March  2011, 15(2): 373-389. doi: 10.3934/dcdsb.2011.15.373

On the stochastic immersed boundary method with an implicit interface formulation

1. 

Department of Mathematics, Pennsylvania State University, University Park, PA 16802

2. 

Department of Mathematics, Pennsylvania Sate University, University Park, PA 16802, United States

Received  June 2009 Revised  October 2009 Published  December 2010

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.
Citation: Qiang Du, Manlin Li. On the stochastic immersed boundary method with an implicit interface formulation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 373-389. doi: 10.3934/dcdsb.2011.15.373
References:
[1]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics,, In, 30 (1998), 139. Google Scholar

[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. doi: 10.1016/j.jcp.2006.11.015. Google Scholar

[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. doi: 10.1016/j.jcp.2007.11.047. Google Scholar

[4]

K. Kassner, T. Biben and C. Misbah, Phase-field approach to three-dimensional vesicle dynamics,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.041921. Google Scholar

[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. Google Scholar

[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. doi: 10.1063/1.1730447. Google Scholar

[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. Google Scholar

[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. doi: 10.1142/S0218202506001212. Google Scholar

[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. doi: 10.1051/m2an:2008013. Google Scholar

[10]

Guiseppe Da Prato and Jerzy Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (2008). Google Scholar

[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., (). Google Scholar

[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. doi: 10.3934/dcdsb.2007.8.539. Google Scholar

[13]

Qiang Du, Chun Liu, Rolf Ryham and Xiaoqiang Wang, Energetic variational approaches to modeling vesicle and fluid interactions,, Physica D, 238 (2009), 923. doi: 10.1016/j.physd.2009.02.015. Google Scholar

[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. doi: 10.3934/cpaa.2005.4.537. Google Scholar

[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. doi: 10.1016/j.jcp.2004.01.029. Google Scholar

[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. doi: 10.1137/050638333. Google Scholar

[17]

Ronald F. Fox and George E. Uhlenbeck, Contributions to non-equilibrium thermodynamics. I. Theory of hydrodynamical fluctuations,, Physics of Fluids, 13 (1970), 1893. doi: 10.1063/1.1693183. Google Scholar

[18]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling,, J. Computational Physics, 155 (1999), 96. doi: 10.1006/jcph.1999.6332. Google Scholar

[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). doi: 10.1103/PhysRevE.76.051907. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/S0036139903422140. Google Scholar

[22]

L. D. Landau and E. M. Lifshitz, "Fluid Mechanics,", vol. \textbf{6}, 6 (1959). Google Scholar

[23]

J. Langer, Dendrites, viscous fingers, and the theory of pattern formation,, Science, 243 (1989), 1150. doi: 10.1126/science.243.4895.1150. Google Scholar

[24]

Jean Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, Acta Math., 63 (1934), 193. doi: 10.1007/BF02547354. Google Scholar

[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. doi: 10.1002/cpa.20102. Google Scholar

[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. doi: 10.1016/0021-9991(88)90002-2. Google Scholar

[27]

G. A. Pavliotis and A. M. Stuart, White noise limits for inertial particles in a random field,, Multiscale Model. Simul., 1 (2003), 527. doi: 10.1137/S1540345903421076. Google Scholar

[28]

Charles S. Peskin, The immersed boundary method,, Acta Numer., 11 (2002), 479. doi: 10.1017/CBO9780511550140.007. Google Scholar

[29]

Samuel A. Safran, "Statistical Thermodynamics Of Surfaces, Interfaces And Membranes,", Westview Press, (2003). Google Scholar

[30]

Udo Seifert, Configurations of fluid membranes and vesicles,, Advances in Physics, 46 (1997), 13. doi: 10.1080/00018739700101488. Google Scholar

[31]

Roger Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Society for Industrial Mathematics, (1983). Google Scholar

[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. doi: 10.1017/S0022112004000370. Google Scholar

show all references

References:
[1]

D. M. Anderson, G. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics,, In, 30 (1998), 139. Google Scholar

[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. doi: 10.1016/j.jcp.2006.11.015. Google Scholar

[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. doi: 10.1016/j.jcp.2007.11.047. Google Scholar

[4]

K. Kassner, T. Biben and C. Misbah, Phase-field approach to three-dimensional vesicle dynamics,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.041921. Google Scholar

[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. Google Scholar

[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. doi: 10.1063/1.1730447. Google Scholar

[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. Google Scholar

[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. doi: 10.1142/S0218202506001212. Google Scholar

[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. doi: 10.1051/m2an:2008013. Google Scholar

[10]

Guiseppe Da Prato and Jerzy Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (2008). Google Scholar

[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., (). Google Scholar

[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. doi: 10.3934/dcdsb.2007.8.539. Google Scholar

[13]

Qiang Du, Chun Liu, Rolf Ryham and Xiaoqiang Wang, Energetic variational approaches to modeling vesicle and fluid interactions,, Physica D, 238 (2009), 923. doi: 10.1016/j.physd.2009.02.015. Google Scholar

[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. doi: 10.3934/cpaa.2005.4.537. Google Scholar

[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. doi: 10.1016/j.jcp.2004.01.029. Google Scholar

[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. doi: 10.1137/050638333. Google Scholar

[17]

Ronald F. Fox and George E. Uhlenbeck, Contributions to non-equilibrium thermodynamics. I. Theory of hydrodynamical fluctuations,, Physics of Fluids, 13 (1970), 1893. doi: 10.1063/1.1693183. Google Scholar

[18]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling,, J. Computational Physics, 155 (1999), 96. doi: 10.1006/jcph.1999.6332. Google Scholar

[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). doi: 10.1103/PhysRevE.76.051907. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/S0036139903422140. Google Scholar

[22]

L. D. Landau and E. M. Lifshitz, "Fluid Mechanics,", vol. \textbf{6}, 6 (1959). Google Scholar

[23]

J. Langer, Dendrites, viscous fingers, and the theory of pattern formation,, Science, 243 (1989), 1150. doi: 10.1126/science.243.4895.1150. Google Scholar

[24]

Jean Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, Acta Math., 63 (1934), 193. doi: 10.1007/BF02547354. Google Scholar

[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. doi: 10.1002/cpa.20102. Google Scholar

[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. doi: 10.1016/0021-9991(88)90002-2. Google Scholar

[27]

G. A. Pavliotis and A. M. Stuart, White noise limits for inertial particles in a random field,, Multiscale Model. Simul., 1 (2003), 527. doi: 10.1137/S1540345903421076. Google Scholar

[28]

Charles S. Peskin, The immersed boundary method,, Acta Numer., 11 (2002), 479. doi: 10.1017/CBO9780511550140.007. Google Scholar

[29]

Samuel A. Safran, "Statistical Thermodynamics Of Surfaces, Interfaces And Membranes,", Westview Press, (2003). Google Scholar

[30]

Udo Seifert, Configurations of fluid membranes and vesicles,, Advances in Physics, 46 (1997), 13. doi: 10.1080/00018739700101488. Google Scholar

[31]

Roger Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", Society for Industrial Mathematics, (1983). Google Scholar

[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. doi: 10.1017/S0022112004000370. Google Scholar

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