American Institute of Mathematical Sciences

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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

Hassib Selmi 1, , Lassaad Elasmi 1, , Giovanni Ghigliotti 2, and  Chaouqi Misbah 2,

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.
Keywords: Fast Multipole Method, red blood cells., Green's functions, Stokes flow.
Mathematics Subject Classification: Primary: 41A58, 65Z05; Secondary: 74F10, 76D0.
Citation: Hassib Selmi, Lassaad Elasmi, Giovanni Ghigliotti, Chaouqi Misbah. Boundary integral and fast multipole method for two dimensional vesicle sets in Poiseuille flow. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 1065-1076. doi: 10.3934/dcdsb.2011.15.1065
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. doi: doi:10.1017/S0022112081003480. Google Scholar

[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. doi: doi:10.1063/1.869703. Google Scholar

[3]

G. Ghigliotti, T. Biben and C. Misbah, Rheology of a dilute two-dimensional suspension of vesicles,, J. Fluid Mech., 653 (2010), 489. doi: doi:10.1017/S0022112010000431. Google Scholar

[4]

L. Greengard and V. Rokhlin, A fast algorithm for particle simulations,, J. Comp. Phys., 73 (1987), 325. doi: doi:10.1016/0021-9991(87)90140-9. Google Scholar

[5]

L. Greengard and V. Rokhlin, "On the Effficient Implementation of the Fast Multipole Algorithm,", Technical report, (1988). Google Scholar

[6]

N. A. Gumerov and R. Duraiswami, Fast multipole method for the biharmonic equation in three dimensions,, J. Comp. Phys., 215 (2006), 363. doi: doi:10.1016/j.jcp.2005.10.029. Google Scholar

[7]

W. Helfrich, Elastic properties of lipid bilayers:l theory and possible experiments,, Z. Naturforschung, 28 (1973), 693. Google Scholar

[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). doi: doi:10.1103/PhysRevLett.103.188101. Google Scholar

[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). doi: doi:10.1103/PhysRevE.77.021903. Google Scholar

[10]

S. R. Keller and R. Skalak, Motion of a tank-treading ellipsoidal particle in a shear flow,, J. Fluid Mech., 120 (1982), 27. doi: doi:10.1017/S0022112082002651. Google Scholar

[11]

M. Kraus, W. Wintz, U. Seifert and R. Lipowsky, Fluid vesicles in shear flow,, Phys. Rev. Lett., 77 (1996), 3685. doi: doi:10.1103/PhysRevLett.77.3685. Google Scholar

[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. doi: doi:10.1073/pnas.0811484106. Google Scholar

[13]

N. Nishimura, Fast multipole accelerated boundary integral equation methods,, Appl. Mech. Rev., 55 (2002), 299. doi: doi:10.1115/1.1482087. Google Scholar

[14]

C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow,", Cambridge University Press, (1992). doi: doi:10.1017/CBO9780511624124. Google Scholar

[15]

C. Pozrikidis, Interfacial dynamics for stokes flow,, J. Comp. Phys., 169 (2001), 250. doi: doi:10.1006/jcph.2000.6582. Google Scholar

[16]

C. Pozrikidis, "Modeling and Simulation of Capsules and Biological Cells,", CRC Press, (2003). doi: doi:10.1201/9780203503959. Google Scholar

[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). Google Scholar

show all references

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. doi: doi:10.1017/S0022112081003480. Google Scholar

[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. doi: doi:10.1063/1.869703. Google Scholar

[3]

G. Ghigliotti, T. Biben and C. Misbah, Rheology of a dilute two-dimensional suspension of vesicles,, J. Fluid Mech., 653 (2010), 489. doi: doi:10.1017/S0022112010000431. Google Scholar

[4]

L. Greengard and V. Rokhlin, A fast algorithm for particle simulations,, J. Comp. Phys., 73 (1987), 325. doi: doi:10.1016/0021-9991(87)90140-9. Google Scholar

[5]

L. Greengard and V. Rokhlin, "On the Effficient Implementation of the Fast Multipole Algorithm,", Technical report, (1988). Google Scholar

[6]

N. A. Gumerov and R. Duraiswami, Fast multipole method for the biharmonic equation in three dimensions,, J. Comp. Phys., 215 (2006), 363. doi: doi:10.1016/j.jcp.2005.10.029. Google Scholar

[7]

W. Helfrich, Elastic properties of lipid bilayers:l theory and possible experiments,, Z. Naturforschung, 28 (1973), 693. Google Scholar

[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). doi: doi:10.1103/PhysRevLett.103.188101. Google Scholar

[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). doi: doi:10.1103/PhysRevE.77.021903. Google Scholar

[10]

S. R. Keller and R. Skalak, Motion of a tank-treading ellipsoidal particle in a shear flow,, J. Fluid Mech., 120 (1982), 27. doi: doi:10.1017/S0022112082002651. Google Scholar

[11]

M. Kraus, W. Wintz, U. Seifert and R. Lipowsky, Fluid vesicles in shear flow,, Phys. Rev. Lett., 77 (1996), 3685. doi: doi:10.1103/PhysRevLett.77.3685. Google Scholar

[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. doi: doi:10.1073/pnas.0811484106. Google Scholar

[13]

N. Nishimura, Fast multipole accelerated boundary integral equation methods,, Appl. Mech. Rev., 55 (2002), 299. doi: doi:10.1115/1.1482087. Google Scholar

[14]

C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow,", Cambridge University Press, (1992). doi: doi:10.1017/CBO9780511624124. Google Scholar

[15]

C. Pozrikidis, Interfacial dynamics for stokes flow,, J. Comp. Phys., 169 (2001), 250. doi: doi:10.1006/jcph.2000.6582. Google Scholar

[16]

C. Pozrikidis, "Modeling and Simulation of Capsules and Biological Cells,", CRC Press, (2003). doi: doi:10.1201/9780203503959. Google Scholar

[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). Google Scholar

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