×

An immersed boundary method for simulating the dynamics of three-dimensional axisymmetric vesicles in Navier-Stokes flows. (English) Zbl 1349.76612

Summary: In this paper, we develop a simple immersed boundary method to simulate the dynamics of three-dimensional axisymmetric inextensible vesicles in Navier-Stokes flows. Instead of introducing a Lagrange’s multiplier to enforce the vesicle inextensibility constraint, we modify the model by adopting a spring-like tension to make the vesicle boundary nearly inextensible so that solving for the unknown tension can be avoided. We also derive a new elastic force from the modified vesicle energy and obtain exactly the same form as the originally unmodified one. In order to represent the vesicle boundary, we use Fourier spectral approximation so we can compute the geometrical quantities on the interface more accurately. A series of numerical tests on the present scheme have been conducted to illustrate the applicability and reliability of the method. We first perform the accuracy check of the geometrical quantities of the interface, and the convergence check for different stiffness numbers as well as fluid variables. Then we study the vesicle dynamics in quiescent flow and in gravity. Finally, the shapes of vesicles in Poiseuille flow are investigated in detail to study the effects of the reduced volume, the confinement, and the mean flow velocity. The numerical results are shown to be in good agreement with those obtained in literature.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M85 Fictitious domain methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76Z05 Physiological flows

Software:

Matlab; FISHPAK
PDFBibTeX XMLCite
Full Text: DOI

References:

[2] Beale, J. T., Partially implicit motion of a sharp interface in Navier-Stokes flow, J. Comput. Phys., 231, 6159-6172 (2012) · Zbl 1277.76056
[3] Biben, T.; Kassner, K.; Misbah, C., Phase-field approach to three-dimensional vesicle dynamics, Phys. Rev. E, 72, 041921 (2005)
[4] Boedec, G.; Leonetti, M.; Jaeger, M., 3D vesicle dynamics simulations with a linearly triangulated surface, J. Comput. Phys., 230, 1020-1034 (2011) · Zbl 1391.76525
[5] Coupier, C.; Farutin, A.; Minetti, C.; Podgorski, T.; Misbah, C., Shape diagram of vesicles in Poiseuille flow, Phys. Rev. Lett., 108, 178106 (2012)
[6] Du, Q.; Liu, C.; Wang, X., A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, J. Comput. Phys., 198, 450-468 (2004) · Zbl 1116.74384
[7] Deschamps, J.; Kantsler, V.; Segre, E.; Steinberg, V., Dynamics of a vesicle in general flow, Proc. Natl. Acad. Sci. USA, 106, 11444-11447 (2009) · Zbl 1203.76005
[8] Doyeux, V.; Chabannes, V.; PrudʼHomme, C.; Ismail, M., Simulation of vesicle using level set method solved by high order finite element, (ESAIM Proc. (2012)) · Zbl 1336.92023
[9] Farutin, A.; Biben, T.; Misbah, C., Analytical progress in the theory of vesicles under linear flow, Phys. Rev. E, 81, 061904 (2010)
[10] Helfrich, W., Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch., C, 28, 693-703 (1973)
[11] Harlow, F. H.; Welsh, J. E., Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface, Phys. Fluids, 8, 2181-2189 (1965) · Zbl 1180.76043
[12] Kantsler, V.; Steinberg, V., Orientation and dynamics of a vesicle in tank-treading motion in shear flow, Phys. Rev. Lett., 95, 258101 (2005)
[13] Kraus, M.; Wintz, W.; Seifert, U.; Lipowsky, R., Fluid vesicles in shear flow, Phys. Rev. Lett., 77, 3685-3688 (1996)
[14] Keller, S. R.; Skalak, R., Motion of a tank-treading ellipsoidal particle in a shear flow, J. Fluid Mech., 120, 27-47 (1982) · Zbl 0503.76142
[15] Kim, Y.; Lai, M.-C., Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method, J. Comput. Phys., 229, 4840-4853 (2010) · Zbl 1305.76075
[16] Lai, M.-C.; Hu, W.-F.; Lin, W.-W., A fractional step immersed boundary method for Stokes flow with an inextensible interface enclosing a solid particle, SIAM J. Sci. Comput., 34, B692-B710 (2012) · Zbl 1255.76026
[17] Lai, M.-C.; Huang, C.-Y.; Huang, Y.-M., Simulating the axisymmetric interfacial flows with insoluble surfactant by immersed boundary method, Int. J. Numer. Anal. Model., 8, 1, 105-117 (2011) · Zbl 1208.76099
[18] Li, Z.; Lai, M.-C., New finite difference methods based on IIM for inextensible interfaces in incompressible flows, East Asian J. Appl. Math., 1, 155-171 (2011) · Zbl 1302.76131
[19] Laadhari, A.; Saramito, P.; Misbah, C., Vesicle tumbling inhibited by inertia, Phys. Fluids, 24, 031901 (2012)
[21] Lebedev, V. V.; Turitsyn, K. S.; Vergeles, S. S., Dynamics of nearly spherical vesicles in an external flow, Phys. Rev. Lett., 99, 218101 (2007)
[22] Maitre, E.; Cottet, G.-H., A level set method for fluid-structure interactions with immersed surfaces, Math. Models Methods Appl. Sci., 16, 415-438 (2006) · Zbl 1088.74050
[23] Maitre, E.; Milcent, T.; Cottet, G.-H.; Raoult, A.; Usson, Y., Applications of level set methods in computational biophysics, Math. Comput. Model., 49, 2161-2169 (2009) · Zbl 1171.76435
[24] Maitre, E.; Misbah, C.; Peyla, P.; Raoult, A., Comparison between advected-field and level-set methods in the study of vesicle dynamics, Phys. D, Nonlinear Phenom., 241, 1146-1157 (2012) · Zbl 1402.92069
[25] Misbah, C., Vacillating breathing and tumbling of fluid vesicles under shear flow, Phys. Rev. Lett., 96, 028104 (2006)
[26] Noguchi, H.; Gompper, G., Swinging and tumbling of fluid vesicles in shear flow, Phys. Rev. Lett., 98, 128103 (2007)
[27] Pozrikidis, C., Introduction to Theoretical and Computational Fluid Dynamics (1997), Oxford University Press · Zbl 0886.76002
[28] Pozrikidis, C., Axisymmetric motion of a file of red blood cells through capillaries, Phys. Fluids, 17, 031503 (2005) · Zbl 1187.76425
[29] Pressley, A., Elementary Differential Geometry (2000), Springer
[30] Salac, D.; Miksis, M., A level set projection model of lipid vesicles in general flows, J. Comput. Phys., 230, 8192-8215 (2011) · Zbl 1408.76614
[31] Secomb, T. W.; Skalak, R.; Ozkaya, N.; Gross, J. F., Flow of axisymmetric red blood cells in narrow capillaries, J. Fluid Mech., 163, 405-423 (1986)
[32] Trefethen, L. N., Spectral Methods in MATLAB (2000), SIAM · Zbl 0953.68643
[33] Tan, Z.; Le, Duc-Vinh; Lim, K. M.; Khoo, B. C., An immersed interface method for the simulation of inextensible interfaces in viscous fluids, Commun. Comput. Phys., 11, 925-950 (2012) · Zbl 1373.76041
[34] Veerapaneni, S. K.; Gueyffier, D.; Zorin, D.; Biros, G., A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D, J. Comput. Phys., 228, 2334-2353 (2009) · Zbl 1275.76175
[35] Veerapaneni, S. K.; Gueyffier, D.; Zorin, D.; Biros, G., A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows, J. Comput. Phys., 228, 7233-7249 (2009) · Zbl 1422.76144
[36] Veerapaneni, S. K.; Rahimian, A.; Biros, G.; Zorin, D., A fast algorithm for simulating vesicle flows in three dimensions, J. Comput. Phys., 230, 5610-5634 (2011) · Zbl 1419.76475
[37] Weiner, J. L., On a problem of Chen, Willmore et al., Indiana Univ. Math. J., 27, 19-35 (1978) · Zbl 0343.53038
[38] Yang, X.; Zhang, X.; Li, Z.; He, G.-W., A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations, J. Comput. Phys., 228, 7821-7836 (2009) · Zbl 1391.76590
[39] Zhao, H.; Shaqfeh, E. S.G., The dynamics of a vesicle in simple shear flow, J. Fluid Mech., 674, 578-604 (2011) · Zbl 1241.76133
[40] Zhou, H.; Pozrikidis, C., Deformation of liquid capsules with incompressible interfaces in simple shear flow, J. Fluid Mech., 283, 175-200 (1995) · Zbl 0836.76107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.