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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 ormnts 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 shearrate 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 Profoezjr21 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 _illing G e nera Fro pr te s i Y 3 Theory of b'irof ce _ f! 1 s 2^ C r' Pdo cst, L  V o 0 D 1 i .t... p_.in Ree an " '. '.'  x xii xv 0OE .,o CHAPTERS: ,TWO (Continued) Newtonian Flow in Cylinders 29 NonNeewtonian Fluids in Rectangular Conduits 31 THREE OPTICAL ATI:AL, IS 38 TwoDimensional Flow 39 ThreeDi ensional Flow 43 Assumnotions 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 SIIEAR 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 Pronorties 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 OVER:HE TANK DECREASES 175 D ALIIG:.ET OF POLAR~IING ARRAYS 177 E C.TAI:P.'TION OF HOPPLE, RHEOVISCOMETER 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 BIOGRAPHIAL SKETCH 229 LIST OF TABLES INTERNAL DIMENSICIS OF CTAIELS TVALLS OF y AID y' FOR SUCCESSIVE ELEMEIITS OF POLLIiZ'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 418 AT WALL RUN L 1i9 AT W.!ALL RUNi 420A AT WTALL RUT 420D rT WALL 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 44 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:STAIS S 145 XXIX. ESINATION OF APPARENT VISCOSITY 154 CI C;:.dI3G IN FLOW RATE AS OVERHEAD TAI1T 176 EI CALIRATIOM: DATA 181 EII C.. Z.1 : iON CHECK (WATER) 184 II TEI.PERATURE COEFFICIE TS 203 I:I FALL TIL.S FOR 11: INCEEI7TS 214 I:II CORRECTED FALL TIZS FOR FIRST 10 .F: OF FAL'L 3Y 1I TI: ..:..:S3 217 CI 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 yzplane. 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 cnd 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$. 15. Fringe order at wall as function of mass 86 flow rate. Runs: 412, 419; 420A. xii LIST OF FIGU3BES (Continued) 16. Fringe ordar at wall as function of mass 87 flow rate. Runs: 420D; 424; 425. 17. Birefringence of original stock solution 95 c:r..:red with an e::traolation of the data of Peebles, Prados, and Honeycutt (1965). 18. Birefringence of fresh stock solution. 97 19. Extinction angles measured by Peebles, 102 Prados, and Honeycutt (1965) replotted to obtain straight lines. 20. Data of Peebles, Prados, and Honeycutt 108 (1965) replotted to obtain a linear relationship. 21. Hoppler RheoViscometer. 112 22. Geometry in annulus. 116 23. Response of viscometer, runs: 42; 420; 425. 141 24. Test of functional form: Pt = k + kg. 143 P 25. Test of functional form: P[Pt(Pt)oo = k + 144 26. Constructed relationship between shear stress 149 and shear rate for Milling Yellow at 250 C. 27. Comparison of equations (.) and (.7), run 150 4243, at 25= 5 . 2. I....::.,.eter readings cor:pared with those 15' exT ected for ewtonian fluid or a poer 29. Schechter's (1961) cofficients plotted as function f powerlaw exponent. or. ball. 1. 3ifunctional material. 138 F2. Flow field within viscometer showing bo .y 18 between regions obeying se.r :te constit;'ive equations. ro. Vr"tn I n ' ar i 194 ziii LIST OF FIGU? D (Continued) Ii. Temperature variation of samples 10330 198 and 101230. 12. Temperature variation of samples 10725' 198 and 101125. 13. Temperature variation of samples 10320 199 and 10920. 14. Temperature variation of samples 10417 199 and 101117. I$. Temperature variation of sample 10414. 200 16. Temperature variation of original stock 200 solution. Jl. Transition in measlirement of apparent 206 viscosity when P is too large. J2. 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 Crosssectional area Constant in PowellEyring 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 curvefitting 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' Ei 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 Rateofdeformation tensor .ass flow rate Manometer reading, difference in fluid levels Unit vector, xdirection (usually direction of the light path) Rms value, light intensity Initial intensity of light Unit vector, ydirection (usually direction alo: which fringe pattern Empirical constant, defined at point of use Unit vector, zdirection (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' (lm)/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 Halfangle 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 bifunctional 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 free 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 nRheo iscoeter 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 xve been usel 2 obtain pressure ad velo ity~ distri jions in trdimensional 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 threedimensional flows in which velocity variations along the light paths are permissible. Further, share rates for which the birefringence and shear stress vary nonlinearly 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 RheoViscometer; 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, thi 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 pattrns. 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 nonviscometric 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 viscoeter. Since neither of these meas urements seem to have been employed previously, analyses are provided to support the present 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 soc.1,lle1 powerlaw 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 stateoftheart 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 RheoViszometer, 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 twodimensional 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 .ethods of using twodimensional 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 twodimensional 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:.cdimensional 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 twodimensional 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 twodimensional 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 twodim.ensional config uatiocns, 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 threedimensional 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 arrIangernt violated the requirement set by the Tennessee studies for twodimensional 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 twodiz.'.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 threedicme nsional_ flows can be considered. It would o helpful if the restriction to e:tre.ely _lo flow ratas 7:.;I be relaxed or eliinatedc 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 monrhs 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 nonsteady shearing flow the optical properties of Milling Yellow have both inphase and outofphase 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 outofphase components are neglected. Two inphase 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 noninteracting 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 oilimmrersion 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 hdro 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 sec1. 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, 20fold 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 102 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 nonlinearity 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 coccrcial 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. Furthr, numer ical solutions are possible only for the very smzll class of substances, notably pazoxyanisole, 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, nonNewtonian* 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 nonsymmetric, and seven or more consti tutive constants are required. The theory has been applied with some success by Atkin and Leslie (1970) and Tseng, *NonNewtonian: 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 medivm 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 replced 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 genrrrl 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 nonlinear 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, threeconstant 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. 5556) Thurstone (1961) measured the acoustic impedance of a 1.39 percent solution in a circular tube whose base was excited by lowamplitude, 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 responses 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. 3239), 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 PouellEyring 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 straightline 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 shearrate distribution is defined by direct differentiation. Once the shearrate distribution is known, the birefringence and orientation angle (or extinction angle) can be calculated. In the present dissertation the shearrate 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 iTavierStokes 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 crosssection, 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 crosssections 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 crosssection. 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 halfwidth and halfdepth 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 crosssections. 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 recetly by jo hnson (1974). "*': r f7'10'."' ^ ^rIs '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 onedimensional form of a powerlaw fluid. Powerlaw fluids are defined by ir = kp Ir2 gij where Tij is the stress tensor, gij is the rateof 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. Powerlaw fluids have been investigated in rectangular pipes by several authors. Schechter (1961) used variational methods to obtain the pressure drop along rectangular pipes for powerlaw 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 powerlaw 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 powerlaw 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 nonlinear equations for the coefficients bst of the polynomial u = Z bst x y2t (2.6) S=D t=o by substitution into the nonlinear partial differential equation governing powerlaw substances in pipe flow: L T2 l u',2 2 l2ynF62u 621 z = I '' S  +  bz 2KL x by/ J Lbx2 + y2] + (()u,\ 2 + 6U U 2 6 2U + 2m' ^(22 ^)'' ru2 \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' = (lm)/2m. The substitution was made at each point (x,y) of a lattice distributed across one quadrant of the crosssection. This set of equations was linearized by substituting into the nonlinear terms the values of bst obtained in the previous iteration. For the first iteration the :avierStokes 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 rsssection. In general the number of lattice points (:,y) was greater than the nT:.ber of bst so that the system of linearz :] equations was overdetermined. The ertra dgrees 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 crosssection differed by less than 1 percent from the previous iteration. Substances other than Iewtonian fluids and powerlaw 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 rectniles, 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.:rg0sgood stress strain law. The fluid analog of a Ramberg0sgood 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 Greenberg 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 onxedIL.I1i.inal flowV he assc...i that u = (6122)(22y2) tb xs 2t s o st and m:nmizc the ,error introduced by the coefficients over the ',._~sssection of his flow. .' a followup 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 crosssection, they found deviations of up to 7 perc'J1t 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 .rnsesection. To test this hypothesis, the authors chose to model the liquid as a Stol:esian fluid* with a constant, but nonzero, 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 twodimensional 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 threedimensional 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 crosssection. TwoDimensional Flow The analysis of twodimensional flows of birefringent fluids occurs in many places. An e::ople is Thurstone and Schrag (1962). Consider Figure 2 in which a twodimensional flow field in the yzplane is observed by polarized light moving through the flow field in the positive xdirection. 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 AI 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 parllel 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) Fli H PH 0 .C 0 P1 / 4 bf3 ."l 0 4 0 H ci I) 4S 0 P to r* c 0 0 lo ci 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(y4) 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(yy') + 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(yT), which is used in twodimensional 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 socalled extinction angle is defined by I I   I_ L.fnochrntcaic iiht ii t is uif icult to Cdistinguih isochromatics fro:i isoclinics. rThis difficulty may be avoid by u sing circularly polarizd. light for which E E sin (an6/A). The derivation of this ::xpre3sion is straightforward, but tedious, and it may be found in many referer.cs includin Dally and Riley (1965, p2. 174179). ThreeDimensional Flow The experimental stress analysis of threedimensional solids is accomplished by locking the deformation in place and then cutting the model into slices for twodimensional 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 threedinensional model. This technique is described by Van DaeleDossche and Van Geen (1969). In liquids the direct threedimensional 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, th?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 threed _i :jnional flows, even if the streamlines are parallel, the principal optic axes will be oriented in threedimensional 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. 139147). 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 threedimensional 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 :rde 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 assL.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 rotatio 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 threedimensional 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 tan1 [ 2 (AA41AAg)_ 2 A12+A 22+A32 2A2A2 and the variables A, through A5 are defined in t'rnrs 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 sohere is expressed in tor:s of cngular coordinaotes 4 and 1. which correspond respectively to latitude :.:.;1 longitude in terres trial nv7igation, "'est"' and "!orth" bing the positive directions. Two points are desigrated on tihe 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. TiLD" 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 xaxis): = 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 yzplane. 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 yzplane. 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.tDrcloclkise 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' = (PB) 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(ei)]; 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 planepolarized (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 twodimensional flow. For more general twodimensional conditions it must be recognized that there will occur two types of fringes: those due to total extinction of planepolarized 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 planepolarized 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(ey') cos ct cos  sin(ey') 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. 3110): 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'IG I dt 2 2 = r( cos2(Ey') 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 twodimensional 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'+y2*) 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 twodimensional 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(xb5i) + 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 ycoordinate 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 (AIANs)/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 X3component 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 = tan1(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(Ey')cos2(a/2) + sin 2(Y )sin2(1/2). 18. Choose a new ycoordinate and repeat steps 2 through 17. 19. '.Jhn the final ycoordinate has been chosen (y = b2) and the final relative intensity has ben 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 nonlinear 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 Fi o0 d4 1< Fi 0 0) c3< 0 oa!^ Fi 4 ) 0 do d H (3) 4l C C4 O0 4 O OFi 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. Scheinatic 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 epoxybound 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 :et 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 cutaway view in Figure 11. 67 \\ FIGURE 11. Cutaway view of rectangular conduit showing roughened walls, gaskets, and spacing wires. The nearly square conduits are constructed by sand wiching two square, 12mm plexiglass rods between two plexiglass strips having widths of 36 mm. Plastic gaskets in Vjoints 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 lrriLnr flow region, and Lindgren (1963), using the present apjratus, 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 crosssectional area which results. Returning to Figure 10, note that each channel was provided with :rn.nom1ter taps on the smoothfaced 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 TriX 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 fi , 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 planepolarized 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 4Ci 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 250ml 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 s5 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 nonNewtonian 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 = [2k112b/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 11/2+ 11/2 21/2 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 