American Institute of Mathematical Sciences

Advanced
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

[1]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167
[2]

Derek H. Justice, H. Joel Trussell, Mette S. Olufsen. Analysis of Blood Flow Velocity and Pressure Signals using the Multipulse Method. Mathematical Biosciences & Engineering, 2006, 3 (2) : 419-440. doi: 10.3934/mbe.2006.3.419
[3]

Hongjie Dong, Seick Kim. Green's functions for parabolic systems of second order in time-varying domains. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1407-1433. doi: 10.3934/cpaa.2014.13.1407
[4]

Mourad Choulli. Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions. Evolution Equations & Control Theory, 2015, 4 (1) : 61-67. doi: 10.3934/eect.2015.4.61
[5]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501
[6]

Ken Abe. Some uniqueness result of the Stokes flow in a half space in a space of bounded functions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 887-900. doi: 10.3934/dcdss.2014.7.887
[7]

Anita T. Layton, J. Thomas Beale. A partially implicit hybrid method for computing interface motion in Stokes flow. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1139-1153. doi: 10.3934/dcdsb.2012.17.1139
[8]

Andy M. Yip, Wei Zhu. A fast modified Newton's method for curvature based denoising of 1D signals. Inverse Problems & Imaging, 2013, 7 (3) : 1075-1097. doi: 10.3934/ipi.2013.7.1075
[9]

Tong Li, Sunčica Čanić. Critical thresholds in a quasilinear hyperbolic model of blood flow. Networks & Heterogeneous Media, 2009, 4 (3) : 527-536. doi: 10.3934/nhm.2009.4.527
[10]

Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333
[11]

Chun Zong, Gen Qi Xu. Observability and controllability analysis of blood flow network. Mathematical Control & Related Fields, 2014, 4 (4) : 521-554. doi: 10.3934/mcrf.2014.4.521
[12]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
[13]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
[14]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791
[15]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307
[16]

Sihem Mesnager, Gérard Cohen. Fast algebraic immunity of Boolean functions. Advances in Mathematics of Communications, 2017, 11 (2) : 373-377. doi: 10.3934/amc.2017031
[17]

Rafael Vázquez, Emmanuel Trélat, Jean-Michel Coron. Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 925-956. doi: 10.3934/dcdsb.2008.10.925
[18]

Yirmeyahu J. Kaminski. Equilibrium locus of the flow on circular networks of cells. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1169-1177. doi: 10.3934/dcdss.2018066
[19]

Macarena Boix, Begoña Cantó. Using wavelet denoising and mathematical morphology in the segmentation technique applied to blood cells images. Mathematical Biosciences & Engineering, 2013, 10 (2) : 279-294. doi: 10.3934/mbe.2013.10.279
[20]

Van Hoang Nguyen. The Hardy–Moser–Trudinger inequality via the transplantation of Green functions. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3559-3574. doi: 10.3934/cpaa.2020155

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences