An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane

Journal of Computational Physics

Volumn 227, Issue 23, 2008, Pages 9955-9983

  Tan, Zhijun ^a   Le, D V ^a   Li, Zhilin ^b   Lim, K M ^a   Khoo, B C ^a  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

Front tracking method; Immersed interface method; Incompressible viscous flows; Piecewise constant viscosity; Projection method; Singular force

Indexed keywords

MEMBRANES; NAVIER STOKES EQUATIONS; NEWTON-RAPHSON METHOD; VISCOUS FLOW;

ELASTIC MEMBRANES; FRONT TRACKING METHOD; IMMERSED INTERFACE METHOD; INCOMPRESSIBLE VISCOUS FLOWS; JUMP CONDITIONS; PIECE-WISE CONSTANTS; PIECEWISE CONSTANT VISCOSITY; PROJECTION METHOD; SECOND ORDERS; SINGULAR FORCE;

VISCOSITY;

EID: 53349162104     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2008.08.013     Document Type: Article

References (34)

1. Beyer R.P. A computational model of the cochlea using the immersed boundary method. J. Comput. Phys. 98 (1992) 145-162
2. Brown D.L., Cortez R., and Minion M.L. Accurate projection methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 168 (2001) 464-499
3. Calhoun D. A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions. J. Comput. Phys. 176 (2002) 231-275
4. Dillon R., Fauci L., Fogelson A., and Gaver D. Modeling biofilm processes using the immersed boundary method. J. Comput. Phys. 129 (1996) 57-73
5. Eggleton C.D., and Popel A.S. Large deformation of red blood cell ghosts in a simple shear flow. Phys. Fluids 10 (1998) 1834-1845
6. Fauci L.J., and Peskin C.S. A computational model of aquatic animal locomotion. J. Comput. Phys. (1988) 85-108
7. Fogelson A.L. A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting. J. Comput. Phys. 1 (1984) 111-134
8. Fogelson A.L. Continuum models of platelet aggregation: formulation and mechanical properties. SIAM J. Appl. Math. 52 (1992) 1089-1110
9. Harlow F.H., and Welch J.E. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8 (1965) 2182-2189
10. Lai M.-C., and Li Z. A remark on jump conditions for the three-dimensional Navier-Stokes equations involving an immersed moving membrane. Appl. Math. Lett. 14 (2001) 149-154
11. Li Z. An overview of the immersed interface method and its applications. Taiwanese J. Math. 7 (2003) 1-49
12. Z. Li, K. Ito, The immersed interface method-numerical solutions of PDEs involving interfaces and irregular domains. SIAM Frontiers in Applied Mathematics 2006, 33, ISBN: 0-89971-609-8.
13. Layton A.T. An efficient numerical method for the two-fluid Stokes equations with a moving immersed boundary. Comput. Meth. Appl. Mech. Eng. 197 (2008) 2147-2155
14. Li Z., Ito K., and Lai M.-C. An augmented approach for Stokes equations with a discontinuous viscosity and singular forces. Comput. Fluids 36 (2007) 622-635
15. Li Z., and Lai M.-C. The immersed interface method for the Navier-Stokes equations with singular forces. J. Comput. Phys. 171 (2001) 822-842
16. Li Z., and Wang C. A fast finite difference method for solving Navier-Stokes equations on irregular domains. Commun. Math. Sci. 1 (2003) 180-196
17. Li Z., Lubkin S.R., and Wan X. An augmented IIM-level set method for Stokes equations with a discontinuous viscosity. Electron. J. Different. Equat. 15 (2007) 193-210
18. D.V. Le, An immersed interface method for solving viscous incompressible flows involving rigid and flexible boundaries, Ph.D thesis, Singapore-MIT Alliance, June 2005.
19. 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) 109-138
20. Lee L., and LeVeque R.J. An immersed interface method for incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 25 (2003) 832-856
21. LeVeque R.J., and Li Z. Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 18 (1997) 709-735
22. 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) 1019-1044
23. 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) 157-192
24. Peskin C.S. The immersed boundary method. Acta Numerica 11 (2002) 479-517
25. Peskin C.S. Numerical analysis of blood flow in the heart. J. Comput. Phys. 25 (1977) 220-252
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) 177-205
27. Stockie J.M., and Green S.I. Simulating the motion of flexible pulp fibres using the immersed boundary method. J. Comput. Phys. 147 (1998) 147-165
28. Stoer J., and Bulirsch R. Introduction to Numerical Analysis. third ed. (2002), Springer-Verlag
29. Z.-J. Tan, D.V. Le, K.M. Lim, B.C. Khoo, An Immersed Interface Method for the Incompressible Navier-Stokes Equations with Discontinuous Viscosity Across the Interface, SIAM J. Sci. Comput., submitted for publication.
30. Tu C., and Peskin C.S. Stability and instability in the computation of flows with moving immersed boundaries: a comparison of three methods. SIAM J. Sci. Stat. Comput. 13 (1992) 1361-1376
31. 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) 827-862
32. Xu S., and Wang Z.J. Systematic derivation of jump conditions for the immersed interface method in three-dimensional flow simulation. SIAM J. Sci. Comput. 27 (2006) 1948-1980
33. 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) 454-493
34. Xu S., and Wang Z.J. A 3D immersed interface method for fluid-solid interaction. Comput. Meth. Appl. Mech. Eng. 197 (2008) 2068-2086