Analysis of a phase field Navier-stokes vesicle-fluid interaction model

Discrete and Continuous Dynamical Systems - Series B

Volumn 8, Issue 3, 2007, Pages 539-556

  Du, Qiang ^a   Li, Manlin ^a   Liu, Chun ^a  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Bending elastic energy; Existence; Fluid interaction; Navier Stokes; Phase field; Uniqueness; Vesicle membrane; Weak solution; Well posedness

Indexed keywords

EID: 35348974079     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2007.8.539     Document Type: Conference Paper

References (32)

1. M. Abkarian, C. Lartigue, and A. Viallat, Tank Treading and Unbinding of Deformable Vesicles in Shear Flow: Determination of the Lift Force, Phys. Rev. Lett., 88 (2002), 068103
2. J. Beaucourt, F. Rioual, T. Sion, T. Biben, and C. Misbah, Steady to unsteady dynamics of a vesicle in a flow, Phys. Rev. E, 69 (2004), 011906
3. T. Biben, K. Kassner and C. Misbah, Phase-field approach to three-dimensional vesicle dynamics, Phys. Rev. E, 72 (2005), 041921
4. T. Biben and C. Misbah, Tumbling of vesicles under shear flow within an advected-field approach, Phys. Rev. E, 67 (2003), 031908
5. Q. Du, C. Liu, R. Ryham and X. Wang, A phase field formulation of the Willmore problem, Nonlinearity, 18 (2005), 1249-1267,
6. Q. Du, C. Liu, R. Ryham and X. Wang, Modeling the Spontaneous Curvature Effects in Static Cell Membrane Deformations by a Phase Field Formulation, Communications in Pure and Applied Analysis, 4 (2005), 537-548.
7. Q. Du, C. Liu, R. Ryham and X. Wang, Modeling Vesicle Deformations in Flow Fields via Energetic Variational Approaches, preprint, 2006.
8. Q. Du, C. Liu, and X. 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.
9. Q. Du, C. Liu, and X. Wang, Retrieving Topological Information For Phase Field Models, SIAM Journal on Applied Mathematics, 65 (2005), 1913-1932.
10. Q. Du, C. Liu, and X. Wang, Simulating the Deformation of Vesicle Membranes under Elastic Bending Energy in Three Dimensions, Journal of Computational Physics, 212 (2006), 757-777.
11. K. de Haas, C. Bloom, D. van den Ende, M. Duits, and J. Mellema, Deformation of giant lipid bilayer vesicles in shear flow, Phys. Rev. E, 56 (1997), 7132.
12. W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C, 28 (1973), 693-703.
13. V. Kantsler and V. Steinberg, Orientation and Dynamics of a Vesicle in Tank-Treading Motion in Shear Flow, Phys. Rev. Lett., 95 (2005), 258101
14. S. Keller and R. Skalak, Motion of a tank-treading ellipsoidal particle in a shear flow, J. Fluid Mech., 120 (1982), 27-47
15. M. Kraus, W. Wintz, U. Seifert, and R. Lipowsky. Fluid Vesicles in Shear Flow, Phys. Rev. Lett., 77 (1996), 3685-3688.
16. H. Li, H. Yi, X. Shan and H. Fang, Shape changes and motion of a vesicle in a fluid using a lattice Boltzmann model, Arxiv preprint physics/0607074
17. F. LIN AND C. LIU, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure. Appl. Math., XLVIII (1995), 1-36.
18. F. LIN, C. LIU AND P. ZHANG, On Hydrodynamics of Viscoelastic Fluids, Comm. Pure Appl. Math., Vol LVIII (2005), 1-35.
19. R. Lipowsky The morphology of lipid membranes, Current Opinion in Structural Biology, 5 (1995), 531-540.
20. Y. Liu, L. Zhang, X. Wang and W.K. Liu, Coupling of Navier-Stokes equations with protein molecular dynamics and its application to hemodynamics, Inter. J. Numerical Methods in Fluids, 46 (2004), 1237-1252
21. H. Noguchi and G. Gompper, Fluid Vesicles with Viscous Membranes in Shear Flow, Phys. Rev. Lett., 93 (2004), 258102
22. H. Noguchi and G. Gompper, Meshless membrane model based on the moving least-squares method, Phys. Rev. E, 73 (2006), 021903
23. Z. Ou-Yang, J. Liu, and Y. Xie, "Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases," World Scientific, Singapore, 1999.
24. C. Pozrikidis, Interfacial dynamics for Stokes flow, Journal of Computational Physics, 169 (2001), 250-301
25. C. Pozrikidis, Numerical Simulation of the Flow-Induced Deformation of Red Blood Cells, Annals of Biomedical Engineering, 31 (2003), 1194-1205
26. U. Seifert, Fluid membranes in hydrodynamic flow fields: Formalism and an application to fluctuating quasispherical vesicles in shear flow, European Physical Journal B, 8 (1999), 405-415
27. U. Seifert, Hydrodynamic Lift on Bound Vesicles, Phys. Rev. Lett., 83 (1999), 876-879
28. U. Seifert, K. Berndl and R. Lipowsky, Configurations of fluid membranes and Vesicles, Physical Rev A, 44 (1991), 1182-1202,
29. R. Temam. "Navier-Stokes Equations, Theory and Numerical Analysis," American Mathematical Society, 2001
30. X. Wang, Optimal Profile of the phase field models for Willmore problems, preprint, 2006.
31. X. Wang and Q. Du, Convergence of numerical approximations to a phase field bending elasticity model of membrane deformations, to appear in Inter. J. Numer. Anal and Modeling, 2006.
32. X. Wang and Q. Du, Modelling and Simulations of Multi-component Lipid Membranes and Open Membranes via a Diffuse Interface Approach, to appear in J. Mathematical Biology, 2006.