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An Immersed Interface Method for the Simulation of Inextensible Interfaces in Viscous Fluids

Published online by Cambridge University Press:  20 August 2015

Zhijun Tan ,
D. V. Le ,
K. M. Lim  and
B. C. Khoo
Zhijun Tan*
Affiliation:
Guangdong Province Key Laboratory of Computational Science & School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China
D. V. Le*
Affiliation:
Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore
K. M. Lim*
Affiliation:
Singapore-MIT Alliance, 4 Engineering Drive 3, National University of Singapore, Singapore 117576, Singapore Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore
B. C. Khoo*
Affiliation:
Singapore-MIT Alliance, 4 Engineering Drive 3, National University of Singapore, Singapore 117576, Singapore Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore
*
Corresponding author.Email:tzhij@mail.sysu.edu.cn
Email:ledv@ihpc.a-star.edu.sg
Email:mpelimkm@nus.edu.sg
Email:mpekbc@nus.edu.sg
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Abstract

In this paper, an immersed interface method is presented to simulate the dynamics of inextensible interfaces in an incompressible flow. The tension is introduced as an augmented variable to satisfy the constraint of interface inextensibility and the resulting augmented system is solved by the GMRES method. In this work, the arclength of the interface is locally and globally conserved as the enclosed region undergoes deformation. The forces at the interface are calculated from the configuration of the interface and the computed augmented variable, and then applied to the fluid through the related jump conditions. The governing equations are discretized on a MAC grid via a second-order finite difference scheme which incorporates jump contributions and solved by the conjugate gradient Uzawa-type method. The proposed method is applied to several examples including the deformation of a liquid capsule with inextensible interfaces in a shear flow. Numerical results reveal that both the area enclosed by interface and arclength of interface are conserved well simultaneously. These provide further evidence on the capability of the present method to simulate incompressible flows involving inextensible interfaces.

Keywords

65N0635R0565M12Inextensible interfaceStokes flowssingular forceimmersed interface methodCG-Uzawa methodfront tracking
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 3 , March 2012 , pp. 925 - 950
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Adams, J., Swarztrauber, P., Sweet, R., FISHPACK: Efficient FORTRAN subprograms for the solution of separable eliptic partial differential equations, 1999. Available on the web at http://www.scd.ucar.edu/css/software/fishpack/.Google Scholar
[2]Beyer, R.P., A computational model of the cochlea using the immersed boundary method, J. Comput. Phys., 98 (1992), pp. 145–162.Google Scholar
[3]Breyiannis, G., Pozrikidis, C., Simple shear flow of suspensions of elastic capsules, Theor. Comp. Fluid Dyn., 13 (2000), pp. 327–47.Google Scholar
[4]Calhoun, D., A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions, J. Comput. Phys., 176 (2002), pp. 231–275.Google Scholar
[5]Christoph, B., Domain imbedding methods for the Stokes equations, Numerische Mathematik, 57 (1990), pp. 435–451.Google Scholar
[6]Dillon, R., Fauci, L., Fogelson, A., and Gaver, D., Modeling biofilm processes using the immersed boundary method, J. Comput. Phys., 129 (1996), pp. 57–73.Google Scholar
[7]Eggleton, C.D. and Popel, A.S., Large deformation of red blood cell ghosts in a simple shear flow, Phys. Fluids, 10 (1998), pp. 1834–1845.Google Scholar
[8]Elman, H.C., Multigrid and Krylov subspace methods for the discrete Stokes equations, Int. J. Numer. Meth. Fluids, 227 (1996), pp. 755–770.Google Scholar
[9]Fauci, L.J. and Peskin, C.S., A computational model of aquatic animal locomotion, J. Comput. Phys., (1988), pp. 85–108.Google Scholar
[10]Fogelson, A.L., A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys., 1 (1984), pp. 111–134.Google Scholar
[11]Fogelson, A.L., Continuum models of platelet aggregation: Formulation and mechanical properties, SIAM J. Appl. Math., 52 (1992), pp. 1089–1110.Google Scholar
[12]Le, D.V., Khoo, B.C., and Peraire, J., An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries, J. Comput. Phys., 220 (2006), pp. 109–138.Google Scholar
[13]Lee, L. and LeVeque, R.J., An immersed interface method for incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 25 (2003), pp. 832–856.Google Scholar
[14]LeVeque, R.J. and Li, Z., Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997), pp. 709–735.Google Scholar
[15]LeVeque, R.J. and Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), pp. 1019–1044.Google Scholar
[16]Li, Z. and Ito, K., The immersed interface method-numerical solutions of PDEs involving interfaces and irregular domains. SIAM Frontiers Appl. Math., (2006), pp. 33.Google Scholar
[17]Li, Z. and Lai, M.C., The immersed interface method for the Navier-Stokes equations with singular forces, J. Comput. Phys., 171 (2001), pp. 822–842.Google Scholar
[18]Li, Z. and Wang, C., A fast finite difference method for solving Navier-Stokes equations on irregular domains, Comm. Math. Sci., 1 (2003), pp. 180–196.Google Scholar
[19]Linnick, M.N. and Fasel, H.F., A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains, J. Comput. Phys., 204 (2005), pp. 157–192.Google Scholar
[20]Oosterlee, C.W. and Lorenz, F.J.G., Multigrid methods for the Stokes system, Comput. Sci. Engr., 8 (2006), pp. 34–43.Google Scholar
[21]Peskin, C.S., Numerical analysis of blood flow in the heart, J Comput. Phys., 25 (1977), pp. 220–52.Google Scholar
[22]Peskin, C.S., The immersed boundary method, Acta Numerica, 11 (2002), pp. 479–517.Google Scholar
[23]Peters, J., Reichelt, V., and Reusken, A., Fast iterative solvers for discrete Stokes equations, SIAM J. Sci. Comput., 27 (2005), pp. 646–666.Google Scholar
[24]Pozrikidis, C., The axisymmetric deformation of a red blood cell in uniaxial straining Stokes flow, J. Fluid Mech. 216 (1990), pp. 231–254.Google Scholar
[25]Stockie, J.M. and Green, S.I., Simulating the motion of flexible pulp fibres using the immersed boundary method, J. Comput. Phys., 147 (1998), pp. 147–165.Google Scholar
[26]Russell, D. and Wang, Z.J., A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. Comput. Phys., 191 (2003), pp. 177–205.Google Scholar
[27]Sadd, Y., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856–869.Google Scholar
[28]Seifert, U., Configurations of fluid membranes and vesicles, Adv. Phys., 46 (1997), pp. 13–137.Google Scholar
[29]Shin, D. and Strikwerda, J.C., Fast solvers for finite difference approximations for the Stokes and Navier-Stokes equations, J. Australian Math. Soc., 38 (1996), pp. 274–290.CrossRefGoogle Scholar
[30]Stevens, M.J., Coarse-grained simulations of lipid bilayers, J. Chem. Phys., 121 (2004), pp. 11942–11948.Google Scholar
[31]Stoer, J. and Bulirsch, R., Introduction to Numerical Analysis, 3rd ed., Springer-Verlag, 2002.Google Scholar
[32]Tau, E.Y., Numerical solution of the steady Stokes equations, J. Comput. Phys., 99 (1992), pp. 190–195.CrossRefGoogle Scholar
[33]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 (2009), pp. 2334–2353.Google Scholar
[34]Veerapaneni, S.K., Gueyffier, D., Biros, G., and Zorin, D., A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows, J. Comput. Phys., 228 (2009), pp. 7233–7249.Google Scholar
[35]Tan, Z.-J., Le, D.V., Li, Z., Lim, K.M. and Khoo, B.C., An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane, J. Comput. Phys., 227 (2008), pp. 9955–9983.Google Scholar
[36]Tan, Z.-J., Lim, K.M. and Khoo, B.C., A fast immersed interface method for solving Stokes flows on irregular domains, Comput. Fluids, 38 (2009), pp. 1973–1983.Google Scholar
[37]Wang, N.T. and Fogelson, A.L., Computational methods for continuum models of platelet aggregation, J. Comput. Phys, 151 (1999), pp. 649–675.Google Scholar
[38]Wiegmann, A. and Bube, K.P., The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal., 37 (2000), pp. 827–862.Google Scholar
[39]Xu, S. and Wang, Z.J., An immersed interface method for simulating the interaction of a fluid with moving boundaries, J. Comput. Phys., 216 (2006), pp. 454–493.Google Scholar
[40]Xu, S. and Wang, Z.J., A 3D Immersed Interface Method for Fluid-Solid Interaction, Comput. Methods Appl. Mech. Engr., 197 (2008), pp. 2068–2086.Google Scholar
[41]Zhou, H. and Pozrikidis, C., Deformation of liquid capsules with incompressible interfaces in simple shear flow, J. Fluid Mech., 283 (1995), pp. 175–200.Google Scholar