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Numerical Simulation of the Motion of Inextensible Capsules in Shear Flow Under the Effect of the Natural State

Published online by Cambridge University Press:  14 September 2015

Xiting Niu ,
Lingling Shi ,
Tsorng-Whay Pan  and
Roland Glowinski
Xiting Niu
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA
Lingling Shi
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA
Tsorng-Whay Pan*
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA
Roland Glowinski
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204, USA
*
*Corresponding author. Email addresses: [email protected] (X. Niu), [email protected] (L. Shi), [email protected] (T.-W. Pan), [email protected] (R. Glowinski)
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Abstract

In this paper, a computational model for the natural state of an inextensible capsule has been successfully combined with a spring model of the capsule membrane to simulate the motion of the capsule in two-dimensional shear flow. Besides the viscosity ratio of the internal fluid and external fluid of the capsule, the natural state also plays a role for having the transition between two well known motions, tumbling and tank-treading (TT) with the long axis oscillates about a fixed inclination angle (a swinging mode), when varying the shear rate. Between tumbling and tank-treading, the intermittent behavior has been obtained for the capsule with a biconcave rest shape. The estimated critical value of the swelling ratio for having the intermittent transition behavior is less than 0.7, i.e., the capsules with rest shape closer to a full disk do not have the intermittent behavior in shear flow. The intermittent dynamics of the capsule in the transition region is a mixture of tumbling and TT with a swinging mode. Just like the motion of TT with a swing mode, which can be viewed as a tank-treading with an incomplete tumbling, the membrane tank-treads backward and forward within a small range during the tumbling motion.

Keywords

76M1076Z0576D0592C05Natural statetumblingtank-treading with a swinging modeintermittent regioncapsuleshear flow
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 3 , September 2015 , pp. 787 - 807
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Haas, K. H. de, Blom, C., van den Ende, D., Duits, M. H. G., and Mellema, J., Deformation of giant lipid bilayer vesicles in shear flow, Phys. Rev. E, 56 (1997), 7132–7137.Google Scholar
[2]Kantsler, V. and Steinberg, V., Orientation and dynamics of a vesicle in tank-treading motion in shear flow, Phys. Rev. Lett., 95 (2005), 258101.Google Scholar
[3]Seifert, U., Fluid membranes in hydrodynamic flow fields: Formalism and an application to fluctuating quasispherical vesicles in shear flow, Eur. Phys. J. B, 8 (1999), 405–415.Google Scholar
[4]Misbah, C., Vacillating breathing and tumbling of vesicles under shear flow, Phys. Rev. Lett., 96 (2006), 028104.Google Scholar
[5]Kraus, M., Wintz, W., Seifert, U., and Lipowsky, R., Fluid vesicles in shear flow, Phys. Rev. Lett., 77 (1996), 3685–3688.Google Scholar
[6]Beaucourt, J., Rioual, F., Séon, T., Biben, T., and Misbah, C., Steady to unsteady dynamics of a vesicle in a flow, Phys. Rev. E, 69 (2004), 011906.Google Scholar
[7]Noguchi, H. and Gompper, G., Fluid vesicles with viscous membranes in shear flow, Phys. Rev. Lett., 93 (2004), 258102.Google Scholar
[8]Noguchi, H. and Gompper, G., Dynamics of fluid vesicles in shear flow: Effect of membrane viscosity and thermal fluctuation, Phys. Rev. E, 72 (2005), 011901.CrossRefGoogle Scholar
[9]Noguchi, H. and Gompper, G., Swinging and tumbling of fluid vesicles in shear flow, Phys. Rev. Lett., 98 (2007), 128103.Google Scholar
[10]Zhao, H. and Shaqfeh, E. S. G.The dynamics of a vesicle in simple shear flow, J. Fluid Mech., 674 (2011), 578–604.CrossRefGoogle Scholar
[11]Keller, S. R. and Skalak, R., Motion of a tank-treading ellipsoidal particle in a shear flow J. Fluid Mech., 120 (1982), 27–47.Google Scholar
[12]Skotheim, J.M. and Secomb, T.W., Red blood cells and other nonspherical capsules in shear flow: oscillatory dynamics and the tank-treading-to-tumbling transition, Phys. Rev. Lett., 98 (2007), 078301.Google Scholar
[13]Kessler, S., Finken, R., and Seifert, U., Swinging and tumbling of elastic capsules in shear flow, J. Fluid Mech., 605 (2008), 207–226.Google Scholar
[14]Vlahovska, P. M., Young, T.-N., Danker, G. and Misbah, C., Dynamics of a non-spherical with incompressible interfacemicrocapsule in shear flow, J. Fluid Mech., 678 (2011), 221–247.CrossRefGoogle Scholar
[15]Dupont, C., Salsac, A.-V., and Barthés-Biesel, D., Off-plane motion of a prolate capsule in shear flow, J. Fluid Mech., 721 (2013), 180–198.CrossRefGoogle Scholar
[16]Fischer, T. M., Stöhr-Liesen, M., and Schmid-Schönbein, H., The red cell as a fluid droplet: tank tread-like motion of the human erythrocyte membrane in shear flow, Science, 202 (1978), 894–896.CrossRefGoogle ScholarPubMed
[17]Tran-Son-Tay, R., Sutera, S. P., and Rao, P. R., Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion, Biophys. J., 46 (1984), 65–72.Google Scholar
[18]Pozrikidis, C., Numerical Simulation of the Flow-Induced Deformation of Red Blood Cells, Ann. Biomed. Eng., 31 (2003), 1194–1205.Google Scholar
[19]Abkarian, M., Faivre, M. and Viallat, A., Swinging of red blood cells under shear flow, Phys. Rev. Lett., 98 (2007), 188302.Google Scholar
[20]Sui, Y., Chew, Y. T., Roy, P., Cheng, Y. P., and Low, H. T., Dynamic motion of red blood cells in simple shear flow, Phys. Fluids, 20 (2008), 112106.Google Scholar
[21]Fedosov, D.A., Caswell, B., and Karniadakis, G. E., A Multiscale Red Blood Cell Model with Accurate Mechanics, Rheology, and Dynamics, Biophys J., 98 (2010), 2215–2225.CrossRefGoogle ScholarPubMed
[22]Tsubota, K. and Wada, S., Effect of the natural state of an elastic cellular membrane on tank-treading and tumbling motions of a single red blood cell, Phys. Rev. E, 81 (2010), 011910.Google Scholar
[23]Tsubota, K., Wada, S., and Liu, H., Elastic behavior of a red blood cell with the membrane??s nonuniform natural state: equilibrium shape, motion transition under shear flow, and elongation during tank-treading motion, Biomech Model Mechanobiol, 13 (2014), 735–746.Google Scholar
[24]Fischer, T. M., Shape Memory of Human Red Blood Cells, Biophys. J, 86 (2004), 3304–3313.Google Scholar
[25]Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), 220–252.CrossRefGoogle Scholar
[26]Peskin, C. S. and McQueen, D. M., Modeling prosthetic heart valves for numerical analysis of blood flow in the heart, J. Comput. Phys., 37 (1980), 113–132.Google Scholar
[27]Peskin, C. S., The immersed boundary method, Acta Numer., 11 (2002), 479–517.Google Scholar
[28]Finite, Glowinski R.element methods for incompressible viscous flow, In Handbook of Numerical Analysis, Vol. IX, Ciarlet, PG and Lions, JL (Eds.). North-Holland: Amsterdam, 2003; 7–1176.Google Scholar
[29]Chorin, AJ, Hughes, TJR, McCracken, MF, Marsden, JE. Product formulas and numerical algorithms. Comm. Pure Appl. Math. 1978; 31: 205–256.CrossRefGoogle Scholar
[30]Dean, EJ, Glowinski, R.A wave equation approach to the numerical solution of the Navier-Stokes equations for incompressible viscous flow. C.R. Acad. Sc. Paris, Série 1 1997; 325: 783–791.Google Scholar
[31]Adams, J., Swarztrauber, P. and Sweet, R., FISHPAK: A package of Fortran subprograms for the solution of separable elliptic partial differential equations, The National Center for Atmospheric Research, Boulder, CO, 1980.Google Scholar
[32]Liu, Y., Liu, W. K., Rheology of red blood cell aggregation by computer simulation, J. Comput. Phys., 220 (2006), 139–154.Google Scholar
[33]Kaoui, B., Harting, J., and Misbah, C., Two-dimensional vesicle dynamics under shear flow: Effect of confinement, Phys. Rev. E, 83 (2011), 066319.Google Scholar
[34]Fischer, T. M., Stöhr-Liesen, M., and Schmid-Schönbein, H., The red cell as a fluid droplet: tank tread-like motion of the human erythrocyte membrane in shear flow, Science, 202 (1978), 894–896.CrossRefGoogle ScholarPubMed
[35]Li, H. B., Yi, H. H., Shan, X. M., and Fang, H. P., Shape changes and motion of a vesicle in a fluid using a lattice Boltzmann model, Europhysics Letters, 81 (2008), 54002.Google Scholar