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April  2011, 4(2): 371-389. doi: 10.3934/dcdss.2011.4.371

Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes

M. Hassan Farshbaf-Shaker 1, and  Harald Garcke 1,

1. 

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany, Germany

Received  June 2009 Revised  October 2009 Published  November 2010

In this paper we derive thermodynamically consistent higher order phase field models for the dynamics of biomembranes in incompressible viscous fluids. We start with basic conservation laws and an appropriate version of the second law of thermodynamics and obtain generalizations of models introduced by Du, Li and Liu [3] and Jamet and Misbah [11]. In particular we derive a stress tensor involving higher order derivatives of the phase field and generalize the classical Korteweg capillarity tensor.
Keywords: fluid interfaces, convection, second law of thermodynamics, dissipation inequality, Phase field model, weak solution., biomembrane, Navier-Stokes equation, momentum equation, bending elastic energy.
Mathematics Subject Classification: Primary: 35K55, 74L15; Secondary: 74K15, 92C0.
Citation: M. Hassan Farshbaf-Shaker, Harald Garcke. Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 371-389. doi: 10.3934/dcdss.2011.4.371
References:
[1]

M. Arroyo and A. DeSimone, Relaxation dynamics of fluid membranes, Phys. Rev. E, 79 (2009), 031915. doi: 10.1103/PhysRevE.79.031915.

[2]

T. Biben, K. Kassner and C. Misbah, Phase field approach to three dimensional vesicle dynamics, Phys. Rev. E, 72 (2005), 041921. doi: 10.1103/PhysRevE.72.041921.

[3]

Q. Du, M. Li and C. Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interaction model, Disc. and Continuous Dyn. Systems. Series B, 8 (2007), 539-556. doi: 10.3934/dcdsb.2007.8.539.

[4]

Q. Du, C. Liu, R. Ryham and X. Wang, Energetic variational approaches in modeling vesicle and fluid interactions, Physica D, 238 (2009), 923-930. doi: 10.1016/j.physd.2009.02.015.

[5]

H. Garcke, B. Niethammer, M. A. Peletier and M. Röger, Mini-workshop: Mathematics of biological membranes. Abstracts from the mini-workshop held September 2008. Organized by H. Garcke, B. Niethammer, M. A. Peletier and M. Röger, Oberwolfach Reports, 5 (2008), 447-486.

[6]

H. Garcke and R. Haas, Modelling of non-isothermal multicomponent, multi-phase systems with convection, in "Phase Transformations in Multicomponent Melts," Wiley-VCH Verlag, Weinheim, (2008), 325-338. doi: 10.1002/9783527624041.ch20.

[7]

M. E. Gurtin, "An Introduction to Continuum Mechanics," Mathematics in Science and Engineering, Volume 158, 2003.

[8]

M. E. Gurtin, D. Polignone and J. Vinals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831. doi: 10.1142/S0218202596000341.

[9]

W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. C, 28 (1973), 693-703.

[10]

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.

[11]

D. Jamet and C. Misbah, Towards a thermodynamically consistent picture of the phase field model of vesicles: Curvature energy, Phys. Rev. E, 78 (2008), 031902. doi: 10.1103/PhysRevE.78.031902.

[12]

I. S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Rat. Mech. Anal., 46 (1972), 131-148. doi: 10.1007/BF00250688.

[13]

I. S. Liu and I. Müller, On the thermodynamics and thermostatics of fluids in electromagnetic fields, Arch. Rat. Mech. Anal., 46 (1972), 149-176. doi: 10.1007/BF00250689.

[14]

J. Lowengrub, A. Rätz and A. Voigt, Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission, Phys. Rev. E, 79 (2009), 031926. doi: 10.1103/PhysRevE.79.031926.

[15]

L. Modica, The gradient theory of phase transitions and minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.

[16]

M. Röger and R. Schätzle, On a modified conjecture of De Giorgi, Math. Z., 254 (2006), 675-714. doi: 10.1007/s00209-006-0002-6.

[17]

U. Seifert, Configurations of fluid membranes and vesicles, Advances in Physics, 46 (1997), 13-137. doi: 10.1080/00018739700101488.

[18]

C. Truesdell and W. Noll, "The Non-Linear Field Theories of Mechanics," Springer Verlag, 1992.

show all references

References:
[1]

M. Arroyo and A. DeSimone, Relaxation dynamics of fluid membranes, Phys. Rev. E, 79 (2009), 031915. doi: 10.1103/PhysRevE.79.031915.

[2]

T. Biben, K. Kassner and C. Misbah, Phase field approach to three dimensional vesicle dynamics, Phys. Rev. E, 72 (2005), 041921. doi: 10.1103/PhysRevE.72.041921.

[3]

Q. Du, M. Li and C. Liu, Analysis of a phase field Navier-Stokes vesicle-fluid interaction model, Disc. and Continuous Dyn. Systems. Series B, 8 (2007), 539-556. doi: 10.3934/dcdsb.2007.8.539.

[4]

Q. Du, C. Liu, R. Ryham and X. Wang, Energetic variational approaches in modeling vesicle and fluid interactions, Physica D, 238 (2009), 923-930. doi: 10.1016/j.physd.2009.02.015.

[5]

H. Garcke, B. Niethammer, M. A. Peletier and M. Röger, Mini-workshop: Mathematics of biological membranes. Abstracts from the mini-workshop held September 2008. Organized by H. Garcke, B. Niethammer, M. A. Peletier and M. Röger, Oberwolfach Reports, 5 (2008), 447-486.

[6]

H. Garcke and R. Haas, Modelling of non-isothermal multicomponent, multi-phase systems with convection, in "Phase Transformations in Multicomponent Melts," Wiley-VCH Verlag, Weinheim, (2008), 325-338. doi: 10.1002/9783527624041.ch20.

[7]

M. E. Gurtin, "An Introduction to Continuum Mechanics," Mathematics in Science and Engineering, Volume 158, 2003.

[8]

M. E. Gurtin, D. Polignone and J. Vinals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831. doi: 10.1142/S0218202596000341.

[9]

W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. C, 28 (1973), 693-703.

[10]

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.

[11]

D. Jamet and C. Misbah, Towards a thermodynamically consistent picture of the phase field model of vesicles: Curvature energy, Phys. Rev. E, 78 (2008), 031902. doi: 10.1103/PhysRevE.78.031902.

[12]

I. S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Rat. Mech. Anal., 46 (1972), 131-148. doi: 10.1007/BF00250688.

[13]

I. S. Liu and I. Müller, On the thermodynamics and thermostatics of fluids in electromagnetic fields, Arch. Rat. Mech. Anal., 46 (1972), 149-176. doi: 10.1007/BF00250689.

[14]

J. Lowengrub, A. Rätz and A. Voigt, Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission, Phys. Rev. E, 79 (2009), 031926. doi: 10.1103/PhysRevE.79.031926.

[15]

L. Modica, The gradient theory of phase transitions and minimal interface criterion, Arch. Rat. Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230.

[16]

M. Röger and R. Schätzle, On a modified conjecture of De Giorgi, Math. Z., 254 (2006), 675-714. doi: 10.1007/s00209-006-0002-6.

[17]

U. Seifert, Configurations of fluid membranes and vesicles, Advances in Physics, 46 (1997), 13-137. doi: 10.1080/00018739700101488.

[18]

C. Truesdell and W. Noll, "The Non-Linear Field Theories of Mechanics," Springer Verlag, 1992.

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