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Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes
1. | Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany, Germany |
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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