A numerical study of a permeable capsule under stokes flows by the immersed interface method

A NUMERICAL STUDY OF A PERMEABLE CAPSULE UNDER STOKES FLOWS BY THE IMMERSED INTERFACE METHOD PAHALA GEDARA JAYATHILAKE NATIONAL UNIVERSITY OF SINGAPORE 2010 A NUMERICAL STUDY OF A PERMEABLE CAPSULE UNDER STOKES FLOWS BY THE IMMERSED INTERFACE METHOD PAHALA GEDARA JAYATHILAKE (B. Sc., University of Moratuwa, Sri Lanka) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHYLOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2010 ACKNOWLEDGEMENTS I wish to express my deepest gratitude to my Supervisors, Professor Khoo Boo Cheong and retired Professor Nihal Wijeysundera, for their invaluable guidance, supervision, patience and support throughout the research work. Their suggestions have been invaluable for the project and for the result analysis. Thanks must also go to Dr Tan Zhijun and Dr Le Duc Vinh, who advised and helped me to overcome many difficulties during the PhD research life. I would like to express my gratitude to the National University of Singapore (NUS) for providing me a Research Scholarship and an opportunity to my PhD study in the Department of Mechanical Engineering. I wish to thank all the staff members and classmates in the Fluid Mechanics Laboratory, Department of Mechanical Engineering, NUS for their useful discussions and kind assistances. I also wish to thank the staff members in the Computer Centre, NUS for their assistance on supercomputing. Also, I would like to thank the office of student affairs, NUS for providing me on campus accommodation due to my special needs. Finally, I wish to thank my dear parents, brothers and sisters for their selfless love, support, patience and continued encouragement during the PhD period. i TABLE OF CONTENTS ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii SUMMARY vi NOMENCLATURE viii LIST OF FIGURES xiv LIST OF TABLES xviii Chapter Introduction 1.1. General aspects of capsule modeling 1.2. Existing numerical methods for capsule modeling 1.2.1. Finite Element Method 1.2.2. Boundary Integral Method 1.2.3. Ghost Fluid Method 1.2.4. Immersed Boundary Method 1.2.5. Immersed Interface Method Literature review 1.3.1. Previous numerical studies of impermeable capsules 1.3.2. Immersed Boundary Method 1.3.3. Immersed Interface Method 1.3. ii 1.4. Objectives and scopes 11 1.5. Outline of the thesis 13 Chapter Immersed Boundary Method and Immersed Interface Method 15 2.1. Immersed Boundary Method 15 2.2. Immersed Interface Method 18 2.2.1. Model formulation 18 2.2.2. Discretization of computational domain 24 2.2.3. Solving the equations of motion 25 2.2.4. Solving the transport equation 29 2.2.5. Summary of the main procedure on the calculation 31 2.3. Comparison between the IB method and IIM for an impermeable capsule 32 Chapter On the Deformation and Osmotic Swelling of an Elastic 40 Membrane Capsule and a Rising Droplet in Stokes Flows 3.1. Introduction 40 3.2. Literature review 41 3.3. Model formulation and numerical method 42 3.4. Results and discussions 43 3.4.1. With semi-permeable elastic membrane 43 3.4.2. With fully permeable elastic membrane 54 iii 3.4.3. Some remarks on application to the biological systems 58 3.5. Application to a rising droplet with mass transfer 60 3.6. Summary and conclusions 63 Chapter On the Effect of Membrane Permeability on Capsule Substrate Adhesion 83 4.1. Introduction 83 4.2. Literature review 83 4.3. Model formulation and numerical method 85 4.3.1. The augmented method for the pressure boundary condition 88 4.3.2. Computing the Laplacian along the boundary 90 Results and discussions 91 4.4.1. Adhesion of an impermeable capsule 92 4.4.2. Adhesion of a semi-permeable capsule 97 4.4.3. Adhesion of a fully permeable capsule 100 Summary and conclusions 103 4.4. 4.5. Chapter On the Capsule-Substrate Adhesion and Mass Transport under Imposed Stokes Flows 116 5.1. Introduction 116 5.2. Literature review 116 5.3. Model formulation and numerical method 118 5.4. Results and discussions 124 5.4.1. Method validations 124 iv 5.4.2. A permeable capsule in a vessel 126 5.4.3. A single RBC/IRBC motion in plasma flows 132 Summary and conclusions 134 Concluding Summary and Recommendations 152 5.5. Chapter 6.1. Conclusions 152 6.2. Recommendations 155 References 156 Publications of the Thesis Work 169 Appendices 171 Appendix A 171 Appendix B 173 v SUMMARY Permeable and deformable capsules and their adhesion are found in many applications in biological and industrial systems such as the circulatory system. However, studies on computational modeling of those capsules are still rather lacking. In this work, the osmotic swelling and capsule-substrate adhesion of a deforming capsule immersed in a hypotonic and diluted binary solution of a non-electrolyte solute under Stokes flows is simulated using the immersed interface method (IIM). The approximate jump conditions of the solute concentration needed for the IIM are calculated numerically with the use of the Kedem-Katchalsky membrane transport relations. The thin-walled membrane of the capsule is considered to be either semi-permeable or fully permeable, and the material of the capsule membrane is assumed to be Neo-Hookean. The used properties of fluid and membrane fall in the range of a typical biological system. The numerical validation tests indicate that the present calculation procedure has achieved good accuracy in modeling the deformation, adhesion, and osmotic swelling of a permeable capsule. The capsule swelling (with mass transfer across the membrane) and deformation in a periodic computational domain (without adhesion) are tested for different solute concentration fields and membrane permeability properties. The numerical investigations show that the initial solute concentration field and the membrane permeability properties have much influence on the swelling and deformation behavior of a permeable capsule under Stokes flow condition. Furthermore, capsule-substrate adhesion in the presence of membrane permeability is simulated and the osmotic inflation of the initially adhered capsule is studied systematically as a function of solute concentration field and the membrane permeability properties. The results demonstrate that the contact length shrinks in vi dimension and deformation decreases as capsule inflates. The equilibrium contact length does not depend on the hydraulic conductivity of the membrane as also theoretically obtained. Further numerical investigations show that the inflation and partial detachment of the initially adhered capsule depend significantly on the solute diffusive permeability and the reflection coefficient of capsule membrane. Finally, the mass transfer of an adhesive capsule flowing in a vessel is simulated for various parameters. The results show that the solute mass transfer between the capsule and the vessel walls is enhanced by introducing adhesion between the capsule and the walls. Moreover, the present numerical approach is employed to simulate the adhesion of a malaria-infected red blood cell and a healthy red blood cell flowing in a capillary in the absence of mass transfer. Keywords: Permeable capsule; Adhesion; Stokes flow; Mass transfer; Simulation; Immersed interface method vii NOMENCLATURE a1 , a , b1 , b2 dimensions of the computational domain Ω, m A capsule enclosed area, m2 AR aspect ratio, AR = rmax/rmin, dimensionless c solute concentration, mol/m3 cˆ average solute concentration across the membrane, cˆ = [c] / ln(c k+ / c k− ) , mol/m3 c characteristic solute concentration, c = [c0 ] / ln(c0+ / c0− ) , mol/m3 cp provisional solute concentration, mol/m3 C1, C2, C3 C1 = C{ } spatial correction terms dm zero-force distance, m D solute diffusivity in the fluid, m2/s DI deformation index, DI = {(width – height)/width} or Taylor deformation Ee L p Vr ,C2 = RT abs c L p V ,C3 = ω RT abs , dimensionless V index, dimensionless Eb bending modulus of the membrane, J Ee shear modulus of the membrane, N/m r f r ∂ force strength, f ( s, t ) = Te ( s, t )τ ( s, t ) + Tb ( s, t ) n( s, t ) + f ad yˆ , N/m2 ∂s ( ) f1, f2 force strengths in the x and y directions, respectively, N/m2 fad adhesive force, f ad = − fN, fT force strengths in the normal and tangential directions, respectively, N/m2 ∂W , N/m ∂y viii Fedkiw, R., Aslam, T., Merriman, B., Osher, S., 1999. A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). Journal of Computational Physics 152, 457-492. Foo, J.J., Chan, V., Liu, K.K., 2003. Contact deformation of liposome in the presence of osmosis. Annals of Biomedical Engineering 31, 1279-1286. Foo, J.J., Chan, V., Liu, K.K., 2006. Coupling bending and shear effects on liposome deformation. Journal of Biomechanics, 39. 2338-2343. Galey, W.R., 1978. Determination of human erythrocyte membrane hydraulic conductivity. Journal of Membrane Science 4, 41-49. Gordon, L.G.M., 1981. Osmophoresis. Journal of Physical Chemistry 85, 1753-1755. Hsu, Y.S., Keh, H.J., 2006. Boundary effects on osmophoresis: Motion of a spherical vesicle perpendicular to two plane walls. Chemical Engineering Science 61, 434448. Huang, H., Sugiyama, K., Takagi, S., 2009. An immersed boundary method for restricted diffusion with permeable interfaces. Journal of Computational Physics 228, 53175322. Imai, Y., Kondo, H., Ishikawa, T., Lim, C.T., Yamaguchi, T., 2010. Modeling of hemodynamics arising from malaria infection. Journal of Biomechanics 43, 1386– 1393. Ito, K., Lai, M-C, Li, Z., 2009. A well-conditioned augmented system for solving Navier–Stokes equations in irregular domains. Journal of Computational Physics 228, 2616-2628. 159 Jiao, G.Y., Tan, K.S.W., Sow, C.H., Dao, M., Suresh, S., Lim, C.T., 2009. Computational modeling of the micropipette aspiration of malaria infected erythrocytes. 13th International Conference on Biomedical Engineering IFMBE Proceedings 23, 1788-1791. Jin, Q., Verdier, C., Singh, P., Aubry, N., Chotard-Ghodsnia, R., Duperray, A., 2007. Migration and deformation of leukocytes in pressure driven flows. Mechanics Research Communications 34, 411-422, Johnson, J.A., Wilson, T.A., 1967. Osmotic volume changes induced by a permeable solute. Journal of Theoretical Biology 17, 304-311. Katkov, I.I., 2000. A two-parameter model of cell membrane permeability for multisolute systems. Cryobiology 40, 64-83. Katkov, I.I., 2002. The point of maximum cell water volume excursion in case of presence of an impermeable solute. Cryobiology 44, 193-203. Kedem, O., Katchalsky, A., 1958. Thermodynamics Analysis of the Permeability of Biological Membranes to Non-electrolytes. Biochimica et Biophysica Acta 27, 229–246. Keh, H.J., Tu, H.J., 2000. Osmophoresis in a dilute suspension of spherical vesicles. International Journal of Multiphase Flow 26, 125-145. Keller, S.R., Skalak, R., 1982. Motion of a tank-treading ellipsoid particle in a shear flow. Journal of Fluid Mechanics 120, 27-42. Khismatullin, D.B., Truskey, G.A., 2004. A 3D numerical study of the effect of channel height on leukocyte deformation and adhesion in parallel plate flow chambers. Microvascular Research 68, 188-202. 160 Kim, J., Moin, P., 1985. Application of a fractional step method to incompressible Navier-Stokes equations. Journal of Computational Physics 59, 308-323. Kim.Y., Peskin,C.S., 2006. 2D parachute simulation by the immersed boundary method. SIAM Journal of Scientific Computing 28(6), 2294–312. Kondo, T., Imai, Y., Ishikawa, T., Tsubota, K. and Yamaguchi, T., 2009. Hemodynamics analysis of microcirculation in malaria infection. Annals of Biomedical Engineering 37, 702-709. Lai, M.C., Tseng, H.C., 2008. A simple implementation of the immersed interface methods for Stokes flows with singular forces. Computers and Fluids 37, 99-106. Layton, A.T., 2006. Modeling water transport across elastic boundaries using an explicit jump method. SIAM Journal on Scientific Computing 28, 2189-2207. Le, D.V., 2005. An Immersed interface method for solving viscous incompressible flows involving rigid and flexible boundaries. Ph.D. thesis, Singapore-MIT Alliance. Le, D.V., Khoo, B.C., Peraire, J., 2006. An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries. Journal of Computational Physics 220, 109-138 Lee, L., LeVeque, R.J., 2003. An immersed interface method for the incompressible Navier–Stokes equations. SIAM Journal on Scientific Computing 25, 832-856. LeVeque, R.J., Li, Z., 1994. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM Journal on Numerical Analysis 31, 1019-1044. 161 LeVeque, R.J., Li, Z., 1997. Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM Journal on Scientific Computing 18, 709– 735. Li, Z., 1994. The immersed interface method – A numerical approach for partial differential equations with interfaces. Ph.D. thesis, University of Washington. Li, Z., 2003. An overview of the immersed interface method and its applications. Taiwanese Journal of Mathematics 7, 1-49. Li, Z., Ito, K., 2006. The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics), Society for Industrial and Applied Mathematics, Philadelphia, PA. Li, Z., Ito, K., Lai, M-C, 2007. An augmented approach for Stokes equations with a discontinuous viscosity and singular forces. Computers and Fluids 36, 622-635. Li., Z, Lai, M-C, 2001. The immersed interface method for the Navier-Stokes equations with singular forces. Journal of Computational Physics 171, 822-842. Li, Z., Lai, M-C, He, G., Zhao, H., 2010. An augmented method for free boundary problems with moving contact lines. Computers and Fluids 39, 1033-1040. Li, Z., Wan, X., Ito, K., Lubkin, S.R., 2006. An augmented approach for the pressure boundary condition in a Stokes flow. Communications in Computational Physics 5, 874-885. Linnick, M.N., Fasel, H.F., 2005. A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains. Journal of Computational Physics 204, 157-192. 162 Liu, K.K., 2006. Deformation behavior of soft particles: a review. Journal of Physics D: Applied Physics 39, R189-R199. Liu, K.K., Chan, V., Zhang, Z., 2002a. Capsule-sunstrate contact deformation: determination of adhesion energy. Medical and Biological Engineering and Computing 40, 491-495. Liu, K.K., Wan, K.T., 2000. New model to characterize cell-substrate adhesion in the presence of osmosis. Cellular Engineering 38, 690-691. Liu, K.K., Wang, H.G., Wan, K.T., Liu, T., Zhang, T., 2002b. Characterizing capsulesubstrate adhesion in presence of osmosis. Colloids and Surfaces B: Biointerfaces 25, 293-298. Liu., Y., Zhaing, L., Wang, X., Liu, W.K., 2004. Coupling of Navier-Stokes equations with protein molecular dynamics and its application to hemodynamics. International Journal of Numerical Methods in Fluids 46, 1237-1252. Lucke, B., Hartline, H.K., McCutcheon, M., 1931. Further studies on the kinetics of osmosis in living cells. The Journal of General. Physiology 14, 405-419. McCabe, W., Smith, J., Harriott, P., 2000. Unit Operations of Chemical Engineering. McGraw-Hill, New York. Nash, G.B., Cooke, B.M., Marsh, K., Berendt, A., Newbold, C., Stuart, J., 1992. Rheological analysis of the adhesive interactions of red blood cells parasitized by Plasmodium falciparum. Blood 79, 798-807. N’Dri, N.A., Shyy, W., Tran-Son-Tay, R., 2003. Computational modeling of cell adhesion and movement using a continuum-Kinetics Approach. Biophysical Journal 85, 2273-2286. 163 Nguyen, D., Fedkiw, R., Kang, M., 2001. A boundary condition capturing method for incompressible flame discontinuities. Journal of Computational Physics 172, 7198 Noguchi, H., Gompper, G., 2005. Dynamics of fluid vesicles in shear flow: effect of membrane viscosity and thermal fluctuations. Physical Review E 72, 011901. Onea, A.A., 2007. Numerical simulation of mass transfer with and without first order chemical reaction in two-fluid flows. PhD thesis, Institut für Reaktorsicherheit. Owen, J.D., Galey, W.R., 1977. Computer simulation of urea permeability in human red cells. Biochimica et Biophysica Acta 466, 517-520. Pain, B.K., Weymann, H.D., 1980. Equilibrium shapes of red blood cells in osmotic swelling. Journal of Biomechanics 13, 105-l12. Pappu, V., Bagchi, P., 2008. 3D computational modeling and simulation of leukocyte rolling adhesion and deformation. Computers in Biology and Medicine 38, 738753. Pappu, V., Doddi, S.K., Bagchi, P., 2008. A computational study of leukocyte adhesion and its effect on flow pattern in microvessels. Journal of Theoretical Biology 254, 483-498. Peres, A., Andrietti, F., 1976. Asymptotic behavior of osmotic volume changes induced by a permeant solute. Mathematical Biosciences 30, 153-158. Peskin, C.S., 1977. Numerical analysis of blood flow in the heart. Journal of Computational Physics 25, 220-252. Peskin, C.S., 2002. The immersed boundary method. Acta Numerica 11(2), 479-517. 164 Peskin, C.S., McQueen, D.M., 1980. Modeling prosthetic heart valves for numerical analysis of blood flow in the heart. Journal of Computational Physics 37, 113132. Pozrikidis, C., 2001. Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. Journal of Fluid Mechanics 440, 269-291. Pozrikidis, C., 2005. Axisymmetric motion of a file of red blood cells through capillaries. Physics of Fluids 17, No 031503. Ramanujan, S., Pozrikidis, C., 1998. Deformation of lipid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities. Journal of Fluid Mechanics 361, 117-143. Rao, P.R., Zahalak, G.I., Sutera, S.P., 1994. Large deformations of elastic cylindrical capsules in shear flow. Journal of Fluid Mechanics 179, 283-305. Rutka, V., 2008. A staggered grid-based explicit-jump immersed interface method for two-dimensional Stokes flows. International Journal for Numerical Methods in Fluid 57, 1527–1543. Seifert, U., 1991. Adhesion of vesicles in the two dimensions. Physical Review A 43, 6803-6814. Servuss, R.M., Helfrich, W., 1989. Mutual adhesion of lecithin membranes at ultralow tensions. Journal de Physique 50, 809-827. Shaafi, R.I., Dakkuri, A., Tomey, G., 1971. Solute and solvent flow across mammalian red cell membrane. Biochimica et Biophysica Acta 249, 260-265. 165 Sherwood, J.D., Risso, F., Collé-Paillot, F., Edwards-Lévy, F., Lévy, M.-C., 2003. Rates of transport through a capsule membrane to attain Donnan equilibrium. Journal Colloid and Interface Science 263, 202-212. Shi, W., Feng, X.Q., Gao, H., 2006. Two-dimensional model of vesicle adhesion on curved substrate. Acta Mechanica Sinica 22, 529-535. Silamut, K., Phu, N.H., Whitty, C., Turner, G.D.H., Louwrier, K., Mai, N.T.H., Simpson, J.A., Hein, T.T., White, N.J., 1999. A quantitative analysis of the microvascular sequestration of malaria parasites in the human brain. American Journal of Pathology 155, 395-410. Smith, A.E., Moxham, K.E., Middelberg, A.P.J., 1998. On uniquely determining cellwall material properties with the compression experiment. Chemical Engineering Science 53(23), 3913-3922. Smith, A.E., Moxham, K.E., Middelberg, A.P.J., 2000. Wall material properties of yeast cells. Part II. Analysis. Chemical Engineering Science 55, 2043-2053. Springer, T.A., 1995. Traffic signals on endothelium for lymphocyte recirculation and leukocyte emigration. Annual Review of Physiology 57, 827-872. Stockie, J.M., 2009. Modelling and simulation of porous immersed boundaries. Computers and Structures 87. 701–709. Sui, Y., Chew, Y.T., Roy, P., Low, H.T., 2009. Inertia effect on the transient deformation of elastic capsules in simple shear flow. Computers and Fluids 38, 49-59. Sukumaran, S., Seifert, U., 2001. Influence of shear flow on vesicles near a wall: a numerical study. Physical Review E 64, 011916-1-11 166 Suresh, S., Spatz, J., Mills, J.P., Micoulet, A., Dao, M., Lim, C.T., Beil, M. and Seufferlein, T., 2005. Connections between single-cell biomechanics and human disease states: gastrointestinal cancer and malaria. Acta Biomaterialia 1, 15-30. Tachev, K.D., Angarska, J.K., Danov, K.D., Kralchevsky, P.A., 2000. Erythrocyte attachment to substrates: determination of membrane tension and adhesion energy. Colloids and Surfaces B: Biointerfaces 19, 61-80. Tan, Z., Le, D.V., Li, Z., Lim, K.M., Khoo, B.C., 2008. An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane. Journal of Computational Physics 227, 9955-9983. Tan, Z., Lim, K.M., Khoo, B.C., 2009. An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries. Journal of Computational Physics 228, 6855–6881. Tu, C., Peskin, C.S., 1992. Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods. SIAM Journal on Scientific and Statistical Computing 13, 1361-1376. Ultman, J.S., 1972. Analysis of osmotic hemolysis of erythrocytes including mechanical and transport effects. Mathematical Biosciences 13, 287-298. Wan, K.T., Liu, K.K., 2001. Contact mechanics of a thin-walled capsule adhered onto a rigid planar substrate. Medical and Biological Engineering and Computing 39, 605-608. Wang, C-H., Popel, A.S., 1993 Effect of red blood cell shape on oxygen transport in capillaries. Mathematical Biosciences 116, 89-110. 167 Wang, C.X., Wang, L., Thomas, C.R., 2004. Modelling the mechanical properties of single suspension-cultured tomato cells. Annals of Botany 93, 443-453. Wang, X, Liu, W.K., 2004. Extended immersed boundary method using FEM and RKPM. Computer Methods in Applied Mechanics Engineering 193,1305-1321. Whiteley, J.P., Gavaghan, D.J., Hahm, C.E.W., 2003. Mathematical modeling of pulmonary gas transport. Journal of Mathematical Biology 47, 79-99. Xu, S., Wang, Z.J., 2006. An immersed interface method for simulating the interaction of a fluid with moving boundaries. Journal of Computational Physics 216, 454-493. Zarda, P.R., Chien, S., Skalak, R., 1977. Elastic deformations of red blood cells. Journal of Biomechanics 10, 211-221. Zhang, J., Johnson, P.C., Popel, A.S., 2008, Red blood cell aggregation and dissociation in shear flows simulated by lattice Boltzmann method. Journal of Biomechanics 41, 47-55. Zhang, L., Gerstenberger, A., Wang, X., Liu, W.K., 2004. Immersed finite element method. Computer Methods in Applied Mechanics Engineering 193, 2051-2067. Zinemanas, D., Nir, A., 1995. Osmophoretic motion of deformable particles. International Journal of Multiphase Flow 21, 787-800. 168 Publications of the Thesis Work P.G. Jayathilake, G. Liu, Zhijun Tan, and B.C. Khoo, 2011. Numerical study of a permeable capsule under Stokes flows by the immersed interface method. Chemical Engineering Science, DOI:10.1016/j.ces.2011.02.005. P.G. Jayathilake, B.C. Khoo, and Zhijun Tan, 2010. Effect of membrane permeability on capsule substrate adhesion: Computation using immersed interface method. Chemical Engineering Science 65, 3567-3578. P. G. Jayathilake, Zhijun Tan, B. C. Khoo, and N. E. Wijeysundera, 2010. Deformation and osmotic swelling of an elastic membrane capsule in Stokes flows by the immersed interface method. Chemical Engineering Science 65, 1237-1252. P.G. Jayathilake and B.C. Khoo, 2011. A two-dimensional numerical study of a permeable capsule under Stokes flow condition. SIAM conference on Computational Science and Engineering, February 28-March 4, 2011, Nevada, USA. P.G. Jayathilake and B.C. Khoo, 2010. Capsule-substrate adhesion, detachment, and mass transport under Stokes flows: Computation using the immersed interface method. Proceedings of the 13th Asian Congress of Fluid Mechanics, December 17-21, 2010, Dhaka, Bangladesh. B.C. Khoo, P.G. Jayathilake, and Z. Tan, 2010. Numerical study of a permeable capsule/cell under Stokes flows by the immersed interface method. Workshop on Fluid Motion Driven by Immersed Structures, August 9-13, 2010, Fields Institute, Toronto, Ontario, Canada. 169 P.G. Jayathilake, B.C. Khoo, and Z. Tan, 2010. Numerical simulation of interfacial mass transfer using the immersed interface method. 6th International Symposium on Gas Transfer at Water Surfaces, May 17-21, 2010, Kyoto, Japan. P.G. Jayathilake, B.C. Khoo, and Zhijun Tan, 2009. Capsule-substrate adhesion in the presence of osmosis by the immersed interface method. International Conference on Computational Fluid Dynamics, December 25-27, 2009, Bangkok, Thailand. 170 Appendices Appendix A: Spatial and temporal correction terms In Fig. A1, the point (i, j) is called an irregular grid point since grid points from both sides of the interface are involved in calculating the first and second order derivatives at point (i, j) using the central finite difference scheme. It is assumed that we are working with the discontinuous model variable u, and the jump conditions [u] and [un] are given. Thus, the derivatives of u with respect to x and y are to be calculated using the generalized finite difference (GFD) at grid point (i, j) (or using Taylor series expansions of u at the intersection points) due to the discontinuity of u across the interface as ux = u xx = (u i +1, j − u i −1, j ) 2h −  (u i +1, j − u i −1, j ) (h +1 )  + [ u ] + h [ u ] + [u xx ] I1  = + C i , j {u x } , (A.1)  I1 x I1 2h  2 h  (u i +1, j − 2u i , j + u i −1, j ) h2 − h2   (h +1 ) + + + [ u ] h [ u ] [u xx ] I1   I1 x I1   (u i +1, j − 2u i , j + u i −1, j ) = + C i , j {u xx } , (A.2) h2 where h1+ = xi +1 − α and h = grid size. The jump conditions [u] and [un] at the intersection point I1 are calculated by cubic interpolation from those known jump conditions at all control points; similarly these values are used to calculate [u x ] I1 and [u xx ] I1 by a coordinate transformation as similarly done by Layton (2006). The correction terms are thus given by 171 Ci , j u x = − (h1+ ) [u xx ] I1   + [ u ] + h [ u ] +  I , x I1 2h   C i , j {u xx } = − h2 { }  (h1+ ) [u xx ] I1  + [ u ] + h [ u ] +  I1 . x I1   (A.3) In the same way, Ci,j{uy} and Ci,j{uyy} are calculated with use of [u y ] I and [u yy ] I . The correction term for the Laplacian of u at (i, j) is calculated using the generalized finite difference for uxx and uyy (Eq. (A.2)) as below: ∇ u = u xx + u yy , ∇ 2u = u i +1, j + u i −1, j + u i , j +1 + u i , j −1 − 4u i , j h2 + C i , j {u xx } + C i , j {u yy }, and hence the correction term for ∇ u is given as C i , j {∇ u} = C i , j {u xx } + C i , j {u yy }. (A.4) The temporal correction term Q = C{ut}is only non-zero at grid points crossed by the interface at a time between tm and tm+1. It is assumed that the interface crosses the grid point (i,j) at time t1 ∈ (t m , t m+1 ) . From the Taylor series expansion of u about t1 , the correction term Qi,j is calculated as ±1 [u ]t1 + (t m − t1 )[u t ]t1 , t m < t1 < t m / 2+1 , ∆t ±1 = [u ]t1 + (t m +1 − t1 )[u t ]t1 , t m / 2+1 < t1 < t m +1 . ∆t Qi , j = { Qi , j { } } (A.5) 172 The + sign is used if the grid point (i, j) enters into Ω- from Ω+, and the – sign is used otherwise. More details on spatial and temporal correction terms can be seen in Le et al. (2006). Figure A1 Sketch for the calculation of correction terms. Appendix B: The analytical solution for the enclosed area of circular swelling capsule Note that the calculation is presented in the dimensional form. The rate of change of enclosed capsule area can be written as dA = − SJ v = SL p {[ p ] − σRT [c]}, dt (B.1) where A = capsule enclosed area, S = total circumferential length of the membrane. Other notations have their usual meaning as defined in the text. 173 For a circular capsule, S ≡ 2πr, A ≡ πr2 and [p] ≡- Ee(ε1.5-ε-1.5)/r, where r ≡ capsule radius, ε = stretch ratio given by ∂s ∂s a ≡ r , and ra≡ unstretched radius of the capsule. If the dimensionless parameter δ>>1, the capsule swelling is primarily governed by the mechanical pressure difference ([p]), and hence the solute concentration difference across the membrane ([c]) is negligible. Thus, the above differential equation can be written in the following form as where dε = − K (ε 1.5 − ε −1.5 ) / ε dt K= L p Ee ra2 . (B.2) By integrating for ε, it is obtained, φ (ε ) − φ (ε ) = − Kt / 2, where φ (ε ) = ε 0.5 (B.3)  3ε 0.5     ε 0.5 −  ε − ε 0.5 +  −1 .  + ln  0.5 − tan   −    ε +  ε + ε 0.5 +  ε    This non-linear equation is solved for ε at a given time t by Newton-Raphson method. Once ε is known, the enclosed area is calculated by π(ra ε)2 at time t. 174 [...]... permeable, and deformable capsules under Stokes flow condition, 2 investigate the swelling and deformation characteristics of a permeable capsule for various physical parameters, 3 extend the IIM approach to study the adhesion and detachment of a permeable capsule adhered onto a rigid planar substrate in the absence of an imposed flow field, 4 apply the IIM to study a permeable capsule (or a drug-loaded capsule) ... review for the present work 1.1 General aspects of capsule modeling Capsules can be generally categorized as natural or artificial The interfaces of natural capsules typically consist of a membrane that is composed of a phospholipid bilayer, and may also host other species such as proteins Artificial capsules are enclosed by a variety of coating materials with various physical and mechanical properties... Figure 3.6 Effect of the osmotic load c0 on the capsule enclosed area, average solute concentration and aspect ratio (a) capsule enclosed area; (b) average solute concentration of the capsule; (c) capsule aspect ratio: γ = 2; α = 0 71 Figure 3.7 Effect of the initial solute concentration ratio γ on the capsule enclosed area and aspect ratio (a) capsule enclosed area; (b) capsule aspect ratio: δ = 3.52x10-5;... hemodynamics The flow induced-deformation of capsules has been studied by many researchers in the past two decades to investigate the effects of the membrane and fluid properties, capsule- substrate adhesion and inertia forces of the flow field on capsule deformation As an extension of capsule simulations, both capsules of permeable membranes and capsules adhesion onto a substrate are very important as their... et al (2004) 1.2.2 Boundary Integral Method One advantage of this method is that it can handle complex geometries easily The main disadvantage of this method is the lack of ability to handle non-linear equations Therefore, this method cannot be applied for the full Navier -Stokes equations directly although some attempts are available for the linearized Navier -Stokes equations (Achdou and Pironneau,... pressure at t = 500 39 Figure 3.1 Schematic diagram for the capsule in the presence of membrane permeability 66 Figure 3.2 Comparison between the analytical and numerical values of the transient of the enclosed area of the circular capsule: δ = 3.45x102; β = 4.3x104; γ = 67 2; Pe = 8.31x10-5; α = 0 Figure 3.3 Comparison between the analytical and numerical values of the capsule enclosed area at the equilibrium... solutions have been investigated for the final equilibrium stage However, the swelling and deformation transient of these capsules have not been investigated in detail 11 3 Although dynamical characteristics of impermeable capsule adhesion onto a rigid planar substrate have been studied in detail, there are currently only a few theoretical studies to calculate the final equilibrium of permeable capsule adhesion... permeable capsule immersed in a binary solution by assuming Stokes flow conditions Also, some comparisons between the IIM and the IB method are given In Chapter 3, the transient deformation and swelling of a permeable capsule is studied numerically Moreover, the mass transfer of a rising droplet is simulated and the results are validated In Chapter 4, capsule- substrate adhesion in the presence of membrane... the presence of osmosis or mass transfer The main objective of the present work is to employ the IIM to simulate a capsule with a permeable membrane under a variety of conditions for a careful and systematic elucidation of the flow physics The specific objectives of the present thesis are to: 1 propose a novel numerical approach based on the IIM to simulate both the semi- permeable and fully permeable, ... deformable and permeable boundary problems 1.4 Objectives and scopes Research gaps for the present numerical study of permeable capsules using the immersed interface method are summarized below: Numerical studies on permeable and deformable capsules are rather lacking due to the difficulty of handling the discontinuity of model variables such as solute concentration and pressure fields 1 It has been . A NUMERICAL STUDY OF A PERMEABLE CAPSULE UNDER STOKES FLOWS BY THE IMMERSED INTERFACE METHOD PAHALA GEDARA JAYATHILAKE (B. Sc., University of Moratuwa, Sri Lanka) A THESIS. A NUMERICAL STUDY OF A PERMEABLE CAPSULE UNDER STOKES FLOWS BY THE IMMERSED INTERFACE METHOD PAHALA GEDARA JAYATHILAKE NATIONAL UNIVERSITY OF SINGAPORE 2010. Schematic diagram for the capsule in the presence of membrane permeability. 66 Figure 3.2 Comparison between the analytical and numerical values of the transient of the enclosed area of the