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New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows

Published online by Cambridge University Press:  28 May 2015

Zhilin Li  and
Ming-Chih Lai
Zhilin Li*
Affiliation:
Center for Research in Scientific Computation & Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA
Ming-Chih Lai*
Affiliation:
Center of Mathematical Modeling and Scientific Computing & Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract

In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the velocity is zero along the interface. Thus, the area enclosed by the interface and the total length of the interface should be conserved during the evolution process. Because of the nonlinear and coupling nature of the problem, direct discretization by applying the immersed boundary or immersed interface method yields complex nonlinear systems to be solved. In our new methods, we treat the unknown surface tension as an augmented variable so that the augmented IIM can be applied. Since finding the unknown surface tension is essentially an inverse problem that is sensitive to perturbations, our regularization strategy is to introduce a controlled tangential force along the interface, which leads to a least squares problem. For Stokes equations, the forward solver at one time level involves solving three Poisson equations with an interface. For Navier-Stokes equations, we propose a modified projection method that can enforce the pressure jump condition corresponding directly to the unknown surface tension. Several numerical experiments show good agreement with other results in the literature and reveal some interesting phenomena.

Keywords

65M0665M1276T05Inextensible interfaceincompressible flowStokes equationsNavier-Stokes equationsimmersed interface methodinverse problemregularizationaugmented immersed interface method
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 2 , May 2011 , pp. 155 - 171
Copyright
Copyright © Global-Science Press 2011

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References

[1]Adams, J., IP Swarztrauber, and R. Sweet. Fishpack: Efficient Fortran subprograms for the solution of separable elliptic partial differential equations. http://www.netlib.org/fishpack/.Google Scholar
[2]Ito, K., Li, Z., and Lai, M-C.. An augmented method for the Navier-Stokes equations on irregular domains. J. Comput. Phys., 228:2616–2628, 2009.Google Scholar
[3]Kim, Y. and Lai, M.-C., Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method, J. Comput. Phys., 229:4840–4853, 2010.Google Scholar
[4]Kraus, M., Wintz, W., Seifert, U., and Lipowsky, R.. Fluid vesicles in shear flow. Physical Review Letters, 95, 2005.Google Scholar
[5]Lai, M.-C. and Li, Z., A remark on jump conditions for the three-dimensional Navier-Stokes equations involving an immersed moving membrane, Applied Mathematics Letters, 14, vol 2, 149–154, (2001).Google Scholar
[6]LeVeque, R. J. and Li, Z.. Immersed interface method for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput., 18:709–735, 1997.Google Scholar
[7]Li, Z., Lai, M-C., He, G., and Zhao, H.. An augmented method for free boundary problems with moving contact lines. Computers and Fluids, 39:1033–1040, 2010.Google Scholar
[8]Li, Z.. IIMPACK: A collection of fortran codes for interface problems. Anonymous ftp at under the directory: /pub/math/zhilin/Package and http://www4.ncsu.edu/zhilin/IIM, last updated: 2008.Google Scholar
[9]Li, Z. and Ito, K.. The Immersed Interface Method - Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. SIAM Frontier Series in Applied mathematics, FR33, 2006.Google Scholar
[10]Li, Z., Ito, K., and Lai, M-C.. An augmented approach for Stokes equations with a discontinuous viscosity and singular forces. Computers and Fluids, 36:622–635, 2007.Google Scholar
[11]Li, Z. and Lai, M-C.. The immersed interface method for the Navier-Stokes equations with singular forces. J. Comput. Phys., 171:822–842, 2001.Google Scholar
[12]Li, Z., Wan, X., Ito, K., and Lubkin, S.. An augmented pressure boundary condition for a Stokes flow with a non-slip boundary condition. Commun. Comput. Phys., 1:874–885, 2006.Google Scholar
[13]Lowengrub, J. S., Ratz, A., and Voigt, A.. Phase-field modeling of the dynamics of multicom-ponent vesicles: Spinodal decomposition, coarsening, budding, and fissio. Physical Review E, 79, 2009.Google Scholar
[14]Markvoort, A. J., Van Santen, R.A., and Hilber, P.A.J.. Vesicle shapes from molecular dynamics simulation. J. Physical Chemistry B, 110, 2006.CrossRefGoogle Scholar
[15]Sohn, J. S., Tseng, Y.-H., Li, S., Voigt, A., and Lowengrub, J. S.. Dynamics of multicomponent vesicles in a viscous fluid. J. Comput. Phys., 229:119–144, 2010.Google Scholar
[16]Song, P.. Numerical simulations of clindrical membranes. PhD thesis, Peking University, 2008.Google Scholar
[17]Veerapaneni, S. K., Gueyffier, D., Zorin, D., and Biro, G.. A numerical method for simulating the dynamics of 3d axismmetric vesicles suspended in viscous flow. J. Comput. Phys., 228:7233–7249, 2009.Google Scholar
[18]Veerapaneni, S. K., Gueyffier, D., Zorin, D., and 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.Google Scholar
[19]Zhou, H. and Pozrikidis, C.. Deformation of capsules with incompressible interfaces in simple shear flow. J. Fluid Mech., 283:175–200, 1995.Google Scholar