On the stochastic immersed boundary method with an implicit interface formulation

Qiang Du; Manlin Li

doi:10.3934/dcdsb.2011.15.373

Article Contents

2011, Volume 15, Issue 2: 373-389. Doi: 10.3934/dcdsb.2011.15.373

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On the stochastic immersed boundary method with an implicit interface formulation

Qiang Du^1, and
Manlin Li^2,

1.
Department of Mathematics, Pennsylvania State University, University Park, PA 16802
2.
Department of Mathematics, Pennsylvania Sate University, University Park, PA 16802

Received: May 31, 2009

Revised: September 30, 2009

Published: August 2011

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Abstract

In this paper, we present a consistent and rigorous derivation of some stochastic fluid-structure interaction models based on an implicit interface formulation of the stochastic immersed boundary method. Based on the fluctuation-dissipation theorem, a proper form can be derived for the noise term to be incorporated into the deterministic hydrodynamic fluid-structure interaction models in either the phase field or level-set framework. The resulting stochastic systems not only capture the fluctuation effect near equilibrium but also provide an effective tool to model the complex interfacial morphology in a fluctuating fluid.

Keywords:

Mathematics Subject Classification: 35R60, 60H15, 76D03, 76Z99.

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References

[1]	D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, In "Annual Review of Fluid Mechanics," volume 30 of Annu. Rev. Fluid Mech., pages 139-165. Annual Reviews, Palo Alto, CA, 1998.
[2]	Paul J. Atzberger, Peter R. Kramer and Charles S. Peskin, A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales, J. Comput. Phys., 224 (2007), 1255-1292.doi: 10.1016/j.jcp.2006.11.015.
[3]	J. Thomas Beale and John Strain, Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces, J. Comput. Phys., 227 (2008), 3896-3920.doi: 10.1016/j.jcp.2007.11.047.
[4]	K. Kassner, T. Biben and C. Misbah, Phase-field approach to three-dimensional vesicle dynamics, Phys. Rev. E, 72 (2005), 041921.doi: 10.1103/PhysRevE.72.041921.
[5]	Yann Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc., 2 (1989), 225-255.
[6]	J. Cahn and J. Hilliard, Free energy of a Nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699.doi: 10.1063/1.1730447.
[7]	Georges-Henri Cottet and Emmanuel Maitre, A level-set formulation of immersed boundary methods for fluid-structure interaction problems, C. R. Math. Acad. Sci. Paris, 338 (2004), 581-586.
[8]	Georges-Henri Cottet and Emmanuel Maitre, A level set method for fluid-structure interactions with immersed surfaces, Math. Models Methods Appl. Sci., 16 (2006), 415-438.doi: 10.1142/S0218202506001212.
[9]	Georges-Henri Cottet, Emmanuel Maitre and Thomas Milcent, Eulerian formulation and level set models for incompressible fluid-structure interaction, M2AN Math. Model. Numer. Anal., 42 (2008), 471-492.doi: 10.1051/m2an:2008013.
[10]	Guiseppe Da Prato and Jerzy Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, 2008.
[11]	Qiang Du and Manlin Li, Analysis of a stochastic implicit interface model for an immersed elastic surface in a fluctuating fluid, Arc. Rational Mech. Anal., to appear.
[12]	Qiang Du, Manlin Li and Chun Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interaction model, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 539-556.doi: 10.3934/dcdsb.2007.8.539.
[13]	Qiang Du, Chun Liu, Rolf Ryham and Xiaoqiang Wang, Energetic variational approaches to modeling vesicle and fluid interactions, Physica D, 238 (2009), 923-930.doi: 10.1016/j.physd.2009.02.015.
[14]	Qiang Du, Chun Liu, Rolf Ryham and Xiaoqiang Wang, Modeling of the spontaneous curvature effect in static cell membrane deformations by a phase field formulation, Comm Pure Appl Anal., 4 (2005), 537-548.doi: 10.3934/cpaa.2005.4.537.
[15]	Qiang Du, Chun Liu and Xiaoqiang Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, Journal of Computational Physics, 198 (2004), 450-468.doi: 10.1016/j.jcp.2004.01.029.
[16]	Xiaobing Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows, SIAM J. Numer. Anal., 44 (2006), 1049-1072.doi: 10.1137/050638333.
[17]	Ronald F. Fox and George E. Uhlenbeck, Contributions to non-equilibrium thermodynamics. I. Theory of hydrodynamical fluctuations, Physics of Fluids, 13 (1970), 1893-1902.doi: 10.1063/1.1693183.
[18]	D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Computational Physics, 155 (1999), 96-127.doi: 10.1006/jcph.1999.6332.
[19]	D. Jamet and C. Misbah, Towards a thermodynamically consistent picture of the phase-field model of vesicles: Local membrane incompressibility, Phys. Rev. E, 76 (2007), 051907.doi: 10.1103/PhysRevE.76.051907.
[20]	Peter R. Kramer, Charles S. Peskin and Paul J. Atzberger, On the foundations of the stochastic immersed boundary method, Comput. Methods Appl. Mech. Engrg., 197 (2008), 2232-2249.
[21]	Peter R. Kramer and Andrew J. Majda, Stochastic mode reduction for particle-based simulation methods for complex microfluid systems, SIAM J. Appl. Math., 64 (2004), 401-422.doi: 10.1137/S0036139903422140.
[22]	L. D. Landau and E. M. Lifshitz, "Fluid Mechanics," vol. 6, London, Pergamon Press, 1959.
[23]	J. Langer, Dendrites, viscous fingers, and the theory of pattern formation, Science, 243 (1989), 1150-1156.doi: 10.1126/science.243.4895.1150.
[24]	Jean Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.doi: 10.1007/BF02547354.
[25]	Andrew J. Majda and Xiaoming Wang, The emergence of large-scale coherent structure under small-scale random bombardments, Comm. Pure Appl. Math., 59 (2006), 467-500.doi: 10.1002/cpa.20102.
[26]	S. Osher and J. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.doi: 10.1016/0021-9991(88)90002-2.
[27]	G. A. Pavliotis and A. M. Stuart, White noise limits for inertial particles in a random field, Multiscale Model. Simul., 1 (2003), 527-533.doi: 10.1137/S1540345903421076.
[28]	Charles S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479-517.doi: 10.1017/CBO9780511550140.007.
[29]	Samuel A. Safran, "Statistical Thermodynamics Of Surfaces, Interfaces And Membranes," Westview Press, 2003.
[30]	Udo Seifert, Configurations of fluid membranes and vesicles, Advances in Physics, 46 (1997), 13-137.doi: 10.1080/00018739700101488.
[31]	Roger Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," Society for Industrial Mathematics, 1983.
[32]	P. Yue, J. J. Feng, C. Liu and J. Shen, A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515 (2004), 293-317.doi: 10.1017/S0022112004000370.