extensive study was reported

by Peebles, Prados, and Honeycutt (1965) who again used

a concentric cylinder polariscope for their measurements.

For shear rates ranging up to 2500 see-" and concentrations

between 1.248 and 1.455 percent by weight, they found

increasing non-linearity as shear rates increased, although

--ey identify a possible "second range of linearity" at

tne highest shear rates. All of their data were taken at

250 C. Although the concentrations, which were measured

very accurately by evaporating samples to dryness, are

reported to four significant figures, dispersion results

when a correlation is attempted between concentration and

the shear rate required to obtain a given fringe order in








the polariscope. Figure 1 shows this dispersion, some of

which may be due to inaccuracies in replotting. The

re~rining variability can be attributed to the use of

the coc-crcial dyestuff and to the difficulty of preparing

a standardized medium as alluded to earlier. In any case

it is clear that birefringence increases markedly with

dye concentration.

Peebles, Prados, and Honeycutt also measured

extinction angles over the same range of concentrations.

They found that the more concentrated solutions exhibit

asymptotic values for the extinction angle in the vicinity

of 200 as the shear rate increases above 40 sec"1. At

lower concentrations a similar as.mptote is reached, but

at higher shear rates.

ilo expressions for either the birefringence or the

extinction angle are advanced by the authors to represent

their findings. Consequently, for the purpose of the

present study it has been necessary to construct empirical

relationships which describe the data of Peebles, Prados,

and Honeycutt. This has been done in Chapter Four.

Swanson and Green (1969) were concerned only with

he minimum concentration at which birefringence could be

observed and not with its magnitude. TIney hypothesize that

the variability of Milling Yellow preparations is due to

a small fraction of the dissolved material, as little as

0.04 percent, which exists in suspension. Their hypothesis

is not evaluated in this dissertation,








- I I 1 I I I I


100







SHEAR
RATE,

see c


o- N = 2
A =
-- -


O

1.3 CO;:CE::TRATION, 14
weight percent


FIGURE 1. Dispersion
Prados, and Honeycutt


in concentration
(1965)


data of Peebles,


SHEAR RATE RE'UIR ED


TO PRODUCE


FE.I:;'G OF ORDER N








Rheological Properties

The theological behavior of media such as Milling

Yellow may be described rigorously in terLs of continuum

mechanics, theoretically, but with less rigor in terms

of hycdrdr.amic models, or empirically, based upon exper-

imental evidence. Each of these methods is discussed

in turn.

Continuum mechanics

With the advent of liquid crystals as a practical

media for electronic display devices (Caulfield and Soref,

1971, is one report among many), there has been a great
increase in publications relating to the constitutive

behavior of anisotropic materials. Not all such reports

have been useful. One reviewer, Kisiel (1968, p. 1043)

spol:e for many when commenting upon a stud- which shall

remain nameless:

This investigation belongs to a class, abundant
at present, of papers dealing with very general problems
with limited applicability to the solving of practical
questions.

An overview of the current state of the art indicates

that rigorous application of the continuum mechanics of

anisotropic media is limited to viscometric flows of the

simple type in which the velocity components are given by

U1 = 0; U2 = 0; u3 = u(x)

where x is a single spatial coordinate. Furt-hr, numer-

ical solutions are possible only for the very smzll class







of substances, notably p-azoxyanisole, for which some, at
least, of the necessary constitutive constants have been

measured and published. Neither of these conditions is

satisfied in the present dissertation; hence, the dis-

cussion of anisotropic continuum mechanics which follows

is succinct and selective.

Oldroyd (1950) established the general procedure by

which constitutive equations must be constructed if the

necessary conditions for tensor invariance were to be

preserved. Noll (1958) introduced the concept of a

"simple fluid": one in which the properties are completely

defined by the temperature and the strain history. The

viscometry of simple, non-Newtonian* fluids was examined

by Coleman, Markovitz, and Noll (1966) in a general

treatise which includes a bibliography of over 350 refer-

ences spanning the period from 1687 to 1965.

Specific constitutive relationships for anisotropic

fluids were formulated by Ericksen (1960a et sea.) and

Leslie (1966 et seg.) who postulate that at each point in

he continuum there is a preferred direction characterized

by a unit vector, or "director," d. On the basis of this

hypothesis it was found that, in general, the constitutive

stress tensor is non-symmetric, and seven or more consti-

tutive constants are required. The theory has been applied

with some success by Atkin and Leslie (1970) and Tseng,

*Non-Newtonian: a substance is Newtonian if and only if the
shear stress is directly proportional to the shear rate.








Silver, and Finlayson (1972) to certain specialized flows.

A more goee:.rl and even less tractable the*r- I Is

been developed by ErinLc- (1964 et sea.) .rL.o postulates

a micromotion of the material points which define the

continuum Associated with this it I-,, i'omoon are corres-

WoIi,-' g micromorments and microiniri i ia. A i r consti-

tutive relationship is obt:. hid at the pr:ce of additional

unl:, -wn constitutive constants. The current literature is

replete with argi'aint concerning the existence of tl.e

various constants, with their signs, and with the rela-

tionships, frequently in the form of inequalities, i. -ig

them. Truesdell (196') has pointed out that the complexity

of modern continuum mechanics is a reflection of nature

and requires no apology, but a hopeful reading of the most

recent review of anisotropic continuum mechanics by A-.rian,

Turk, and Sylvester (1973) leads only to the conclusion

that the theory is not yet useful.

Model construction

As an alternative to the utilization of rigorous, but

complex constitutive relationships for a continuum, :.: ny

uithors have elected to model anisotropic behavior in terr.s

of the effect which the presence of microscopic -rticles

in a -- -ttonio,.n mediv-m has urnn the microscopic pronieties

of the mixture. The success of such theories, of 'rJl.:.h

Einstein's (1906) calculation of the intrinsic viscosity

of a suspension of rigid spheres is the classic e-:':,r.!le,








has led not only to the analysis of particles whose shape
is less well defined, as in colloidal suspensions, but

also to inferences about the microstructure when an ill-

defined or poorly understood medium is found to obey the

predictions of a particular theory.
The most influential body of analysis has grown from

Jeffery's (1922) solution for the periodic motion of

rigid ellipsoids suspended in a viscous fluid undergoing

uniform shearing motion. Jeffery's solution was open-
ended, consisting of an infinite set of permissible orbits.

Other authors, notably Peterlin (1938), calculated the
distribution of orbits which would result from pertur-

bations of the particles due to Brownian motion as

expressed by the rotational diffusivity constant. Inte-

gration of such distributions leads to an estimate of the

viscosity. Kuhn and Kuhn (1945), Scheraga (19"5), and

Leal and Hinch (1971) are among those -.-ho have perforrred

this integration.

Cylindrical particles have been treated by Boeder

(1932), who re-plced the cylinders by ellipsoids of high
axial ratio, Burgers (1938), who obtained the torques due

to shears fcr true cylinders, and Broersma (1960), w.;`
included end effects. Still later Bretherton (1962)

demonstrated that any rigid particle having an axis of

revolution can be replaced by an ellipsoid of appropriate
dimensions and incorporated into the genr-rrl theory.








A common assumption of these theories is that there
is no interaction between the particles. When interaction

is permitted, as in Ziegel (1970) or Batchelor (1971), the

analysis is greatly complicated.
Although rigid spheres, ellipsoids, and rods have
served as the primary models for the analysis of non-linear

theological behavior, other shapes also play an important

role. A sampling of investigations which have served as
alternate models might include the work of Taylor (1934)
on drops, Debye (1946) on swarms and porous spheres, Kuhn

and Kuhn (1943) and Kirkwood and Riseman (1948, 1949) on

chains and necklaces, Simha (1950) on dumbbells, and
Frohlich and Sack (1946) on elastic spheres.
The practical value of these theories is that they

permit the replacement of a complex constitutive rela-
tionship with many unknown constants by a relatively simple

constitutive equation; however, the coefficients of this

equation will exhibit an involved (though theoretically
explicit) dependence upon the various material parameters,
and these parameters may prove as difficult to measure as

"he constitutive constants which the3; replace. An example
is the rigorous, three-constant constitutive equation

1 i1 '1ij
Cij = -Pbij 'gij '(8n+i)gikgkj + 28n t

where the constants are the intrinsic viscosity 4, and the
primary and secondary normal stress functions Onand p.








For the model of rigid ellipsoidal particles in suspension,
rr has been calculated by Saito (1951) and Scheraga (1955),

and the stress functions have been obtained by Giesekus

(1962). The study by Scheraga tabulates its results in

terms of the rotational diffusivity constant and the ratio

of the lengths of the major and minor axes of the

ellipsoid. In practice it has been commoner to infer

these properties from the macroscopic properties rather

than the reverse. Thus the model, even when it is valid,

may not be predictive,

Experimental measurements of rheology

Amenenhet's (1540 B.C.) boastful account of his water-

clock, which was capable of compensating for seasonal vari-

ations in viscosity (due to temperature changes), begins

the written record of rheology. It is clear that Amenechet

did his work without the benefit of continuum mechini.3.

The present knowledge of the rheology of Hilling
-ellow is also founded upon experiment. Although the

empirical relationships which describe these data may

iolate conditions of invariiance prescribed by continuum

-:l.-.-.s and include constants which cannot be obtained
fron mcde construction, they may be employed with care

provided that the flows to which they are applied do not

differ too greatly from those in which the experi.Lontal

data were obtained, A more len,,thy discussion of the

feasibility of employing empirical relationships to

describe the rheology of "i*lir;- Yellow will be found at








the beginning of Chapter Five.
When the feasibility of Milling Yellow as a bire-

fringent medium was established by Peebles, Garber, and

Jury (1953), measurements of the apparent viscosity were
made in a rolling ball viscometer at various temperatures
For solutions varying in concentration from 1.46 to 2.02

percent, a sharp exponential rise in viscosity was

observed as the temperature decreased. In the 2 percent

solution, the viscosity doubled as the result of a two-
degree temperature drop. Above a certain critical temper-

ature the optical activity of the solution ceased and the
viscosity approached that of water. It was recognized

that the apparent viscosity had a shear rate dependence

which was not obtained from the measurements.

Prados (1957) and Peebles and Prados (1959) measuredd
the viscosity of a 1.3 percent solution but did not report
the results. For shear stresses less than 5 dyne/cm2,
Frados assumed that the viscosity was constant, citing

Honeycutt and Peebles (1955) as his authority that Milling
yellow solutions:

...exhibit marked non-'ewtonian behavior when
subjected to shearing stresses greater than five to ten
dynes per square centimeter. (Prados, 1957, pp. 55-56)
Thurstone (1961) measured the acoustic impedance of
a 1.39 percent solution in a circular tube whose base was

excited by low-amplitude, axial, oscillatory vibrations.
At an uncontrolled temperature between 220 and 260 C, he







found that Milling Yellow exhibited viscoelastic properties.

That is, the local stresses were a function of both the

shear and the shear rate. As the frequency increased

from 10 to 300 Hz, the viscous term of the complex viscos-

ity coefficient dropped from 66 to 13 centipoise, the

elastic component having about the same magnitude as the

viscous component over this range.

Thurstone and Schrag (1964) and Cerf and Thurstone

(1964) did not report the viscous and elastic terms

separately. Thurstone and Schrag found that the comply:

viscosity coefficient is approximately the same for both

axial and transverse oscillations of the medium. Cerf

and Thurstone found that elastic forces predominate at

frequencies below 0.3 Hz, but that viscous forces are

dominant above 10 Hz for oscillatory shear waves. At

very high frequencies a limiting viscosity of 45 centi-

poise was obtained for their 1.73 percent solution at 250 C.

In steady flows the most extensive examination of

Milling Yellow theology is Peebles, Prados, and Honeycutt

(1965) who measured apparent viscosities with a capillary
'viscometer at 25 C over a concentration r.nge of 1.25

to 1.50 percent. For each sample they obtained smooth
monotonic curves for calculated values of wall shear

stress .trzus shear rate with linear respons-es when the

shear rate exceeded about 2500 sec-, 1.hen the wall shear

stress corresponding to a given h--.ur rate is replott-:







versus concentration, there is a significant scatter of
the data, just as a similar replotting of the optical
data (Figure 1) also resulted in dispersion. This
confirms a difficulty experienced in all investigations
including the current one: specification of the commer-
cial dye concentration is insufficient to define the
properties of the medium even at a fixed temperature.

An important result of the investigation of Peebles,
Prados, and Honeycutt was the demonstration that plots of
apparent viscosity versus wall shear stress are independent
of the diameter (and Lc/D ratio) of the capillary in which
the measurement is made. As Skelland (1967, pp. 32-39),
among others, has pointed out, this coincidence of curves
indicates the absence of inlet effects of the type
described by Naude and Whitmore (1956) or of wall effects
such as slippage or the radial migration of microscopic
elements as measured by Goldsmith and Mason (1961, 1962,
1964) and Gauthier, Goldsmith, and Mason (1971). In the
absence of a reliable explanation of Milling Yellow's
exceptional properties, the elimination of such effects
from consideration is welcome.

Peebles, Prados, and Honeycutt (1965) conclude that
Chilling Yellow is well represented, though not uniquely,
by the Pouell-Eyring equation:
A
P = o + q(Po Poo) sinh-'(g/Ap)
where p is the viscosity at shear rate g, and Po, Ioo, and

Ap are constants. The authors provide straight-line pl)ts







of these constants versus concentration, and for the four

concentrations plotted the agreement is excellent. Based

upon these plots, the following numerical relations can
be obtained:

logo Po = 10.72 q 12.92,

logo Poo = 1.38 q 1.63,
loglo Ap = -10.87 q + 16.16.
The units of p, Po, and Poo are centipoise, shear rates

g and A are in sec'-, and q is the weight percent of

Milling Yellow in the solution.

The data of Peebles, Prados, and Honeycutt are not

compelling with regard to the prediction of the Powell-

Eyring equation that the viscosity will approach a

constant value at low shear rates. Since most of the

quantitative studies reported in the literature were

conducted at very low shear rates, the absence of

conclusive data in this range is of major concern.

Fortunately, Peebles and Liu (1965) have provided a plot

from Hirsch's (1964) dissertation which indicates clearly

that the viscos ity does approach a constant value for
shear rates below about 5 sec"- These data were obtained

in a capillaryr -iscometer and replicate closely data from
a "Rotovisco" instrument when the two methods are compared

at shear rates around 50 sec"-. The latter instrument

shows the upper range of l-' --r responses. It should be

recalled that Prados (19?7) ~: surely, though he did not

report, constant viscosities at lower shear rates.







Velocity Distribution in Rectangular Conduits
Whenever a differentiable expression for the velocity
distribution is known, the shear-rate distribution is
defined by direct differentiation. Once the shear-rate
distribution is known, the birefringence and orientation
angle (or extinction angle) can be calculated. In the
present dissertation the shear-rate distribution is
required in a rectangular conduit. This distribution
has not been calculated for a fluid with Milling Yellow's
theological properties. The review which follows includes
those studies which show a potential usefulness in the
development of such a distribution.
-!ewtonian Flow in Cylinders

The determination of velocity distributions in pipes
dates from the experimental studies of Hagen (1839) and
Poiseuille (1840). In modern derivations the equation
which bears their names

u(r) = (R2 2)
4)iN dz

where u(r) is the speed at a distance r from the center-
line, R is the pipe radius, WN is the viscosity, and dp/dZ
is the pressure gradient, is obtained directly from the
iTavier-Stokes equation for incompressible fluids,

-vp + PN^u + pF = p du/dt,
by recognizing that

u = 0, U9 = 0, uz = u(r),
and integrating. The density p is implicit in the dp/dz








term which, in gravitational fields, is simply

dp/dz = -pgc

where g is the gravitational constant.
For a pipe which is not circular in cross-section,

the assumption

S= 0, = 0 u3 = U(X1,X2), (2.1)

yields

pN"2u = dp/dz. (2.2)
Exact solutions for this equation have been obtained

for cross-sections in the shape of concentric circles,

ellipses, and equilateral triangles. Lamb (1945), in

reviewing these solutions, points out that the analysis

of laminar flow in a cylindrical conduit is identical in

mathematical form to the analysis of torsion in a uniform

cylindrical bar and of fluid motion in a rotating cylin-

drical case, the cylinders in each case having the same

cross-section. Tiedt (1969) adds the reminder that the

analog is valid without modification only if the boundary

of the cylinder is simply connected.

Davies and 'hite (1928) obtained the relationship
bet:;e3n the pressure gradient dp/dz along a rectangular

duct and the Reynolds number

Re = 162
e P (b1+b2)
where 61 and 52 are the half-width and half-depth of the

duct.








In the same year Cornish (1928) published the solution

to equation (2.2) in a rectangular conduit in the form of

a Fourier series:

u = -- ((12 x2) + (2.3)
2uN dz
S+ i
32512 (-1) 2 FsTx cosh(sTry/256
co rs coh(sY/
T3 .s3 L 2I cosh(sTb2/261)

Cornish successfully related the corresponding volumetric

flow rate to experimental pressure gradients. Further

data which support this relationship are those of Nikuradse

(1930) and Lea and Tadros (1931), which also show the
predicted dependence of pressure gradient upon average

flow rate. The accuracy of the velocity profile must be

inferred from Eckert and Irvine (1956) who obtained

excellent agreement between local velocity IL suremenrts

and the FouZrier series solution to equation (2,2) for

triangular cross-sections. The Cornish solution is clearly
inaccurate wher. violations of equation (2.1) occur due to

secondary flows arising from convective effects. Such

flows are common even in circular pipes as dc'onstrated
most rec-etly by jo hnson (1974).
"*': r f-7'10'."' ^ ^-r-Is '1r. p e t 1~r .i r Conduits

Of the various forms of the theological equation

which have been or will be suggested for Milling Yellow

in this dissertation, only one has been investigated in

pipes. Christiansen, Ryan, and Stevens (19! ) related

pressure gradients to average flow rates for a Powell-








Eyring fluid, but their analysis was limited to circular

pipes.
It will be shown in Chapter Five that at low flow
rates two of the empirical expressions for Milling Yellow

reduce to the form
T = K g 1"3

which is the one-dimensional form of a power-law fluid.

Power-law fluids are defined by

ir = kp Ir2 gij

where Tij is the stress tensor, gij is the rate-of-
deformation tensor, II is the second invariant of gijp
and kp and m are constants. For the limited range over

which the power law applies to Milling Yellow, m = 1/3.
Power-law fluids have been investigated in rectangular
pipes by several authors.
Schechter (1961) used variational methods to obtain
the pressure drop along rectangular pipes for power-law
fluids having m = 0.5, 0.75, and 1.0 when the pipe aspect
ratio b1/62 was 0.25, 0.5, 0.75, and 1. He tabulates the
coefficients to be used in the series solution

u = u A, sin(Ns Tx/2& ) sin(Ns'ry/2&2) (2.4)
to obtain local velocity values.

Wheeler and Wissler (1965) elected to solve the same

problem by finite difference methods. Taking the solution

for a Newtonian fluid (m = 1) as a starting point,
successive approximations for the velocity distribution




33



were obtained for values of m between 0.4 and 1. Both the

stability of the solution and the rate of convergence

decreased with m. Below m = 0., stability was a serious

problem, and several hundred iterations were required for

convergence at the lowest value of m. For square pipes

it was found that:
-1
-dp/dz = 7.4942 (1.7330 m- + (2.5)

5.8606f K Um/281+m

For 0.4< m<1.0, the constants in this relationship were

accurate to four significant figures. Velocity profiles

obtained by this method were not published, but Wheeler

and Wissler state that the profiles obtained by their

method could be differentiated numerically several tires.

In contrast, differentiation of Schechter's profiles led

to erratic results. The empirical expression given above

was verified experimentally using power-law constants

obtained for their nmdium (sodium carboxymethylcellulose

solutions of various concentrations) by averaging the

measurements made in a circular pipe and a Couette visco-

reter. For Reynols numbers less than 2000 there was

excellent agreement between the predicted pressure drop

;-r. the corresponding Reynolds ; 'ber

Arai and Toyoda (1968) considered short rectangular
conduits with power-law fluids having values of m from

0.3 to 1. They provide average wall sheir rates in terms

cf an effective radius:

Requiv = 28162/(si+6).







The velocity distribution obtained for m = 0.4, b1/52 = 2,
is also provided together with the corresponding shear-
rate distribution.
A return to variational methods was provided by
Rbtheneyer (1970) who obtained a series of non-linear
equations for the coefficients bst of the polynomial

u = Z bst x y2t (2.6)
S=D t=o
by substitution into the non-linear partial differential
equation governing power-law substances in pipe flow:
L T2 l u',2 2 l2ynF62u 62-1
z = I '-' S -- + -
bz 2KL x by/ J Lbx2 + y2]

+ (()u,\ 2 + 6U U 2 6 2U
+ 2m' ^(22 ^)'' r-u2
\ybx] Ty J Ox/ 5,2
6u bu b2u 1/6u 2 u 2u7
+ 2 2+ 7- 1 7y
ox by 6xby \ '' y2j'
where m' = (l-m)/2m. The substitution was made at each
point (x,y) of a lattice distributed across one quadrant
of the cross-section. This set of equations was linearized
by substituting into the non-linear terms the values of
bst obtained in the previous iteration. For the first
iteration the :-avier-Stokes solution obtained by Cornish
(1928) was used, The boundary condition was met by setting
u = 0 in equation (2.6) for lattice points along the wall
and adding the resultant set of linear equations to the
linearized set obtained by substitution. The decision
not to write equation (2.6) in a form which satisfied the
boundary conditions, as was done by Schechter (1961), was








dictated by Rit!r.ce: r's intent to provide a method which
was appropriate for cylinders of arbitrary c rss-section.
In general the number of lattice points (-:,y) was greater

than the nT:.ber of bst so that the system of linear-z -:]

equations was overdetermined. The ertra d-grees of freedom
were used to minimize the error due to the bgt estimate, a

least squares fit being employed. The iterative process

ended when the computed flow rate through the cross-section

differed by less than 1 percent from the previous iteration.

Substances other than Iewtonian fluids and power-law

substances have received little attention in flows through

rectangular conduits. Sokolovskii (1966) considered a

dilatant material with the response:

S T / Io r rJc
g c >I c

For rectangular pipes he considered only freely dilantant

movement (Tc 0) for -..;:'ich the lines of constant velocity

form a set of rect-niles, one inside the next.

Greenberg, Dorn, and Uetherell (1960) solved by finite

difference methods the torsion problem for a square cylinder

composed of a material obeying the RS.t-.:rg-0sgood stress-

strain law. The fluid analog of a Ramberg-0sgood solid is

a DeHaven fluid, defined by the relationship:
S= po g/(l + ).

The values of n for which DeHaven (19r9a, 1ZI" ) esployed

this relationship were much smaller than the values

preferred by Greenberg, Dorn, and Ietherell (1960).









Hanzawa and Tlshi-.v:a (1970) investigated the problem of
Gree-nberg etal. after greatly simplifying the boundary

coniditi-.!ns ., rel,.laciin' the straight walls b. c.ave

surfi ':. C. ,~,j'.:.l rult. were obtain i'z1' where the

s ;:;;" e s e.; ,:... e '- s t.

LitviaL-.:-. (15'.8) used --iiation methods upon empir-

ical rheol .cal data for polypropylene. After e' p eis ji

r and g in the f ..,

jdl g + de g2 + d3 g3 g < g

d4 g + d5 g c

where T(g) was determined in onxe-dIL.I1i.inal flowV he

assc...i that

u = (612-2)(22y2) tb xs 2t
s o st
and m:nmizc the ,error introduced by the coefficients

over the ',-._~ss-section of his flow.

.' a follow--up of their 1965 study, Wheeler and

Wissler (1966) .:e.:sured the velocity distribution of a

0.9 percent solution of sd.:i, carboxymethy1cellulose

flowing in a square pipe. By obseri-,g the movement of

sus.cS.ded partcles at 12 locations in and surrounding

one quadrant of the cross-section, they found deviations

of up to 7 perc'J-1t from the velocity profile obtained; by

Wheeler ;..u Wissler (1965). The direction of the v-:1i-

ations was consistent with ihe L \, r'othesis that :.l.- was

secondary flow within the .rnse-section. To test this

hypothesis, the authors chose to model the liquid as a








Stol:esian fluid* with a constant, but non-zero, normal

stress function. The velocity distributions thus

obtained gave qualitative support to the hypothesis

that secondary flows were present. The method by which

the distribution was calculated is stated in general

terms and the constants which were obtained for the

Stokes equation were not published.


*Stokesian fluids are discussed at considerable length
in Chapter Five.













CHAPTER THREE

OPTICAL ANALYSIS


As stated in the Introduction, when a birefringent

liquid flows between two polarizers, a pattern of fringes

is observed. The purpose of this chapter is to derive a

method for determining the amount of light which emerges

from a given location on the second polarizer (hereafter

called the analyzer). The form of the resultant rela-

tionships will determine the parameters and functions

which are necessary to compute the location of fringes

for a given flow field.

The discussion is in three parts. The first considers

flows in which the velocity may be regarded as constant

along any given light path. Such flows will be designated

as two-dimensional and have a direct parallel in the two-

dimensional models analyzed by the traditional methods of

photcelasticity. The second part of the dis~. ssion will

consider steady flows in which the velocity v'.-ies along

the light paths. It will be shown that the results of this

three-dimensional analysis reduce to those of the two-

dimensional case when the limiting case of negligible
v'.ri-.tion long each light path is considered. The final

part of this chapter lists the successive steps to be








carried out in turn to obtain the fringe patt':rn from a

hypothetical velocity distr'ibulion in the cross-section.

Two-Dimensional Flow
The analysis of two-dimensional flows of birefringent
fluids occurs in many places. An e::-ople is Thurstone and

Schrag (1962). Consider Figure 2 in which a two-dimensional
flow field in the yz-plane is observed by polarized light

moving through the flow field in the positive x-direction.

Neglecting attenuation, the electric field* has the form

E = Eo cos(2nx/A) (-sin y j + cos y k)
where Eo is the amplitude of the wave, A is the wavelength,
and y is the angle which the polarizer makes with k.

The propagation of the vector E is a function of its
direction in the birefringent mediua-. Specifying the E

directions for maximum and minimum proji.1tion speeds by rn
and 6^2 respectively and assuming that A-I 62 = 0, the vector

E can be resolved into components parallel with Li and E2:

E = Eo [cos(2Tn~x/A) cos(y--.) n^ -

cos(2Tn2x/A) sin(y-\ ) ^2]
Here r is the orientation angle shown in Figure 2, and nl

and n2 are the refractive indices of the medium when E is

par-llel with nG a.nd 1 2 respectively.
".,,jn the light emerges from the flow field at : = i
the only light which will pass through the analyzer is

*In this dissertation vectors are designated by boldface (E)
and unit vectors are denoted by a circumflex (i5).































N















H ci-

cdd *H *H
HlO p

OC)









F-li
H -P-H

0













.C

0
P-1


/


4--

bf3
."l





0

4-






0
H
ci













I)
4-S















0
-P








to
r*






c











-0
0

















lo
c-i




Or



'-

a0








o cj


- i4
.-i c







A
that component which is parallel with na where
ni na = cos (y'- )

2 na = sin (y'- ).
In that case the emergent amplitude is
Ee = E fa
or
Ee = Eo [cos(2Tni"/A) cos(y-4) cos(y'-') +
cos(2nn2E/A) sin(y- ) sin(y'-)] .
Setting
n = (n + n2)/2
an = n2 nl
and expanding the cosines yields
Ee = Eo { [cos(2ri,-/A) cos(,7rn/A) +

sin(2nb/A) sin(rThn/) ] cos(y-*)cos(y'-*)
+ [cos(2nr /A) cos(Tan/A) -
sin(2nrW'/A) sin(Tbn/A)] sin(y-*)sin(y'-*) }.
This simplifies to
Ee = E [ cos(2n~T"/A) cos(nTban/A) cos(y-y') +
sin(2n~iT/A) sin(nabn/x) cos(y+y'-2)]
For a dark polarizing field, y y' = n/2 with the
familiar result upon substitution,
Ee = Eo sin(2rnb/A) sin(abAn/A) sin 2(y-T),
which is used in two-dimensional optical stress analysis
of solids (e.g. Dally and Riley, 1965, p. 171).
For a fringe to be observed it is necessary and
sufficient that any of the sine factors in this equation








equals zero. Each of these conditions will be discussed

individually.

The first factor reflects the periodic variation in

Ee due to the wave nature of light. The associated

frequency, about 1014 Hz, is too rapid for eye or camera.

For this reason the constant amplitude is usually replaced

by Eo, where
Eo = Eo sin(2TT/A).

The middle factor is responsible for the colored

fringes known as isochromatics which are seen when the

flow field is illuminated with white light. In mono-

chromatic light the same name is retained for those

fringes resulting from the condition

hnl&A = K

where iT is an integer known as the fringe order.

The final factor is responsible for the black fringes

iknow as isoclinics along which the principal optical

asr:es are parallel with tIhe polarizer This condition

is k:non as extinction ,nd the so-called extinction angle

is defined by

I I | -


I_ L.fnochrntcaic iiht ii t is uif icult to Cdistinguih

isochromatics fro:i isoclinics. rThis difficulty m-ay be

avoid by u sing circularly polarizd.- light for which

E E sin (an6/A).








The derivation of this ::xpre3sion is straightfor-ward, but

tedious, and it may be found in many referer.c-s includin-

Dally and Riley (1965, p2. 174-179).

Three-Dimensional Flow

The experimental stress analysis of three-dimensional

solids is accomplished by locking the deformation in place

and then cutting the model into slices for two-dimensional

analysis. The availability of this technique, which

provides local stress distributions along any desired

path, has inhibited interest in the study of three-

dimensional fringe patterns as such. Some studies have

been carried out using scattered light polariscopes which

have the effect of placing a temporary analyzer or polar-

izer at a selected plane within the three-dinensional

model. This technique is described by Van Daele-Dossche

and Van Geen (1969).

In liquids the direct three-dimensional analysis of

birefrirnent patterns to obtain velocity fields is not

feasible due to the variety of conditions which, in

principle, could lead to the same fr',n configuration.

On the other hand, t-h-?re seems no theoretical objection

to the invrerse method: assuming a flor distribution and

determining the resultant fringe pattern. The corres-

ponding anal:.sis follows.








In three-d _i :jnional flows, even if the streamlines

are parallel, the principal optic axes will be oriented in

three-dimensional space. As a result, the directions ri

and 62 must be recognized as lying parallel to the principal

axes of the ellipse formed by the intersection of a three-

dimensional ellipsoid with the plane orthogonal to the

light path at the point of interest (Sommerfeld, 1964, pp.

139-147). Care must be taken not to confuse the directions

ni1 and r2- with the projected axes of the ellipsoid. The
former are orthogonal; the latter, in general, are not. In

three-dimensional flows the directions An and P2 will vary

along the light paths as will nI and n2, the magnitudes of

their respective refractive indices.

Assumptions

In the analysis which follows three assumptions are

:r-de regarding the optical ellipsoid. These are discussed

separately.

Assumption 1. The properties of the optical ellipsoid

are completely defined by three characteristics: the diff-

erence between the length of the longest aris and the two

shorter axes (assumed equal), the magnitude of the incli-

nation of the longest ar:is to the principal flow directic

and the direction of that inclination. These variables are

shown in Figure 3 and are denoted by an, *, and 9 respec-

tively. The first two are easily identified with the bire-

fringence and the orientation angle of the previous section.

The third parameter, will be identified as the rotation
































x- x '





X \ I













FIG;RE 3. Optical ellipsoid, showing parameters 0 ,
sand rn.








angle and is necessary to describe variation of the optical

properties as the direction of the flow changes along the

light path. The magnitudes of the ellipsoidal axes vary

so little, an << that it can be assumed, as usual, that

the mean value of the refractive index is constant. The

previous assumption that the two shorter optical axes are

equal in length las been made principally in the interest

of economy since it reduces the number of optical para-

meters to a .:- a'=eable number.

Assumption 2. The optical properties are uniquely

defined by the local shear rate. iHumerous authors, among

them Truesdall and Noll (1965), have pointed out the

critical importance of history in the description of the

properties of a material. In the present case the flow is

steady. If a fading memory is ass-L.rd for the material,

then after a short time the history of the flow may be

neglected. The neglect of strain in the definition of the

optica. properties follows from e-: erimental "-r,: with the

medium ;-.:.ic:i indicates that the elastic properties are

eligible except at very low rates of shear Work in
this are. principally by Thurstone rnd his associates

(e.g. Thurston.e -.nd Schrag_ 1962), .-,s discussed in the

previoucc chapter. A unique deencdence of the optical

properties upon the shear rate is consistent with those

theories ulwhich c-,plin birefringence in terms of the

continuous rot-atio of microe elements within the fluid







as described by, say, Boeder (1932) and Kuhn and Kuhn
(1943). Acceptance of these theories is not necessary
for acceptance of Assumption 2 vhich is not contradicted
by any experimental evidence.
Assumption 3. The inclination of the longest axis
of the ellipsoid occurs in the direction of the local
velocity gradient. That is, the rotation angle is given by




-.;here g is the magnitude of the shear rate. The coincidence
of principal directions for certain optical and theological
properties has been suggested by Lodge (1956) and has
strong heuristic appeal.
Definition of Effective Optical Prorerties
The properties of the optical ellipse at any point on
a light path within the three-dimensional flow field follow
immediately from the assumptions. As shown in Appek:i: A,
the effective birefringence will be
WA = [(A1 2+A22+A32)Cos2q- 2(AlA4+A3A-)sin'PcosC
+ (A, ,A2 sin2 ]-
[(A1 2+22+A32)sin2*+ 2(A1A4+A3A5)sint cos'"
+ (A +A.2)cos2] -1/2
".*:re 'P is the effective orientation angle,

S- tan-1 [ 2 (AA41A-Ag)_
2 A12+A 22+A32 2-A2-A2

and the variables A, through A5 are defined in t'r-nrs of
the optical parameters an, 4r, and (. For example,








A1 = cos 4' sin 4 / ( -)

Values for A2 through A, are given in Appendix A.

It is appropriate to consider conditions for light

naths parallel with and close to the side walls where 0 has

a limiting value of wT/2. Substitution of this result leads

after considerable m .-nipulation to the result

S= *t
AIw = an sin2 2.

At low flow rates, when i approaches w/4,
AIJL = An.


To analyze flows in which AN and k vary along each

light path, it is convenient to employ the Poincar6 (1839)

model in which any change in the condition of a polarized

light beam Iv:7 be represented by the movement of a point on

the surface of a unit sphere. Procedures for use of the

Poincare sphere are found in many texts. The present sign

convention follows Hartshorne and Stuart (1970) and is

illustrated _in Figure 4. 3ch point on the s-ohere is

expressed in tor:-s of cngular coordinaotes 4 and 1. which

correspond respectively to latitude :.:.;1 longitude in terres-

trial n-v7igation, "'est"' and "!orth" b-ing the positive

directions.

Two points are desigr-ated on ti-he Pin.cre sphr e:

F, whic. r presentss th c Wndition of tie polarized light,


"ne: ... "re m l iS S5,
....... R w h o c n tecpal.,_ optic ::---is of the

to,71. Ti-LD" o .... dte rz inC d s "o















































=- 172


FIGURE 4. Poincare sphere showing P, the condition of
the polarized light beam, R, the principal (fast) optic
axis of the mediira, and AP, an arc on the surface repre-
senting the change in polarization which occurs.








Point P. In the general case polarized light is

elliptically polarized having the general form (for propa-

gation along the x-axis):
= As cos (ot + ) i + s cos (Wt + ) 2

where a', al", As, and Bs are constants, only two of which
A A
are independent, and XK and X2 are orthogonal cartesian

unit vectors in the yz-plane. In the present case the

form selected is

E = [i cos (C t ) + W2 cOS (t + j)] ,


4.2
which yields on rotation through the angle (e n/4):

Eo S (G COS (co d
E = -[sin ( ) cos (t ) -


cos (6 6) cos (ct + )] ji


+ [cos (e ") cos (ot 6) +

sin ( ) sin (cot + ) k .

Further trigonometric manipulation yields:

S= Eo [-(sine cos cos ot + cos e sin sin ot)
A
+ (coos Cs a- cos ot sin e sin a sin ot)k].

This form of representing E has two useful properties

(Figure 5). The variable e can be identified with the

angle which the principal axis of the elliptically polar-
A
ized light makes with the principal flow direction k. The

variable a reflects the eccentricity of the ellipse for

which the major and minor axes have magnitudes:







Eax = Eo cos ,

Ein = Eo sin .
Further, the variables e and a are simply related to

the coordinates of the Poincare sphere. Specifically,

P(Un,<) = P(2e,c).

The dependence of the electric field vector upon

time is usually neglected in studies utilizing polarized

light,and texts such as Hartshorne and Stuart (1970)

utilize the parameters e and a directly without stating

E(t) explicitly.
Point R. Point R on the Poincare sphere represents

the optical properties of the medium. If, as has been

assumed in all previous studies using Milling Yellow as

the birefringent medium, the solution acts as a linear

wave plate, then the point R is located on the "equator"

of the Poincare sphere at

R(i1,>) = R(29,0)

where IP is the effective orientation angle. In this case

point R represents the location of the "fast" optical

axis parallel to nl. If there were evidence of isomerism
or helicity in the molecular structure of Milling Yellow,

or if a degree of circular polarization were observed

under conditions of zero shear, then these effects would
require redesignation of point R at that value of 1

corresponding to the eccentricity of the elliptical wave

plate (see, for e::arple, Shurcliff and Ballard, 1964).




















Emax
N in




FIGURE F. Elliptically polarized light in the yz-plane.


A

Ir \\



A\+ *


FIGURE 6. Projection of OP
on M and definition of r.


FIGURE 7. Definition
of Ar. Angle is
27,ANxA .








Once points P and R have been designated the chL:-.: e in

::,larization follows directly. The Poncare sphere is

constructed such that the ca-'ge in P due to a mediun with

properties represented by R follows a cou.-t-Dr-cloclkise

circular path about the axis OR with an included angle of

2TTr&AX/A relative to this axis.

To obtain the associated vector equations, designate

the cartesian coordinates of the sphere by xl, X2, and x3

with corresponding unit vectors i, 21 and 3. Locate

P and R by the unit vectors P and R where

S= ER = i sin 2%9 + X2 cos 2?

and

S= GP = A cos a sin 2e + 22 cos a cos 2e +

Z3 sin a. (3.1)
Referring to Figure 6, the projection of OP on CO

is (P.)R and the radius of the path is
^ A ) A
= (P.R) R.

ihe change in P is AP, a circular arc with chord ar, where

A6 = P Po.

Since the chord subtrnds an angle of 2,nlA /: it follows

fro Figure 7 that the length of Ai is

AT 2r s in( A a-:/A)

and its direction is deterrrined by the conditions:

R*.Ar = 0;

r(.& = -r (r sin(&rA':!A)
(lxir)*M' = (P-B) r COS(:AO:!A).








When the indicated vector operations are carried out,
three simultaneous equations result from which the
components of Ar are found to be: (3.2)
Lrl = 2 sin(rWNax/A) cos 2T[sin a cos(TANax/A)
cos a sin(TANAx/A) sin 2(--)] ;

Ar2 = -2 sin(LANax/A) sin 2If[sin a cos(rrANAx/A)
cos a sin(wANrx/A) sin 2(e-i)];

Tr3 = -2 sin(rANAx/A) [sin a sin(rANAx/A)
+ cos a cos(rANAx/A) sin 2(E-~r)].

Consider a homogeneous flow of thickness x =
having optical properties AN = an and *k = r which are
constant along any given light path. Upon this two-
dimensional flow let light fall which is plane-polarized
(a = 0) by passage through a polarizer oriented such that
y = E. If

An'/ A =
where N is an integer, or

Y = +-
it can be seen that
zrl = Ar2 = Ar3 = 0
and the polarization of the light is the same when it
leaves the flow as when it enters. In a "dark polarizing
field" the analyzer is oriented at right angles to the
polarizer, and for such a field, if Ar = 0, a fringe will
result. Necessarily the conditions for which Ac = 0 are
identical with those for isochromatics and isoclinics in








the previous section on two-dimensional flow.
For more general two-dimensional conditions it must

be recognized that there will occur two types of fringes:

those due to total extinction of plane-polarized light

and those due to partial extinction of elliptically

polarized light. Besides the cases already considered,

there exists only one other condition for which the

light is plane-polarized and completely extinguished as

it leaves the flow. This occurs when

ani/A = N+ 7

where N is an integer, and simultaneously


where N is again an integer and Ly is the angle between

the principal directions of the polarizer and analyzer.

Note that both conditions must be satisfied for a fringe

to be observed.

In general, light emerging from a flow field will be

elliptically polarized. In the discussion of how point P

is determined on the Poincare sphere, the electric field

vector was expressed in the form:
E = Eo -(sin cos cos cos cos os sin sin 'At) j
2 a
+ (os e cos cos ,U sin sin -I sin )

Upon passage through an analyzer, the er.-erent amplit'-d

of such light would be Ee = r.ai where

a = -j sin y' + cos y',








Ee = Eo [cos(e-y') cos ct cos -

sin(e-y') sin t sin 2 .

The intensity of the light varies with the square of

the electric field (see, for example, Feynman, Leighton,

and Sands, 1963, p. 31-10):
I = kEe2

and the average value of the intensity for periods of

time much larger than the period of the light waves will

be:
lin k rt
I = t'I-G- I dt

2 2
= r-( cos2(E-y') cos2 j +

sin2(-y' ) sin2 j]

The conditions for a fringe when the light is plane-

polarized (a = 0) have already been discussed. If the

light is circularly polarized (o = + -), then

S= kEco2/2,

result which, predictably, does not depend upon the
A
characteristic direction ma of the analyzer. For any

th'er condition of light leaving the fluid there Will

be sone analyzer angle y' 'which will minimize I. This

condition may be obtained formally by differentiating
"(y') with respect to y' and setting the result equal

to zero. The result,
E y' = --
SY 2 1







is consistent with what would be predicted from looking
at Figure 5 and can be used to determine the necessary
condition for a fringe to occur when the emergent light
is elliptically polarized.
For two-dimensional flows, since the entrance
conditions are a = 0, E = y, the initial polarization
is given by substitution into equation (3.1) to obtain

Po = X1 sin 2y + 2 cos 2y.
The change in P is Ar, for which when a = 0, e = y,
AN = An, ? = and Ax = 5:
Arl = -2 sin2(wanb/A) cos 21 sin 2(y-)),
Ar2 = 2 sin2(~an /A) sin 2* sin 2(y-*),
Ar3 = -2 sin(kanb/A) cos(1ran/A) sin 2(y-*).
On emergence the condition for a fringe requires that
P(e,a) = 2( + r ,).

Setting
P(- + r',1) = P(y,o) + Ar
and solving for a yields
a = sin" [-sin(2Kn~/A) sin 2(y-*)],
which defines the ellipticity of the emergent light, and

tan2(nai/A) = -sin 2(y'-y)
sin 2(y'+y-2*)

which is the condition for a minimum to occur when the
light is elliptically polarized.

It has been shown in the previous paragraph that
the Poincar6 sphere can be used to obtain a complete
description of the polarization of light passing through








a two-dimensional flow. These relationships will now be
used in an iterative scheme to determine the polarization

of a light beam passing along a path for which the

effective optical properties are known explicitly but are

no longer constant.

Integration to Obtain Fringe Patterns

The polarization of a light beam moving through a

birefringent medium has been obtained in terms of a

position vector P which designates a point on the surface

of the Poincare sphere. The incremental change in P may

be expressed in terms of the chord Ar:

FP (x+ax;e+ AfE) 6AeC;N,') =

P(x;,(;h n,?) + A=r(A;e ,crAN)t .
By choosing values of Ax small enough so that AN and 9

may be regarded as constant for the increment, the change

in P along the entire light path may be obtained by

summation:

P(x=5i) =P(x--b5i) + fC-.
The effective parameters AlN and I are functions of an,

*, and 9 which vary in turn with the local shear rate

g(x,y), itself a function of the spatial coordinates x
and y. Figure 8 provides a schematic representation of

the means by which the summation is to be performed. The

corresponding steps are tabulated below:

1. Choose the y-coordinate of the light beam. Set
x = -b1"






























START L

-TI
62? <

sI E:FD
FIGURE 8. Schematic representation of successive deter-
minations of the variables. The subscript s has been
omitted from the variable nanes. The notation (0) indicates
that all variables except y are reset at their initial
(x = -6) values. Question marks indicate decisions.








2. Determine the polarization P as the light beam
enters the flow by setting 6 = y and a = 0 in equation

(3.1).
3. Obtain the shear rate gs(xs,y) from a distribution
calculated from those in Chapter Six or elsewhere. The
subscript indicates that this is the st iteration.
4. It has been assumed that the rotation angle 0Q

is defined by

es = tan IIgs/Y *
ssC X /

Obtain s (gs).

5. Obtain ans(gs) from the optical relationships
of Chapter Four or elsewhere. Obtain *s(gs) in a similar
manner.
6. *^s(Ans,,~sos)and AhLs(anst,,Os) are defined in
Appendix A. Obtain E "Y and A1 .

Z. Choose a trial value for Ayx.
8. Repeat steps 3 through 7 with Xs+1 = xs + axs
to obtain the corresponding terms with subscripts s+1.
9. Set = (= (s+ +1 s )/2.
10. Set ALT = (ANs+1 + ANs)/2.
11. Determiile the fractional varia L....s (Y--_IK)/4
and (AI-ANs)/A', of the optical coefficients for the interval

xs. If either variation exceeds a prescribed level, say
f = 0.01, reduce the value of Axs and repeat steps 8 through

11. Omit this sten if AN or 1'= 0.








12. Obtain Ars from equation (3.2).
A A
13. Set Ps+1 = Ps + ^se
A
14. From the definition of P the X3-component is

P3 = sin a.
Hence,
sin1
as+1 = sin-' [(P3)s+1] *

Obtain as+i

I From the definition of P it is also clear that

+i = tan-1(P1/P2)s+1

Obtain Es+1.

16. If Axs + xs+1 < b1, repeat steps 8 through 15.

If Axs + xs+l > bl, set axs = 51 xs+i and repeat steps

8 through 15. If Axs = 0, go on to step 17.

12. Calculate the relative intensity of the light

emerging from the flow field and analyzer at coordinate y.

Recall that:

I = cos2(E-y')cos2(a/2) + sin 2(-Y )sin2(1/2).

18. Choose a new y-coordinate and repeat steps 2
through 17.

19. '.Jhn the final y-coordinate has been chosen

(y = b2) and the final relative intensity has be-n calcu-
lated, plot I/-o as a function of y to obtain the frin7s

pattern predicted by the relationships chosen in steps 3

and 5.












CHAPTER FOUR

DETiERI!:lTION OF OPTICAL PROPERTIES


This chapter discusses the determination of two

optical properties of Milling Yellow: the birefringence,

An, and the orientation angle '. Both of these quantities

have been defined in Chapter Three. They will be discussed

separately, the orientation angle only briefly at the end.
Birefringence

The birefringence is that property of an optically

active fluid which is responsible for that part of the

fringe pattern known as isochromatics. The necessary

condition for these fringes is that

Anb/A = N

where An is the birefringence, & is the thickness of the

flow field through which the light passes, A is the

wavelength of the incident light, and N is an integer

known as the fringe order. It is clear that, except for

a constant of proportionality, the birefringence and the

fringe order may be used interchangeably for a given

experimental arrangement.

The data of Peebles, Prados, and Honeycutt (1965)

indicate that at a given concentration and temperature

Milling Yellow has a birefringence (fringe order) which








shows a progressive non-linear increase with shear rate.

This increase suggested that the data be replotted with

the square of the fringe order as the dependent variable.

Figure 9 shows the result when this is done for the four

most concentrated solutions reported by Peebles, Prados,

and Honeycutt. The hypothetical relationship
N2 = k1g + k2

appears adequate for the range of shear rates shown.

An experimental verification of this form was carried

out with the Milling Yellow used in the present disser-

tation. In contrast to the data of Peebles, Prados, and

Honeycutt, which were measured in a concentric cylinder

polariscope at nearly constant shear rates, the data for

the present study were obtained near the wall of a nearly

square rectangular conduit which is described in the next

section. In the succeeding sections will be found an

analysis of Milling Yellow's optical response when flowing

slowly through such a channel, some experimental data, and

a discussion thereof.

.Aparatus

The Milling Yellow solution was prepared as described

in Appendix B. 7 : nearly square conduits used in th.se

experiments have been described previously by Lindgren

(1962, 1963) who constructed them to observe the transition

between lrir~nar and turbulent flows in bentonite sus-

pensions. A schematic of the apparatus is shown in

F ..rc 10.




64















O F,.
mo '
*O






'O 0 *
S 0 -.0

H
*0
0\ d r-





t -i
rlr



0 0 4
F-i

o0 d4-
1-<






F-i 0
0)
c3<- 0 oa-!^
F-i 4 -)


0 do



d H
(3) 4--l
C C4





O0

4- O


OF-i
0 Pi
-P ; G )


or1 .-n




r
0* *!-P

0 H O
HH0 r
wH H (P


o 0 o0
0 C
1 CN













Overhead tank


\Rectangular
\conduit
Mano-
\N meter
\ ^ Light
Camera source
Camera


Control valve




Lower tanki





3 Pump and motor


FIL'URE 10. Sche-inatic of experimental anuaratus. The
polarizer and analyzer are represented I:y vertical lines
to the right and left of the rectangular conduit between
the light source and the camera.








The Milling Yellow was normally stored in a lower tank

which is lined with epoxy-bound fiberglass. From the tank

there is a gravity feed into a Moyno E.-2304 special appli-

cation pump which provides positive displacement with

minimum shearing of the fluid.

At intervals, between experimental runs, the stopcock

above the pump was opened and the covered, polystyrene,

overhead tank was filled. When pumping was complete,

normally a matter of seconds, the pump was turned off and

the stopcock closed to prevent siphoning through the pump

or the introduction of air into the pipe.

The fluid level in the overhead tank can be maintained

at a constant head by pumping continuously and permitting

the excess fluid to return to the lower tankI through an

overflow pipe; however, this procedure results in undesir-

able temperature rises and to the entri :-e-t of air in

the Milling Yellow. When it was found that the fluid level

in the overhead tank had a negligible effect upon flow rate

through the rectangular conduit (see Appendix C), use of the

overflow pipe wO.s restricted to providing protection agirLcs

accidental overfilling of the upper tank.

A gravity feed from the overhead tank leads to a set

of parallel, vertical conduits, only one of which was used

at a given time. The construction of these conduits is

shown in a cut-away view in Figure 11.




67
















\\

















FIGURE 11. Cut-away view of rectangular conduit
showing roughened walls, gaskets, and spacing wires.








The nearly square conduits are constructed by sand-
wiching two square, 12-mm plexiglass rods between two

plexiglass strips having widths of 36 mm. Plastic gaskets
in V-joints seal the flow channel, and the assembly is

bolted together along the length of the conduit. Thin

wires run parallel with but outside the gaskets to
maintain constant internal dimensions. On certain of

the channels two facing surfaces were covered with
grinding cloth to provide a known roughness. The conduits
are 4.87 meters in length with the internal dimensions

shown in Table I.
TABLE I

INTERNAL DIE;,SIONS OF CHANNELS (Lindgren, 1963)

Channel Height of Roughness Distance, mm, between:
Number Elements Strips, 25, Rods, 252

1 Polished plexiglass 13.43 13.26

2 0.035 to 0.044 mm 13.43 12.&0
6 0.59 to 0.70 mm 13.42 11.92

Because of the roughness elements, the flow can be
viewed through the side walls only in the clear channel,

and even in this case the view is unsatisfactory due to
the gaskets which prevent a view of flow along the walls.

Through the front wails there is an unimpeded view of the
flcw in all of the channels. Nikuradse (1933) and Moody
(1944) found that surface roughness plays no significant

role in the l-rriLnr flow region, and Lindgren (1963),








using the present apj-ratus, reported no significant

difference in flows through tubes with two walls roughened

and flows through tubes with four walls roughened when

the transition region was studied. Hence, the principal

effect of the roughinoss elements in the present dissertation

is the reduction in cross-sectional area which results.

Returning to Figure 10, note that each channel was

provided with :rn-.nom1ter taps on the smooth-faced sides of

the channel. The manometer fluid was carbon tetrachloride

(specific gravity: 1.f84).

Th? flow was normally illuminated by a sodium vapor
'.-:p from which light passed through an array of polarizers,

through the trans :rent channel and flow field, and through

an array of analyzers. The orientation of the sets of

elements in the three array configurations used during the

exprerimentss is given in Figure 12 and Table II. The method

of alignment is given in Appendix D. On four occasions

(rns 45, 418, 419, and 424) the arrays were replaced by

circular polarizers differing in phase by /2 radians.

Thoe fringes .:erae -oto=,graphod iith an E:acta Varex lac

Sr; coa'.era .i. th a Jena 58 ur:,, lens. A bayonet :.tension
+.,*.s n:l fc.ct**-ed .'f the lens per;.iii "-g n ...ob tct 23 cm
from the fil. plane to be focused -p an. phtographedt.

Kodak Tri-X film- a.s used and cor,.ercially developed at an
















/
O /

/>
I


/
/
/





/
/


/ /
/ /
/ /


0



o N


*H 0
0

*ro
4- 4
tl1i 0 cj0

*H H

ro r


(Ci











OH


0d 0
0





















-o
*H
4Or1
Oi
0 h



















F!
ur


p
N i>
cJk
c I


0 /
-- /

/


cdi










ci)











-(X
F-
N t>
>.n


to
0)







NM

dB
-p





f-i ,
0i
- 'H

























0 -4
N
0t-?















H03
cri
0 tH
ri


* *H






Ha





rl -H

5P-:








ci







HH
RC)









TABLE II


VALUES OF


y AMID y' FOR SUCCE SSIiVE EL ::IT3
OF POLA IZTP, 'G ARRAYS


Configuration for
runs 123 & 218


Y'
Tr/2

0


-n/2

0


-n/2


Configuration for
runs 130 & 131


rr/2


0 1/4


7/4

n/2


- 7/2


TT/4


- 17/2


-n/2


-nT/4


Configuration for
all other runs with
plane-polarized
r~- ys


1/2

T/4-


2/2

T7/2

- 'T/2

- /2



C


77/4


7/4


- F/4

rr/2



C


0 r-/2
0 TT/2


r~izoticri~ ttscd: dur~irl 'nnz 45,t 41,'1'. k


4-Ci cular noli
424.








equivalent exposure index of ASA 1600. A typical exposure

was a lens setting of f3.5 with a shutter speed of 1/150

second.

The choice of flow channels and the rate of flow

were controlled by a stopcock at the base of the flow

channel. Flow rates were measured by collecting the

fluid at the outlet above the lower storage tank and

timing the efflux to the nearest 0.1 second with a

Junghans timer. In a few instances the efflux volume

was measured to the nearest milliliter with a 250-ml

graduate, but in general the efflux was caught in a clean,

dry vessel and its mass measured on a Harvard Trip

Balance to the nearest 0.1 gm. Prior to each weighing

the temperature of the efflux was measured to 0,1 Co with

an ordinary laboratory thermometer.

Procedure

The fluid was circulated in the system for about

twenty minutes to insure uniformity of temperature and

concentration between the upper and lower tanks. When

this was accomplished, as indicated by constant temper-

ature readings in the lower container, the camera was

cocked so that the shutter would release 12 seconds after

the preset mechanism was actuated. The lower stopcock

was adjusted until the desired flow was obtained as indi-

cated by the birefringent fringe order at the wall of the

channel. The efflux container was placed under the fluid







outlet and simultaneously the timer was started. After

an appropriate interval the preset mechanics on the

camera was tripped so as to release the shutter at the

midpoint of the efflux interval. The time at which the

shutter was heard was noted, and at approximately twice

this time the container was removed and the timer stopped.

The temperature of the efflux was measured, and the flow

rate checked by confirming that the fringe order at the

wall remained unchanged. This done, the flow was stopped

and the mass of the efflux measured. Finally, the fluid

container was cleaned and dried, and the camera and timer

readied for the next run. For the last two runs manometer

readings were taken just prior to cutting off the flow.

As necessary, the overhead tank was refilled.

Prediction of Results

Consider the steady flow of a fluid in a rectangular

conduit having dimensions of 2b5 and 252. A force balance

yields

(261)(262) dp = w(45, + 462) dL

where dp/dL is the pressure gradient along the conduit

and 7, is the average shearing stress along the perimeter.

For a Newtonian fluid in a rectangular conduit, Cornish

(1928) showed that

dp/dL = 265pip/82So

where i is the mean flow velocity, p~ is the viscosity,

and So is a geometric constant having the dimensions of








area:
2b1b2 128522
So= 2 -2 -1 2 s-5 tanh(s7b1/252) .
3 T5 S ,3..

The dimensions b1 and b2 may be interchanged in the last
two equations without affecting anything but the rate of
convergence of the series. Numerical values for So are
tabulated along with some other constants in Appendix G.
Define the apparent viscosity of a non-Newtonian
fluid by

a = (62So/265i) dp/dL.
Substitution into the force balance yields

w = 2b 1 29a/So ( 1+2)
Replacing the average shear stress at the wall by the
product of the average shear rate and the average viscosity
there,

g = 2b12J a/So(b1+82))!W
The viscosity is lowest at the wall where the shear
rate is highest. Replacing )a by j, Ay, obtain

w = L2612u/So(51+s2) (1 "pw ).
B~ hypothesis,
N2 = kig + k2a

so at the wall:
N = [2k-112b/So(1+62) (1- Z) + k (4.1)

If, as Peebles and Liu (1965) have shown, the viscosity is
nearly constant at low flow rates, A1 may be neglected and
N2 = 2k1 12U/S0(51+52) + k2.
w








The mass flow rate is

G = 4pb1b2U,

where p is the density, so at low flow rates:

N2 = k1b1G/2So52P(5bi+2) + k2. (4.2)

As G rises the assumption that At is negligible becomes

invalid and the slope, dNw2/dG, should decrease.

In the vicinity of the wall the flow approaches two-

dimensionality, for which isochromatic fringes will be

observed when
S=

Substitution and rearrangement lead to the result

an 2Gk + 22
n862So (bt+62) 412

where ki and k2 are to be determined experimentally at

the low flow rates for which the relationship is valid.

experimentall Data

Tests of the equation just given were conducted with

the results shown in Tables III to XVI and Figures 13 to

16. It will be noted that the temperature varied during

these tests.

The run numbers reflect the date in 1973 upon which

the data were taken. Run 420A, for example, was the

first run conducted on April 20. Run 420D was the fourth

run on the same day. Although data were obtained for two

other runs, 420B and 420C, these data are not presented

because the flow rates were too high for the fringe cr.er

at the wall to be accurately determined.







TABLE III

FRINGE ORDER AT WALL RUN 123


Fringe
Order*

2+

2

1-1/2+

1-1/2

2-1/2

1+

1/2+


Efflux,
gm

1590.8

1505.2

1289.9

1453.2

1076.7

1321.2

1436.3

1391.5

1322.1


Time,
sec

15.6

15.3
18.0

20.8

9.6

29.1

59.9
46.3

151.1


TABLE IV


FRINGE ORDER AT WALL RUN 130


Time,
sec

136.2

36.7

45.4

15.8


Flow Rate,
gm/sec

8.19

25.53

27. 56

65.37


Temp,
oC

20.0

19.7

20.3

20.1


"Halved fringe orders were so identified when a fringe
occurred at the wall in a light polarizing field
(analyzer parallel with polarizer). A plus (+) indicates
distinct separation of the fringe from the wall.


Flow rate,
gm/sec

102.0

98.4

71.7

69.9

112.2

45.4

23.98

30.05

8.75


1/2


Temp,
oC

22.0

21.9

22.0

22.3

21.8

22.2

22.2

22.1

22.2


Fringe
Order*


Efflux,
gm

1115.7

937.0
1251.1

1032.8




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