# American Institute of Mathematical Sciences

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

 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.
Citation: M. Hassan Farshbaf-Shaker, Harald Garcke. Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes. Discrete & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] M. E. Gurtin, "An Introduction to Continuum Mechanics," Mathematics in Science and Engineering, Volume 158, 2003.  Google Scholar [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.  Google Scholar [9] W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. C, 28 (1973), 693-703. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] U. Seifert, Configurations of fluid membranes and vesicles, Advances in Physics, 46 (1997), 13-137. doi: 10.1080/00018739700101488.  Google Scholar [18] C. Truesdell and W. Noll, "The Non-Linear Field Theories of Mechanics," Springer Verlag, 1992.  Google Scholar

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##### References:
 [1] M. Arroyo and A. DeSimone, Relaxation dynamics of fluid membranes, Phys. Rev. E, 79 (2009), 031915. doi: 10.1103/PhysRevE.79.031915.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] M. E. Gurtin, "An Introduction to Continuum Mechanics," Mathematics in Science and Engineering, Volume 158, 2003.  Google Scholar [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.  Google Scholar [9] W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments, Z. Naturforsch. C, 28 (1973), 693-703. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] U. Seifert, Configurations of fluid membranes and vesicles, Advances in Physics, 46 (1997), 13-137. doi: 10.1080/00018739700101488.  Google Scholar [18] C. Truesdell and W. Noll, "The Non-Linear Field Theories of Mechanics," Springer Verlag, 1992.  Google Scholar
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