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Entry flow problem of a liquid body into a suction pipette Yeung, Anthony Kwok-Cheung 1987

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Entry Flow Problem of a Liquid Body into a Suction Pipette by Anthony Kwok-Cheung Yeung B.A.Sc, The University of British Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1987 © Anthony Kwok-Cheung Yeung, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of _ The University of British Columbia 1956 Main Mal l Vancouver, Canada V6T 1Y3 Date DcAoUs , DE-6(3/81) i i ABSTRACT The m a t h e m a t i c a l problem o f t h e p i p e t t e a s p i r a t i o n o f a l i q u i d s p h e r e i s s t u d i e d i n t h e low Re y n o l d s number l i m i t . Two d i s t i n c t models a r e proposed f o r t h e d e f o r m i n g body. They a r e : 1) a l i q u i d d r o p l e t o f c o n s t a n t v i s c o s i t y , and 2) a v i s c o e l a s t i c c o r t e x e n c a p s u l a t i n g an i n v i s c i d i n t e r i o r . These models r e p r e s e n t energy d i s s i p a t i o n d i s t r i b u t e d i n t h e i n t e r i o r and on t h e s u r f a c e o f t h e body, r e s p e c t i v e l y . Because t h e i n - f l o w r a t e s v a r y d i f f e r e n t l y w i t h t h e p i p e t t e s i z e f o r t h e two models, t h i s i s s u g g e s t e d as a means o f e x p e r i m e n t a l l y i d e n t i f y i n g t h e dominant r e g i o n o f v i s c o u s d i s s i p a t i o n , and t h u s p r o v i d e i n s i g h t i n t o t h e i n t e r n a l s t r u c t u r e o f t h e t e s t sample. F o r t h e d r o p l e t problem, t h e l i n e a r S t o k e s e q u a t i o n s a r e s o l v e d i n t h e i n t e r i o r o f t h e d e f o r m i n g body. The s o l u t i o n s , f o r some s p e c i f i e d s t r e s s boundary c o n d i t i o n s on a s p h e r e , can be e x p r e s s e d as i n f i n i t e sums o f Legendre p o l y n o m i a l s . I n s o l v i n g t h e s u r f a c e f l o w problem, t h e c o m p l e x i t i e s o f t h e e q u a t i o n s n e c e s s i t a t e a p p r o x i m a t e s o l u t i o n s by c o m p u t a t i o n a l means. A n u m e r i c a l p r o c e d u r e i s d e v e l o p e d w h i c h compares w e l l w i t h a n a l y t i c a l r e s u l t s when t h e l a t t e r i s a v a i l a b l e . i i i T A B L E O F C O N T E N T S Abstract i i L i s t of Figures i v Acknowledgement v I. Introduction 1 I I . Pipette Aspiration of a Droplet 8 2.1) The Creeping Motion Equations 9 2.2) Problem Formulation and Solution 12 2.3) Surface Tension E f f e c t s 16 2.4) Results and Discussion 19 I I I . Two-Dimensional Membrane Mechanics 23 3.1) Analysis of St r a i n 26 3.2) Balanace of Forces i n a Two-Dimensional Membrane 3 0 3.3) Constitutive Relations 34 3.4) The Linear Quasi-Elastic Solution 41 3.5) Implementation of Numerical Method 45 3.6) Results and Discussion 47 IV. Summary 55 Appendix A. Solution to Creeping Flow Equations 58 Appendix B. Pipette Aspiration of C o r t i c a l S h e l l i n the absence of In-Plane Shearing Stresses 63 Appendix C. F i n i t e Difference Equations 69 Appendix D. Matrix Solution to F i n i t e Difference Equations 73 L i s t of References 79 I V L I S T O F F I G U R E S Number T i t l e Page 1 Photograph of an Aspirated Granulocyte 3 2 Schmatic I l l u s t r a t i o n of Droplet Aspiration Problem 13 3 Equilibrium Pressure vs. Projected Length for Droplet with I n t e r f a c i a l Tension 17 4 Droplet In-Flow Rate vs. Pipette Radius 20 5 Hydrostatic Pressure on a Droplet Surface 22 6 The L i p i d Bilayer 25 7 The Two Modes of Membrane Deformation 28 8 D e f i n i t i o n of Tension Resultants and Coordinate Variables 31 9 Two Models of V i s c o e l a s t i c i t y 38 10 Matrix Structure of Fini t e - D i f f e r e n c e Formulation 48 11 Time Evolution of C e l l Aspiration with *>{/>£= 0 50 12 V e l o c i t y F i e l d s of Surface Flow Problem with -vf/OL = o 51-52 13 In-Flow Rate vs. Pipette Size for Surface Flow Model 53 14 C e l l Shapes for Different Values of ^/X. 54 15 D e f i n i t i o n of Dimensions for the Problem with t^/yc = 0 66 16 F i n a l Form of Upper Triangular Matrix 74 V ACKNOWLEDGEMENT I would l i k e to thank Dr. Evan Evans, my the s i s advisor, for introducing me to the f i e l d of s c i e n t i f i c research, and for a l l the encouragement and support he has given me. 1 I . INTRODUCTION Micropipette aspiration, which involves the manipulation and deformation of t e s t samples with a suction pipette, has become an important technique i n studying the mechanical properties of b i o l o g i c a l c e l l s . Since i t s f i r s t a p p l i c a t i o n by Mitchison and Swann (1954) to t h e i r work on sea urchin eggs, the technique has developed extensively, p a r t i c u l a r l y i n the research area of red blood c e l l membranes (Rand and Burton, 1964; Evans, 1973; Evans and Hochmuth, 1977; Skalak et a l . , 1973). Recently, the micropipette technique has been adapted to the d i r e c t measurements of weak adhesive interactions between surfactant b i l a y e r s (Evans and Needham, 1986). A l l these applications involve measurements of s t a t i c forces i n equilibrium configurations. Time dependent behaviour of bodies (leukocytes) i n a micropipette has also been investigated for small deformations (Schmid-Schonbein, et a l . , 1981; Chien and Sung, 1984). In these cases, the c e l l i s treated as a "standard v i s c o e l a s t i c " material (see Fung, 1965), which, i n essence, i s a s o l i d body with an e l a s t i c l i m i t to deformation. The aim of t h i s present work i s to analyze the pipette a s p i r a t i o n of a l i q u i d - l i k e body i n the context of continuum mechanics. The body of i n t e r e s t i s assumed to be s u f f i c i e n t l y large that i t must deform upon entering the pipette. Models of the material properties of the body can then be used to predict i t s deformation and flow under the given external forces. 2 As the name implies, the theory of continuum mechanics requires the deforming body to have, even when i t s s i z e i s reduced to the l i m i t i n g resolution of the experiment, a s u f f i c i e n t number of p a r t i c l e s to enable a thermodynamic characterization of i t s macroscopic properties. The experiment that motivates t h i s present analysis i s the as p i r a t i o n of human white blood c e l l s (granulocytes) into micropipettes, as shown i n figure 1, i n an attempt to understand t h e i r r heological behaviour (Evans and Kukan, 1984). Here, the c e l l s of in t e r e s t are on the order of IO - 3 cm i n diameter - a length scale that i s much larger than molecular dimensions. U l t r a s t r u c t u r a l evidence (Bessis, 1973; Schmid-Schonbein et a l . , 1980) has shown that the granulocyte i s encapsulated i n a plasma membrane that has a res e r v o i r of excess surface area i n the form of evenly d i s t r i b u t e d r u f f l e s . This membrane, when i n the f l a c c i d state, i s l i k e l y to o f f e r l i t t l e or no mechanical resistance to c e l l deformations (Evans and Skalak, 1980). Anchored to the underneath of the plasma membrane i s a meshwork of densly packed, randomly oriented a c t i n filaments (Southwick and Stossel, 1983; Amato et a l . , 1983). When stimulated to contract, these filaments are believed to provide for the c e l l ' s a c t i v e locomotory. Because we are only interested i n c e l l s that are not "turned on", the c o r t i c a l meshwork i s considered here as a passive gel with i t s s p e c i f i c rheological properties. The small s i z e of the a c t i n filaments (8 nm i n diameter) enables us to t r e a t the gel as a continuum; moreover, because the filaments are randomly oriented on the c e l l surface, isotropy and F i g u r e 1: A s p i r a t i o n o f a human g r a n u l o c y t e a t 23 C. The p i p e t t e i n n e r r a d i u s i s 2 m i c r o m e t e r s and t h e s u c t i o n p r e s s u r e i s 500 dynes/cm . 4 homogeneity i n t h e s u r f a c e p l a n e ( e x c l u d i n g t h e t h i c k n e s s d imension) can be assumed. I n c o n t r a s t t o t h e c o r t e x , t h e c e l l i n t e r i o r i s h i g h l y heterogeneous i n s t r u c t u r e ; t h a t i s , i t c o n t a i n s o r g a n e l l a r b o d i e s l i k e t h e n u c l e u s and g r a n u l e s . T h i s i s a c o m p o s i t e s t r u c t u r e w i t h each component b e i n g a continuum i t s e l f . To u n d e r s t a n d t h e o v e r a l l p r o p e r t y o f t h i s s t u r c t u r e , we w i l l model t h e c e l l i n t e r i o r as a t h r e e - d i m e n s i o n a l l y i s o t r o p i c and homogeneous s u b s t a n c e . I n d o i n g t h i s , i t i s u n d e r s t o o d t h a t t h e m a t e r i a l p r o p e r t i e s a s s i g n e d r e f e r t o t h e b u l k o f t h e i n t e r i o r , and s h o u l d n o t be a t t r i b u t e d t o any one o f t h e o r g a n e l l a r components. F o r t h e a n a l y s i s i n t h i s t h e s i s , t h e c o m p l i c a t e d s t r u c t u r e o f a g r a n u l o c y t e has been i d e a l i z e d as a t h r e e d i m e n s i o n a l l y i s o t r o p i c s u b s t a n c e s u r r o u n d e d by a two d i m e n s i o n a l l y i s o t r o p i c g e l . The n e x t s i m p l i f i c a t i o n i s t o i d e n t i f y t h e c e l l as e s s e s t i a l l y a l i q u i d body. T h i s i s e v i d e n c e d by t h e c o n t i n u o u s f l o w o f such c e l l s i n t o m i c r o p i p e t t e s when t h e s u c t i o n p r e s s u r e i s i n e x c e s s o f a c e r t a i n t h r e s h o l d v a l u e . T h i s t h r e s h o l d p r e s s u r e i n t u r n i s e s t a b l i s h e d by a c o r t i c a l s t r e s s much l i k e t h e i n t e r f a c i a l t e n s i o n between two l i q u i d s . The f a c t t h a t t h e f l o w i s c o n t i n u o u s w i t h o u t any approach t o s t a t i c e q u i l i b r i u m i n d i c a t e s t h a t no l i m i t i n g e l a s t i c f o r c e s e x i s t i n t h e c e l l i n t e r i o r . F u r t h e r , t h e c o r t i c a l t e n s i o n can be shown t o be in d e p e n d e n t o f d e f o r m a t i o n by l o w e r i n g t h e s u c t i o n p r e s s u r e t o t h e i n i t i a l t h r e s h o l d . I n such a c a s e , t h e f l o w c e a s e s and t h e a s p i r a t e d p r o j e c t i o n o f t h e c e l l i n s i d e t h e p i p e t t e remains 5 stationary. In general, both the c o r t i c a l s h e l l and the i n t e r i o r may contribute to the viscous resistance to flow. The two l i m i t i n g models proposed here are ones that have the viscous d i s s i p a t i o n dominated by one of the regions, with the other region being e s s e n t i a l l y i n v i s c i d . In p a r t i c u l a r , the models are: 1) a l i q u i d droplet with a constant v i s c o s i t y , r e s u l t i n g i n energy d i s s i p a t i o n d i s t r i b u t e d throughout the i n t e r i o r , and 2) a viscous c o r t i c a l s h e l l encapsulating an i n v i s c i d f l u i d . The condition of volume conservation, as well as a c o r t i c a l tension are incorporated into both models. Also, the viscous drag of the e x t e r i o r aqueous solution on the c e l l i s neglected because of the extremely slow response observed for c e l l entry into pipettes compared with the rapid in-flow of water at the same suction pressures ( i e : flow rates that d i f f e r by a factor of 10 5). Because the d i s s i p a t i o n of mechanical energy i s volumetrically d i s t r i b u t e d i n the former case and two dimensionally confined i n the l a t t e r , the functional dependence of any flow related quantity (eg., the c e l l entry flow rate) on any c h a r a c t e r i s t i c length scale (eg., the pipette radius) should be d i s t i n c t l y d i f f e r e n t for the two models. These functional behaviours can be obtained by solving the above mentioned mechanical problems i n the enti r e region that the c e l l occupies. The r e s u l t s w i l l provide an experimenter with a means to discriminate the d i f f e r e n t viscous d i s s i p a t i o n zones by performing a s p i r a t i o n tests with various sized pipettes. 6 Continuum mechanical analyses are composed of three independent and d i s t i n c t developments. They are: 1) the quantitation of deformation and rate of deformation i n r e l a t i o n to changes i n the body's geometry; 2) the balance of forces within the body, as dictated by Newtonian mechanics; and 3) the modelling of the material properties of the substance. Knowledge of any two of these aspects can be used to predict the t h i r d one. For example, the deformation and rate of deformation of a body i n response to controlled forces can be analyzed to give the material properties (eg: e l a s t i c and viscous c o e f f i c i e n t s ) . I t i s important to note that these three developments are formulated i n terms of intensive quantities, i e : quantities that do not depend on the s i z e of the sample. For instance, deformation i s measured by s t r a i n , which i s a dimensionless r a t i o of the material displacement to some i n i t i a l length. Likewise, the d i s t r i b u t i o n of forces i s measured by i t s i n t e n s i t y ( i e : on a per unit area basis) c a l l e d s tress. Material properties, which are c o e f f i c i e n t s r e l a t i n g the stresses and the deformations, are therefore based on l o c a l functions as well . The v i r t u e of such formulations i s that i n t r i n s i c properties of the substance can be defined independent of the nature of the experiment. In the following chapters, these three developments w i l l be followed i n a r r i v i n g at the equations that govern the flow f i e l d . A n a l y t i c a l solutions to these equations are not available except f o r a few s i m p l i f i e d cases. For the more general problem, we must be content with 7 a p p r o x i m a t e s o l u t i o n s by n u m e r i c a l means. These methods w i l l a l s o be d i s c u s s e d i n t h e n e x t c h a p t e r s and t h e r e s u l t s w i l l be p r e s e n t e d . 8 I I . P I P E T T E A S P I R A T I O N O F A D R O P L E T The two models proposed i n the introduction a t t r i b u t e d the c e l l ' s stress bearing component to the i n t e r i o r and to the c o r t i c a l region, respectively. To address the f i r s t problem, the pipette a s p i r a t i o n of a l i q u i d droplet i s analysed i n t h i s chapter. The droplet i s to have a spherical i n i t i a l geometry due to an i n t e r f a c i a l tension, and the i n t e r i o r i s modelled as an incompressible newtonian f l u i d . A s i m i l a r problem i s solved by Schmid-Schonbein et a l . (1981) with the c e l l i n t e r i o r treated as a "standard v i s c o e l a s t i c s o l i d " . Because the authors have used the l i n e a r i z e d s t r a i n tensor i n t h e i r analysis, the r e s u l t s are v a l i d only for small deformations. Two s i m p l i f y i n g assumptions are made here: 1) during aspirat i o n , the portion of the body exterior to the pipette can be approximated as a spherical segment; and 2) viscous d i s s i p a t i o n inside the pipette can be neglected. The f i r s t assumption i s equivalent to the s i t u a t i o n where the Laplace pressure (created by the i n t e r f a c i a l tension) greatly exceeds the dynamic stress normal to the surface boundary. This c r i t e r i o n can be v e r i f i e d a p o s t e r i o r i from the f i n a l solution. The second approximation i s introduced to represent the free-s l i p condition between the c e l l surface and the pipette wall (Evans and Kukan, 1984). By putting i n proper boundary conditions, the present problem w i l l have a unique solu t i o n i n the form of a v e l o c i t y f i e l d . We s t a r t our analysis by f i r s t 9 summarizing some equations relevant to the theory of continuum mechanics. 2 . 1 : T H E C R E E P I N G MOTION EQUATIONS Let x^ (i=l,2,3) be a coordinate system set up within the continuous body that locates every material point. I f the v e l o c i t y v^ (i=l,2,3) i s continuous everywhere, then the rate of s t r a i n tensor i s defined as x a v i avj v i j = 2 ( air + ex. } (2.1) with the property that ^ (As 2 - A s Q 2 ) = 2 Ax ± Ax. (2.2) Here, Ax^ i s an instantaneous p o s i t i o n vector connecting two points that are i n f i n i t e s i m a l l y close, As i s the absolute length of Ax., and As_ i s the distance between the same two material l o points i n the i n i t i a l configuration. Because equation 2.2 involves only the difference between absolute lengths, the rate of s t r a i n tensor excludes a l l r i g i d body displacements and i s therefore a true measure of deformation. The trace of the rate of s t r a i n tensor represents volume 1The r e p e t i t i o n of an index w i l l imply summation with respect to that index over i t s range. 10 d i l a t a t i o n . Because t h e f l u i d i s assumed i n c o m p r e s s i b l e i n t h i s c a s e , "V\^ v a n i s h e s . E x p r e s s e d e q u i v a l e n t l y i n terms o f v e l o c i t y components, t h e i n c o m p r e s s i b i l i y c o n d i t i o n i s dv . ax" 1 = 0 < 2' 3> i The s t r e s s v e c t o r T^ ( n ) c o r r e s p o n d i n g t o a u n i t v e c t o r n i s d e f i n e d as f o l l o w s : C o n s i d e r a c r o s s s e c t i o n a l a r e a AA t h a t i s normal t o n. L e t AF^ be t h e t o t a l f o r c e e x e r t e d on t h e p o s i t i v e s i d e o f AA ( i e : t h e s i d e on w h i c h n p o i n t s o u t w a r d ) . O b v i o u s l y , t h e amount o f f o r c e w i l l d e c r e a s e as t h e a r e a s h r i n k s . I n t h e l i m i t as AA v a n i s h e s , t h e s t r e s s v e c t o r i s g i v e n by t h e r a t i o o f A F i t o AA: T ( n \ = l i m A F i (2.4) i K " ' AA*o ~AA~ The s t r e s s t e n s o r , 0 ^ j , i s d e f i n e d as t h e j t h component o f t h e s t r e s s v e c t o r on a p l a n e whose normal i s i n t h e x^ d i r e c t i o n : CTij S T j ( e i } ( 2 ' 5 ) Here, t h e u n i t v e c t o r s a r e t h e b a s i s s e t v e c t o r s . F o r a n e w t o n i a n l i q u i d , t h e s t r e s s t e n s o r i s r e l a t e d t o t h e r a t e o f s t r a i n t e n s o r by a p r o p o r t i o n a l i t y c o n s t a n t : ID = -p 5 ID 2 T) V, (2.6) 11 ^ i s c a l l e d t h e c o e f f i c i e n t o f s h e a r v i s c o s i t y . The h y d r o s t a t i c p r e s s u r e p i s i n t r o d u c e d so t h a t t h e t r a c e o f rem a i n s n o n - z e r o . I t can be v i e w e d as a Lagrange m u l t i p l i e r a s s o c i a t e d w i t h t h e i n c o m p r e s s i b i l i t y c o n s t r a i n t . The b a l a n c e o f f o r c e s w i t h i n a continuum i s e x p r e s s e d by t h e e q u a t i o n P dv^ da d t a X j r i i ^ - + F. (2.7) where p i s t h e mass d e n s i t y o f t h e body, v ^ t h e v e l o c i t y , and F^ t h e body f o r c e p e r u n i t volume. Because t h e t e s t samples ( g r a n u l o c y t e s ) a r e f r e e l y suspended d u r i n g e x p e r i m e n t , t h e r e a r e no body f o r c e s ( o r r a t h e r t h e y a r e n e g l i g i b l e i n c o m p a r i s o n t o t h e s u c t i o n f o r c e s ) , and hence F^= 0. The l e f t hand s i d e o f e q u a t i o n 2.7 r e p r e s e n t s t h e i n e r t i a l f o r c e s w h i l e t h e t e r m f o r a n e w t o n i a n f l u i d , as i n d i c a t e d i n t h e l a s t p a r a g r a p h , r e p r e s e n t s t h e v i s c o u s f o r c e s . The r a t i o o f t h e f o r m e r t o t h e l a t t e r i s known as t h e R e y n o l d s number (Landau and L i f s h i t z , 1982). From t h e c h a r a c t e r i s t i c s i z e s and f l o w r a t e s o f t h e m i c r o p i p e t t e e x p e r i m e n t s , t h e Reyn o l d s number i s e s t i m a t e d t o have an upper bound o f 10" 6, t h u s l e a v i n g t h e l e f t hand s i d e o f e q u a t i o n 2.7 c o m p l e t e l y n e g l i g i b l e . The e q u a t i o n o f m e c h a n i c a l e q u i l i b r i u m f o r o u r pu r p o s e s i s t h e r e f o r e da . i l = 0 (2.8) ax D 12 Combining equations 2.1, 2.3, 2.6, and 2.8, we a r r i v e at the l i n e a r Stokes equation for creeping motions: v p = n v 2 v (2.9) By taking the c u r l of equation 2.9 and using the vector i d e n t i t y v x vp = o the creeping motion equation can be further s i m p l i f i e d to contain only the v e l o c i t y term: v 2 (v x v ) = o (2.10) By defining the quantity vxv as the v o r t i c i t y vector, we see that the creeping motion equations have been reduced to a homogeneous Laplace equation of v o r t i c i t y . 2 . 2 : PROBLEM FORMULATION AND SOLUTION Equation 2.10 has to be s a t i s f i e d at every point inside the droplet. In addition, proper boundary conditions have to be prescribed f o r the problem to be well posed. Consider the s i t u a t i o n depicted i n figure 2. Because of the body's geometry, i t i s natural to use spherical coordinates (^?,e). Here, the azimuthal angle $ drops out due to axisymmetry while e, the polar angle, can be a l t e r n a t i v e l y represented by i t s cosine: 13 z Figure 2: D e f i n i t i o n of coordinates and various dimensions for the pipette a s p i r a t i o n problem. Note that there i s a f i n i t e region of contact between the droplet and the pipette, as defined i n eqn. 2.11. 14 £ = cos e In terms of £ , the outer surface of the sphere can be divided into three regions that are subjected to d i f f e r e n t stresses. F i r s t , there i s the part inside the pipette (-1 < £ < C^, ) that experiences a suction pressure A P . The region i n contact with the pipette ( Cj> < £ < + e ) i s under a uniform compressive load 7[ /£ ( being a small quantity) , while the remaining spherical portion (£J, + € < £ < 1 ) i s stress free. The magnitude of A can be related to A P by requiring the t o t a l a x i a l force on the body be zero. The externally applied normal stresses are c o l l e c t i v e l y c a l l e d (X ( £ ) such that AP - X / e 0 -1 < c < r s. p r +£ < r < i i p a (2.11) with AP and ^ both p o s i t i v e quantities. The stress boundary conditions are therefore o* (C) PP V J > ' p=R = « ( f ) (2.12) for the normal stress, and p=R = 0 (2.13) for zero tangential stress. In an axisymmetric problem, the quantities a and o „ are the only possible non-zero stresses PP P 15 on t h e c o o r d i n a t e s u r f a c e p = R. The s o l u t i o n t o t h e L a p l a c e e q u a t i o n f o r v o r t i c i t y ( e q u a t i o n 2.10) can be e x p r e s s e d as i n f i n i t e sums o f p o l y n o m i a l s i n P and a n g u l a r h a r m o n i c s i n L\ , as d e v e l o p e d by Happel and B r e n n e r , 1973 (see a p p e n d i x A ) . By m a t c h i n g boundary c o n d i t i o n s 2.12 and 2.13 t o t h e g e n e r a l s o l u t i o n s o f e q u a t i o n 2.10, t h e f i n a l form o f t h e s o l u t i o n i s o b t a i n e d . I n p a r t i c u l a r , on t h e boundary p = R, t h e y a r e 00 v (O = _ v ° r - X ( n - l ) (2n-l> p 2 f I J „ 2 R a n P n - l ^ > (2.14a) n = 3 2(n-2) (2rr+l ) n n l 00 v e(D = v z° s i n e + J 3n(n-l) R a V P (2.14b) n = 3 2 ( n - 2 ) ( 2 n 2 + l ) n s i n e n=3 where P n ( £ ) i s t h e Legendre p o l y n o m i a l o f degree n, and I (£. ) t h e Gegenbauer p o l y n o m i a l g i v e n by pn-2<n - V " n 2n - 1 l n ( C ) =-w i t h t h e d e g e n e r a t e c a s e s d e f i n e d as I O (C ) s P O (C ) = l ; i 1 ( C ) = - P^O = - C (2.16) The c o n d i t i o n t h a t ^ = 0 a t t h e p o i n t where t h e sphere t o u c h e s t h e p i p e t t e i s s a t i s f i e d by s u p e r i m p o s i n g an a x i a l v e l o c i t y 16 onto the series solutions. The Legendre c o e f f i c i e n t s a are n functions of AP, ^, and geometry. The expression f o r a n , as well as a derivation of equations 2.14 are given i n appendix A. 2.3: SURFACE TENSION EFFECTS The normal stress boundary condition, equation 2.11, does not include any i n t e r f a c i a l tension contribution. Adding the Laplace pressure term -2TQ/R (T Q being the i n t e r f a c i a l tension with units of force/unit length) to the r i g h t hand side of 2.11 i n f a c t leaves the f i n a l expression for a n , and hence the v e l o c i t y f i e l d , unaltered. This i s because the geometry of a droplet ( i e : sphere) i s the equilibrium configuration created by the i n t e r f a c i a l forces. The v e l o c i t y f i e l d s i n equations 2.14, representing the balance of viscous stresses against forces that deform the sphere, are therefore independent of T Q. Any deviation from t h i s equilibrium shape, however, w i l l be r e s i s t e d by the i n t e r f a c i a l tension. By c a l c u l a t i n g t h i s resistance to shape changes, i t i s shown that despite of the surface tension e f f e c t , the r e s u l t s from section 2.2 are s t i l l useful i f the quantity Ap i s interpreted as a pressure i n excess of some thershold value. Consider the a s p i r a t i o n of a l i q u i d droplet that has a constant i n t e r f a c i a l tension T Q. The pressure P e g required to hold the droplet at s t a t i c equilibrium at a projected length d i s calculated, keeping the volume of the drop fixed. Figure 3 17 Figure 3 : Suction pressure required to hold a l i q u i d droplet at s t a t i c equilibrium versus the projected length for d i f f e r e n t i n i t i a l c e l l s i z e s . 18 shows t h e p l o t o f t h e d i m e n s i o n l e s s p r e s s u r e , p e q R p / T Q f as a f u n c t i o n o f d/R f o r v a r i o u s c e l l s i z e s . The common f e a t u r e t o P a l l t h e s e c u r v e s i s t h e s t e e p r i s e i n p r e s s u r e f o r p r o j e c t i o n s l e s s t h a n one p i p e t t e r a d i u s ( i e : b e f o r e a h e m i - s p h e r i c a l cap i s formed i n t h e p i p e t t e ) , f o l l o w e d by an e s s e n t i a l l y c o n s t a n t o r even d e c r e a s i n g p r e s s u r e l e v e l . The p r e s s u r e r e q u i r e d t o form t h e h e m i s p h e r i c a l cap can be c a l c u l a t e d from t h e e q u a t i o n * - 2 T o Hr " ib (2.17) p T h i s i s t h e t h r e s h o l d t h a t must be exceeded f o r f l o w t o commence. F o r p r e s s u r e s above t h e t h r e s h o l d , t h e amount i n A e x c e s s o f P ( a p p r o x i m a t e l y ) i s t h e e f f e c t i v e p r e s s u r e t h a t i s b a l a n c e d a g a i n s t t h e v i s c o u s f o r c e s . Assuming t h e r e i s no p r e s s u r e drop a l o n g t h e a s p i r a t e d l e n g t h ( t h e p l u g f l o w a s s u m p t i o n ) , t h i s e x c e s s p r e s s u r e i s j u s t t h e q u a n t i t y AP i n e q u a t i o n 2.11. To be e x a c t , we w r i t e AP = P - P (2.18) where P i s t h e a p p l i e d s u c t i o n p r e s s u r e . W i t h t h i s new i n t e r p r e t a t i o n o f A P , t h e v e l o c i t y f i e l d s (eqns. 2.14a & 2.14b) r e m a i n unchanged, w h i l e t h e q u a n t i t y p(©) i n e q u a t i o n 2.14c s h o u l d be r e p l a c e d by p(9) - 2T Q/R , w h i c h i s t h e c o r r e c t i o n f o r t h e L a p l a c e p r e s s u r e . 19 2.4: R E S U L T S A N D D I S C U S S I O N The volumetric flow rate into the pipette i s derived from the r e l a t i o n 6 7 P Q = - 2ir R X v s i n e de (2.19) 0 p where 9 i s the value of 0 at the pipette entrance. Equation 2.19 can be integrated using the formula to obtain S P (x) dx = - I (O 0 n o 3 Q = i R_ The rate of growth of the projection inside the pipette i s calculated from the volumetric flow rate: Q = TT R p 2 L (2.21) A l l quantities i n the above equations are made dimensionless by sc a l i n g with Rp, ">j , and A p. Figure 4 shows a p l o t of the dimensionless rate of entry as a function of the pipette radius. As the sphere radius approaches i n f i n i t y , the dimensionless flow rate, L ^ / C A P R ^ ) , has a l i m i t i n g value of 0 . 25 . The s i m i l a r problem of viscous flow from an i n f i n i t e half-space into an o r i f i c e i s solved by Happel & Brenner (1973) and Torzeren et a l . (1984) with d i f f e r e n t v e l o c i t y boundary conditions along the 2 0 Figure 4: The dimensionless flow rate as a function of the pipette radius f o r a newtonian l i q u i d droplet. septum: t h e forme r used t h e no s l i p boundary c o n d i t i o n w h i l e t h e l a t t e r assumed t h e v e l o c i t y be d r i v e n by a r a d i a l l y c o n v e r g e n t membrane ( i e : r a d i a l v e l o c i t y p r o p o r t i o n a l t o 1 / r ) . Because t h e p r e s e n t problem i n v o l v e s t h e f r e e - s l i p boundary c o n d i t i o n (eqn. 2.13), we e x p e c t e d t h e f l o w r a t e t o be g r e a t e r t h a n t h e n o - s l i p c a s e , and l e s s t h a n t h a t f o r t h e membrane d r i v e n f l o w . Indeed, w i t h t h e same s c a l i n g as used above, t h e d i m e n s i o n l e s s f l o w r a t e s a r e r e s p e c t i v e l y (0.212) and 1/2. On t h e s u r f a c e n = R, our i n i t i a l a s s u m p t i o n o f n e g l i g i b l e normal s t r e s s i n comparison t o t h e L a p l a c e p r e s s u r e ( c r e a t e d by t h e i n t e r f a c i a l t e n s i o n ) can now be v e r i f i e d . U s i n g e q u a t i o n 2.14c, t h e dynamic p r e s s u r e i s c a l c u l a t e d on t h e s u r f a c e o f t h e s p h e r e . The r e s u l t , as shown i n f i g u r e 5, i s t h a t t h e r e i s no a p p r e c i a b l e p r e s s u r e d i f f e r e n c e ( r e l a t i v e t o t h e e x t e r n a l medium) on t h e segment e x t e r i o r t o t h e s u c t i o n p i p e t t e . T h i s i m p l i e s t h a t a l l t h e p r e s s u r e drop must be c o n c e n t r a t e d i n a s m a l l r e g i o n a t t h e o r i f i c e e n t r a n c e where t h e v e l o c i t y g r a d i e n t s a r e l a r g e . 2 2 0 AP - 4 -8 -F i g u r e 5: The d i m e n s i o n l e s s h y d r o s t a t i c p r e s s u r e v s . t h e c u r v i l i n e a r d i s t a n c e on t h e s u r f a c e f = R. The s h a r p drop i n p r e s s u r e a t e/tc = 0.83 c o r r e s p o n d s t o t h e p i p e t t e c o n t a c t r e g i o n . 2 3 I I I . TWO-DIMENSIONAL MEMBRANE MECHANICS The t h e o r y o f t w o - d i m e n s i o n a l t h i n s h e l l mechanics has been e x t e n s i v e l y d e v e l o p e d by Evans and S k a l a k (1980), w h i c h forms t h e t h e o r e t i c a l b a s i s o f t h i s c h a p t e r . The p r i n c i p l e s w i l l be a p p l i e d t o t h e a x i s y m m e t r i c p r oblem o f t h e p i p e t t e a s p i r a t i o n o f a c o r t i c a l s h e l l . The f l u i d i n s i d e t h e c e l l i s assumed t o be i n v i s c i d and i n c o m p r e s s i b l e , r e s u l t i n g i n a u n i f o r m i n t e r n a l p r e s s u r e . M a t e r i a l p r o p e r t i e s o f t h e c o r t e x , on t h e o t h e r hand, can be q u i t e g e n e r a l . I n a d d i t i o n t o h a v i n g an i s o t r o p i c i n t e r f a c i a l t e n s i o n t h a t a c c o u n t s f o r t h e c e l l ' s s p h e r i c a l shape, t h e c o r t e x can a l s o be " v i s c o e l a s t i c " . T h i s i s a g e n e r a l d e s c r i p t i o n o f a f a m i l y o f models t h a t a r e c o m b i n a t i o n s o f two b a s i c i d e a l i z a t i o n s : t h e l i n e a r e l a s t i c body and t h e l i n e a r v i s c o u s body. These w i l l be d i s c u s s e d i n more d e t a i l s i n l a t e r s e c t i o n s . Because o f t h e s h a r p bend o b s e r v e d f o r a s p i r a t e d c e l l s a t t h e edge o f t h e p i p e t t e e n t r a n c e (see f i g u r e 1 ) , we a n t i c i p a t e a n e g l i g i b l e b e n d i n g r i g i d i t y i n t h e c o r t i c a l l a y e r . As s u c h , t h e p r o b l e m i s r e d u c e d t o c o n s i d e r a t i o n o f f o r c e s t h a t a c t o n l y i n t h e s u r f a c e p l a n e . The development o f t h i s c h a p t e r w i l l t h u s be based on two s i m p l i f i c a t i o n s : axisymmetry and t h e n e g l e c t o f b e n d i n g moments. I n c o n t r a s t t o a t h i n s h e e t o f r u b b e r , f o r example, w h i c h has s m a l l s c a l e s t r u c t u r e even a c r o s s t h e t h i c k n e s s d i m e n s i o n , a t w o - d i m e n s i o n a l membrane can have i s o t r o p y c h a r a c t e r i z e d o n l y i n t h e s u r f a c e p l a n e . Though i t may seem u n u s u a l as an e n g i n e e r i n g 24 m a t e r i a l , such a c o n c e p t f i n d s i t s e l f q u i t e common i n t h e f i e l d o f b i o l o g i c a l membranes. As an example, c o n s i d e r t h e fundamental component o f t h e b i o l o g i c a l c e l l membrane - t h e l i p i d b i l a y e r . I t i s composed o f two l a y e r s o f l i p i d mosaic as shown i n f i g u r e 6. * Because each l i p i d m o l e c u l e o n l y o c c u p i e s an a r e a o f a p p r o x i m a t e l y 100 A , on a c e l l u l a r s c a l e ( i e : m i c r o m e t e r s ) , t h e s u r f a c e p l a n e o f t h e b i l a y e r can be c o n s i d e r e d a continuum. A c r o s s t h e t h i c k n e s s , however, t h e r e a r e p r e c i s e l y two m o l e c u l e s . The p r i n c i p l e s o f continuum mechanics a r e o b v i o u s l y i n v a l i d i n t h i s d i m e n s i o n . T h i s a n i s o t r o p i c s t r u c t u r e i s r e f l e c t e d i n t h e m e c h a n i c a l p r o p e r t i e s o f t h e b i l a y e r membrane, as measured by Kwok and Evans (1981). F o r example, b i l a y e r s above t h e a c y l c h a i n c r y s t a l l i z a t i o n t e m p e r a t u r e e x h i b i t a v e r y s t r o n g s t a t i c r e s i s t a n c e t o a r e a e x p a n s i o n s , b u t has no su c h r e s i s t a n c e t o shape changes under c o n s t a n t a r e a . These c o n f l i c t i n g v a l u e s o f s t a t i c r i g i d i t i e s a r e u n c h a r a c t e r i s t i c o f a t h i n membrane t h a t i s t h r e e - d i m e n s i o n a l l y i s o t r o p i c . I t can however be r a t i o n a l i z e d by r e c o g n i z i n g t h e d i s c o n t i n u i t y i n t h e t h i r d d i m e n s i o n . The a s s u m p t i o n o f a t w o - d i m e n s i o n a l membrane i s a g e n e r a l -i z a t i o n r a t h e r t h a n a r e s t r i c t i o n . I n g e n e r a l , d e f o r m a t i o n s o f a t h i n s h e e t can be e x p r e s s e d as a s u p e r p o s i t i o n o f two fund a m e n t a l modes: a r e a d i l a t a t i o n and i n - p l a n e s h e a r . The r e s i s t a n c e o f t h e m a t e r i a l t o t h e s e two modes o f d e f o r m a t i o n a r e e x p r e s s e d n u m e r i c a l l y as t h e " m o d u l i " o f e l a s t i c i t y ( o r t h e m o d u l i o f v i s c o s i t y f o r r e s i s t a n c e t o r a t e s o f d e f o r m a t i o n ) . Figure 6 : Schematic i l l u s t r a t i o n of a section of the l i p i d b i l a y e r , which comprises of two layers of l i p i d molecules arranged i n two-dimensional arrays. F o r a t h r e e d i m e n s i o n a l l y i s o t r o p i c s h e l l m a t e r i a l , t h e m o d u l i a s s o c i a t e d w i t h t h e two modes o f d e f o r m a t i o n a r e u n i q u e l y r e l a t e d whereas i n t h e case o f a t w o - d i m e n s i o n a l m a t e r i a l , t h i s r e s t r i c t i o n i s removed and t h e two m o d u l i can be c o m p l e t e l y i n d e p e n d e n t o f each o t h e r (see Evans, 1973 f o r a more d e t a i l e d d i s c u s s i o n ) . The f i r s t p a r t s o f t h i s c h a p t e r w i l l be d e v o t e d t o t h e t h e o r e t i c a l a s p e c t s o f t w o - d i m e n s i o n a l t h i n s h e l l m echanics. A n u m e r i c a l a l g o r i t h m t h a t a p p l i e s t h e p r i n c i p l e s t o t h e p i p e t t e a s p i r a t i o n o f a c o r t i c a l s h e l l w i l l t h e n be d i s c u s s e d . 3.1: ANALYSIS OF STRAIN To a n a l y z e t h e d e f o r m a t i o n o f a t h i n membrane, we c o n c e p t u a l i z e t h e i n i t i a l (undeformed) s u r f a c e as a g r i d o f many e l e m e n t a l s q u a r e s , w i t h each r e g i o n s m a l l enough t h a t l o c a l l y i t can be t r e a t e d as a f l a t s u r f a c e . By comparing t h e s i z e and shape o f each i n s t a n t a n e o u s element t o i t s i n i t i a l c o n f i g u r a t i o n , we can have complete i n f o r m a t i o n on t h e body's d e f o r m a t i o n f i e l d . Each d i f f e r e n t i a l element can be o r i e n t e d so t h a t a sq u a r e maps s i m p l y i n t o a r e c t a n g l e i n t h e deformed s t a t e . T h i s e s p e c i a l l y c o n v e n i e n t o r i e n t a t i o n i s s a i d t o be i n t h e p r i n c i p a l c o o r d i n a t e system. F o r an a x i s y m m e t r i c s u r f a c e ( i e : a s u r f a c e g e n e r a t e d by r e v o l v i n g a m e r i d i a n c u r v e about an a x i s ) w i t h d e f o r m a t i o n s symmetric about t h e same a x i s , t h e p r i n c i p a l c o o r d i n a t e s a r e i m m e d i a t e l y g i v e n - t h e y a r e i n t h e 27 meridional and the azimuthal d i r e c t i o n s . Two quantities are needed to r e l a t e the instantaneous rectangle to the i n i t i a l square. The simplest are the extension r a t i o s and A$> , which are the r a t i o s of the deformed length to the o r i g i n a l length i n the meridional and the azimuthal d i r e c t i o n s , respectively. In the case of axisymmetry, any quantity can be expressed as a function of one s p a t i a l variable alone - namely the c u r v i l i n e a r p o s i t i o n s of a material point along the meridian. This variable i n turn can be uniquely rel a t e d to the i n i t i a l c u r v i l i n e a r distance of the same material point, s Q . In t h i s manner, the extension r a t i o s are given by X (s ) = mv o' ds (3.1) o * ' X = r * r Q (3.2) where r ( s Q ) and ^ 0 ( s 0 ) are respectively the r a d i a l distances from the axis of symmetry i n the instantaneous and the i n i t i a l configurations. Although the extension r a t i o s provide a complete description of the s t r a i n f i e l d , there are other deformation v a r i a b l e s that have more relevance to the physics of the membrane material. Consider the uncoupling of a general deformation into the two fundamental modes: 1) expansion or contraction without shape changes; and 2) deformation at constant area. Figure 7 shows 2 8 L Figure 7 : The uncoupling of a general deformation into the two independent modes. such a case i n the p r i n c i p a l coordinate system. The actual deformation that involves the two extension r a t i o s can be broken down into two steps - the f i r s t one characterized by DC , the f r a c t i o n a l area change, and the second one by "A , which measures the amount of extension. These deformation variables can e a s i l y be related to the extension r a t i o s : a = A A , - 1 _ _ m <t> (3.3) x = <VV* <3-4> The two variables, OL and ^ , form a set of l i n e a r l y independent functions (with respect to ^ and ) that completely s p e c i f i e s the geometric features of the deformation. Likewise, the time rates of deformation can be seperated into the same two modes. In an Eulerian formulation ( i e : with the instantaneous coordinates as the reference geometry), the rate of deformation variables are not the time derivatives of OC and 'X / but of t h e i r logarithms (see Evans and Skalak,1980): Va " d f l n< 1 + a) ( 3' 5 ) d , . <3-6> V s - dF l n X Thus, the deformation and rate of deformation of an axisymmetric body i s quantitated i n terms of four intensive va r i a b l e s - each of which can be expressed as a function of the 30 i n i t i a l c u r v i l i n e a r distance s o 3.2: BALANCE OF FORCES IN A TWO-DIMENSIONAL MEMBRANE In Newtonian mechanics, the sum of a l l forces and the sum of a l l moments acting on a body must be zero i n the absence of acc l e r a t i o n . As mentioned e a r l i e r , the bending moments i n the c o r t i c a l layer i s assumed n e g l i g i b l e i n t h i s study. The membrane force resultants therefore must act tangent to the plane of the surface. For an axisymmetric problem, these forces are expressed i n terms of two tension resultants, T m and T^ ,as shown i n figure 8a. These are intensive quantities defined as the force per unit length i n the meridional and the azimuthal d i r e c t i o n s respectively. From the c h a r a c t e r i s t i c s i z e s and flow rates of the micropipette experiment, i n e r t i a l e f f e c t s are n e g l i g i b l e i n comparison to other forces. Thus, mechanical equilibrium requires the balance of the i n t e r n a l forces i n the t h i n s h e l l (eg: e l a s t i c and viscous forces) against the applied stresses (eg: the suction pressure). In an axisymmetric configuration, the equilibrium equations f o r a membrane are (see Evans and Skalak, 1980) d_ ds T . 9r 4> ds + a s 0 (3.7a) T. (3.7b) m + R. a m n 31 Figure 8 : The tension resultants and coordinate variables are i l l u s t r a t e d f o r an axisymmetric geometry. (a) The tension resultants act only i n the meridional and azimuthal d i r e c t i o n s , which are also the p r i n c i p a l d i r e c t i o n s . (b) D e f i n i t i o n of the coordinate v a r i a b l e s . R_ • • • III the meridional curvature, i s not shown. 32 A l l variables i n equations 3.7 are as defined i n figure 8. Note that and 0^ are the externally applied stresses that have dimensions of force per unit area (unlike the tension resultants, which have dimensions of force/length). The quantities and R^ are the r a d i i of curvature i n the meridional and azimuthal d i r e c t i o n s . They can be related to the coordinate variables by R = d 6 m ds ( 3 < 8 ) R * s i n 9 (3..9) where 0 i s defined as the angle between the outward normal vector and the axis of symmetry. I t follows from t h i s d e f i n i t i o n that S l n 9 = "If (3-10) cos 0 = d r ds (3.11) Equations 3.7 are the d i f f e r e n t i a l equations of mechanical equilibrium. I t i s also possible to cumulate the a x i a l component of the external stresses and equate to the meridional tension, r e s u l t i n g i n a set of integrated equations which are al t e r n a t i v e s to 3.7. They are F T = z_,_ n (3.12a) m 2ir r s i n 0 33 F = ^-VT (c - — — . „ (3.12b) 9 s i n 9 * s 2TT r s i n 9 d s ' where s F z ( s ) == 2ir S r ds ( a n cos 9 - a g s i n 9 ) (3.13) E q u a t i o n s 3.12 a r e v a l i d f o r an a x i s y m m e t r i c geometry w i t h t h e c u r v i l i n e a r c o o r d i n a t e s o r i g i n a t i n g from a p o l e . They can be shown t o be e q u i v a l e n t t o 3.7. Because t h e f l u i d s b o t h i n t e r i o r and e x t e r i o r t o t h e c o r t i c a l s h e l l a r e assumed i n v i s c i d , t h e r e can be no s h e a r s t r e s s e s on t h e membrane, and hence = 0. A l s o , f o r t h e same r e a s o n , t h e r e i s a u n i f o r m p r e s s u r e 0*n i n s i d e t h e c o r t i c a l s h e l l ( r e l a t i v e t o t h e e x t e r n a l medium) t h a t i s ind e p e n d e n t o f t h e c o o r d i n a t e s. W i t h t h e s e s i m p l i f i c a t i o n s , we can r e a d i l y e v a l u a t e t h e a x i a l f o r c e i n e q u a t i o n 3.13 and s u b s t i t u t e i n t o 3.12. The e q u a t i o n s o f e q u i l i b r i u m t h e n become £ = 4T - a r (3 r £ - $2) = 0 (3.14a) a 4 1 n s i n 9 ^ J s i n 9 ds ; u r , r d9 e = 4T — a -rtr-R (rrrr-fl Si - D = 0 ( 3 . i 4 b ) s n s i n 6 ' s i n 9 ds where T and T a r e r e s p e c t i v e l y t h e i s o t r o p i c and d e v i a t o r i c s t e n s i o n s d e f i n e d as T s XA (T + T ) m 9 (3.15) T S S * <Tm - V (3.16) The above equations of equilibrium express the balance of forces (an e x t r i n s i c quantity) i n a t h i n membrane that has no bending r i g i d i t y . The equations have to hold regardless of the structure of the membrane. For example, the questions of whether the material i s i s o t r o p i c , or whether i t i s s o l i d or l i q u i d , are i r r e l e v a n t . 3.3: CONSTITUTIVE RELATIONS Any r e l a t i o n that describes the property of a material can be c a l l e d a c o n s t i t u t i v e r e l a t i o n . In t h i s present work, we are interested i n mathematical functions that r e l a t e the intensive deformation variables ( t>C , /\) to the intensive stress variables (T,T g) at constant temperature. Much information about the material i s revealed i n these r e l a t i o n s . For instance, an e l a s t i c s o l i d i s one that has conservative i n t e r n a l forces ( i e : forces that can be represented as the gradient of a s c a l a r function). E l a s t i c stresses can only depend on the body's deformation and not on i t s time rate. A l i q u i d , on the other hand, i s characterized by i t s i n a b i l i t y to sustain shear stresses i n a state of r e s t . In the process of deforming, however, there i s i n e v i t a b l y i n t e r n a l molecular f r i c t i o n and s t r u c t u r a l changes that appear macroscopically as a resistance to flow. The stresses of a viscous l i q u i d w i l l therefore depend only on the rate of deformation variables. 35 The simplest way to model s o l i d and l i q u i d behaviours i s to take the f i r s t order approach. For an e l a s t i c s o l i d undergoing isothermal deformations, the mechanical work done can be shown to be equivalent to the change i n the Helmholtz free energy of the body (the body i s considered here as a closed system). I t follows from the d e f i n i t i o n of mechanical work that the stresses are the derivatives of the Helmholtz free energy with respect to the corresponding s t r a i n variables. To obtain a f i r s t order s t r e s s - s t r a i n r e l a t i o n , the free energy i s written as a Taylor expansion i n terms of the deformation variables up to the quadratic terms (see Evans and Skalak, 1980). In t h i s manner, the e l a s t i c stresses are given by T e = T Q + K a ( 3 > 1 7 ) e - x 2 -,-2 T = 'A u. ( X~X) (3.18) s v 1 + a > where the superscripts "e" denote e l a s t i c stresses. T Q i s a constant i s o t r o p i c tension which may a r i s e from i n t e r f a c i a l e f f e c t s . K a n d ^ are respectively the i s o t r o p i c and shear moduli of e l a s t i c i t y . For a three dimensionally i s o t r o p i c material, i t can be shown that K = 3y£* , whereas f o r a material that i s anisotropic i n the thickness dimension, these two moduli are completely unrelated. The viscous forces i n a l i q u i d a r i s e from i n t e r n a l f r i c t i o n and h e a t d i s s i p a t i o n t h a t a r e t h e r m o d y n a m i c a l l y i r r e v e r s i b l e . Because o f t h e i r c o m p l e x i t i e s , t h e s e n o n - c o n s e r v a t i v e f o r c e s a r e b e s t d e s c r i b e d by p h e n o m e n o l o g i c a l r e l a t i o n s . To f i r s t o r d e r , s u c h r e l a t i o n s a r e - v (3.19) T v = K V a T V = 2 T) V (3.20) s s where t h e s u p e r s c r i p t s "v" denote v i s c o u s s t r e s s e s . T<- and ~*f a r e r e s p e c t i v e l y t h e i s o t r o p i c and s h e a r m o d u l i o f v i s c o s i t y . L i k e t h e e l a s t i c s o l i d , t h e m o d u l i o f v i s c o s i t y a r e r e l a t e d by a 3:1 r a t i o f o r a t h r e e - d i m e n s i o n a l l y i s o t r o p i c m a t e r i a l , and a r e u n r e l a t e d i n t h e t w o - d i m e n s i o n a l c a s e . I n r e a l i t y , most m a t e r i a l s do n o t e x h i b i t p u r e l y e l a s t i c o r p u r e l y v i s c o u s c h a r a c t e r i s t i c s . The two f i r s t o r d e r models i n t r o d u c e d can be t h o u g h t o f as t h e i d e a l i z e d "extremes" o f m a t e r i a l b e h a v i o u r s : 1) t h e e l a s t i c s o l i d t h a t i s c a p a b l e o f i n s t a n t a n e o u s d e f o r m a t i o n when s u b j e c t t o i m p u l s i v e f o r c e s , w i t h m e c h a n i c a l work f u l l y r e c o v e r e d upon u n l o a d i n g , and 2) t h e v i s c o u s l i q u i d t h a t has a f i n i t e r a t e o f d e f o r m a t i o n and w i t h no t e n d e n c y o f r e t u r n i n g t o t h e i n i t i a l c o n f i g u r a t i o n upon removal o f e x t e r n a l f o r c e s . Work done by a v i s c o u s l i q u i d i s c o m p l e t e l y i r r e c o v e r a b l e . These two f i r s t o r d e r i d e a l i z a t i o n s can be combined t o model " i n t e r m e d i a t e " c a s e s o f mixed b e h a v i o u r s . A l t h o u g h i n p r i n c i p l e t h e r e can be an i n f i n i t e number o f c o m b i n a t i o n s , we w i l l i n t r o d u c e t h r e e models t h a t a c c o u n t v e r y w e l l f o r most m a t e r i a l b e h a v i o u r s . The f i r s t two models a r e i l l u s t r a t e d m e t a p h o r i c a l l y i n f i g u r e 9, u s i n g a s p r i n g t o denote t h e e l a s t i c component and a p i s t o n t o denote t h e v i s c o u s c o u n t e r p a r t . I n t h e f i r s t c a s e , t h e s o - c a l l e d V o i g t model, an e l a s t i c element i s p l a c e d i n p a r a l l e l w i t h a v i s c o u s element t o r e p r e s e n t s o l i d s t h a t have a p p r e c i a b l e i n t e r n a l energy d i s s i p a t i o n . A l t h o u g h t h e m a t e r i a l i s c a p a b l e o f r e t u r n i n g t o t h e o r i g i n a l c o n f i g u r a t i o n upon removal o f e x t e r n a l f o r c e s (which c h a r a c t e r i z e s i t as a s o l i d ) , t h e l o a d - u n l o a d c u r v e w i l l a l w a y s show h y s t e r e s i s due t o v i s c o u s d i s s i p a t i o n . F o r t h i s model, t h e t o t a l f o r c e i s t h e sum o f t h e two c o n t r i b u t i o n s : T = T e + T V = T + T v (3.21) (3.22) and t h e d e f o r m a t i o n i s by d e f i n i t i o n common t o b o t h e l e m e n t s : a = a = a X = = X (3.23) (3.24) U n l i k e t h e V o i g t m a t e r i a l , w h i c h can be c o n s i d e r e d as e s s e n t i a l l y a s o l i d , t h e r e a r e o t h e r s u b s t a n c e s w h i c h show i n t e r n a l s t r e s s r e l a x a t i o n mechanisms. These m a t e r i a l s a r e c h a r a c t e r i z e d by two o b s e r v a t i o n s : 1) an e l a s t i c r e s p o n s e t o 3 8 (a) (b) F i g u r e 9 : M e t a p h o r i c a l r e p r e s e n t a t i o n s o f v i s c o e l a s t i c models u s i n g c o m b i n a t i o n s o f s p r i n g s ( e l a s t i c component) and d a s h p o t s ( v i s c o u s component). (a) t h e V o i g t model; (b) t h e M a x w e l l model. i m p u l s i v e f o r c e s , and 2) l i q u i d f l o w b e h a v i o u r under c o n s t a n t s h e a r s t r e s s e s . Such m a t e r i a l s can a d e q u a t e l y be d e s c r i b e d by a f i r s t o r d e r M a x w e l l model, as shown i n f i g u r e 9b. Because o f t h e s e r i a l c o u p l i n g , t h e s t r e s s e s a r e common t o b o t h e l e m e n t s : T ~ T (3.25) T s = T s e = T s V (3.26) and t h e t o t a l r a t e o f d e f o r m a t i o n i s d e f i n e d as t h e sum o f t h e two i n d i v i d u a l p a r t s : v s = V + V s V < 3' 2 8> F o r t h e g e n e r a l f i r s t o r d e r c o n s t i t u t i v e r e l a t i o n s g i v e n by 3.17 t o 3.20, t h e r a t e o f d e f o r m a t i o n v a r i a b l e s f o r a M a x w e l l model can be e x p r e s s e d i n terms o f t h e s t r e s s e s as d l n ( l + a ) = 1 9T _T_ (3.29) a t a dt K — dT T 3 i n X = ; 1 ( T 3T _ s _ s d t 2(T 2 a2 + K 2 u2)* ( S 9 t a t l 2T» (3*30) s where o = T - T + K (3.31) o 4 0 The r a t i o o f the v i s c o u s modulus t o the e l a s t i c modulus i n a Maxwell model d e f i n e a r e l a x a t i o n time r t h a t determines i t s s e m i - s o l i d behaviour. In p a r t i c u l a r , the model p r e d i c t s an e l a s t i c response t o f o r c e s t h a t are a p p l i e d f o r a time d u r a t i o n much s h o r t e r than t , and a l i q u i d response t o d u r a t i o n s t h a t a r e l a r g e compared t o X . i n the l i m i t t h a t the r e l a x a t i o n time approaches zero, the Maxwell model becomes an i d e a l l i q u i d . While the Maxwell body i s c h a r a c t e r i z e d by a time co n s t a n t r e l a t e d t o i t s i n t e r n a l s t r e s s r e l a x a t i o n , t h e r e i s another c l a s s o f s e m i s o l i d - the v i s c o p l a s t i c m a t e r i a l (or the Bingham m a t e r i a l ) t h a t i s c h a r a c t e r i z e d by a s t r e s s magnitude c a l l e d the y i e l d shear. In t h i s case, the body behaves e l a s t i c a l l y under s t r e s s l e v e l s below the y i e l d shear. Beyond t h i s p o i n t , the m a t e r i a l b egins t o show l i q u i d flow w i t h the r a t e of deformation p r o p o r t i o n a l t o the amount of s t r e s s i n excess of the y i e l d . The c o n s t i t u t i v e r e l a t i o n f o r a two d i m e n s i o n a l l y i n c o m p r e s s i b l e i d e a l p l a s t i c i s T = 14 M (X 2 ~ X" 2) ; | T J < T e (3.32a) | T s ' ~ *s = 2 7 7 | V s ' '* | T s ' > *s (3.32b) where T i s the y i e l d shear which i s a p o s i t i v e q u a n t i t y . In s e q u a t i o n 3.32b, the r a t e o f deformation V g i s t o have the same s i g n as T . The e l a s t i c c o e f f i c i e n t U. can be made v e r y l a r g e t o a c c o u n t f o r t h e n e g l i g i b l e y i e l d s t r a i n i n most p l a s t i c m a t e r i a l s . I n t h e l i m i t t h e y i e l d s h e a r v a n i s h e s , t h e i d e a l p l a s t i c becomes a l i q u i d . 3.4: THE LINEAR QUASI—ELASTIC SOLUTION I n t h e p r e v i o u s s e c t i o n s , we have i n t r o d u c e d e q u a t i o n s t h a t d e s c r i b e t h e t i m e e v o l u t i o n o f a two d i m e n s i o n a l membrane under e x t e r n a l s t r e s s e s . I n p a r t i c u l a r , t h e b a l a n c e o f m e c h a n i c a l f o r c e s , as e x p r e s s e d i n e q u a t i o n s 3.14, must be s a t i s f i e d . The q u a n t i t i e s T and T g i n t h e s e e q u a t i o n s a r e c h a r a c t e r i s t i c o f t h e m a t e r i a l model chosen, w i t h t h e c o n s t i t u t i v e r e l a t i o n s as o u t l i n e d i n e q u a t i o n s 3.17 t o 3.32. The d e f o r m a t i o n and r a t e o f d e f o r m a t i o n v a r i a b l e s i n t h e c o n s t i t u t i v e r e l a t i o n s a r e i n t u r n q u a n t i f i e d i n 3.1 t o 3.6. The g e o m e t r i c f e a t u r e s o f t h e d e f o r m i n g body a r e e x p r e s s e d i n terms o f t h e c o o r d i n a t e v a r i a b l e s i n e q u a t i o n s 3.8 t o 3.11. To model t h e p i p e t t e a s p i r a t i o n o f a c o r t i c a l s h e l l , a l l t h e above e q u a t i o n s have t o be a c c o u n t e d f o r o v e r t h e e n t i r e s u r f a c e a t any i n s t a n t o f t i m e . F o r a g e n e r a l c o n s t i t u t i v e model, i t i s n o t s u r p r i s i n g t h a t o n l y a p p r o x i m a t e s o l u t i o n s can be o b t a i n e d by n u m e r i c a l means. However, a c l o s e d form s o l u t i o n does e x i s t i n t h e c a s e o f membranes w i t h no r e s i s t a n c e ( s t a t i c o r dynamic) t o i n - p l a n e s h e a r . Here, because t h e s h e a r r e s u l t a n t v a n i s h e s , t h e p o r t i o n o f t h e c e l l e x t e r i o r t o t h e p i p e t t e must remai n s p h e r i c a l (see e q u a t i o n 3.14b). I t f o l l o w s t h a t t h e i s o t r o p i c t e n s i o n , and hence o£ , i s u n i f o r m o v e r t h e d e f o r m i n g s u r f a c e a t any t i m e . A d e t a i l e d o u t l i n e o f t h e s o l u t i o n t o t h i s i s o t r o p i c problem i s g i v e n i n a p p e n d i x B. These r e s u l t s can be used t o v e r i f y t h e v a l i d i t y o f t h e n u m e r i c a l approach t o t h e more g e n e r a l c a s e , where t h e s h e a r v i s c o s i t y and t h e s h e a r e l a s t i c i t y can be non-z e r o . We w i l l now d e s c r i b e t h e n u m e r i c a l method f o r s o l v i n g t h e g e n e r a l c o r t i c a l s h e l l problem w i t h b o t h i s o t r o p i c and i n - p l a n e s h e a r r e s u l t a n t s . Here, t h e t i m e e v o l u t i o n o f t h e f l o w p r o c e s s i s a p p r o x i m a t e d by a s e r i e s o f s m a l l d i s p l a c e m e n t s , each o v e r a s h o r t t i m e i n t e r v a l A t . A t each t i m e s t e p , t h e v i s c o u s component o f t h e c o n s t i t u t i v e r e l a t i o n i s t r e a t e d " q u a s i -e l a s t i c a l l y " by d i v i d i n g t h e v i s c o u s m o d u l i by A t . The r e s u l t i n g s o l u t i o n , i n t h e form o f a d i s p l a c e m e n t f i e l d , i s t h e n added t o t h e o r i g i n a l geometry t o a p p r o x i m a t e t h e c e l l shape a t t h e end o f t h e t i m e i n t e r v a l . L e t t h e shape a t t i m e t be s p e c i f i e d by c y l i n d r i c a l c o o r d i n a t e s ( r , z ) , and t h a t a t t i m e t+At by ( r ' , z ' ) . The i n c r e m e n t i n g f u n c t i o n s a r e d e f i n e d as A r ( s Q ) = r ' - r ( 3 . 3 3 a ) A z ( s Q ) = z« - z ( 3 . 3 3 b ) A l l g e o m e t r i c a l l y r e l a t e d v a r i a b l e s (eg: Tin, s i n 6, R m , e t c ) can now be w r i t t e n i n t h e same manner; i e : f o r any such q u a n t i t y x, l e t x* be i t s v a l u e a t t i m e t+At, x be t h e v a l u e a t t , and 4 3 I n g e n e r a l , A x can be expanded i n a T a y l o r s e r i e s i n terms o f A r and Az. F o r s m a l l 4t, t h e problem i s l i n e a r i z e d by t r u n c a t i n g t h e s e r i e s a f t e r t h e f i r s t powers. The r a t e s o f s t r a i n a r e ap p r o x i m a t e d by d i v i d i n g t h e i n c r e m e n t s by A t . F o r example, t h e d i l a t o r y s t r a i n r a t e , wh i c h was p r e v i o u s l y w r i t t e n as v = 1 5 a a 1 + a at can now be a p p r o x i m a t e d by V A a a (l+a)At where ADC i s l i n e a r i n A r ( s ) and A z ( s ). W i t h t h e s e o o a p p r o x i m a t i o n s , c o n s t i t u t i v e r e l a t i o n s t h a t i n v o l v e v i s c o s i t i e s now become q u a s i - e l a s t i c . F o r i n s t a n c e , l e t T* be t h e i s o t r o p i c t e n s i o n i n a V o i g t m a t e r i a l a t t i m e t+At, t h e n T * = K (a + Aa) + K V = K a + = T + AT K + ( 1 + a ) A t Aa where T = Koi i s t h e cumulated e l a s t i c s t r e s s up t o t i m e t , and AT = K + (i+a) At ] A a * s t h e i n c r e i n e n t i n s t r e s s r e s u l t a n t ( a l s o l i n e a r i n A r and Az) due t o p e r t u r b a t i o n i n geometry. The e f f e c t i v e e l a s t i c modulus i s K+ (2.+a)At ' w h ^ - c h ' f o r s m a l l A t , i s dominated by t h e q u a s i - e l a s t i c t e rm. 44 We see t h a t the numerical scheme i s a s y s t e m a t i c way of w r i t i n g a l l q u a n t i t i e s i n terms of t h e i r v a l u e s a t time t , p l u s an i n c r e m e n t a l amount due t o geometric p e r t u r b a t i o n over a time i n t e r v a l A t . The increments are l i n e a r i z e d t o c o n t a i n o n l y the f i r s t powers o f the b a s i c p e r t u r b a t i o n f u n c t i o n s A r ( s Q ) and A Z ( S Q ) . The r e s i d u a l f u n c t i o n s , as d e f i n e d i n e quations 3.14, a r e a l s o w r i t t e n i n the same manner: & 1 = e + Ae = 0 (3.34a) a a a v ' e + Ae. = 0 ,„ s s (3.34b) These are the equations of mechanical e q u i l i b r i u m t h a t must h o l d everywhere on the c e l l s u r f a c e . Note t h a t by r e q u i r i n g t h e primed q u a n t i t i e s be zero, we are s a t i s f y i n g the f o r c e balance c o n d i t i o n i n the p e r t u r b e d geometry. Here l i e s the reason f o r u s i n g the l i n e a r q u a s i - e l a s t i c approach: For the p i p e t t e a s p i r a t i o n o f a s p h e r i c a l s h e l l ( i e : the s t a r t i n g geometry) wi t h non-zero shear v i s c o s i t y , t h e r e i s no s o l u t i o n c o n s i s t e n t w i t h t h e volume c o n s e r v a t i o n requirement. To a v o i d v i o l a t i n g the i n c o m p r e s s i b i l i t y c o n d i t i o n , the e q u i l i b r i u m e quations must be a p p l i e d t o a s l i g h t l y p e r t u r b e d sphere. T h i s p e r t u r b a t i o n , however, i s not a r b i t r a r y s i n c e i t d e f i n e s an average v e l o c i t y f i e l d i n the time i n t e r v a l At, from which the v i s c o u s s t r e s s e s a r e c a l c u l a t e d . 4 5 The object of the q u a s i - e l a s t i c approach i s to obtain incrementing functions (Ar and Az) for the enti r e c e l l surface. This i s done by seperating the c e l l into two regions: For the portion e x t e r i o r to the suction pipette, equations 3 . 3 4 are used to solve f o r the two functions Ar(s ) and Az(s ). The method to * o o do t h i s w i l l be discussed i n the next section. The problem inside the pipette i s much simpler because A r = 0 . Further, i t can be seen from equation 3 . 7 a that the a x i a l tension i s uniform along the tube. There i s therefore only one force balance equation (in the a x i a l direction) from which A Z ( S q ) can be evaluated. The two solutions are matched at the pipette entrance, subject to the constraint of volume conservation. The c e l l shape (r,z) at time t i s then incremented and the procedure i s repeated again. In t h i s manner, the viscous flow i s approximated by a series of small q u a s i - e l a s t i c displacements. 3.5: IMPLEMENTATION OF NUMERICAL METHOD So f a r , a l l the variables we have introduced (eg: tensions, curvatures, extension r a t i o s , etc.) are continuous i n space. Because the problem i s axisymmetric, they can be written as functions of the instataneous c u r v i l i n e a r distance s alone. An al t e r n a t i v e approach, known as the Lagrangian formulation, i s to express a l l quantities (including s i t s e l f ) i n terms of the i n i t i a l , time-independent c u r v i l i n e a r distance S q . This can be done because "mapping" between the instantaneous space and the i n i t i a l space are assumed to be one-to-one. The next step i s to d i s c r e t i z e the sQ space into a number of g r i d points. A l l continuous functions w i l l now be represented by a seri e s of nodal values located along the meridional curve. A f i r s t d e r i v a t i v e with respect to S q i s approximated by d i v i d i n g the difference between two adjacent nodal values by the l o c a l g r i d s i z e . Such i s the simplest form of numerical d i f f e r e n t i a t i o n c a l l e d the forward-difference formula. In dealing with curvatures, i t i s necessary to evaluate second derivatives of the coordinate variables. These quantities can be represented by applying the forward difference formula to the f i r s t d e rivatives of the variables. Additional equations are then needed to approximate the f i r s t derivatives themselves. Thus, by r e s t r i c t i n g a l l derivatives to be at most f i r s t order (at the expense of additional equations), we can cast the two equilibrium equations (eqns. 3 . 3 4 ) into a f i n i t e difference form that involve nodal values of only two neighboring points. As w i l l be summarized i n appendix C, there are four FDEs ( f i n i t e d i fference equations) i n t o t a l : two from equations 3 . 3 4 and two to represent the f i r s t derivatives of Ar and Az. The four unknowns at each node are Ar, Az, ^ i r and 5 f 4Az. Because the FDEs are written between two neighboring points, l o c a l l y there are four equations that involve eight unknowns (four nodal values at each point). Futher, because the equations are l i n e a r i n the incrementing functions Ar and Az, they can be put into a matrix form. For a g r i d of N points, there w i l l be N-l sets of four algebraic equations i n eight unknowns. These equations can be solved g l o b a l l y i f the four boundary conditions (the number o f boundary c o n d i t i o n s must e q u a l t h e number o f FDEs) a r e p u t i n . The m a t r i x o r g a n i z a t i o n f o r a g e n e r a l s e t o f M d i f f e r e n c e e q u a t i o n s i s shown i n f i g u r e 10. F o r c l a r i t y , t h i s i s done f o r a g r i d w i t h o n l y t h r e e nodes, a l t h o u g h i n t h e a c t u a l i m p l e m e n t a t i o n , N i s t y p i c a l l y s e v e r a l hundred. N i s t h e number o f boundary c o n d i t i o n s a t t h e f i r s t node, and N g i s t h a t f o r t h e end node. The c o n d i t i o n o f N + N = M c s i s r e q u i r e d f o r a u n i q u e s o l u t i o n . I n our c a s e , M=4 and N can be as l a r g e as 500. I t i s o b v i o u s l y i m p r a c t i c a l t o s o l v e a s e t o f 2000 l i n e a r e q u a t i o n s by G a u s s i a n e l i m i n a t i o n o r by any o t h e r d i r e c t means. Appendix D d e s c r i b e s a method, due t o P r e s s e t a l . (1986), t h a t t a k e s f u l l advantage o f t h e s p a r s i t y o f t h e m a t r i x i n f i g u r e 10. S u b r o u t i n e s coded i n FORTRAN language t h a t implement t h e s e methods w i l l a l s o be g i v e n . 3.6: RESULTS AND DISCUSSION The q u a s i - e l a s t i c method can be a p p l i e d t o any s h e l l m a t e r i a l model d i s c u s s e d i n s e c t i o n 3.3. As a f i r s t s t e p , t h e f l o w b e h a v i o u r o f a l i q u i d s h e l l w i l l be i n v e s t i g a t e d . The a l g o r i t h m i s s e t up f o r t h e c o n s t i t u t i v e r e l a t i o n s g i v e n i n e q u a t i o n s 3.19 and 3.20, where X. and ^ a r e t h e v i s c o u s p a r a m e t e r s . By n o r m a l i z i n g w i t h t h e f o l l o w i n g t h r e e q u a n t i t i e s : t h e d i l a t o r y v i s c o s i t y X. , t h e s u c t i o n p r e s s u r e P, and t h e p i p e t t e r a d i u s R , t h e problem has o n l y two d i m e n s i o n l e s s p a r a m e t e r s . They a r e ^/X. and where ^ i s t h e d e v i a t o r i c 4 8 N cxM M x 2 M M x 2 M N s * M Figure 10: Global matrix structure f o r a set of M f i n i t e -difference equations. The dimensions of each block are as l a b e l l e d . For s i m p l i c i t y , t h i s i s done for a g r i d with only three nodal points. Matrix elements that l i e outside of the blocks are zeros. 49 v i s c o s i t y and R the i n i t i a l c e l l radius. The v a l i d i t y of the numerical method i s then checked by comparing the r e s u l t s with that of the a n a l y t i c a l solution for the case -y = 0 (see appendix B). Figure 11 shows the time evolution of the c e l l p r ojection inside the tube for two d i f f e r e n t pipette r a d i i . The instantaneous v e l o c i t y f i e l d s are shown i n figure 12. These p l o t s c l e a r l y show the agreement of the r e s u l t s obtained by three e n t i r e l y d i f f e r e n t means (the a n a l y t i c a l r e s u l t s a c t u a l l y involve two methods). Another feature of the numerical method not shown here i s the that by s e t t i n g yj/-*- to zero, the c e l l shape remains p e r f e c t l y spherical, even a f t e r 30 to 40 time steps. I t i s also i n t e r e s t i n g to look at how the i n i t i a l flow rates vary with pipette radius, as shown i n figure 13 f o r d i f f e r e n t values of ^ / K . Note that i n order to be consistent with actual data reduction, which i s plotted as L/R vs. R /R, P P we are normalizing the flow rates with a time constant defined as (C T = - P - R since i t i s the c e l l radius R that remains unchanged as the parameter R /R v a r i e s . From t h i s p l o t , i t i s evident that very small shearing stresses can have large e f f e c t s on the o v e r a l l flow rates. The c e l l shapes are also strongly affected by the shear v i s c o s i t y , as shown i n figure 14 for R /R = 0.4. I t i s P seen from these computed shapes that c e l l f l a t t e n i n g towards the pipette i s i n d i c a t i v e of the presence of shearing stresses. 5 0 F i g u r e 11: Comparing n u m e r i c a l r e s u l t s t o t h e a n a l y t i c a l s o l u t i o n o f t h e t i m e e v o l u t i o n o f a s u r f a c e f l o w p r o c e s s . 51 F i g u r e 1 2 ( a ) : Comparing t h e n u m e r i c a l l y c a l c u l a t e d r a d i a l v e l o c i t y f i e l d t o t h e a n a l y t i c a l s o l u t i o n . S i s t h e p o s i t i o n a t t h e p i p e t t e e n t r a n c e . 52 F i g u r e 1 2 ( b ) : Comparing t h e n u m e r i c a l l y c a l c u l a t e d a x i a l v e l o c i t y f i e l d t o t h e a n a l y t i c a l s o l u t i o n . S i s t h e p o s i t i o n a t t h e p i p e t t e e n t r a n c e . 53 F i g u r e 13: The i n i t i a l i n - f l o w r a t e i s p l o t t e d as a f u n c t i o n o f t h e d i m e n s i o n l e s s p i p e t t e r a d i u s f o r s e v e r a l v a l u e s o f ">7 / x . 54 F i g u r e 14: C e l l shapes p r e d i c t e d by t h e n u m e r i c a l method f o r v a r i o u s v a l u e s o f . The i n i t i a l c e l l r a d i u s i s 2.5 R , and t h e c e l l p r o j e c t i o n s a r e 3 R i n a l l t h r e e c a s e s . 5 5 IV. SUMMARY AND DISCUSSION The work i n t h i s t h e s i s i s an attempt t o b e t t e r u n d e r s t a n d t h e p i p e t t e a s p i r a t i o n o f a l i q u i d body w i t h a d i f f e r e n t i a t e d c o r t i c a l s h e l l i n t h e low R e y n o l d s number s i t u a t i o n . A l t h o u g h p r e s e n t e d as a m a t h e m a t i c a l problem, t h e a n a l y s i s i s m o t i v a t e d by o u r i n v e s t i g a t i o n i n t o t h e m e c h a n i c a l p r o p e r t i e s o f g r a n u l o c y t i c w h i t e b l o o d c e l l s . As mentioned i n t h e i n t r o d u c t i o n , because t h e s e c e l l s do n o t show any e l a s t i c l i m i t t o d e f o r m a t i o n , t h e y can be t r e a t e d , t o f i r s t o r d e r , as l i q u i d b o d i e s . Based on b i o l o g i c a l c o n s i d e r a t i o n s , one a n t i c i p a t e s two p o s s i b l e r e g i o n s i n w h i c h d i f f e r e n t l e v e l s o f v i s c o u s d i s s i p a t i o n may o c c u r : t h e y a r e i n t h e c e l l p e r i p h e r y and i n t h e i n t e r i o r . Two continuum m e c h a n i c a l problems a r e posed t h a t model t h e c e l l as h a v i n g i t s v i s c o u s d i s s i p a t i o n dominated e n t i r e l y i n each o f t h e two r e g i o n s . Only f i r s t o r d e r c o n s t i t u t i v e r e l a t i o n s a r e used i n b o t h c a s e s . These a r e e q u a t i o n s t h a t r e l a t e t h e v i s c o u s s t r e s s e s t o t h e r a t e s o f d e f o r m a t i o n by a c o n s t a n t o f p r o p o r t i o n a l i t y . F o r t h e i n t e r i o r model, t h i s amounts t o s o l v i n g t h e f a m i l i a r c r e e p i n g m o t i o n e q u a t i o n s i n a sphere s u b j e c t t o p r e s c r i b e d s t r e s s boundary c o n d i t i o n s . N o t a b l e r e s u l t s a r e t h e v a r i a t i o n i n f l o w r a t e s as a f u n c t i o n o f p i p e t t e r a d i u s as shown i n f i g u r e 5, and t h e f a c t t h a t t h e n e t p r e s s u r e drop i s c o n c e n t r a t e d i n t h e v i c i n i t y o f t h e o r i f i c e e n t r a n c e . As s u c h , f l o w r a t e s i n t o t h e t u b e w i l l be i n s e n s i t i v e t o t h e geometry o f t h e segment e x t e r i o r t o t h e p i p e t t e - i m p l y i n g t h e p o s s i b i l i t y o f r e p r e s e n t i n g t h e f l o w 56 p r o c e s s as a p r e s s u r e drop a t t h e p i p e t t e e n t r a n c e t h a t i s p r o p o r t i o n a l t o t h e i n - f l o w r a t e . I n s o l v i n g t h e d r o p l e t p r o b l e m , v i s c o u s d i s s i p a t i o n (and hence t h e p r e s s u r e drop) i n s i d e t h e s u c t i o n p i p e t t e i s n e g l e c t e d . T h i s a s s u m p t i o n , a l t h o u g h appears r e a s o n a b l e based on e x p e r i m e n t a l o b s e r v a t i o n ( i e : t h e f a c t t h a t t h e c e l l s l i d e s f r e e l y on t h e p i p e t t e w a l l ) , s h o u l d be examined c r i t i c a l l y e s p e c i a l l y i n c a s e s o f membrane d r i v e n f l o w s . T h i s can be done by s o l v i n g t h e c r e e p i n g m o t i o n e q u a t i o n s i n s i d e t h e c y l i n d e r w i t h e i t h e r s t r e s s - o r v e l o c i t y -boundary c o n d i t i o n s on t h e s u r f a c e a d j a c e n t t o t h e p i p e t t e w a l l . The c o r t i c a l model i s f u n d a m e n t a l l y d i f f e r e n t from t h e d r o p l e t model because energy d i s s i p a t i o n i s c o n f i n e d w i t h i n t h e s u r f a c e p l a n e . By n e g l e c t i n g b e n d i n g moments i n t h e c o r t i c a l l a y e r (as j u s t i f i e d by t h e s h a r p bend around t h e p i p e t t e edge i n f i g u r e 1 ) , t h e t e n s i o n r e s u l t a n t s must a c t t a n g e n t t o t h e d e f o r m i n g s u r f a c e , t h u s s i m p l i f y i n g t h e e q u a t i o n s o f m e c h a n i c a l e q u i l i b r i u m c o n s i d e r a b l y . These e q u a t i o n s a r e s o l v e d by a q u a s i - e l a s t i c n u m e r i c a l method w h i c h a c c o u n t s f o r t h e v i s c o u s s t r e s s e s b o t h i n s i d e and e x t e r n a l t o t h e s u c t i o n p i p e t t e . S a t i s f a c t o r y agreement i s o b t a i n e d between t h e n u m e r i c a l r e s u l t s and t h e a n a l y t i c a l s o l u t i o n f o r t h e c a s e T g = 0 (see f i g u r e s 11,12,13). The n u m e r i c a l p r o c e d u r e i s t h e n e x t e n d e d t o f l o w s i t u a t i o n s where s h e a r v i s c o s i t y can be n o n - z e r o . A k i n e m a t i c consequence o f s u r f a c e f l o w i n t o a t u b e i s t h a t v e r y l a r g e m agnitudes o f i n - p l a n e s h e a r o c c u r ( i e : s q u a r e s become h i g h l y e x t e n d e d r e c t a n g l e s ) , e s p e c i a l l y i n r e g i o n s n e a r t h e p i p e t t e 57 e n t r a n c e . T h i s i s r e f l e c t e d i n t h e s i g n i f i c a n t changes i n f l o w r a t e s w i t h t h e i n t r o d u c t i o n o f v e r y s m a l l v a l u e s o f s h e a r v i s c o s i t y i n t o t h e c o n s t i t u t i v e r e l a t i o n s , as shown i n f i g u r e 13. The f l o w r a t e s i n f i g u r e 5 and 13 a r e n o r m a l i z e d t o be c o n s i s t e n t w i t h a c t u a l d a t a r e d u c t i o n , w h i c h i s p l o t t e d as d ( L / R p ) / d ( t A P ) . I n comparing t h e two t h e o r e t i c a l p l o t s , i t i s e v i d e n t t h a t t h e e n t r y f l o w r a t e s f o r t h e d r o p l e t model has l e s s v a r i a t i o n w i t h p i p e t t e r a d i u s . T h i s may be a t t r i b u t e d t o t h e l o a c l i z a t i o n o f t h e v i s c o u s d i s s i p a t i o n r e g i o n n e a r t h e o r i f i c e e n t r a n c e , w h i c h , f o r s m a l l v a l u e s o f R /R, becomes i n s e n s i t i v e t o t h e e x t e r n a l b o u n d a r i e s . I t i s t h e r e f o r e p o s s i b l e t o i d e n t i f y t h e dominant r e g i o n o f v i s c o u s d i s s i p a t i o n by p e r f o r m i n g a s p i r a t i o n t e s t s w i t h d i f f e r e n t s i z e d p i p e t t e s . T h i s i n f o r m a t i o n can be combined w i t h u l t r a s t r u c t u r a l e v i d e n c e i n c r e a t i n g a more complete p i c t u r e o f b i o l o g i c a l c e l l s . 58 APPENDIX A: SOLUTION TO CREEPING MOTION EQUATIONS I n t h i s a p p e n d i x , we propose t o s o l v e t h e c r e e p i n g f l o w e q u a t i o n s 2 -•> v (v x v ) = 0 (2.10) i n t h e c a s e o f a s p h e r i c a l newtonian d r o p l e t s u b j e c t t o t h e s t r e s s boundary c o n d i t i o n s o (C) PP v w % B CO p=R p=R = «(f) = 0 (2.12) (2.13) The a p p l i e d s t r e s s o( ( L,) i s d e f i n e d as AP ; -1 < f < C P / p P p * (2.11) w h i c h r e p r e s e n t s a p i p e t t e s u c t i o n p r e s s u r e . The g e n e r a l a x i s y m m e t r i c s o l u t i o n t o e q u a t i o n 2.10 i s d e v e l o p e d by Happel and B r e n n e r (1973). I n s p h e r i c a l components, t h e v e l o c i t y f i e l d s a r e g i v e n by v o - " I < An » n ~ 2 + C n ^ Pn-1<£> ( A . l ) n=2 59 V 6 = 1 [n A n p I 1 " 2 + < n + 2> C n p n n=2 s i n 0 (A.2) where t h e c o n s t a n t s A and C a r e a r b i t r a r y . P (/ ) i s t h e n n 1 n v ^ ' Legendre p o l y n o m i a l o f degree n, and T-n(£ ) i s t h e Gagenbaur p o l y n o m i a l g i v e n by e q u a t i o n 2.15. By c o m b i n i n g t h i s s o l u t i o n w i t h e q u a t i o n 2.9, an e x p r e s s i o n f o r t h e p r e s s u r e f i e l d i s o b t a i n e d . S i n c e 2.9 i n v o l v e s t h e g r a d i e n t o f p, t h e p r e s s u r e d i s t r i b u t i o n can o n l y be d e t e r m i n e d up t o an a d d i t i v e c o n s t a n t (say ff ) : D = . „ \ f 2(2n+l) „ n-1 1 P 7 7 2 [ "Vr C n P p n - l ^ ) + ff (A.3) n=2 The v i s c o u s s t r e s s e s can be e x p r e s s e d i n terms o f t h e p r e s s u r e and v e l o c i t y f i e l d s a c c o r d i n g t o eqn. 2.6, w h i c h i s w r i t t e n i n t e n s o r form. I n terms o f s p h e r i c a l p o l a r c o o r d i n a t e s , we have (see Landau and L i f s h i t z , 1982) dv °pp = " P + 2 (A-4) dp T 3v 6V v a = 7 j ( I _ _ P _ + _ § - - - 9 - ) (A.5) P 0 p ae dP P By s u b s t i t u t i n g A . l and A . l i n t o A.5, and u s i n g t h e i d e n t i t y d P n - l ( C ) J n ( C ) n 1 = - n ( n - l ) — (A.6) d9 s i n 9 t h e f r e e - s l i p boundary c o n d i t i o n (eqn. 2.13) becomes 60 2n(n-2) R n" 3 A + 2(n 2-l) R n _ 1 C = 0 ; n > 2 n n (A.7) For n=2, we must have C = 0 (A.8) while A 2 can be a r b i t r a r y . In a s i m i l a r fashion, we substitute equations A . l ,A.3 into A.4 and equate to the external normal stresses according to 2.12. The r e s u l t i n g expression i s I [ i (»-2, R - 3 A n + 2 B _ = | 2n -6n-2_ Rn-1 c n Pn-1^> n=2 1 V L IT + a(C) (A.9) The r i g h t hand side of A.9 can be expanded as a sum of Legendre polynomials with c o e f f i c i e n t s a n : I an Pn-l ' f > S "I [" + «<"] (A.10) n=l Using the orthogonal property,the c o e f f i c i e n t s a n are given by n 1 2n-l T? 2 1 J" -1 Pn-1^) ^ (A.11) The se r i e s i n A.9 excludes terms associated with n=l to avoid i n f i n i t e v e l o c i t i e s at the poles. Accordingly, a^ ^ must be set 61 to zero. This r e s u l t s i n the expression 217 + AP(1+C ) - X = 0 (A.12) hr Also, because C2=0, we see from A.9 that a 2 has to vanish as wel l . This i s i n fact equivalent to the balance of a x i a l forces on the spherical body, which leads to the equation C P 2 - 1 X = AP —& (A. 13) 2C p + e Combining equations A.12 and A.13, the integration constant Tf can be rela t e d to the suction pressure A P by = AP_ d + C p ) ( l + C p + e ) 2 2f + e (A.14) For n > / Z , A. 11 can be integrated using the r e l a t i o n P - P S P n , dC = — ^ + C (A. 15) 2n -1 to obtain V - Pn-2<Vj ln - " W { A P [ P n [p n(C p+e) - P n . 2 ( C p + e ) - P n (C p ) + P n . 2 ( C p ) ] } ( A . x_ 16) The normal stress boundary condition can now be matched at each separate harmonic according to A.9. For n^-3, we have 2(n-2) R n" 3 A + 2 n 2 - 6 n ~ 2 Rn"! c = n n-1 * u n 62 (A.17) Equations A.17 and A.7 can now be solved simultaneously. In terms of the Legendre c o e f f i c i e n t s , the constants A and C are: n n A = (n+1)(n-1) 2 a n ^ n 2(n-2)(2n 2+l) R n~ 3 n ^ 3 (A.18) C n(n-l) a n n 2(2n 2+l) R 1 1 - 1 Substituting A.18 back into A . l - A.3, the f i n a l solutions for p = R are 00 v = - A C - ^ (n-1)(2n-l) , r ) 2 C n £ 3 2(n-2)(2n 2 +l) R a n P n - l ( C ) (A.19) CO v = A s i n 9 + ^ 3n(n-l) I n ( C ) v e A 2 s i n 9 + ^ 2 ' 7 R a n — (A. 20) 3 2(2n^+l)(n-2 n s i n 9 v ; n=3 < 2 n + 1> (A.21) where the a r b i t r a r y constant A 2 can be interpreted as an a x i a l v e l o c i t y superimposed onto the e x i s t i n g v e l o c i t y f i e l d . This i s allowed because neither the pressure nor the rate of s t r a i n tensor are affected. 63 APPENDIX B: PIPETTE ASPIRATION OF CORTICAL SHELL IN THE  ABSENCE OF IN-PLANE SHEARING STRESSES The problem of the aspiration of a l i q u i d membrane can be solved a n a l y t i c a l l y when there are no surface shear stresses. In t h i s case, the co n s t i t u t i v e r e l a t i o n i s T = K Va (3.19) while ^ ,the shear v i s c o s i t y i n equation 3.20, i s i d e n t i c a l l y zero. I t i s important to recognize that since there are no shearing stresses, the c e l l geometry must remain spherical at a l l times. There are act u a l l y two seperate problems involved, they are: 1) the solution of the instantaneous v e l o c i t y f i e l d f o r a given geometry, and 2) the time evolution of the c e l l . These w i l l be dealt with by two d i f f e r e n t methods. To obtain expressions for the v e l o c i t y p r o f i l e , we f i r s t rewrite the d i l a t o r y s t r a i n rate i n terms of the v e l o c i t y components (see Evans and Skalak, 1980): ^ v s 1 1 v = -dr + -fr + vnHr" + HH (B'1) a a s r n Rm * where v and v are the tangential and normal v e l o c i t y s n components, respectively. Because the c e l l i s spherical with radius R, equation B.l can be written, using the r e l a t i o n s=R9, as 64 V a = RT I f re M <Vs S i n 9> + (B.2) Since the c e l l i s to remain spherical, the normal v e l o c i t y , r e l a t i v e to the centre of the sphere, must be a constant (say * R). We substitute t h i s , along with equation 3.19 into B.2 to obtain 3 ! (v s i n 9) = a s i n 9 (B.3) aa s where the quantity a =" 2_5_ - 2 R <B-4> K i s independent of 0. Equation B.3 can now be integrated to obtain the v e l o c i t y f i e l d . In a reference frame fix e d at the centre of the sphere v 1 ~ cos 9 (B.5a) s s i n 9 v n = R (B.5b) Using the transformations vr = v g cos 9 + v n s i n 9 (B.6a) v = v s i n 9 - v cos 9 (B.6b) z s n 65 t h e v e l o c i t y f i e l d i n B.5 can be r e w r i t t e n i n terms o f t h e r a d i a l and a x i a l components. A l s o , t o be c o n s i s t e n t w i t h t h e n u m e r i c a l s o l u t i o n , an a x i a l v e l o c i t y i s added t o t h e v e l o c i t y f i e l d so t h a t t h e base o f t h e sphere (s=0) remains s t a t i o n a r y . The f i n a l r e s u l t s a r e V r " a °°S 6lin"e C° S 9 ' + B « n 8 (B.7.) v„ = (a+R) (1 - cos 9) (B.7b) z The two p a r a m e t e r s a and R a r e u n i q u e l y r e l a t e d i f volume c o n s e r v a t i o n i s a c c o u n t e d f o r . The t i m e e v o l u t i o n o f t h e c e l l can be e x p r e s s e d as a c l o s e d form s o l u t i o n by r e a r r a n g i n g e q u a t i o n 3.19: da_ = T (1+a) ( B > 8 ) d t K. where T and ot a r e b o t h u n i f o r m o v e r t h e d e f o r m i n g s u r f a c e . The o b j e c t i s t o r e l a t e T t o 66, and hence i n t e g r a t e e q u a t i o n B.8. C o n s i d e r an a s p i r a t e d c e l l w i t h tounge l e n g t h L and a s p h e r i c a l segment o f r a d i u s R, as shown i n f i g u r e 15. The volume and s u r f a c e a r e a i n t h i s c o n f i g u r a t i o n a r e g i v e n by V = , R p 2 L + | R 3 ( l + C p ) 2 ( 2 - C p ) + § . R p 3 ( B - 9 » 66 F i g u r e 15: D e f i n i t i o n o f v a r i o u s d i m e n s i o n s f o r t h e problem posed i n a p p e n d i x B. A = 2TT R L + 2TT R 2 ( i + c ) + 2TT R 2 P P P 67 (B.10) where r = C O S 6 L p P ( B . l l ) i s a p o s i t i v e q u a n t i t y . G i v e n V and A, £ can be s o l v e d f o r from t h e f o l l o w i n g c u b i c e q u a t i o n : 1+c 2 r 3 _ ( 1 + c ) 2 f 2 2 k p 2 k p + 3 (3+c) _ (1+c)(5+c) _ c 2 P -2 - 0 where t h e q u a n t i t y (B.12) 6V - 3R A c = f — + 1 (B.13) 2, R p 3 f o r c o n s t a n t volume, i s a f u n c t i o n o f A a l o n e . Thus, g i v e n A ( o r (X. ) , we can s o l v e f o r X u s i n g B.12. The q u a n t i t i e s R and L f o l l o w i m m e d i a t e l y : L + R - - h - . ^ ' ^ P ' ( B . 1 5 ) By r e q u i r i n g t h e m e r i d i o n a l t e n s i o n t o be c o n t i n u o u s a c r o s s t h e p i p e t t e e n t r a n c e , t h e i s o t r o p i c t e n s i o n can be e x p r e s s e d i n terms o f t h e s u c t i o n p r e s s u r e P and t h e i n s t a n t a n e o u s r a d i u s R 68 as _ P R T = 2(1 - R p/R) (B.16) S i n c e t h e s u c t i o n p r e s s u r e i s c o n s t a n t , T i s o n l y dependent on R. Thus, g i v e n t h e i n i t i a l c e l l s i z e (which e s t a b l i s h e s t h e t o t a l volume) and t h e i n s t a n t a n e o u s v a l u e o f oC , we can d e t e r m i n e t h e c o n s t a n t c i n B.13. Z, ^  i s t h e n s o l v e d f o r from e q u a t i o n B.12, from w h i c h we can o b t a i n R, L, and T. I n t h i s manner, e q u a t i o n B.8 can be i n t e g r a t e d n u m e r i c a l l y (eg: by t h e Runge-Kutta method) t o v e r y h i g h a c c u r a c y . 69 A P P E N D I X C : F I N I T E D I F F E R E N C E EQUATIONS A s e t o f f o u r f i n i t e d i f f e r e n c e e q u a t i o n s i s w r i t t e n between e v e r y p a i r o f n o d a l p o i n t s . A t each node, s a y node number i ( i = l , 2 , . . . , N , w i t h i = l c o r r e s p o n d i n g t o s=0), t h e r e a r e f o u r unknowns: A r ( i ) , A z ( i ) , ^ A r ( i ) , ^ ; A z ( i ) . The f i r s t two d i f f e r e n c e e q u a t i o n s a r e t o a p p r o x i m a t e t h e d e r i v a t i v e s o f A r and Az: 9 i A r ( i ) - A r ( i - l ) •=— Ar ( l ) = — » — 1 — r 1 1 dso h i ( C D * A z ( i ) = A z ( i ) - A z ( i - l ) 5 s o h i ( C 2 ) where i ra n g e s from 2 t o N, and h^ i s t h e l o c a l g r i d s p a c i n g d e f i n e d as h i - s o ( i ) " s o ( i - 1 ) ( C * 3 ) The o t h e r two d i f f e r e n c e e q u a t i o n s a r e based on e q u a t i o n s 3.34. By e x p r e s s i n g t h e e i g h t unknowns a t t h e two nodes c o l l e c t i v e l y as a ^ ( k = l , 2 , . . . , 8 ) , e q u a t i o n 3.34 can be r e w r i t t e n as a d a k a a k £ a (C4.a) g f - A e s • a k = - e s (C4.b) T h i s i s v a l i d because A£oc and A£ s a r e l i n e a r i z e d t o c o n t a i n o n l y t h e f i r s t powers o f t h e s o l u t i o n v e c t o r . From t h e d e f i n i t i o n s o f t h e r e s i d u a l f u n c t i o n s (eqns. 3.14), i t f o l l o w s t h a t a 4AT + o- (2R./R - 3) AR . + a R 2 A(—^-) n x 9' m ' 9 n 9 v R ' m (C5.a) Ae 4AT s + a n ( l - 2R^/R m) AR^ - a ^ 2 A m (C5.b) The c u r v a t u r e s and t h e i r i n c r e m e n t s a r e g i v e n as f o l l o w s : R 9 s i n G (C.6) AR 9 s i n 6 A r _ r cos 9 A e s i n 9 (C.7) 1/R. ( dr 8 z dz a r m . 3 d s ^ _ 2 X o os m o a s o as 2 o (C.8) A m -3 R X m m AX + m a Ar a z a r a Az m as a 2 o as o as _ 2 o as o a Az a^r as a 2 o ds o az a*Ar  5 s o as 2 (C.9) F o r t h e s t r e s s e s , t h e forms o f T and T depend on t h e p a r t i c u l a r m a t e r i a l model chosen. As a s i m p l e example, c o n s i d e r t h e t e n s i o n r e s u l t a n t i n a l i q u i d (eqns. 3.19-3.20): ( C I O ) T = 0 s ( C . l l ) 71 A T = AT" ^ n / N a + A W (C.12) A T s = A ? " ( A V X m " A W (C.13) The t e n s i o n s T and T g a r e z e r o because t h e y r e p r e s e n t e l a s t i c s t r e s s e s a c c u m u l a t e d up t o t h e p r e s e n t c o n f i g u r a t i o n . A l l t h e i n c r e m e n t s i n t r o d u c e d above can be e x p r e s s e d i n terms o f t h e b a s i c d e f o r m a t i o n v a r i a b l e s u s i n g t h e f o l l o w i n g r e l a t i o n s : AA_ = cos G m d Ar 9s + s i n 0 a Az 3s (C.14) (C.15) A9 = i , a a Az — cos 6 m o a 2 A r as ( i ) _1_ h, 3 Ar a s _ ( i ) 0 9 A r . s i n 6 —= ) 3s _ ' o (C.16) (C.17) a 2 A z as ( i ) 1_ h. 1 L a_Az_ a s _ ( 1 } O J (C.18) S i n c e t h e r e a r e f o u r e q u a t i o n s , t h e r e must a l s o be f o u r boundary c o n d i t i o n s t o e n s u r e a u n i q u e s o l u t i o n . I f we choose t o l o c a t e t h e r e f e r e n c e frame a t t h e base o f t h e c e l l (s=0), two boundary c o n d i t i o n s i m m e d i a t e l y f o l l o w : A r ( l ) = o (C.19) A z ( l ) = 0 72 (C.20) A l s o , 9 a t t h e base must be always z e r o . The r e q u i r e m e n t o f Ae v a n i s h i n g a t s=0 l e a d s t o a n o t h e r boundary c o n d i t i o n a t t h e f i r s t node: a A z - ( i ) = 0 dsQ" ' (C.21) A t t h e o t h e r end o f t h e g r i d , a t a n g e n t i a l v e l o c i t y ^ can be p r e s c r i b e d . U s i n g t h e t r a n s f o r m a t i o n r e l a t i o n v = v cos 0 + v s i n 0 (C.22) s r z t h e l a s t boundary c o n d i t i o n becomes Ar (N) cos 0 + Az (N) s i n G - v e P At = 0 (C.23) I n a c t u a l i m p l e m e n t a t i o n , \ j f i s an i t e r a t i v e p arameter t h a t i s u s e d t o s a t i s f y t h e volume c o n s e r v a t i o n c o n d i t i o n . 7 3 APPENDIX D: MATRIX SOLUTION TO F I N I T E DIFFERENCE EQUATIONS F i g u r e 10 shows t h e m a t r i x s t r u c t u r e o f a t y p i c a l f i n i t e d i f f e r e n c e scheme, where a system o f M d i f f e r e n t i a l e q u a t i o n s a r e a p p r o x i m a t e d by d i f f e r e n c e f o r m u l a s t h a t i n v o l v e n o d a l v a l u e s o f two a d j a c e n t p o i n t s . The method o f s o l u t i o n , as o u t l i n e d by P r e s s e t a l . (1986), i s t o reduce t h e g i v e n m a t r i x t o a s p e c i a l upper t r i a n g u l a r form, as shown i n f i g u r e 16. To do t h i s , o n l y m a t r i x elements from two b l o c k s need be m a n u p i l a t e d a t any t i m e . We s t a r t w i t h t h e t o p N xM b l o c k i n c f i g u r e 10. T h i s b l o c k , a l o n g w i t h t h e f i r s t N el e m e n t s on t h e c r i g h t hand s i d e , i s Gauss reduced u n t i l t h e f i r s t N c columns form an i d e n t i t y m a t r i x . A t t h e end o f t h i s p r o c e s s , o n l y t h e l a s t N columns and t h e c o r r e s p o n d i n g p o r t i o n o f t h e r i g h t hand v e c t o r need be s t o r e d . T h i s i n f o r m a t i o n i s t h e n used t o e l i m i n a t e t h e f i r s t N columns i n t h e second b l o c k , w h i c h l i e s c ' d i r e c t l y u n d e r n e a t h t h e i d e n t i t y m a t r i x . The r e m a i n i n g elements o f t h i s b l o c k a r e t h e n Gauss reduced u n t i l t h e n e x t M columns (columns N + 1 t o N +M) form an i d e n t i t y m a t r i x . A g a i n , o n l y t h e l a s t N columns, and t h e c o r r e s p o n d i n g p o r t i o n o f t h e r i g h t hand s i d e a r e s t o r e d . T h i s p r o c e d u r e i s r e p e a t e d u n t i l we g e t t o t h e l a s t b l o c k , w h i c h has d i m e n s i o n s N xM. As b e f o r e , t h e f i r s t N s c columns a r e e l i m i n a t e d u s i n g i n f o r m a t i o n from t h e p r e v i o u s b l o c k . The r e m a i n i n g p a r t o f t h e b l o c k i s t h e n r e d u c e d t o an i d e n t i t y m a t r i x , t h u s a t t a i n i n g t h e d e s i r e d form i n f i g u r e 16. Note t h a t a t t h i s s t a g e , o n l y t h e s u b - b l o c k s l a b e l l e d "S", and t h e a l t e r e d r i g h t hand s i d e , a r e s t o r e d . From h e r e , t h e 7 4 N c t M 1 H N s h -I I I I M 1 N. F i g u r e 16: The d e s i r e d form o f t h e upper t r i a n g u l a r m a t r i x w h i c h m i n i m i z e s s t o r a g e space f o r t h e f i n i t e -d i f f e r e n c e s o l u t i o n scheme. Only b l o c k s l a b e l l e d 1 1S" a r e s t o r e d . The s q u a r e b l o c k s l a b e l l e d " I " a r e i d e n t i t y m a t r i c e s , and a l l t h e r e m a i n i n g e n t r i e s a r e z e r o s . 75 s o l u t i o n f o l l o w s q u i c k l y by back s u b s t i t u t i o n . S u b r o u t i n e s coded i n FORTRAN a r e g i v e n w h i c h implement t h e above t a s k s . GJPP p e r f o r m s Gauss-Jordan e l i m i n a t i o n ( w i t h p a r t i a l p i v o t i n g ) on a g i v e n m a t r i x u n t i l an i d e n t i t y m a t r i x i s formed. The e l i m i n a t i o n o f t h e f i r s t N columns i n t h e sub-c b l o c k s i s done by s u b r o u t i n e REDUCE. UPTRI i s a d r i v e r r o u t i n e w h i c h u s e s GJPP and REDUCE t o form t h e upper t r i a n g u l a r m a t r i x shown i n f i g u r e 16. The u s e r has t o s u p p l y s u b r o u t i n e s BC1, FDE, and BC2, w h i c h g e n e r a t e t h e f i r s t (N xM) , t h e i n t e r m e d i a t e (Mx2M), and t h e l a s t (N xM) b l o c k s i n f i g u r e 10, r e s p e c t i v e l y . The f i n a l s o l u t i o n i s o b t a i n e d u s i n g BKSUB w h i c h p e r f o r m t h e n e c e s s a r y b a c k - s u b s t i t u t i o n s . 76 SUBROUTINE GJPP(A,MA,NA,NCI,IP) C I n p u t i s m a t r i x A o f di m e n s i o n s MA by NA (NA.GE.MA+NC1). C The f i r s t NCI columns a r e i g n o r e d . The r e s t o f t h e m a t r i x C i s Gauss r e d u c e d ( w i t h p a r t i a l p i v o t i n g ) u n t i l t h e n e x t C MA columns become an i d e n t i t y m a t r i x . IMPLICIT REAL*8(A-H,O-Z) DIMENSION A ( 4 , 9 ) , I P ( 4 ) , S ( 4 ) DO 20 1=1,MA S ( I ) = 0 . DO 10 J=1,MA IF(DABS(A(I,J+NC1)).GT.S(I)) S(I)=DABS(A(I,J+NC1)) 10 CONTINUE IF(S(I).EQ.0.D0) GOTO 80 20 I P ( I ) = I DO 70 ID=1,MA JD=ID+NC1 BIG=0. DO 30 I=ID,MA DUM=DA B S ( A ( I P ( I ) , J D ) / S ( I P ( I ) ) ) IF(DUM.LE.BIG) GOTO 30 BIG=DUM IMAX=I 30 CONTINUE IF(BIG.EQ.O.DO) GOTO 80 IDUM=IP(ID) IP(ID)=IP(IMAX) IP(IMAX)=IDUM DUM=A(IP(ID),JD) IF(DUM.EQ.l.DO) GOTO 45 DO 40 J=JD,NA 40 A ( I P ( I D ) , J ) = A ( I P ( I D ) , J ) / D U M 45 DO 60 1=1,MA IF ( ( I . E Q . I P ( I D ) ) . O R . ( A ( I , J D ) . E Q . 0 . D 0 ) ) GOTO 60 DUM=A(I,JD) DO 50 J=JD,NA 50 A ( I , J ) = A ( I , J ) - D U M * A ( I P ( I D ) , J ) 60 CONTINUE 70 CONTINUE GOTO 100 80 WRITE(*,90) 90 FORMAT(' M a t r i x s i n g u l a r i n GJPP, program t e r m i n a t e d ' ) STOP 100 CONTINUE RETURN END C SUBROUTINE REDUCE(A,B,NE,NCI,IPT,NPTS) IMPLICIT REAL*8(A-H,0-Z) DIMENSION A(4,9),B(4,2,510) NS=NE-NC1 IF(IPT.GT.NPTS) GOTO 10 MA=NE 7 7 NA=2*NE+1 GOTO 20 10 MA=NS NA=NE+1 2 0 ID0=NS IF(IPT.EQ.2) ID0=0 DO 50 J=1,NC1 ID=ID0+J DO 40 1=1,MA IF(A(I , J ) . E Q . 0 . D 0 ) GOTO 40 DO 30 K=1,NS 30 A(I,NC1+K)=A(I,NC1+K)-A(I,J)*B(ID,K,IPT-1) A(I,NA)=A(I,NA)-A(I,J)*B(ID,NS+1,IPT-1) 40 CONTINUE 50 CONTINUE RETURN END C SUBROUTINE UPTRI(B,NE,NC1,NPTS) IMPLICIT REAL*8(A-H,0-Z) DIMENSION A ( 4 , 9 ) , B ( 4 , 2 , 5 1 0 ) , I P I V ( 4 ) DIMENSION S0(510),R0(510),R(510),Z(510) NS=NE-NC1 NS1=NS+1 10=0 NA=2*NE+1 NABC=NE+1 CALL BC1(A,NE,NC1) CALL GJPP(A,NC1,NABC,I0,IPIV) DO 10 1=1,NCI DO 10 J=1,NS1 10 B ( I , J , 1 ) = A ( I P I V ( I ) , N C 1 + J ) DO 30 IPT=2,NPTS CALL FDE(A,NE) CALL REDUCE(A,B,NE,NCI,IPT,NPTS) CALL GJPP(A,NE,NA,NC1,IPIV) DO 20 1=1,NE DO 20 J=1,NS1 20 B(I,J,IPT)=A(IPIV(I),NC1+NE+J) 30 CONTINUE IPT=NPTS+1 CALL BC2(A,NE,NC1) CALL REDUCE(A,B,NE,NCI,IPT,NPTS) CALL GJPP(A,NS,NABC,NC1,IPIV) DO 40 1=1,NS 40 B(I+NC1,1,1)=A(IPIV(I),NABC) RETURN END C SUBROUTINE BKSUB(B,NE,NCI,NPTS) C S o l u t i o n X ( I ) a t t h e J - t h g r i d p o i n t i s s t o r e d i n B ( I , 1 , J ) IMPLICIT REAL*8(A-H,0-Z) DIMENSION B(4,2,510),X(2) NS=NE-NC1 NS1=NS+1 DO 10 1=1,NS 10 X(I)=B(I+NC1,1,1) DO 50 IPTDUM=2,NPTS IPT=NPTS+2-IPTDUM DO 30 IDUM=1,NE I=NE+1-IDUM DUM=0. DO 20 J=1,NS 20 DUM=DUM+X(J) * B ( I , J , IPT) 30 B(I,NS1,IPT)=B(I,NS1,IPT)-DUM DO 40 1=1,NS 40 X(I)=B(I,NS1,IPT) 50 CONTINUE DO 70 IDUM=1,NC1 I=NC1+1-IDUM DUM=0. DO 60 J=1,NS 60 DUM=DUM+X(J)*B(I,J, 1) 70 B(I,NS1,1)=B(I,NS1,1)-DUM DO 80 1=1,NS 80 B(I+NC1,1,NPTS)=B(I+NC1,1,1) DO 90 1=1,NCI 90 B(I,1,1)=B(I,NS1,1) DO 120 IPT=2,NPTS DO 100 1=1,NS 100 B(I+NC1,1,IPT-1)=B(I,NS1,IPT) DO 110 1=1,NCI 110 B(I,1,IPT)=B(I+NS,NS1,IPT) 120 CONTINUE RETURN END 79 LIST OF REFERENCES Amato P h i l i p A., Unanue E m i l R., T a y l o r D. L a n s i n g . , 1983 D i s t r i b u t i o n o f A c t i n i n S p r e a d i n g Macrophages: A Comparative Study on L i v i n g and F i x e d C e l l s The J o u r n a l o f C e l l B i o logy,96:750-761 B e s s i s M., 1973 L i v i n g B l o o d C e l l s and t h e i r U l t r a s t r u c t u r e S p r i n g e r , B e r l i n C h e i n Shu, Sung K u o - L i P a u l , 1984 E f f e c t o f C o l c h i c i n e on V i s c o e l a s t i c P r o p e r t i e s o f N e u t r o p h i l s B i o p h y s i c a l J o u r n a l 46:383-386 Evans Evan A., 1973 A New M a t e r i a l Concept f o r t h e Red C e l l Membrane B i o p h y s i c a l J o u r n a l 13:926-940 Evans Evan A., Hochmuth R. M., 1977 A S o l i d - L i q u i d Composite Model o f t h e Red C e l l Membrane J o u r n a l o f Membrane B i o l o g y 30:351 Evans Evan A., S k a l a k R i c h a r d , 1980 M e c h a n i c s and Thermodynamics o f Biomembranes CRC P r e s s , Roca Raton, F l o r i d a Evans Evan A., Kukan B., 1984 L a r g e D e f o r m a t i o n , Recovery a f t e r D e f o r m a t i o n , and A c t i v a t i o n o f G r a n u l o c y t e s B l o o d 64:1028-1035 Evans Evan A., Needham D., 1986 P h y s i c a l P r o p e r t i e s o f L i p i d B i l a y e r Membranes: C o h e s i o n , E l a s t i c i t y , and C o l l o i d a l I n t e r a c t i o n s J o u r n a l o f P h y s i c a l C h e m i s t r y ( s u b m i t t e d ) Fung Y. C., 1965 F o u n d a t i o n s o f S o l i d M echanics P r e n t i c e - H a l l , Englewood C l i f f s , New J e r s e y Happel J o h n , B r e n n e r Howard, 1973 Low R e y n o l d s Number Hydrodynamics P r e n t i c e - H a l l , Englewood C l i f f s , New J e r s e y Kwok R., Evans Evan A., 1981 T h e r m o e l a s t i c i t y o f L a rge L e c i t h i n B i l a y e r V e s i c l e s B i o p h y s i c a l J o u r n a l 35:637-652 Landau L. D., L i f s h i t z E. M., 1982 F l u i d M e c h a n i c s Pergamon P r e s s , O x f o r d 80 M i t c h i s o n J . M., Swann M. M., 1954 The M e c h a n i c a l P r o p e r t i e s o f t h e C e l l S u r f a c e . I . The C e l l E l a s t i m e t e r J o u r n a l o f E x p e r i m e n t a l B i o l o g y 31:443 P r e s s W. H., F l a n n e r y B. P., T e u k o l s k y S. A., V e t t e r l i n g W. T., 1986 N u m e r i c a l R e c i p e s Cambridge U n i v e r s i t y P r e s s , Cambridge Rand R. P., B u r t o n A. C , 1964 M e c h a n i c a l P r o p e r t i e s o f t h e Red C e l l Membrane. B i o p h y s i c a l J o u r n a l 4:115 Schmid-Schonbein G. W., S h i h Y. Y., C h e i n S., 1980 Morphometry o f Human L e u k o c y t e s B l o o d 56:866-875 Schmid-Schonbein G. W., Sung K. L. P., T o z e r e n H., S k a l a k R., 1981 P a s s i v e M e c h a n i c a l P r o p e r t i e s o f Human L e u k o c y t e s B i o p h y s i c a l J o u r n a l 36:243-256 S k a l a k R., T o z e r e n A., Zarda R. P., C h i e n S., 1973 S t r a i n Energy F u n c t i o n o f Red B l o o d C e l l Membranes B i o p h y s i c a l J o u r n a l 13:245 S o u t h w i c k F. S., S t o s s e l T. P., 1983 C o n t r a c t i l e P r o t e i n s i n L e u k o c y t e F u n c t i o n Seminars i n Hematology 20:(4)305-321 T o z e r e n H., C h e i n S., T o z e r e n A., 1984 E s t i m a t i o n o f V i s c o u s D i s s i p a t i o n i n s i d e an E r y t h r o c y t e d u r i n g A s p i r a t i o n a l E n t r y i n t o a M i c r o p i p e t t e B i o p h y s i c a l J o u r n a l 45:1179-1184 

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