# American Institute of Mathematical Sciences

June  2011, 15(4): 1065-1076. doi: 10.3934/dcdsb.2011.15.1065

## Boundary integral and fast multipole method for two dimensional vesicle sets in Poiseuille flow

 1 Laboratoire d'Ingénierie Mathématique, Ecole Polytechnique de Tunisie, Université de Carthage, B.P. 743 - 2078 La Marsa, Tunisia 2 Laboratoire Interdisciplinaire de Physique, 140, rue de la Physique, 38402 Saint Martin d'Hères, France, France

Received  January 2010 Revised  March 2010 Published  March 2011

Two dimensional numerical simulations of sets of vesicles in a Poiseuille flow are presented. Vesicles are a simple model to describe the dynamics of red cells in blood flow. At the scale of vesicles, the hydrodynamics is well described by the Stokes equation, whose linearity allows the use of Green's functions via the boundary integral method. This is coupled with the fast multipole method to acheive optimal scaling with respect to the number of discretization points. Results are presented for sets of different number of vesicles, showing their spatial organization. Vesicles assume a parachute-like shape and align one to the other in the centre of the parabolic profile. The relative distances depend on the total number of vesicles and on the position in the set.
Citation: Hassib Selmi, Lassaad Elasmi, Giovanni Ghigliotti, Chaouqi Misbah. Boundary integral and fast multipole method for two dimensional vesicle sets in Poiseuille flow. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 1065-1076. doi: 10.3934/dcdsb.2011.15.1065
##### References:
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##### References:
 [1] D. Barthès-Biesel and J. M. Rallison, The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech., 113 (1981), 251-267. doi: doi:10.1017/S0022112081003480. [2] C. D. Eggleton and A. S. Popel, Large deformation of red blood cell ghosts in a simple shear flow, Phys. Fluids, 10 (1998), 1834-1845. doi: doi:10.1063/1.869703. [3] G. Ghigliotti, T. Biben and C. Misbah, Rheology of a dilute two-dimensional suspension of vesicles, J. Fluid Mech., 653 (2010), 489-518. doi: doi:10.1017/S0022112010000431. [4] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comp. Phys., 73 (1987), 325-348. doi: doi:10.1016/0021-9991(87)90140-9. [5] L. Greengard and V. Rokhlin, "On the Effficient Implementation of the Fast Multipole Algorithm," Technical report, Yale University, Department of Computer Science, 1988. [6] N. A. Gumerov and R. Duraiswami, Fast multipole method for the biharmonic equation in three dimensions, J. Comp. Phys., 215 (2006), 363-383. doi: doi:10.1016/j.jcp.2005.10.029. [7] W. Helfrich, Elastic properties of lipid bilayers:l theory and possible experiments, Z. Naturforschung, 28 (1973), 693-703. [8] B. Kaoui, G. Biros and C. Misbah, Why do red blood cells have asymmetric shapes even in a symmetric flow? Phys. Rev. Lett., 103 (2009), 188101. doi: doi:10.1103/PhysRevLett.103.188101. [9] B. Kaoui, G. H. Ristow, I. Cantat, C. Misbah and W. Zimmermann, Lateral migration of a two-dimensional vesicle in unbounded Poiseuille flow, Phys. Rev. E, 77 (2008), 021903. doi: doi:10.1103/PhysRevE.77.021903. [10] S. R. Keller and R. Skalak, Motion of a tank-treading ellipsoidal particle in a shear flow, J. Fluid Mech., 120 (1982), 27-47. doi: doi:10.1017/S0022112082002651. [11] M. Kraus, W. Wintz, U. Seifert and R. Lipowsky, Fluid vesicles in shear flow, Phys. Rev. Lett., 77 (1996), 3685-3688. doi: doi:10.1103/PhysRevLett.77.3685. [12] J. L. McWhirter, H. Noguchi and G. Gompper, Flow-induced clustering and alignment of vesicles and red blood cells in microcapillaries, PNAS, 106 (2009), 6039-6043. doi: doi:10.1073/pnas.0811484106. [13] N. Nishimura, Fast multipole accelerated boundary integral equation methods, Appl. Mech. Rev., 55 (2002), 299-324. doi: doi:10.1115/1.1482087. [14] C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow," Cambridge University Press, Cambridge, UK, 1992. doi: doi:10.1017/CBO9780511624124. [15] C. Pozrikidis, Interfacial dynamics for stokes flow, J. Comp. Phys., 169 (2001), 250-301. doi: doi:10.1006/jcph.2000.6582. [16] C. Pozrikidis, "Modeling and Simulation of Capsules and Biological Cells," CRC Press, 2003. doi: doi:10.1201/9780203503959. [17] E. Sackmann, Physical basis of self-organization and functions of membranes: Physics of vesicles, chapter 5, pages 213-302. Elsevier Science B.V., 1995.
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