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STUDIES OF ARTERIAL BRANCHING IN MODELS USING FLOW BIREFRINGENCE By WILLIAM JOSEPH CROWE, JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1969 CIRCLE OF WILLIS INTERNAL CAROTID ARTERY EXTERNAL CAROTID ARTERY COMMON CAROTID ARTERY ARCH OF AORTA  THORACIC AORTA / / i , SUPERIOR MESENTERIC ARTERY  ABDOMINAL AORTA  COMMON ILIAC ARTERY  FEMORAL ARTERY POPLITEAL ARTERY Reprinted with permission. by Scientific American, Inc. VERTEBRAL ARTERY  SUBCLAViAN ARTERY SBRACH:AL ARTERY S COELiAC AXiS S. PENAl ARTERY Copyright (c) 1961 All rights reserved. Copyright by William Joseph Crowe, Jr. 1969 To my lovely wife Doreen whose enthusiasm is matched only by her imagination ACKNOWLEDGMENTS Inevitably, in any work such as this there are those people who through their continuing support make it both possible and meaningful. It is a pleasure to acknowledge my indebtedness to them. I should like to express my genuine appreciation to Dr. R. W. Fahien, who provided much of my fundamental training and introduced me to the field of biomedical engineering, and to Dr. L. J. Krovetz, who provided most of the training, support and facilities for this project. The logistic support of Dr. G. L. Schiebler has been invaluable. Technical assistance provided by Mr. Horace Brown was the major reason why the experiments proceeded smoothly. Thanks are also extended to Dr. Michael DeBakey and Scientific American, Inc. for permission to use their figure as a frontispiece. Grateful appreciation is extended to the Chemical Engineering Department and National Science Foundation for the opportunities to pur sue this work under first an Engineering College Fellowship and later a National Science Foundation Traineeship. The Departments of Medical Illustrations and Photography deserve much credit for their prompt and courteous service of high quality. Finally, I should like to thank Mrs. Lawrence Herrin for her careful checking and typing of this manuscript. TABLE OF CONTENTS ACKNOWLEDGMENTS . LIST OF TABLES .. .. LIST OF FIGURES . LIST OF SYMBOLS . ABSTRACT. . SECTION I. INTRODUCTION ......... A. Problem Statement . B. Prior Branching Studies. II. THEORETICAL CONSIDERATIONS . A. Fundamental Equations . B. Flow Birefringence . III. EXPERIMENTAL EQUIPMENT . A. The FluidPerfusion System B. The Flow Test Sections . C. The Optical System . IV. EXPERTMENTAL TECHNIQUE . A. Preparation and Storage of Suspension . .... . . . . * . * . . . e . * . the Dye B. Determination of Fluid Physical Properties Flow Regulation and Collection . Photography. . . Page iv vii viii xi xvii * . . ""' ' SECTION V. THE ANALYSIS OF FLOWPATTERN PHOTOGRAPHS AND EXPERIMENTAL RESULTS . . A. Isochromatic ShearStress Calibration for Laminar Tube Flow . . B. SteadyFlow Experiments . . C. PulsatileFlow Experiments .. ... .. VI. CONCLUSIONS AND RECOMMENDATIONS . APPENDICES............. ........... A. THE STEWARTHAMILTON TECHNIQUE. . B. APPARENTVISCOSITY AND REYNOLDSNUMBER DATA FOR 1.4% MILLINGYELLOW DYE . . C. PHOTOGRAPHIC DATA . . Page . 52 . 67 . 104 . 111 BIBLIOGRAPHY . . . BIOGRAPHICAL SKETCH . . . . . LIST OF TABLES Table Page 1. FLOW TEST SECTIONS. .................. 41 2. LIGHTPATH LENGTH AND AVERAGE SHEAR STRESS ALONG THIS PATH AS DIMENSIONLESS FUNCTIONS OF FRACTIONAL DISPLACEMENT IN THE FLUID MEDIUM. . .. 66 3. APPARENTVISCOSITY AND REYNOLDSNUMBER DATA FOR 1.4% MILLINGYELLOW DYE . ... .. 126 4. PHOTOGRAPHIC DATA ................... 127 vii LIST OF FIGURES Page Figure 1. Dyestreamline experiment (Evans Blue in water) by Krovetz for steady flow through a glass branch. .... 10 2. Dyestreamline experiment (Evans Blue in water) by Krovetz for steady flow through a plastic branch molded from a dog's artery. . .... ..11 3. Schematic diagram of the experimental setup .. 35 4. Flow wave generated by the rotary bellows pump used for pulsatileflow experiments . .. 39 5. Photograph of the flow test sections. .... ..42 6. Apparent viscosity and Reynolds number as functions of flow rate for 1.4% MillingYellow dye. .. 48 7. Isochromaticband distribution for fully developed laminar flow through a circular tube (I.D. = 3.00 mm) 54 8. Schematic cross section of a circular tube, illustra ting the refraction of a typical light ray passing from the fluid to the observer. . ... .56 9. Schematic flow cross section, illustrating a typical light path along which the average shear stress is required . . ... 63 10. Isochromaticband distribution for steady flow past a cardiac catheter (O.D. = 1.67 mm) inserted axially into a circular tube (I.D. = 3.00 mm) ..... .70 11. Isochromaticband distribution for steady flow past a beveled needle (O.D. = 0.92 mm) inserted radially through the wall of a circular tube (I.D. = 3.00 mm). 12. Isochromaticband distribution for steady flow through a 30 branch of circular cross section (area ratio = 1.0) . . . 75 S. 77 13. Flow ratio versus upstream Reynolds number, with area ratio as a parameter, for 30 branches of circular cross section . . .. .. 79 viii 14. Flow ratio versus upstream Reynolds number for Milling Yetllow dye and water in a 300 branch of circular cross section (area ratio = 1.0). . . 82 15. Branch entrance lengths versus downstream Reynolds number for steady flow through a 30 branch of circular cross section (area ratio = 1.0). . 86 16. Isochromaticband distribution for steady flow through a 300 branch of rectangular cross section (area ratio = 1.0). . ... ... 89 17. Flow ratio versus upstream Reynolds number for 300 branches of circular and rectangular cross section (area ratios= 1.0). . .... ... 90 18. Isochromaticband distribution for steady flow through a 600 branch of circular cross section (area ratio = 1.0) 92 10. Branch entrance lZengths versus downstren Reyld number for steady flow through a 600 branch of circular cross section (area ratio = 1.0). . 93 20. Isochromaticband distribution for steady flow through a 900 branch of circular cross section (area ratio = 1.0) 95 21. Branch entrance lengths versus downstream Reynolds number for steady flow through a 90 branch of circular cross section (area ratio = 1.0) ... 97 22. Isochromaticband distribution for steady unseparated flow through a + 450 wye branch of circular cross section (area ratio = 1.0). . . 99 23. Isochromaticband distribution for steady separated flow through a + 450 wye branch of circular cross section (area ratio = 1.0). . ... 100 24. Flow ratio versus upstream Reynolds number, with branching angle as a parameter, for branches of circular cross section (area ratios= 1.0) .. 101 25. Branch entrance lengths versus flowpulse phase angle (time), with area ratio as a parameter, for pulsatile flow through 300 branches of circular cross section 107 Figure Page Figure Page 26. Branch entrance lengths versus flowpulse phase angle (time), with branching angle as a parameter, for pulsatile flow through branches of circular cross section (area ratios = 1.0). . .. 108 27. Branch entrance lengths versus flowpulse phase angle (time), with internal geometry and type of branch as parameters, for pulsatile flow through branches having an area ratio of unity .. 109 28. The StewartHamilton experiment, illustrating a typical indicatordilution curve . .... 118 LIST OF SYMBOLS A identifies the main downstream stem of a branch denotes the arbitrary position of an isochromatic band seen in a flow photograph indicator injection site mass of indicator injected, (mg) AA crosssectional area of the flowing stream in the main downstream stem A Ag crosssectional area of the flowing stream in the side arm B a perpendicular displacement of the light path in the fluid from the center of the tube amplitude of the component vibration parallel to the optic axis of the fluid. amax maximum value of the perpendicular displacement a, (aax = R) B the side arm of a branch the point on the outer surface of the circular tube where the observed light ray emerges b lateral displacement of an isochromatic band in the flow photograph, (b = ang/n1) amplitude of the component vibration perpendicular to the optic axis of the fluid bma, maximum value of the band displacement b, (bma = Rn3/n1) C the point on the inner surface of the circular tube where the observed light ray emerges from the fluid C' position of point C after rotation Ci mass concentration of indicator at point i, (mg/1) D inside diameter of a circular tube the point on the inner surface of the circular tube where the observed light ray enters the fluid D' position of point D after rotation e subscript denoting the entrance to a branch F arbitrary function of integration Fbody total body force acting onthe macroscopic branchingflow system drag total drag fcrce acting on the macroscopic branchingflow system G arbitrary function of integration g vector sum U of bdy forces per uni.t mus of fluid i subscript denoting an arbitrary downstream point subscript denoting an arbitrary inlet to the macroscopic branchingflow system j subscript denoting an arbitrary outlet from the macroscopic branchingflow system K diameter ratio for a circular tube, defined as (outside diameter/ inside diameter) L length of the circular tube LE entrance length 4 length of the light path in the fluid M total mass contained within the macroscopic branchingflow system xii N an arbitrary integer order of an isochromatic band N1 normal to the surface of the tube at point B N2 normal to the surface of the tube at point C n outwardly directed unit vector normal to the surface of the macroscopic branchingflow system n, index of refraction of medium 1 (air) n2 index of refraction of medium 2 (glass or Plexiglas) n3 irdex of refraction of medium 3 (MillingYellow dye) nx index of refraction for the component vibration parallel to the optic axis of the fluid ny index of refraction for the component vibration perpendicular to the optic axis of the fluid 0 coordinate origin P pressure of the fluid at any point P total momentum of the macroscopic branchingflow system Q volumetric flow rate QA volumetric flow rate at the injection site A, (1/sec) Qi volumetric flow rate at the sampling site i, (I/sec) R inside radius of the circular tube Re Reynolds number based on apparent viscosity and tube diameter, (Re = D7poli) r radial coordinate ? position vector denoting the center of the light path in the fluid xiii S denotes the surface of the macroscopic branchingflow system T period of vibration for the incident light beam t time t' defined as t + gT ts sampling time V denotes the volume occupied by the macroscopic branchingflow system v fluid velocity at any point Vs velocity of the moving surface of the macroscopic branchingflow system at any point v magnitude of the fluid velocity v Vr radial component of the fluid velocity v Ve angular component of the fluid velocity v vz axial component of the fluid velocity v V crosssectional average of v 2 crosssectional average of v2 w subscript denoting the inside wall of the tube x displacement of the component vibration parallel to the optic axis of the fluid at any time arbitrary coordinate perpendicular to the observer's external line of sight and passing through the center of the tube y displacement of the component vibration perpendicular to the optic axis of the fluid at any time arbitrary coordinate parallel to the observer's external line of sight and passing through the center of the tube xiv y^ limit of integration, (y* = (R2 a )) z axial coordinate Greek Symbols a fractional displacement of the light path in the fluid from the center of the tube, (az = a/amax = a/R) B fractional displacement of the isochromatic band in the flow photograph, (B = b/bmax = a/R =a) AP pressure drop over the length L of the circular tube 6 path difference between the interfering vibrations in the fluid e angle of rotation between the abscissa describing the incident polarized light and the optic axis of the fluid 0 angular coordinate 01 angle between the light ray in medium 1 and the normal NI 02 angle between the light ray in medium 2 and the normal N1 Q3 angle between the light ray in medium 2 and the normal N2 04 angle between the light ray in medium 3 and the normal N2 A difference operator meaning the sum over all outlet streams minus the sum over all inlet streams of the macroscopic branchingflow system X wavelength of incident light (in vacuo) \X wavelength of the component vibration parallel to the optic axis of the fluid xy wavelength of the component vibration perpendicular to the optic axis of the fluid 4 pp apparent viscosity of the MillingYellow dye (app = PR  app app 8QL V fluid viscosity (p = papp for MillingYellow dye) T irrational number, 3.14159... p fluid density Summation over either inlet or outlet streams of the macroscopic branchingflow system shearstress tensor T component of the shearstress tensor T Tz component of the shearstress tensor = rz component of the shearstress tensor T T magnitude of shear stress (at upstream positions, T = prz) S wall stress, (rT = () ~ D dimensionless average shear stress, (c = T(avg)/T~) Miscellaneous Symbols underscore denotes a vector quantity  overscore denotes an integral average over a cross section overscore also denotes a line segment  denotes a tensor quantity II 11 denotes a tensor quantity (avg) denotes an integral average along the light path (up) denotes the value at the upstream position (down) denotes the value at the downstream position xvi Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STUDIES OF ARTERIAL BRANCHING IN MODELS USING FLOW BIREFRINGENCE By William Joseph Crowe, Jr. March, 1969 Chairman: Dr. R. W. Fahien Cochairman: Dr. L. J. Krovetz Major Department: Chemical Engineering The optical technique of flow birefringence (double refraction) has been used to investigate steady and pulsatile flows through drilled Plexiglas branches simulating those of the arterial system. This technique was also used to trace flow disturbances created by inserting needles and cardiac catheters radially and axially into a laminar flow through a circular tube. The birefringent liquid used was a 1.4% aqueous suspension of the aniline dye Milling Yellow, and birefringence patterns were photographed on 16 mm motionpicture film. The experiments covered a Reynoldsnumber range from 0 to 250. A scheme has been proposed for calibrating the isochromaticband distribution obtained for steady laminar flow through a circular tube against the average shear stress along the light path through the fluid. The optical effects arising from curved boundaries and differences in refractive index have been included in this calibration. xvii Analysis of birefringence photographs has revealed that regions of flow stasis and boundarylayer separation occur along the outer walls of branches for upstream Reynolds numbers between 20 and 50, depending on the angle of branch. In addition, it has been shown that regions of locally increased shear, where stresses exceed the upstream wall stress, occur along the inner walls of branches if 1. the branching angle is greater than 30, 2. the upstream Reynolds number is greater than 150 and 3. the downstream arms of the branch are of approximately the same diameter. Birefringence films for pulsatile flows have shown that these regions do not move with the incoming pulse but remain centered approxi mately one diameter downstream from the point of bifurcation. It has been suggested that locally increa;ePd shpering stress on the arterial intima at sites of branching represents that physical mechanism for initiating wall trauma and atherosclerotic disease which is most closely supported by pathologic data. Photographic evidence has been presented for the occurrence of spiral secondary flows in branches, and the importance of the resultant mixing effect has been discussed with respect to the StewartHamilton technique for measuring cardiac output. The propagation of flow disturbances has been characterized by defining branch entrance lengths in terms of distorted birefringence patterns. Entrancelength data and birefringence photographs have shown that as upstream Reynolds number increases,flow disturbances in branches are generated and propagated by the following three processes: 1. the readjustment of a laminar velocity profile, 2. the onset of boundary xviii layer separation and 3. the onset of secondary flow. Both MillingYellow dye and water Lave been used to study the flowdistribution characteristics of branches for steady flows in the Reynoldsnumber range from 0 to 2100. These experimental results have shown that upstream fluid momentum per unit volume is the principal dynamic variable affecting branching flow distributions and that area ratio is a much more significant geometric parameter than is branching angle. It has also been shown that wall attachment occurs in branches to the extent that sidearm flow will exceed that in the main downstream stem providing that 1. the branching angle is less than 600, 2. the upstream Reynolds number is below a certain value which is dependent on the fluid and 3. the diameter of the side arm is approximately equal to that of the main downstream stem. xix SECTION I INTRODUCTION During the past twenty years, engineering research has undergone a transition characterized by continual erosion of once welldefined boundaries separating individual disciplines. One need look no farther than the modern electronics industry, with its increasing dependence on such fields as metallurgy, physical chemistry and chemical engineering, to appreciate this fact. What might be called the interdisciplinary revolution has grown both in scope and intensity, affecting not only engineering disciplines but also creating new interfaces with the life and social sciences. Undoubtedly, several factors have contributed to this process but perhaps two of the most significant have been first, the development of new materials and devices having broad application (e.g., the high speed electric computer and the laser) and second, the refinement of existing measuring instruments and techniques, coupled with the ever present desire of researchers to extend known theoretical and experimen tal results to new areas of research. The field of biomedical engineering has evolved as a direct consequence of this process and is now enjoying rapid growth both in this country and internationally. Significantly, this growth has occurred at both academic and industrial levels and is reflected by new curricula and publications in the field. Among those topics receiving attention in the biomedical engineering literature, none has received more than the cardiovascular system. This is not surprising when one considers that diseases of the circulation are among the most common yetunsolved problems in medicine and that the circulatory system, featuring unsteadystate heat, mass and momentum transport within materials possessing nonNewtonian rheology and functioning under a closedloop automatic control system, involves many of the most interesting and challenging engineering prob lems of the day. The great majority of biomedicalengineering studies dealing with the cardiovascular system fall into one or both of the following catego ries: those presenting experimental data obtained from in vivo, in vitro or model experiments, and those proposing mathematical models and analogs to incorporate existing experimental data. The studies of McDonald, Taylor, Hardung, Womersley, Attinger, Bergel, Noordergraaf and Gessner are a few examples of these. However, with some notable exceptions,13141622424748495054 few studies have attempted to correlate either experimental data or theoretical predictions with abnormal clinical and pathological findings. That is to say, little attention has been given to providing physical bases or mechanisms for the formation of observed pathological phenomena, as opposed to providing models simply to explain observed experimental data. The recent work of Fry demonstrating that excessive shearing stress causes walltissue erosion is a particularly significant step in this direction. The present work represents an attempt to further this approach by focusing attention on experiments with models simulating arterial branches and interpreting experimental results in the light of known findings from the medical literature. The approach here is then in contrast with an engineeringscience approach to the problem, wherein one would be concerned with solving the equations of change for various types of fluids and wall conditions. A. Problem Statement The problem may be posed generally in two parts: First, in the flow range common to the arterial system in man, what physical phenomena may be attributed to the presence of a branch in the flowing stream? Second, what are the implications of these phenomena with respect to wall trauma and atherosclerotic disease, known to have a predilection for formation at such sites in the arterial system? More specifically, the present study seeks to investigate three major areas of branching dynamics under conditions of both steady and pulsatile flow by using the optical technique of flow birefrigence. These three areas are: 1. the distri bution of wall shearing stresses in branches insofar as determining whether local concentrations might occur which could damage the vessel wall, 2. branchingflow distributions as functions of branching angle and upstream Reynolds number and 3. convectivemixing effects at branches and the propagation of disturbances downstream as these relate to suspended particles in the blood stream and to the StewartHamilton indicatordilution technique957 for determining cardiac output. With the exception of one wye branch, this study has been restricted to branches of the snglesidearm type fabricated from drilled or milled Plexiglas block. Though model branches of this type cannot imitate branches of the arterial tree in every detail, particu larly their distensibility and wall characteristics, at least some indications of possible flow behavior in the arterial branches can be obtained in this way; and for those branches rigidly bound by surrounding tissue and muscle, such a model represents at least a good first approxi mation. The approach here is primarily experimental and considers only mechanical interactions between the flowing stream and the branch. Though biochemical processes do serve a vital function in the cardiovas cular system, there is no reason to suppose a priori that biochemical activity at a bifurcation would be any different from that occurring in unbranched arterial segments. On the other hand, there is every reason to suppose that flow behavior at two such sites would be different, per haps even to the extent of either altering normal biochemical function resulting in the development of arterial disease, as Rodbard's work 42 suggests, or simply causing trauma to the vessel wall, such as that shown in Fry's work.6 This point is further emphasized by the studies of Texon, Imparato, Helpern and Lord, who have shown experimentally 22 49,50 4950that atherosclerosis can be produced artificially in dogs by altering bloodvessel geometry. There are four important reasons why flow birefringence was chosen as the flowvisualization technique used in the present study: 1. Since flow birefringence is an optical technique, there is no need to introduce probes or other devices into the flow field thereby causing external disturbances. 2. It is possible to study flow behavior through out the entire branch, not just along a single streamline as with the technique of dye injection. 3. Prior twodimensional studies36'37'40,41 have indicated that flowbirefringence patterns can be correlated with the shearingstress distribution, so the possibility of obtaining shearstress information in branches directly seemed fruitful. 4. It was felt that flow birefringence would provide a convenient way of tracing flow distur bances and entrance effects downstream. As early as 1964, Wayland had suggested the possibility of using flow birefringence to study flow disturbances in branches,51 though this was unknown to the author until recently. Fortunately, the present study has incorporated most of Wayland's suggestions. B. Prior Branching Studies Interest in the problem of branching flows is not new. In fact, problems of flow distribution have been of engineering concern since ancient times, and, today, relationships between vessel branching and flow behavior are receiving considerable attention in connection with the design of fluid amplifiers and fluidic digitallogic devices. On the other hand, physiologic interest in branching has arisen only within the last fifty years and has been sporadic. Recent studies,47'54 demonstrating that atherosclerosis shows a predilection for formation at sites of branching and curvature in the arterial system, have increased the interest in branching. There now seems little doubt that problems associated with vessel branching on a small scale consti tute significant research areas both in engineering and medicine. Surprisingly little work, either theoretical or experimental, has been published on the branching problem. Poisson's work,39 concern ing reflected and refracted waves at a branch site was among the earliest theoretical studies of branching. Poisson's results were later repeated by Lord Rayleigh 46 in his discussion of the propagation of sound through branched tubes. The earliest physiologic interest in vessel branching appears to have been that of Moens34 and Frank,15 who were concerned with the propagation of pulsewaves through the arterial tree. Though Frank's study has often been cited, it apparently has never been criticized, which is surprising in view of the fact that it contains numerous mathe matical errors. Frank did discuss the problem of a pulsewave encounter ing a bifurcation but only to the extent of repeating Poisson's formula tions (incorrectly) as they appeared in Rayleigh's work. More recent studies of branching have emphasized 'oth theoretical and experimental approaches. Womersley's study56 represents the most comprehensive theoretical attack on the problem of mathemati cally modeling pressure and flow relationships in the arterial tree. His proposed model was based on the linearized NavierStokes equations and the equations for an ideally elastic wall. Womersley also considered wave reflectionsat branches and investigated the frequency dependence of wavereflection coefficients and input impedence. Acrivos, Babcock and Pigford considered the problem of flow into an idealized manifold consisting of uniformly spaced side ports of equal area. They proposed a simplified mathematical model based on friction factors and pressurerecovery coefficients and were able to show good agreement with their pressure and flow data for both blowing and sucking manifolds. Knox24 considered the more general problem of measuring pressure drops and flow distributions for branches of varying branch angle. He considered branches of the singlesidearm type and determined pressure recoveries at bifurcations. He defined a recovery length for branches as the distance downstream from the branch necessary for the pressure gradient to regain linearity, and his data showed that recovery lengths for the side arms were significantly greater than those for the main stems. The technique of dye injection was used by Copher and Dick,11 Hahn, Donald and Grier18 and by Barnett and Cochrane5 to study particle and flow distribution in glass models simulating venous flow. Distinct 8 flow channeling was noted in these experiments. Helps and McDonald2021 investigated venousflow patterns both in vivo and in a glass model and found that at Reynolds numbers less than 1000 injected dye formed a parabolic profile in the tributary, but at the junction circulating movement developed. Two sets of secon dary flow were set up in each pipe half, and this persisted "for some distance down the parent trunk until a new parabola was formed across the single trunk". 33 McDonald33 used the dyeinjection method to show that marked disturbances were present at the aortic bifurcation in a rabbit at Reynolds numbers less than 1000. He reported that dye could be seen to impinge on the wall, and eddies were set up near the orifices of the two branches. Krovetz25'27'28 conducted similar studies using glass models and Silastic (silicone rubber) molds of dogs' arteries35 to simulate arterial branching. He observed and photographed secondary flows which dis tributed dye particles across the cross section as if the flow were turbulent, even though the upstream flow was well within the laminar range. He found that dye particles were distributed unevenly between the downstream branches if the incoming dye front was laminar but were distributed evenly if the incoming front was turbulent. He also found that the critical upstream Reynolds number for the onset of secon dary flow in branches ranged from 58 to 89% of the critical Reynolds number for the onset of true turbulence in a straight tube. Another study by Stehbens,45 reporting a series of experiments on turbulence in glass models simulating arterial bifurcations and the curvature of the carotid artery, found critical Reynolds numbers rang ing from 306 to 1473 "calculated for the lowest rate of flow at which turbulence could be induced in the models". Typical experimental results obtained by Krovetz27 are shown in Figures 1 and 2. Figure 1 shows a laminar dye front encountering a branch in a glass tube, and Figure 2 is the equivalent experiment for flow through a plastic mold of a dog's artery. In both cases, pronounced secondary flow occurs immediately distal to the bifurcation, and significant mixing occurs in this area. In the glass branch (Figure 1), this mixing proceeds downstream approximately five diameters before any dye enters the side arm; and in the mold (Figure 2), no dye at all enters the upper downstream branch, indicating a definite channeling effect similar to that observed by Barnett and Cochrane.5 These figures also show that most of the mixing activity occurs near the inner walls of the branches (the medialcrotch region) and that little, if any, mixing occurs near the outer walls in the immediate region of the branches. Other studies by Attinger3 and by Fox and Hugh14 have confirmed that the regions along the outer walls in branches represent areas of boundarylayer separation and local stasis. Attinger used suspended dust particles to trace flow disturbances in branch models for Reynolds REYNOLDS NUMBER 1440 ... .... . 2 t 7 AISM Figure 1. Dyestreamline experiment (Evans Blue in water) by Krovetz for steady flow through a glass branch. Pronounced secon dary flow and mixing occur at the branch even though upstream flow is laminar. 0 4) 1I p 0rX 4J 4 N 0 O c >1 0 >*cri r4 co 0 c . 0) CO 4U *r4 0 0 C64 0 Zr 0 M 11 ri i ,C I *U d) 0 11 3 0) numbers up to 10,000. In addition to observing helical flow, he also observed cavitation (which he attributed to air dissolved in the stream) at the highest Reynolds numbers in the regions of boundarylayer separation along the outer walls. Fox and Hugh considered openchannel flows and used aluminum powder sprinkled on the flowing surface to trace disturbances in branched configurations. They postulated that the boundarylayer separation and stasis along the outer branch walls are contributing factors for the deposition of atheromatous placque there. SECTION II THEORETICAL CONSIDERATIONS In this section, we present the basic fluiddynamic and optical theory underlying the birefringence technique used and derive the macroscopic balances of mass and momentum for a "generalized" branching flow system. Although the intent here is primarily to develop those mathematical relations needed later for analyzing the flow photographs, a brief discussion of flow birefringence, together with a review of certain previous studies employing this technique, has been included. A. Fundamental Equations We start by considering the equations of continuity and motion in their Lagrangian form8: 2 = p V v continuity (1) Dt D = t VP V + pg motion (2) Here, p is the fluid density, v the fluid velocity, P the pressure, T the shearstress tensor and the vector sum of body forces per unit mass acting on the fluid. D/Dt is the substantial 14 derivative defined as + v V. It is well known that these equations apply in general for any continuous medium and in Lagrangian form apply specifically to elements of fluid moving with the flow. We wish to consider the set of equations resulting from (1) and (2) for flow through a uniform cylindrical tube of inside radius R. Anticipating the experimental flow conditions to be encountered later, we use the cylindrical coordinates r, 0 and z and impose the following assumptions upon Equations (1) and (2). 1. = 0 at steady flow 2. p = constant Trr 3. v = 0, T = 0 rz 4. z =, = 0 3z 7z  Trz 0 , 0 a2  0 0 zz 5. g = 0 incompressible flow angularly symmetric flow fullydeveloped flow no body forces The symmetry of the stress tensor is recognized in assumption 3, as is the possible existence of normal stresses (such as would occur in a Stokesian fluid). From the continuity equation (1), we have by assumption 2 that V* v = 0. That is, in cylindrical coordinates 1 3(rvr) 1 DvQ r ar r a Since the second and the third term zero becomes simply 3z az term of Equation (3) is zero by assumption 3 by assumption 4, the continuity equation (rvr) = 0 ar Integrating Equation (4) directly, we obtain rvr = G where the function of integration G is independent of r. If Equation (5) is solved for vr, then it becomes apparent that G must be zero on the physical basis that the velocity component vr cannot be infinite along the axis r = 0. We therefore conclude that vr itself is zero and that the above assumptions describe a parallel flow with vz the only remaining velocity component. The components of the equation of motion (2) in accordance with assumptions 1 through 5 are P 1 3(rTrr) Br r ar rcomponent Ocomponent aP = _1 3(rrz) az r 3z zcomponent Equation (7) shows that the pressure P is independent of 0 and therefore dependent only on r and z. If Equation (6) is differentiated partially with respect to z and the order of differentiation interchanged on the righthand side, one has a aP 1 a a(rTrr) az ar r ar az Since the right side of (9) is zero by assumption 4, it follows that a ap 0 a T 0 Sz 3r (10) That is, the radial pressure gradient is independent of axial position. Because the cross partial derivatives of P are equal, it follows that a a 0 ar az (11) That is, the axial pressure gradient is independent of radial position. (6)  = 0 According to Equation (8), the axial pressure gradient must also be independent of both 0 and z, since differentiation of that equation with respect to either 0 or z and interchanging the order of differen tiation on the right side renders the right side zero by virtue of P either assumption 3 or assumption 4 above. Thus, is independent of 3z all three coordinates, and we have the important result that the axial pressure gradient is constant. Equation (8) may now be integrated directly to give rTrz = () + F (12) where the function F is independent of r. Solving Equation (12) for I and noting that T cannot be rz rz infinite at r = 0 leads to the conclusion that F must be zero, and therefore r z= ( ) (13) rz az 2 That is, the shear stress Trz is a linear function of radial position. Equation (13) represents the starting point for the isochromatic shearstress calibration to be discussed in Section V. There we shall find it convenient to drop the subscripts on irz and to consider Equation (13) in terms of the wall stress Twas follows: wR W R where w = ( ) The preceding derivation involved integrating the equations of continuity and motion over an arbitrary cross section of a circular tube. We now consider the more general problem of integrating these equations over an arbitrary volume in space to obtain the socalled macroscopic balances of mass and momentum. This problem has been treated by Bird7 for a system consisting of a "generalized" chemical plant having a single inlet flow and a single outlet flow. We consider the equivalent problem for a "generalized" branchingflow system in which no chemical reactions are occurring but which may have any number of inlet and outlet streams. The arbitrary region of interest consists of that volume V bounded laterally by the walls of the branch and bounded at the inlets and outlets by imaginary control surfaces which are assumed fixed in space and time and oriented perpendicular to the direction of mean flow through the inlet or outlet. It is convenient to start from the equations of continuity and motion in Eulerian form: cE = V. p v continuity (15) at S= V Pvv Vp7V* + Pg motion (16) at These equations are easily derived from the corresponding Lagrangian relations (1) and (2), but when written in the above form, they apply to stationary elements of volume. Integrating the continuity equation over V we have formally fff dV = fff V P v dV (17) V V By an extension of the Leibnitz rule for differentiating an integral,8 the left side of (17) may bo written as ffJ P dV = fff p dVff pv ndS (18) V V S where S refers to the surface bounding the branch, ys is the velocity of this surface at any point and n is the outwardly directed unit vector normal to the surface. Note that vs is zero for a rigid branch, but would not be so in the cardiovascular system. By the Gauss divergence theorem, the righthand integral of (17) becomes ff V. pv dV= ff p v* n dS (19) V S where the surface integral is again taken over the bounding surface S, and v is the velocity of the fluid. Substituting Equations (18) and (19) into (17) and rearranging terms gives d f PdV = .ff (v .) n dS (20) S The left side of Equation (20) is the time rate of change of the total mass M within the branch. The integrand on the right side vanishes on all solid surfaces, and v vanishes on the control surfaces at all inlets and outlets. If we now assume that the density p does not vary across the inlet and outlet surfaces, and note that the normal n is opposite in sense to v for inlet streams but in the same sense as v for the outlet streams we may carry out the above integration to obtain dt PiiFi j j j (21) IN OUT where the first summation is taken over all inlet streams and the second over all outlet streams. The subscripts i and j refer to the ith inlet stream and jth outlet stream respectively. Si and Sj are the cross sectional areas of these streams, and the average velocity V is by definition ff S vdS I dS S where v is the magnitude of v at the cross section of interest. If we now define the difference operator A as meaning the sum over all inlet streams minus the sum over all outlet streams, then Equation (21) may be conveniently rewritten as d = A(p V S) (22) dt Equation (22) is the macroscopic or overall unsteadystate mass balance for the branchingflow system. Under steadyflow conditions, dM = 0. IF Integrating the equation of motion we have fff 2() dV = fff(V p v v)dV fff (VP)dV fff (V7. )dV (23) V V V V + fff (pg)dV V The left side of this expression may be transformed as before to give fff a(P) dV = d fff (Pv) dV ff(Pv)v n dS (24) V t dt V S The first three integrals on the right side of (23) may be transformed to surface integrals by modifications of the divergence theorem7 as follows: fff (V. p v v) dV = ff (p v)v n dS (25) V S Jff (V P)dV = (f P n dS (26) S (ff (V X ) dV = f5 n dS (27) V S Substituting Equations (24) through (27) into (23) and rearranging terms we have d fff pv dV = ff pv (v V) n dS ff P n dS (28) dt  V S S ff (c n)dS + fff p dV S V The left side of Equation (28) is the time rate of change of the total momentum P within the branch. Again, the first integrand on the right side vanishes on all solid surfaces, and ys vanishes at all inlets and outlets. The remaining contribution from this term points in the direction of the flow both at the inlets and outlets. The second term on the right contributes to the influx and efflux of momentum at the inlets and outlets respectively and to the drag force on all solid surfaces. The third term represents the viscous contribution to drag on solid surfaces and gives a usually negligible contribution to momentum transport at flow inlets and outlets. Finally, the last term on the right represents the vector sum of all the body forces body acting on the fluid. Using the previously defined A notation, we may write the macroscopic momentum balance for the branchingflow system as d = A (pvS + PS)  F + y (29) dt drag body where the vector S has a magnitude equal to the crosssectional area of the inlet or outlet of interest, but points in the direction of the flow. Fdrag and v2 are defined as follows: ;rag ^drag = Pn dS+ffl *.ds Ssolid Ssolid ffS 2dS vfZ fdS S dP Again, for steadyflow conditions = 0. dt B. Flow Birefringence According to Jerrard,23 "the first published observations on 30 flow birefringence were made in 1873, when Mach30 observed double refraction in extremely viscous substances such as strong metaphos phoric acid and Canada balsam, when poured into a beaker." Soon there after, Maxwell31 reported that he had experimented with birefringence as early as 1866 and had devised a concentriccylinder apparatus for studying the phenomenon in Canada balsam. Since that time, flow birefringence (also called streaming birefringence and flow double refraction) has been observed in many liquids, and numerous theories of flow birefringence have been proposed. From the extensive reviews of these theories presented by Jerrard23 and by Cerf and Scheraga,9 it is evident that no single theory has been successful in predicting all birefringence phenomena for liquids. Though most of the early studies of flow birefringence were directed toward using this as a technique for investigating molecular size and shape, within the last forty years, many investigators have reported using birefinngence techniques to study the properties of flows themselves. Attinger's recent studies3'4 of eddy formation and flow disturbances both in distensible tubes and in rigid tubes of circu lar and elliptical cross section, are typical examples of the qualitative use of flow birefringence. The most significant quantitative work has been that by Prados41 and Prados and Peebles,40 who showed that the bire fringence technique could be used to determine shearstress distributions and velocity profiles for flows through various types of flat channels. They used aqueous suspensions of MillingYellow dye and obtained good agreement between the flow rates calculated from birefringence measurements and the experimentally determined values. Essentially no quantitative work has been attempted for threedimensional flow systems, and,apparently, the first attempt to use flow birefringence for studying unsteady or pul satile flow was the 1957 work of Thurston and Hargrove cited by Prados and Peebles.40 It is probable that the present study represents the first attempt to use flow birefringence for studying pulsatile flows in branches. As is the case for photoelastic solids, flow birefringence in liquids results from the optical anisotropy induced by a shearing field. The phenomenon is most pronounced in solutions or suspensions containing asymmetric particles, although it has been observed at high shear rates in pure liquids (and Newtonian fluids), where the anisotropy presumably arises because of molecular deformation rather than molecu lar reorientation. A typical flowbirefringence experiment requires that an incident beam of planepolarized light (from the polarizer) be passed through the flowing liquid in a direction perpendicular to the direction of flow. The observer then "analyzes" the emerging light by viewing this light through a second polarizer (the analyzer) which is oriented so that its plane of polarization is crossed 90 with respect to that of the incident light. It is found that when the liquid is at rest, the incident beam of polarized light emerges unchanged from the liquid and is extinguished by the analyzer. Thus, the entire field appears dark to the observer under zeroflow conditions. However, when the liquid is set in motion, the observer sees a series of interference bands dis tributed through the flow field. These bands appear dark when monochro matic light is used and colored when white light is used, and their presence indicates that only under certain conditions does the incident beam emerge unchanged from the flowing liquid. Although the above description pertains to the use of plane polarized light, a similar set of results is obtained when circularly polarized light is used. An incident beam of circularly polarized light can be produced by inserting a quarterwave plate between the polarizer and the liquid such that the optic axis of the quarterwave plate makes an angle of 450 with the plane of polarization of the polarizer. The effect of this quarterwave plate is to decompose the incident planepolarized vibration into two components vibrating at right angles to one another and 90 out of phase. These are pre cisely the properties of circularly polarized light. When circularly polarized light is used, it is also necessary to insert a quarterwave plate between the fluid and the analyzer. This plate is oriented so that its optic axis is crossed 90* with respect to that of the first quarterwave plate. If a light beam emerges unchanged from the fluid (that is,circularly polarized), the second quarterwave plate changes the vibration back to a single planepolarized beam vibrating perpen dicular to the plane of polarization of the analyzer. The beam is then extinguished. Regardless of the type of polarized light used, it is true that the observer will see an interference band in the flow field only at positions where the incident light emerges unchanged from the liquid. The effects of passing beams of plane and circularly polarized light through doubly refracting materials and the conditions under which such beams will emerge unchanged have been treated analytically by Rosenberg.43 Basically, if a beam of planepolarized light enters a birefringent liquid, this beam will be decomposed into two component vibrations, one vibrating parallel to the optic axis of the fluid and the other perpendicular to this axis. Initially, these two vibrations are in phase and may be represented by x = a cos (21t) (30) y = b cos(2nA) (31) T where x is the displacement of the component vibration parallel to the optic axis of the fluid, y the displacement of the component vibration perpendicular to this axis, a the amplitude of the xvibration, b the amplitude of the yvibration, and T is the period of vibration. Owing to the optical anisotropy of the flowing liquid, these two vibrations will in general propagate through the liquid at different speeds and will emerge out of phase by an amount dependent upon the path difference (or relative retardation) 6 between them. The emerging vibrations may therefore be expressed as x = a cos(2wt) (32) T y = b cos 2ir( + ) (33) where X is the wavelength of the incident beam. If the emerging components are now recombined by eliminating t, the resulting equation is + 2 (22) x2 + 2xy cos (2i) sin2 4) ay yZ ab sin ( (34) Since Equation (34) describes an ellipse, the emerging beam will in general be elliptically polarized. However, if one compares Equa tion (34) with Equations (30) and (31), it is apparent that under the following conditions the beam will emerge unchanged 6 = NX, N = 0, 1, 2.... (35) a = 0 or b = 0 (36) The first of these conditions governs the appearance of the socalled isochromatic bands (or isochromatics), while the second con dition specifies the appearance of the socalled isoclinics. Thus, when planepolarized light is used, dark fringes can appear at the analy zer from both isochromatics and isoclinics. Equations (35) and (36) show that the isochromatic bands are related to the magnitude of the path difference 6, while the isoclinics appear only when the incident planepolarized vibration is parallel (b = 0) or perpendicular (a = 0) to the optic axis of the fluid. It has been found experimentally that the path difference 6 is dependent upon the length of the light path and upon the average shearing stress along this path while the orientation of the optic axis is dependent upon the direction of shearing stress along the light path. Thus, it is clear that the isochromatics are related to both the magnitude of the shearing stress and the length of the light path, while the isoclinics are related to the direction of shearing stress along the light path. The analysis is similar for circularly polarized light. However, since circularly polarized light consists of two mutually perpendicular vibrations which are of equal amplitude but 90 out of phase, each of these vibrations will be decomposed into components vibrating parallel and perpendicular to the optic axis of the fluid. When the appropriate contributions from the incident beam are added, the components propagating through the fluid may be represented by x = a cos 2t' (37) T y = a sin 2t' (38) T where a is the amplitude of the incident beam, T the period of vibra tion and t' = t + y with e being the angle of rotation between the 27r abscissa describing the incident beam and the optic axis of the fluid. Again, these components will undergo a relative retardation resulting in a corresponding phase change. Upon emerging from the fluid the components can be represented by x = a cos 2'T (39) y T a sin 2 y = a sin 2w( + 6) (40) T Equations (39) and (40) can be combined by eliminating t' to give x2 + y2 2xy sin(2i ) = a2 cos2(2ra) (41) A A Equation (41) also describes an ellipse, and,hence, when circularly polarized light is used, the emerging beam will in general be elliptically polarized. However, comparison of Equation (41) with Equations (37) and (38) shows that the circularly polarized beam will emerge unchanged only if 6 = NX. Therefore, when circularly polarized light is used, only the isochromatic bands can appear in the flow field. It has been shown that the condition 6 = NA corresponds to con structive interference of the emerging vibrations for both plane and circularly polarized light. Since the two vibrations are known to travel at different speeds through the fluid, constructive interference also implies that the difference between the number of waves of each vibration in the fluid is an integer N. Because the difference in wave speeds of the two vibrations occurs from changes in wavelength and not changes in frequency,52 constructive interference requires that (A/Xx y) = N (42) where k is the length of the light path, Ax the wavelength associated with the vibration parallel to the optic axis and y the wavelength associated y with the vibration perpendicular to the optic axis. Since the frequency of vibration f is unchanged in the fluid, the wave speeds must obey the following equations Cx fx (43) Cy = fXy (44) The wave speed of any light ray in vacuo is given by c = fX (45) where c is the speed of light, f the frequency of vibration and X the wavelength of the vibration in vacuo. If Equation (45) is now divided by Equations (43) and (44) the following relations are obtained: X/Ax = c/cg = nx (46) X/Xy = c/y = ny (47) where nx and ny are by definition the indices of refraction for the two vibrations. Equation (42) may now be rewritten in terms of the indices of refraction nx and ny as follows: (nx ny)S = N. (48) This expression gives the important relation between the socalled amount of birefringence (nx n ) and the isochromaticband order N. The quantities 2nx and by are know as the optical path lengths for vibra tions x and y respectively.52 According to Equation (48) the phase difference in radians between the two vibrations is given by A = 27r(nrny) (49) This is the result given without derivation by Jerrard.23 The quantitative application of flow birefringence begins with Equation (48). The usual assumption employed states that the amount of birefringence (nx ny) is a linear function of the average shear stress T(avg) along the light path. The results obtained by Prados40 indicate that this is a good assumption for suspensions of MillingYellow dye. This assumption may be stated mathematically as (nx ny) = MT(avg) (50) where the coefficient M is known as the Maxwell constant. If the assumption given by Equation (50) is substituted into Equation (48) one has NX = MAR(avg) (51) Defining M* as M/X, and rewriting Equation (51) in terms of the 33 isochromaticband order N we have finally N = M* T(avg) (52) This expression represents the starting point for the analysis of the flow photographs to be presented in Section V. SECTION III EXPERIMENTAL EQUIPMENT The production of flowbirefringence patterns requires a relatively simple experimental setup. Basically, the birefringent liquid must be caused to flow through a transparent test section located between two crossed polarizing filters (it is common practice to refer to the incidentlight filter as the polarizer and to the second filter as the analyzer). A light source, which may or may not be mono chromatic, must be oriented such that the incident unpolarized light passes through the polarizer, test section and analyzer respectively. The observer, looking through the analyzer in the directionof the light .source, sees optical interference patterns wherever shear stresses are present and a dark field elsewhere. The experimental setup used in the present study was designed with simplicity and ease of operation in mind and is pictured schematically in Figure 3. For purpose of discussion, this setup may be conveniently subdivided into three major components: A. The fluidperfusion system, B. The flow test sections and C. The optical system. Where pertinent, the important pieces of peripheral equipment are also discussed. A. The FluidPerfusion System Since flow birefringence occurs under dynamic conditions, C: 24 4< u 4 z w w U) u v, I3 U. L I 2: j J ma C a4 4J 0 4 0 frt o4 0). o 0 a) 44 cU I 04 0 0 OH co 0 * X e 0 S4 *rd 0 w w) ct 1 *c4 00 4*N CY) C4 143 00n M 0 Ocd H 4) 1 CA D: 60 i < ll a) P^ mf provision must be made for either recirculating the birefrigent fluid through a test section or collecting it after a single pass. It was decided that for branchingflow studies, and in particular for deter mining branch flow ratios, the best approach was to provide for a single pass through the test sections with fluid collection and metering at the outlets. Direct unobstructed flow metering using graduated cylin ders was thus possible, assuring the experimenter that observed flow ratios were due to branch configuration rather than side effects of flow metering. The singlepass approach provided the additional advan tage of eliminating the need for a constanttemperature bath to remove viscous heat caused by continuous recirculation. Change in fluid vis cosity due to viscous heating was a difficulty reported by Peebles, Prados and Honeycutt in a previous study.37 General requirements were that the fluidperfusion system be capable of producing both steady and pulsatile flows over the same range of flow:.ates, that the fluid be maintained at constant temperature and isolated from outside vibration during experimental runs and that the flow system be leveled so as to eliminate gravitational effects in the flow test sections. Steady flows were generated using the flow system illustrated in Figure 3. This system consisted of a glass tank approximately 30 cm high which could be sealed with a rubber stopper and pressurized using an external compressedair supply. Though a larger constanthead tank was available in the laboratory, accurate flow control over a much wider range of flow rates was possible using the pressurized system. A pressure gauge permitted tank air pressure to be monitored to better than 1 mm of mercury over the range 0 to 200 mm of mercury. The fluid tank was equipped with a tapering outlet nozzle converging to an outside diameter of 9 mm, just sufficient to match the outside diameter of the glass connecting tube joining the fluid tank to the flow test section. This provided for a smooth transition flow from tank to connecting tube. The connecting tube used for all experiments was a 3.00 mm diameter section of precisionbore glass tubing, 96 cm in length. This length was sufficient to insure that fully developed flow entered the test sections. A tightfitting rubber sleeve over the nozzle and connecting tube served both as a seal against leaks and a structural support, thus assuring a secure flush connection between nozzle and connecting tube. In addition, this sleeve provided a simple means for adapting the flow system from steadyto pulsatileflow operation. To do this, it was simply necessary to slide the rubber sleeve off the connecting tube while pinching it to arrest any flow due to fluid in the tank. The glass tube could then be connected directly to the pulsatileflow source using a section of rubber tubing. Pulsatile flows simulating heart action were generated using a positivedisplacement rotary bellows pump (Model 1000, manufactured by Research Appliance Company, Pittsburgh, Pennsylvania). This pump employed a rotating eccentric cam to displace fluid through a ball valve. The resulting flow wave was measured independently using a squarewave electromagnetic flowmeter (Model 301, manufactured by Carolina Medical Electronics Company, Winston Salem, North Carolina) and is shown in Figure 4. Whether operating under steadyor pulsatileflow conditions, the entire fluidperfusion system was mounted on sturdy laboratory tables and maintained in an airconditioned laboratory at a constant ambient temperature of 250C. External vibration was small but could be mini mized by using felt pads for supports and by turning off the aircondi tioning system during experimental runs. B. The Flow Test Sections There are several desirable properties for flow test sections to be used in birefrigence work. Among these are that they should be as nearly transparent as possible and of reasonable cost. They should also be easily fabricated from a material in which residual stresses can be removed. If the last requirement is not met, the resulting defor mation of crystal planes inthe material may render it optically active with respect to polarized light, hence interfering with experimental results. While there may be no ideal material in every respect, previous 40 41 studies441 have reported success using Plexiglas test sections. This material can be fabricated with relative ease and is reasonably priced, 0 *1 4J1 0l VI 0  a LEl '0  o ix 0 pr 10 *O 0 o *0 M 2 z V IL 0 u V) a A 0 I' oo fa ~ 0 WCu ( > although the cost of labor for fabrication may be considerable. Plexiglas is optically active in polarized light when residual stress is present, but this can be removed by proper temperature curing. Such test sections can be polished both inside and out to a high degree of transparency, rendering them ideal from the optical point of view.. The flow test sections used in the present study have been listed in Table 1. With the exception of sections 1, 2 and 7, all test sections were constructed of drilled Plexiglas block and were fabricated in various branched configurations in an attempt to simulate typical branches found in the cardiovascular system. Fabrication cost imposed a hard limit on the number and variety of branches which could be investigated. Figure 5 is a photograph of the branched test sections listed in Table 1. Overall size of these branches was limited by the lengths of the drills available. Test section 7 was milled to a 3 mm wide by 2 mm deep rectangular cross section so that an experimental comparison could be made regarding the effects of internal geometry upon optical and flow ratio behavior. Great care was required in drilling the Plexiglas block, as eccentric motion of the drill caused internal burring which could not be removed by later polishing. Care was also necessary to insure that branches met at the proper angles and lay in a plane paralleling that of the broad face of the Plexiglas block. The upstream faces of all blocks were drilled to a depth of 0 41 0 i H w w r4 U) u 0 0 0 00 a 0 o o 04 a 4 0 0 b rl U U) H H5 m & < AM e (l 3 L3 $4 $4 HP 0 O H 0 0 P4 r15 g 4 r o S 0 0 Z Z 0 .H 0 0 0 0 cv cv z 0 U0 0 0 Sco u0 0 V c o F T o o q 0 0 0 0 * . 0 0 0 0 o 0 o o) * *; o 0 0 0 0 0 n ,. ,, o H C4 cn o r o H 0 r,. o rf cY C4 0 0 0 0 C>0 0 0 n ND 0o1o,+' 00+1 ,0 1 I I I I I I 4 U) a) o ( a) il ) So 0 0 4 H 0 q H H H q H H q H H cc)a 0 CO Ca a O (0 * I m in o r o. * H 4 r4 4 H H  z 0 H S H 0 w E4 01 42 Co 0 41 m w 40 400 M. 041 0 p 4.1 1 4 4 00a 4I Cfl *r U4 44 0 41 OJO 4 "OC 00 r O)O t~o)0 i'P) tO approximately 1 cm on center with the upstream branch stem to accommodate the glass connecting tube with a snug and flush internal fit. Duco cement was used as a leak seal around the connecting tube on the upstream block faces. Figure 5 shows the connecting tube joined to test section 6. Branch outlets were mounted in a similar fashion and very satisfactory results were obtained using Plexiglas tubing for these. C. The Optical System There are three basic requirements to be considered in designing the optical system. First, this system must be oriented such that the light path is perpendicular to the direction of flow being studied. Secondly, it should be possible to produce incident circularly polarized light as well as planepolarized light of arbitrary angular orientation. Finally, it should be possible to photograph the resulting birefrigence patterns. Figure 3 illustrates the optical scheme adopted. The lightdiffusing screen, consisting of a 60 cm square plate of smoked glass, was mounted horizontally on runners between the two laboratory tables. Mounted in this fashion, the screen provided both a uniformly lighted background and a rigid level support for the polarizer and test sections. Two Sylvania flood lamps (type R32, color temperature 32000K, 375 watt), constituting a white.light source, were mounted on ring stands which rested on the floor. These lamps could be easily raised to compen sate for intensity loss due to aging, and heat could be dissipated by a fan. A whitelight source was chosen, because it was felt that the resulting colored bands would be easier to distinguish in the final photographs than the light and dark bands obtained with monochromatic light. As Figure 3 shows, the polarizer was mounted directly on the lightdiffusing screen. Planepolarized light of various orientations was produced using the edge of the screen as an azimuthal angular reference. A second polarizing filter mounted directly on the camera lens served as the analyzer. Circularly polarized light could be produced by inserting quarterwave plates (not shown) between the polarizer and test section and between the test section and analyzer respectively. Again, the edge of the diffusing screen was used as an angular reference for properly inserting these. The camera used was a Bolex Paillard H16 (16 mm) Reflex motion picture camera equipped with a 25 mm lens (1 : 1.4 Switar Rx). This lens in conjunction with 5 mm and 10 mm extension tubes permitted photography from a distance of a few centimeters. The film used was Ektachrome (ER type B) and a Weston light meter (Model 745) was used to check light intensity at the camera. The camera was mounted on a tripod taped securely to the floor. This arrangement permitted easy focusing using the tripod crank. Once focused, the camera could be locked in position and the reflex viewing used to accurately check on proper analyzer orientation by light extinction under zeroflow conditions. SECTION IV EXPERIMENTAL TECHNIQUE The experimental work was composed of four parts: A. Preparation and storage of the dye suspension, B. Determination of fluid physical properties, C. Flow regulation and collection and D. Photography. A. Preparation and Storage of the Dye Suspension The doubly refracting liquid used was a 1.4% aqueous suspension of the aniline dye Milling Yellow (obtained from National Aniline Division, Allied Chemical Company). The suspension was prepared directly from the commercial product, a yellow powder, using the method suggested 36 by Peebles, Garber and Jury.3 Distilled water, about 50% in excess of that desired for the final volume, is heating to boiling. To this is added a thick paste of dye powder and distilled water, sufficient to make a 1% suspension by weight. Boiling and agitation are allowed to proceed until excess water has evapo rated. Since birefringent sensitivity is markedly affected by dye concen tration, the final suspension should contain 1.3 to 1.6% dye for best results. The doublerefraction property may be tested during evaporation by withdrawing a small sample into a test tube, cooling to room temperature and observing whether or not interference patterns are produced when the tube is agitated between illuminated crossed polarizing plates. After the dye suspension was prepared in sufficient quantity, it was stored in sealed polyethylene tanks until needed for use. It was found that suspensions stored in this fashion were very stable with respect to particle agglutination over a period of two weeks, after which some coagulation occurred. B. Determination of Fluid Physical Properties To date, the most extensive physicalproperty studies of Milling Yellow suspensions are those of Peebles, Prados and Honeycutt37 estab lishing the birefringent and theological behavior. Their studies have shown that viscosity data for MillingYellow suspensions are accurately correlated by the PowellEyring equation for pseudoplastic materials. Little work has been done to establish other physical properties for such suspensions. The physical properties required for the present study were viscosity, density and static refractive index of a 1.4% dye suspension at 250C. Viscosity data were obtained by using the 96 cm length of precisionbore 3 mm glass tubing as a capillary viscometer. The dye suspension was permitted to flow through the tube under steadyflow con ditions for several different but known pressure gradients. The corres ponding flow rates were then measured and the HagenPoiseuille equation used to calculate apparent viscosity (that is, the viscosity which a New tonian fluid would have flowing through the tube at the same flow rate under the same pressure gradient). Typical experimental results are listed in Table 3, Appendix B, and Figure 6 shows apparent viscosi ties and the corresponding Reynolds numbers plotted as functions of flow rate Q in the 3 mm diameter tube. These data are in strong agree ment with the data reported by Peebles, Prados and Honeycutt.37 Fluid density was determined by weighing 25 ml aliquots of the 3 dye suspension at 25"C. The density was found to be 1.00 gm/cm3 Staticrefractiveindex measurements were made using a Bausch and Lomb refractometer. The static refractive index at 250C was found to be 1.338 for the 1.4% suspension. Static refractive index, however, should not be confused with the indices of refraction associated with the ordinary and extraordinary rays when the fluid is in motion. It is not clear just what relationship the static index bears to these two, though Peterlin and Stuart38 have taken it to be the mean value of the dynamic indices. C. Flow Regulation and Collection The procedure for steadyflow experiments was as follows: Prior to filling the pressure tank with fluid, short rubber tubes were fitted over each flow outlet and clamped to prevent flow. During the experi ments, these tubes were unclamped and served to divert outlet flows to large collecting beakers without altering flow distributions in the test sections. ( 3SIOdllN33 ) AIIS03SIA q? co cuj o 0 IN3SVddV (0 o 0 0 0 0 13N OO CO (N 83SWnN So1oNA13 E Q) C0 L&J o C 0 o 0 4 Co 0 0 *rI 04Jl cm t u 'I M *C 0 0 $1O a) m (l Q) *r r4 a n Cu 03 l ri 4 r0 .*l 01 *cr Fr{ t*4 The pressure tank and test section were positioned for photography and the tank filled to a depth of 20 cm with the Milling Yellowdye suspension. The laboratory compressedair supply was then connected to the tank via a threeway valve opened initially to the atmosphere. The pressure gauge in series between the valve and tank permitted tank air pressure to be monitored directly. For an experimental run,the outlet flows were opened and the valve partially closed until the desired driving pressure was registered on the pressure gauge. Successive experimental runs were made by incre menting the driving pressure in steps of 10 mm of mercury. Before and after each run, the threeway valve was opened to the atmosphere, thus reducing excess tank pressure and lowering the outlet flows. This also permitted the liquid level in the tank to be returned to the 20 cm level. Depending upon the flow rate, experimental runs lasted for 30 seconds or 1 minute. However, the tank was of sufficient capacitance that the liquid level did not drop more than a few millimeters during any experimental run. Outlet flows were measured with graduated cylinders. For the pulsatileflow experiments, the pressure tank was replaced by the bellows pump described previously. Since the flow output of this pump was limited to the flow wave already shown in Figure 4, no flow regulation was possible here. D. Photography Prior to actually photographing the experimental birefringence patterns it was necessary to standardize several important photographic variables such as light intensity, framing speeds, exposures, depth of field, the film used and processing. Several preliminary runs were made to test these factors using a color standard made from strips of Kodak red, green and yellow Wratten filters. It was found that Ektachrome film (ER type B) with a correspon ding light source having a color temperature of 3200K gave very good color reproduction when proper exposures were used. All films were pro cessed by the same company so as to minimize any differences due to processing. Experimental birefringence patterns were photographed with the camera shooting down at the test section mounted in the horizontal plane. After the test section was positioned, the desired exposure and depth of field were set and the lights turned on. The camera was then focused and locked into position. The reflex viewing was used to assure that polarizer and analyzer were crossed by checking for maximum light extinc tion under zeroflow conditions. Light intensity was checked at the camera using a Weston light meter (Model 745). The flow was started, and after the desired flow conditions were achieved, the resulting birefringence pattern was photographed. Steady flows were photographed at 16 frames/sec and pulsatile flows at 64 frames/sec. The pertinent photographic data for the flow photographs appearing in the next section have been listed in Table 4, Appendix C. SECTION V THE ANALYSIS OF FLOWPATTERN PHOTOGRAPHS AND EXPERIMENTAL RESULTS In this section, selected frame enlargements of the 16 mm film are presented together with experimental data obtained from flow measurements and framebyframe analysis of the film. The section is divided into three parts. In Part A, the isochromaticband pattern obtained for a fully developed, angularly symmetric laminar flow through a tube of circular cross section is considered, and a method is proposed for calibrating this pattern against shearing stress, including the effects of curved boundaries and differences in refractive index in the analysis. In Part B, steady flow data and isochromatic photographs are presented, and birefringence theory together with the calibration of Part A is applied to estimate shearstress distributions for flows around a cardiac catheter and through a 90 branch. Experimental results are then compared with results from the medical literature concerning wall trauma and athero sclerotic disease at sites of branching in arteries. In Part C, branch entrancelength data for the pulsatileflow experiments are presented, and discussion is focused on the areas of stasis, boundarylayer sepa ration and increased shear readily observed in the films. A. Isochromatic ShearStress Calibration for Laminar Tube Flow The problem of isochromatic calibration in flowbirefringence work is that of associating an appropriate value of shear stress with any given isochromatic band in the flow field. In general, this requires that a birefringence experiment be run under conditions of known geometry, shearstress distribution and isochromaticband distri bution. A correspondence or calibration may then be drawn between the shear field and isochromatic field and this calibration used to determine the shear field in geometries where only the isochromatic field is known. It will be recalled from Section II that previous studies have demonstrated experimentally that fringe order N is directly proportional to the product of T(avg), the average shear stress along the light path, and X, the length of this path in the fluid. According to Equation (52) that relationship was given as N = M*ZT(avg) where M* is a characteristic of the particular fluid. Although M* is not constant for all fluids, the data of Prados41 show that this is the case for MillingYellow suspensions. It is evident from Equation (52) that for a threedimensional flow system in which the lightpath length (fluid depth) associated with a given band N is constant, the shear stress T(avg) associated with this band will also be constant. However, if this lightpath length varies along the band, then the corresponding shear stress will vary inversely as the path length, and there will no longer be only one appropriate shear stress associated with the given band. A reasonable approach for this situation would be to establish some reference point along the band (perhaps upstream from the position of interest), and to refer the stress at other positions on the band to the value at this point. This, of course, requires that the lightpath length be known at the downstream position. Previous experimentalists utilizing flow birefringence were able to avoid this difficulty associated with test sections of varying depth by using essentially twodimensional systems (such as highaspectratio channels). As will be seen in Part B, the isochromatic bands obtained in the present study represent a combination of the above two cases, with the undisturbed upstream state corresponding to constant lightpath length in the fluid and the distorted downstream state to variable length. For this reason,we shall establish the upstreamflow condition as a reference state for branching flows and turn now to the problem of calibrating the isochromaticband distribution obtained for laminar tube flow. We consider a steady, incompressible, fully developed laminar flow through a cylindrical tube. A typical isochromaticband distribu tion observed for such a flow is that shown in Figure 7 for test section 1. The flow here is from left to right at a Reynolds number of 50 (based on apparent viscosity and inside tube diameter), and the incident light is plane polarized perpendicular to the axis of the tube. It is 54 a, 0 0 $4.0 o 14 0w Co cd d r Cu .H H 0 02 rlo u 0 W~H *rt acu HH ar N 14 COCu $4 H 0 10 014 4.4 Co 1.4 H 4 oa, d 0 4Jt4 4i ri2 r Hr *GJ C5 u 934 woa wo A O *H t v u0 0 00 H I *'T O c 0 cd ed & ca 4 H 4i C3u 41 rl. *$ POo 0 U 4 ucu, aCu ~HH,~ gZ4 4 seen that the bands parallel the direction of flow and are not displaced from their original radial position as they pass downstream. In seeking to associate a characteristic value of shear stress with each band, the observer must remember two things: First, successive bands seen in the flow photograph are the net result of viewing the flow through a curved wall, and,hence, through varying liquid depths in the radial direction. Second, since curved boundaries are involved, it is necessary to consider the possibility of optical distortion arising in the flow photograph due to differences in refractive index between the different media involved. A crosssectional representation of the overall situation is shown in Figure 8. For convenience, the coordinate origin 0 has been chosen at the center of the tube with lines ON1 and 0N2 representing normals to the tube surface at points B and C respectively. Each of the three media involved is characterized optically by a static index of refraction (nl, n2, n3). The path of a typical light ray as it enters the fluid from below and passes to the observer is indicated by line DCBA, and it will be noted that with the exception of the ray passing through the center of the tube, all rays seen by the observer pass obliquely from one medium to the next, thereby undergoing refraction. This refraction is characterized by the angles 01, 02, 03 and 04 which are predictable from geometrical considerations and Snell's law of refraction. These relationships will be considered presently. The Y OBSERVER LIGHT DIFFUSING SCREEN Figure 8. Schematic cross section of a circular tube, illustrating the refraction of a typical light ray passing from the fluid to the observer. Light path DCBA is characterized by angles 01,02, 03 and 04. These angles are determined by the parameter K and Snell's law of refraction. nl, n2 and n3 are indices of refraction. N1 and N2 are normals. observer (camera) viewing from above sees only those light rays emerging parallel to the yaxis, and this implies that chords such as DC which are the light paths in the fluid corresponding to isochro matic bands seen the the flow photograph, will not in general be parallel to the yaxis. For the present analysis, we shall choose chord DC as being a typical chord associated with an isochromatic band seen at point A in the photograph and shall identify this chord by its perpendicular dis placement from the origin. The lateral displacement from the yaxis of the band at point A is, of course, dependent on the chord's displace ment from the origin, and this dependence will now be established by considering the abovementioned relationships. It should be mentioned that there is some question as to the justification for using Snell's law of refraction for a doubly refrac ting material, since it is known that the ordinary ray obeys this law but the extraordinary ray does not. In addition, each of these rays has its own dynamic index of refraction, so the use of the static index must also be justified. Actually, these questions pose no difficulty here, because both rays travel the same path, and, as Peterlin and Stuart have pointed out,38 the difference between the three indices is very small. Therefore, for the purpose of determining the light path, the static index and Snell's law may be used. Referring again to Figure 8, one may write the following relations between angles 01, 02, 03 and 04; n1 sin 01 = n2 sin 02 by Snell's law (53) n2 sin 03 = n3 sin 04 by Snell's law (54) R cos(0302) + BC cos 02 = KR Geometry (55) R sin(0302) = BC sin 02 Geometry (56) If 81 is considered the independent variable (this can be done because of the principle of optical reversibility52), then these equations may be solved simultaneously to give 1 n1 02 = sin (nH sin 01) (57) Sn1 83 = sin('a2sin 1) (58) 1 n, 94 = sin (K. sin 01) (59) where the parameter K is simply (outside diameter)/(inside diameter). For the special case of a hole drilled parallel to a flat face of a Plexiglas block, these equations still apply providing K is given a numerical value of unity and nI is replaced by n2. This is equivalent to immersing the observer in the Plexiglas, since perpendicular viewing would create no refraction at the airblock interface. One notes from Equation (59) that the angle 04 is completely independent of medium 2 (that is n2). This is a wellknown result in optics. Another interesting and important observation is that the diameter ratio K plays a central role in determining how much of the flow field is actually visible to the observer. This fact may be appreciated by considering the following relationships, which may be deduced from equations (57) through study n2>n3>nl. (59) and the fact that in this THEN K = 1.0 and n1 = n2 (Plexiglasblock case) n3 1 n3 K n n1 "3 n2 n3 < K <  n, n1 n1 n2 K >  n "1 04 > 01 3 = 02 01 > 04 > 03 > 02 e4 = 81 > 83 > e2 04 = 01 > 3 > 02 04 > 01 > 03 > 02 04 > 01 = 03 > 02 04 > 03 > E1 > 82 These relationships show that with the exception of the second case (61), 04 is always greater than or equal to 01. This is important, because as shown in Figure 8, the magnitude of 04 must range from 0 to 90 if the observer is to see the entire flow field. At the same time, it will be noted that 01 physically can be no greater than 90. If case (61) holds, that is, if K falls between 1.0 and n3/n1, then 04 will be less than 01, and the observer will not be able to see the entire flow field but only 1 i 1 n1 the part of it for which 4 ranges from sin (Kn) to + sin (K as dictated by Equation (59). Thus, an experimenter conducting (60) (61) (62) (63) (64) (65) flowbirefringence experiments employing circular tubes and wishing to visualize the entire flow field must consider the geometrical range 1 < K < n3/n1 a forbidden zone when choosing experimental test sections. For the flow pictured in Figure 7, the appropriate numerical values are K = 3.0, nI = 1.00, n2 = 1.5 and n3 = 1.338. Since this value of K is greater than n3/nl, one is assured that the entire flow field is visible in that photograph. In fact, the entire flow field is visible in all photographs appearing in this study. For the branchingflow figures the appropriate numerical values are K = 1.0, n1 = n2 = 1.55 and n3 = 1.338. It can be seen from Figure 8 that the perpendicular displacement of the typical chord DC from the origin is given by R sin 04, and that the lateral displacement of the corresponding band at point A is KR sin 01. Defining a distortion ratio as (band displacement)/(chord dis sin 01 placement), one obtains a distortion ratio of K From sin 04 Equation (59) the value of sin 01/ sinO4 is n3/Kn1. Substituting this above gives a distortion ratio of n3/n1. Thus, our analysis shows that for circular geometry the distortion ratio is constant and independent of both the dimensions and the material of medium 2 (the tube). Furthermore, this distortion represents a magnification effect with the distortion ratio n3/n1 as the magnification factor. In the present study, this factor has the value 1.338 for the glass tubes; but since nI must be replaced by n2 for the Plexiglasblock studies, the factor becomes 0.865 for them. In either case, the only optical distortion arising from curvature is magnification or shrinking of the original image by a con stant known factor. We can use this information to obtain one more important result. Let the band displacement at point A be denoted by "b" and the corresponding chord displacement by "a". Then, according to the above discussion, b = an3/n1. The displacement b will have its maximum value b(max) when a has its maximum value a(max) = R. Thus, b(max) = Rn3/nl. If both displacements are now made dimensionless by normalizing them to their maximum values and the fractional displacements "8" and "a" defined as 8 = b/b(max and a = a/a(max), we obtain the important result that B = (an3/nl)/(Rn3/nl) = a/R = a. This means that the fractional displacement dimensionlesss radius) of an isochromatic band measured in the flow photograph is precisely equal to that of the corresponding chord (light path) measured inside the tube. Now, having related the geometrical properties of the band in the flow photograph to those of the corresponding light path in the fluid, we are in a position to derive the expression relating the fractional displacement a (or 8) to the average shear stress along the chord DC. Up to this point, no mention has been made of the nature of the shear~stress distribution in the fluid. From here on, however, it will be assumed that the flow field in the tube is angularly symmetric, which implies that the stress field is also angularly symmetric. The assumption of angular symmetry was verified experimentally for the band distribution shown in Figure 7. In Section II it was shown that for the steady, incompressible, angularly symmetric and fully developed flow being considered here the equations of motion could be integrated to give the shearstress field as a linear function of radial position r. That function was given by T r (66) aP R where the wall stress Tw is ()  The problem of determining the average value of T over the light path DC can be considerably simplified by taking advantage of the angular symmetry. Consider the enlarged view of the flow cross section shown schematically in Figure 9. Though the light path DC might in general occupy a position such as that indicated by the position vector r, the assumption of angular symmetry permits this chord to be rotated about the origin to an arbitrary new position without in any way altering the shearstress distribution along the chord. In particular, if one chooses to rotate chord DC to the position D'C' parallel to the yaxis, then the average stress from D' to C' is the same as the average stress from y = 0 to y = y*, owing to symmetry with respect to reflection in the xaxis. Thus, the average shear stress along the chord can be obtained by integrating from y = 0 toy = y* = (R2 2 obtained by integrating T from y = 0 to y = y* = (R a ) where Y* O a r Figure 9. Schematic fiow cross section, illustrating a typical light path along which the average shear stress is required. For angularly symmetric flow, the average stress along light path DC is equal to that along path D'C'. the parameter a is the perpendicular displacement of the chord from the origin. Along the path D'C',Equation (66) takes the form (a2+ y2 ) T = Tw (67) T= R and the average shear stress T(avg) is given by fTdy (avg) =, y* = (R a (68) I dy Substituting d from Equation (67) into Equation (68) and performing the indicated integration gives 2 2 2 +4 w a (R2a ) +R r(avg) =  1 + ] In I[ a R(R2 a) If we now define the dimensionless variable ( as T(avg) /Tw and rewrite Equation (69) in terms of the fractional displacement a, we obtain finally 1 2 (1i_2) + 1i T(avg) = 1+ a In (1 (70) W 2 (1a2)5 a This equation gives the desired average shear stress along the light path as a fraction of the wall stress in the tube. On the basis of physical intuitionone would expect D to converge to the values of 0.5 and 1.0 for a = 0 and a = 1 respectively. Although inspection of Equation (70) shows that P is indeterminate at both endpoints, a single application of L'Hospital's rule is sufficient to demonstrate that 4 does indeed converge to the expected values. The length Z of the chord D'C' (or DC) may be written in terms of a by considering the geometry of Figure 9. The relationship between A and a is I /2R = (1 a ) (71) The right side of Equation (71) thus gives the lightpath length in the fluid as a fraction of the inside tube diameter, and, as expected, this length has a value of one diameter for a = 0 and zero for a = 1. Numerical results from Equations (70) and (71) have been summarized in Table 2. The first column in the table represents either the fractional displacement of the light path from the center of the tube or the fractional displacement of an isochromatic band as measured in the flow photograph, The second column gives the length of the corres ponding light path in the fluid in diameters, and the third column gives the average shear stress along the light path as a fraction of the wall stress in the tube. As an example of the use of this table, consider the green band which was seen in Figure 7. In the photograph, the center of this band is displaced 10.4 mm from the center of the tube. The edge of the birefringence pattern is displaced 13.0 mm from the center. The fractional Table 2 LIGHTPATH LENGTH AND AVERAGE SHEAR STRESS ALONG THIS PATH AS DIMENSIONLESS FUNCTIONS OF FRACTIONAL DISPLACEMENT IN THE FLUID MEDIUM* Fractional Displacement, (a or B) 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 Dimensionless Path Length, (/2R) 1.000 0.999 0.995 0.989 0.980 0.968 0.954 0.937 0.917 0.893 0.866 0.835 0.800 0.760 0.714 0.661 0.600 0.527 0.436 0.312 0.000 Dimensionless Average Shear Stress, D 0.500 0.505 0.515 0.529 0.547 0.567 0.588 0.612 0.637 0.663 0.690 0.718 0.747 0.777 0.807 0.838 0.870 0.902 0.934 0.967 1.000 *NOTES: a. This calibration applies only for fully developed angularly symmetric laminar flow in a circular tube. b. Fractional displacement a measures the perpendicular displacement between the center of the tube and the chord representing the light path. Fractional dis placement B measured in the flow photograph for the corresponding isochromatic band is exactly equal to a. c. A/2R is calculated from Equation (71). d. i is calculated from Equation (70). displacement of this band is therefore 8 = 10.4/13.0 = 0.8. Consulting Table 2, one finds that the lightpath length in the fluid corresponding to this displacement is 0.600 diameters. Since the tube is known to have an inside diameter of 3.00 mm, the light path is 1.80 mm long. Finally, from Table 2, the average shear stress associated with this band is 0.870 times the know wall stress of 90 dynes/cm2. Therefore, the shear stress associated with the green band is 78.3 dynes/cm2. It should be emphasized that the calibration given by Table 2 applies only to isochromatic bands obtained for tubes of circular cylindrical geometry under the steady laminarflow conditions described earlier. These are the upstream conditions prevailing in the steadyflow experiments of this study. The problem of determining the shear stress at downstream entranceflow positions in terms of the upstream conditions will be considered in Part B with numerical examples for flows around a cardiac catheter and through a 900 branch. B. SteadyFlow Experiments To facilitate discussion of the following figures, it would be appropriate to discuss first the conventions which have been followed regarding the orientation of the flow photographs and general terminology. The photographs appearing in this section show typical isochroma ticband patterns for flows through both straight and branched test sections. The observer should keep in mind that for each photograph the flow is steady and from left to right and for branched test sections this flow is such as to impinge on the bifurcation (that is, all branched flows are of the diverging type). The convention followed in labeling branches is that side arms have been designated "B" and downstream main stems "A", with the term "upstream" referring to the prebranch segment. All Reynolds numbers have been based on apparent viscosity and the pertinent tube diameter as a characteristic length (for the single branch of rectangular cross section, this length has been taken as four times the hydraulic radius, defined as the crosssectional area of the stream divided by the wetted perimeter). The term "flow ratio" applies only to branched test sections and is defined as the ratio of the volumetric flow rate in the side arm to that for the downstream main stem (i.e., QB/QA). Similarly, the term "area ratio" is defined as the ratio of the crosssectional area of the stream in the side arm to that for the downstream main stem (i.e., AB/AA) . Unless otherwise noted, the values of wall stress and Reynolds number given in the figures refer to the upstream (prebranch) values of these quantities. In each case, upstream wall stress has been experimentally determined as the negative of the pressure gradient mul" tiplied by half the inside radius at the upstream position. Finally, all entrancelength measurements have been expressed in dimensionless form by normalizing them to the diameter of the branch for which they apply (i.e., sidearm entrance lengths have been normalized with respect to sidearm diameter and mainstem entrance lengths with respect to mainstem diameter). Prior to the actual branching studies, it was decided to employ the birefringence technique to obtain isochromatic patterns for steady flows around various needles and cardiac catheters inserted radially and axially into a circular tube. Since the birefringence technique had not been used previously for this purpose, it was hoped that useful information would be obtained regarding the creation and propagation of flow disturbances caused by these devices. Figure 10 shows an isochromaticband distribution for steady flow around a catheter (O.D. = 1.67 mm) inserted axially into a cylin drical tube (I.D. = 3.00 mm). Here, planepolarized light has been used with the plane of polarization perpendicular to the axis of the tube. The upstream Reynolds number for the experiment was 10, and the wall stress was 75 dynes/cm2 In the upstream region, the bands are distributed much as they were for the fully developed flow in Figure 7, but as they encounter the catheter, these bands become noticeably displaced toward the tube wall. The flowinterference front caused by the catheter extends approximately one diameter upstream, and the region between the catheter and the tube wall is recognized as one of accelerated flow owing to the restriction of flow area. The band crowding seen in this region is typical of the 70 ,i 0 .0 V4 . u n0 0 0 4 w0 t cu 0 i 44 W 0 00 4i M 0 0 4* 4j a Z4 a 0 00 0 U '0 C:4 7O 404 co r c SH0 ca O O4 (D 'I 0 ,4 *1 c Or ., .74 5' r4 vI 0 *e0 ?4I QJ .L 0 740r 5i *5. lU P 0U patterns seen for regions of accelerated flow and will also be seen in photographs of branching flows. This crowding of bands provides a clue to areas where one would expect to find increased velocity gradients and shearing stress. It is apparent from Figure 10 that the flow in the entrance region surrounding the catheter does not correspond to the fully developed flow discussed in Part A. This means that the shearstress calibration given by Equation (70) in Part A cannot be applied directly to calculate the stress distribution for the entrance region. In fact, since the lightpath length varies along the bands in this region, there is no unique stress which can be associated with any band. However, it was intimated in Part A that the fundamental Equation N = M*jT(avg) could be used to relate the stress distribution in the entrance region to that in the upstream region of fully developed flow, where the cali bration of Part A does apply. In particular, since the coefficient M* has been found to be independent of shear rate for MillingYellow sus pensions, one can apply the above equation to an upstream and downstream point on the same band (or on two different bands of the same order N) to obtain T(avg, down) = k (up) T(avg, up) (72) k (down) From Equation (72) it is evident that the average shear stress along the light path at the downstream position can be calculated from the upstream value for the given band providing that the lightpath length at each of the two points is known. It was shown in Part A that for flow through tubes the lightpath length k is related directly to the observed dimensionless displacement 3 of the isochromatic band at the point in question by Equation (71), 1/2R = (182) where we have used the fact that B = a. Since this equation was derived solely from geometrical considerations, it may be applied accurately to both upstream and downstream positions and Equation (72) written as (i 2(up))% T(avg, down) = 2 T(avg, up) (73) (18 (down)) Here (up) and 8(down) are the dimensionless displacements of the isochromatic band measured in the flow photograph at the upstream and downstream points respectively. Rewriting Equation (73) in terms of the upstream wall stress T,(up) and the dimensionless average shear stress D(up) one has 2 u T(avg, down) = (1 (up) (p) p) (74) (1_ (down)) All the quantities on the right side of Equation (74) are known, either from the birefringence experiment or from the shearstress cali bration of Part A. The quantities D(up), (l2 (up)) and (l 2(down)) can be obtained either from Table 2 in Part A for the appropriate value of 8 or by direct calculation. As an example, consider the dark band appearing upstream in the center of the tube shown in Figure 10. Encountering the catheter, this band is displaced from its original position, 0 = 0, through intermediate values of 8 to a final position, B = 0.5, between the tip of the catheter and the tube wall. Choosing this point as the downstream point of interest and consulting Table 2, one finds the following: 8(up) = 0, (l82(up))4 = 1.000 and D(up) = 0.500 s(down) = 0.5, (l 2(down))2 = 0.866 The known wall stress for this experiment is given in Figure 10 as 75 dynes/cm2. Inserting these values in Equation (74) gives t(avg, down) = (1.000/0.866) (0.500) (75) = 43.3 dynes/cm2. For comparison, the average shear stress corresponding to this band in the upstream position is 2 t(avg, up) = 4(up) Tw(Up) = (0.500) (75) = 37.5 dynes/cm2 As expected, the region of accelerated flow is also one of higher shear. It should be emphasized that the above calculation gives only the average shear stress along the light path at a single downstream position and applies only when the reference (upstream) flow is fully developed as described earlier. By repeating this calculation for bands falling close to the wall of the tube one can obtain an estimate of the shear stress acting on the wall. Figure 11 shows another experiment in which a beveled needle (shank diameter = 0.92 mm) has been inserted through the wall of the cylindrical tube (I.D. = 3.00 mm) and oriented with the bevel facing upstream. The Reynolds number for this experiment was 40, and the upstream wall stress was 80 dynes/cm2. Again, planepolarized light has been used with the plane of polarization perpendicular to the axis of the tube. The photograph has been underexposed to emphasize the two distinct vortex patterns seen trailing behind the needle. These patterns persist for approximately two diameters downstream where the disturbance becomes damped out because of fluid viscosity. The region of disturbed flow extends less than one diameter upstream. Though the vortex patterns are similar in size and shape in this photograph, motion pictures reveal that when the needle is rota ted 900, the lower vortex disappears almost entirely while the upper one remains essentially unchanged. This is evidence that the edge of the bevel creates far less disturbance than does the face. When the needle is rotated further until the bevel faces downstream, the lower vortex reappears but is not as prominent as the upper one. The same experiment using a cylindrical needle having no bevel produces only a single vortex pattern which appears in the center of the tube and is much more streamlined than those in Figure 11. This disturbance also .C I 4 ft mC e o c ,C 0 a0 04*i Si 0 41 C = 41CO 0 r qC 1 Co 02 CU E4 0 0 10 1) ) 01 Q 41 0 0) M .4 OH C 0) 11 0 *r o ta ori 3 0 0: A C! 0 0) I % 41 .0 SH C W *l 0) ocuw 44 oH t Ca S r l S) 0 r CU CU H 4 m C (W *ri a p, a CH 0W 0 .N CO C S01 0(C 0< *r1 0 0C r 0 CO < 5 persists downstream about two diameters. The most significant finding from these experiments was that when the needles were withdrawn to a position flush with the wall of the tube no disturbances could be seen in the birefringence patterns. This strongly suggests that lateral pressure measurements could be made in smalldiameter models as well as in vivo without appreciably disturbing the flow. Figure 12 shows an isochromaticband distribution obtained using circularly polarized light for the first of the branchingflow experiments. The branch shown (test section 6) has a branching angle of 300 and an area ratio of unity (the side arm and main downstream stem both having an inside diameter of 3.17 mm). The upstream Reynolds number for this experiment was 116 with a corresponding wall stress of 110 dynes/cm2. Although the bands are parallel to the axis of the tube in the upstream segment, they tend to converge toward the inner walls and distribute themselves concavely about the bifurcation point in the entranceflow region. This region of disturbed flow extends downstream for a few diameters in both arms until the bands return to the parallel disturbution indicative of laminar flow. The fact that the bands tend to be distributed concavely about a stagnation point, such as the point of bifurcation, is helpful in locating zones of stasis at other positions in a flow test section. This will be more apparent in later photographs. The displacement of bands toward the inner walls of the .0 r ) 4 41 0 W 0 a *4 0o a o a) 00 C4 O0 *w 0 04 d) 00 4 1 r *)1 ^ 00 QHO 4. 4.1 Cd 10  C0 0 0 Ii W 4 a 4J4*  *1 0 41C 0 .0 (1 *0 m ?O con ca Cd44J O (00 S' E4 P .4 C P . 0 II U 00$ r0 M oc~ n Tl Tl &( Et, U (*i branch indicates that the regions directly adjacent to the bifurcation represent areas of accelerated flow and increased shearing stress, while the absence of bands toward the outer walls indicates boundary layer separation in those areas. Aside from the regions immediately adjacent to the bifurcation, there is no evidence for increased stress at any other site in this branch. Figure 13 gives a comparison of the flowdistribution characteristics for three different 30" branches. Here, flow ratio (QB/QA) has been plotted against upstream Reynolds number with area ratio (AB/AA) as a parameter. The upper curve applies to the branch just discussed, and the lower two apply to branches having sidearm diameters of 2 mm and 1 mm respectively. TFrom the overall decrease of flow ratio with increasing Reynolds number shown by all three branches, it is evident that the upstream flow increasingly favors a straight path into the main down stream stem A as the upstream fluid momentum per unit volume increases. The marked separation between successive pairs of curves in Figure 13 simply verifies that reducing the sidearm diameter is an effective means of controlling sidearm flow over the entire range of upstream Reynolds numbers. It will be seen later that this control is much better than that achieved by increasing the branching angle. The disproportionate increase of flow ratio which occurs below the Reynolds number of 25 is undoubtedly a manifestation of wall attachment by the flow, similar to that observed by Fox and Hugh14 in their experiments with openchannel 0 U I4J1 ooO p li 0 Cd S H1 0 m < 0 0 I S coo"I I' I w H e f U ) *)0 a j0 4j3.3 0r r<0 f0 4 4 , I1 aW W I =C (1) 000 W 44 CO < O M 0 0) m L co O 0 4 4Q ) 14 io en o cu Ib1w 1 ccO 4 Jl I I iOl0 O.18 f V0/90r Oo~dAA~ O o 0 \ flows. The data of Figure 13 indicate that for 30 branches wall attachment can cause the sidearmflow to greatly exceed the main stem flow, but only if the branch area ratio approaches unity and only below a certain upstream Reynolds number. The phenomenon of wall attachment (sometimes called the Coanda effect) is known to arise because of a lowpressure region generated along the outer sidearm wall owing to fluid moving along curved streamlines concave toward this wall. In branches, this lowpressure region tends to divert a greater fraction of the incoming flow into the side arm until the upstream momentum forces can overcome the pressure forces causing the attachment. At still higher flow rates, the boundary layer separates from the outer wall and a vortex is generated there. The data of Knox24 indicate that there is a sharp drop in pressure associated with this vortex and a corresponding rise of pressure in the main stem due to the exchange of incoming fluid kinetic energy for pres sure energy there. It has been found that very reliable fluid amplifiers and fluidic digitallogic devices can be produced by carefully designing branches with control ports at the vortex regions. Fluid can be introduced through these ports to destroy a given lowpressure zone, thus causing the main flow to switch from one arm of the branch to the other. Unfor tunately, very little has been published concerning the branchingflow characteristics of fluidics devices, since most of this information has been considered proprietary. 