Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T09:22:00.878Z Has data issue: false hasContentIssue false
  • English
  • Français

A finite element approach to incompressible two-phase flow on manifolds

Published online by Cambridge University Press:  08 August 2012

I. Nitschke ,
A. Voigt  and
J. Wensch
I. Nitschke
Affiliation:
Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, 01062 Dresden, Germany
A. Voigt*
Affiliation:
Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, 01062 Dresden, Germany Center for Advanced Modeling and Simulation, Technische Universität Dresden, 01062 Dresden, Germany
J. Wensch
Affiliation:
Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, 01062 Dresden, Germany
*
†Email address for correspondence: axel.voigt@tu-dresden.de
Get access Rights & Permissions [Opens in a new window]

Abstract

A two-phase Newtonian surface fluid is modelled as a surface Cahn–Hilliard–Navier–Stokes equation using a stream function formulation. This allows one to circumvent the subtleties in describing vectorial second-order partial differential equations on curved surfaces and allows for an efficient numerical treatment using parametric finite elements. The approach is validated for various test cases, including a vortex-trapping surface demonstrating the strong interplay of the surface morphology and the flow. Finally the approach is applied to a Rayleigh–Taylor instability and coarsening scenarios on various surfaces.

JFM classification

Biological Fluid Dynamics: Membranes Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 708 , 10 October 2012 , pp. 418 - 438
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Aland, S. & Voigt, A. 2012 Benchmark computations of diffuse interface models for two-dimensional bubble dynamics. Intl J. Numer. Meth. Fluids 69, 747–761.CrossRefGoogle Scholar
2. Arroyo, M. & DeSimone, A. 2009 Relaxation dynamics of fluid membranes. Phys. Rev. E 79, 031915.CrossRefGoogle ScholarPubMed
3. Bertalmio, M., Cheng, L. T., Osher, S. & Sapiro, G. 2001 Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174, 759–780.CrossRefGoogle Scholar
4. Bonito, A., Nochetto, R. & Pauletti, M. S. 2010 Parametric FEM for geometric biomembranes. J. Comput. Phys. 229, 3171–3188.CrossRefGoogle Scholar
5. Bothe, D. & Pruess, J. 2010 On the two-phase Navier–Stokes equations with Boussinesq–Scriven surface fluid. J. Math. Fluid Mech. 12, 133–150.CrossRefGoogle Scholar
6. Bowick, M. J. & Giomi, L. 2009 Two-dimensional matter: order, curvature and defects. Adv. Phys. 58, 449–563.CrossRefGoogle Scholar
7. Cao, C. S., Rammaha, M. A. & Titi, E. S. 1999 The Navier–Stokes equations on a rotating 2-d sphere: Gevrey regularity and asymptotic degrees of freedom. Z. Angew. Math. Phys. 50, 341–360.CrossRefGoogle Scholar
8. Dziuk, G. 1991 An algorithm for evolutionary surfaces. Numer. Math. 58, 603–611.CrossRefGoogle Scholar
9. Dziuk, G. & Elliott, C. M. 2007a Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, 262–292.CrossRefGoogle Scholar
10. Dziuk, G. & Elliott, C. M. 2007b Surface finite elements for parabolic equations. J. Comput. Math. 25, 385–407.Google Scholar
11. Ebin, D. G. & Marsden, J. 1970 Groups of diffeomorphisms and motion of an incompressible fluid. Ann. Maths 92, 102.CrossRefGoogle Scholar
12. Elliott, C. M. & Stinner, B. 2009 Analysis of a diffuse interface approach to an advection diffusion equation on a moving surface. Math. Models Meth. Appl. Sci. 19, 787–802.CrossRefGoogle Scholar
13. Elliott, C. M. & Stinner, B. 2011 A surface phase field model for two-phase biological membranes. SIAM J. Appl. Math. 70, 2904–2928.CrossRefGoogle Scholar
14. Fan, J., Han, T. & Haataja, M. 2010 Hydrodynamic effects on spinodal decomposition kinetics in planar lipid bilayer membranes. J. Chem. Phys. 133, 235101.CrossRefGoogle ScholarPubMed
15. Feng, X. 2006 Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44, 1853–1866.CrossRefGoogle Scholar
16. Fraenkel, T. 1997 The Geometry of Physics: An Introduction. Cambridge University Press.Google Scholar
17. Gomez, H. & Hughes, T. J. R. 2011 Provable unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models. J. Comput. Phys. 230, 5310–5327.CrossRefGoogle Scholar
18. Heine, C.-J. 2004 Isoparametric finite element approximation of curvature on hypersurfaces. Preprint, No. 26, Fakultät für Mathematik und Physik, Universität Freiburg.Google Scholar
19. Hohenberg, P. C. & Halperin, B. I. 1977 Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479.CrossRefGoogle Scholar
20. Hu, D., Zhang, P. W. & E, W. 2007 Continuum theory of moving membrane. Phys. Rev. E 75, 041605.CrossRefGoogle ScholarPubMed
21. Kwon, Y., Thornton, K. & Voorhees, P. W. 2010 Morphology and topology in coarsening of domains via non-conserved and conserved dynamics. Phil. Mag. 90, 317–335.CrossRefGoogle Scholar
22. Laradji, M. & Kumar, P. B. 2006 Anomalously slow domain growth in fluid membranes with asymmetric transbilayer lipid distribution. Phys. Rev. E 73, 040901.CrossRefGoogle ScholarPubMed
23. Li, S., Lowengrub, J., Rätz, A. & Voigt, A. 2009 Solving PDEs in complex geometries: a diffuse domain approach. Commun. Math. Sci. 7, 81–107.CrossRefGoogle ScholarPubMed
24. Lowengrub, J. S., Rätz, A. & Voigt, A. 2009 Phase-field modelling of the dynamics of multicomponent vesicles: spinodal decomposition, coarsening, budding, and fission. Phys. Rev. E 79, 031926.CrossRefGoogle ScholarPubMed
25. Lowengrub, J. & Truskinowsky, L. 1998 Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454, 2617–2654.CrossRefGoogle Scholar
26. MacDonald, C. B. & Ruuth, S. J. 2009 The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM J. Sci. Comput. 31, 4330–4350.CrossRefGoogle Scholar
27. Meyer, M., Desbrun, M., Schröder, P. & Barr, A. H. 2003 Discrete differential-geometry operators for triangulated 2-manifolds. In Visualization and Mathematics III (ed. Hege, H.-C. & Ploltier, K. ), pp. 59–68. Springer.Google Scholar
28. Mitrea, M. & Taylor, M. 2001 Navier–Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann. 321, 955–987.CrossRefGoogle Scholar
29. Neamtan, S. M. 1946 The motion of harmonic waves in the atmosphere. J. Meteorol. 3, 43–56.2.0.CO;2>CrossRefGoogle Scholar
30. Ramachandran, S., Komura, S. & Gompper, G. 2010 Effects of an embedding bulk fluid on phase separation dynamics in a thin liquid film. Eur. Phys. Lett. 89, 56001.CrossRefGoogle Scholar
31. Ramachandran, S., Laradji, M. & Kumar, P. B. 2009 Lateral organisation of lipids in multi-component liposomes. J. Phys. Soc. Japan 78, 041006.CrossRefGoogle Scholar
32. Rätz, A. & Voigt, A. 2006 PDEs on surfaces – a diffuse interface approach. Commun. Math. Sci. 4, 575–590.CrossRefGoogle Scholar
33. Rätz, A. & Voigt, A. 2007 A diffuse-interface approximation for surface diffusion including adatoms. Nonlinearity 20, 177–192.CrossRefGoogle Scholar
34. Ruuth, S. J. & Merriman, B. 2008 A simple embedding method for solving partial differential equations on surfaces. J. Comput. Phys. 227, 1943–1961.CrossRefGoogle Scholar
35. Saeki, D., Hamada, T. & Yoshikawa, K. 2006 Domain-growth kinetics in a cell-sized liposome. J. Phys. Soc. Japan 75, 013602.CrossRefGoogle Scholar
36. Safmann, P. G. & Delbrueck, M. 1975 Brownian motion in biological membranes. Proc. Natl Acad. Sci. 72, 3111–3113.CrossRefGoogle Scholar
37. Scriven, L. E. 1960 Dynamics of fluid interfaces. Equations of motion for Newtonian surface fluids. Chem. Engng Sci. 12, 98–108.CrossRefGoogle Scholar
38. Stöcker, C. & Voigt, A. 2008 Geodesic evolution laws – a level set approach. SIAM J. Imag. Sci. 1, 379–399.CrossRefGoogle Scholar
39. Taniguchi, T. 1996 Shape deformation and phase separation dynamics of two-component vesicles. Phys. Rev. Lett. 76, 4444–4447.CrossRefGoogle ScholarPubMed
40. Temam, R. 1988 Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer.CrossRefGoogle Scholar
41. Turner, A. M., Vitelli, V. & Nelson, D. R. 2010 Vortices on curved surfaces. Rev. Mod. Phys. 82, 1301–1348.CrossRefGoogle Scholar
42. Veatch, S. L. & Keller, S. L. 2003 Separation of liquid phases in giant vesicles of ternary mixtures of phospholipids and cholesterol. Biophys. J. 85, 3074–3083.CrossRefGoogle Scholar
43. Vey, S. & Voigt, A. 2007 AMDIS – adaptive multidimensional simulations. Comput. Vis. Sci. 10, 57–66.CrossRefGoogle Scholar
44. Wang, X. Q. & Du, Q. 2008 Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol. 56, 347–371.CrossRefGoogle ScholarPubMed
45. Yanagisawa, M., Imal, M., Masui, T., Komura, S. & Ohta, T. 2007 Dynamics of domains in ternary fluid vesicles. Biophys. J. 92, 115–125.CrossRefGoogle ScholarPubMed
46. Zienkiewicz, O. C. & Zhu, J. Z. 1987 A simple error estimator and adaptive procedure for practical engineering analysis. Intl J. Numer. Meth. Engng 24, 337–357.CrossRefGoogle Scholar