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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 |
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., ().
|
[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. |
show all references
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., ().
|
[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. |
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