On a phase-field model for electrowetting

Interfaces and Free Boundaries

Volumn 11, Issue 2, 2009, Pages 259-290

  Eck, C ^a   Fontelos, M ^b   Grun G ^c   Klingbeil, F ^c   Vantzos, O ^d  

^a UNIVERSITY OF STUTTGART   (Germany)

^b CSIC   (Spain)

^c UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

^d UNIVERSITY OF BONN   (Germany)

Author keywords

Cahn Hilliard model; Electrowetting; Existence of weak solutions; Free boundary problem in PDE; Navier Stokes system; Phase field model

Indexed keywords

EID: 71049194958     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/211     Document Type: Article

References (38)

1. ABELS, H. Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Preprint, Max-Planck-Institute for Mathematics in the Sciences (2008).
2. ALLEN, S. M., & CAHN, J. W. A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metal. 27 (1979), 1085-1095.
3. BADALASSI, V. E., CENICEROS, H. D., & BANERJEE, S. Computation of multiphase systems with phase field models. J. Comput. Phys. 190 (2003), 371-397. Zbl 1076.76517 MR 2013023
4. BENSOUSSAN, A., & FREHSE, J. Regularity Results for Nonlinear Elliptic Systems and Applications. Springer, Berlin (2002).
5. BERGE, B., & PESEUX, J. Variable focal lens controlled by an external voltage: An application of electrowetting. Eur. Phys. J. E: Soft Matter Biol. Phys. 3 (2000), 159-163.
6. BLAKE, T. D., CLARKE, A., & STATTERSFIELD, E. H. An investigation of electrostatic assist in dynamic wetting. Langmuir 16 (2000), 2928-2935.
7. BÜHRLE, J., HERMINGHAUS, S., & MUGELE, F. Interface profiles near three-phase contact lines in electric fields. Phys. Rev. Lett. 91 (2003), 86101, 4 pp.
8. CAHN, J. W., & HILLIARD, J. E. Free energy of a nonuniform system, I. Interfacial free energy. J. Chem. Phys. 28 (1958), 258-267.
9. CIARLET, P. G. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). Zbl 0999.65129 MR 1930132
10. COONEY, C. G., CHEN, C. Y., EMERLING, M., NADIM, A., & STERLING, J. D. Electrowetting droplet microfluidics on a single planar surface, Microfluidics and Nanofluidics 2 (2006), 435-446.
11. DACOROGNA, B. Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals. Lecture Notes in Math. 922, Springer (1982). Zbl 0484.46041 MR 0658130
12. DIBENEDETTO, E. Degenerate Parabolic Equations. Springer (1993). Zbl 0794.35090 MR 1230384
13. ELMAN, H., SILVESTER, D., & WATHEN, A. Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Oxford Univ. Press, New York (2005) Zbl 1083.76001 MR 2155549
14. FENG, X. Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44 (2006), 1049-1072. Zbl pre05167765 MR 2231855
15. FENG, X., HE, Y., & LIU, C. Analysis of finite element approximations of a phase field model for two-phase fluids. Math. Comput. 76 (2007), 539-571. Zbl 1111.76028 MR 2291827
16. GILBARG, D., & TRUDINGER, N. S. Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983). Zbl 0562.35001 MR 0737190
17. GIRAULT, V., & RAVIART, P. A. Finite Element Methods for Navier-Stokes Equations. Springer (1986). Zbl 0585.65077 MR 0851383
18. DE GROOT, S. R., & MAZUR, P. Non-equilibrium Thermodynamics. North-Holland (1962). MR 0140332
19. JACKSON, J. D. Classical Electrodynamics. 3rd ed., Wiley (1999). Zbl 0920.00012
20. KRÖNER, D. Numerical Schemes for Conservation Laws. Wiley and Teubner (1997). Zbl 0872.76001 MR 1437144
21. KUIPER, S., & HENDRIKS, B. H. W. Variable-focus liquid lens for miniature cameras. Appl. Phys. Lett. 85 (2004), 1128-1130.
22. LANDAU, L. D., LIFSHITZ, E. M., & PITAEVSKII, L. P. Electrodynamics of Continuous Media. 2nd ed., Academic Press (1984).
23. LIPPMANN, G. Ann. Chim. Phys. 5 (1875), 494.
24. LOWENGRUB, J. S., & TRUSKINOVSKY, L. Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. Roy. Soc. London A 454 (1998), 2617-2654. Zbl 0927.76007 MR 1650795
25. LU, H. W., GLASNER, K., BERTOZZI, A. L., & KIM, C. J. A diffuse interface model for electrowetting drops in a Hele-Shaw cell. J. Fluid Mech. 590 (2007), 411-435. Zbl 1141.76482 (Pubitemid 350066517)
26. MUGELE, F., & BÜHRLE, J. Equilibrium drop surface profiles in electric fields. J. Phys. Condens. Matter 19 (2007), 375112, 20 pp.; doi: 10.1088/0953-8984/19/37/375112.
27. MUGELE, F., & HERMINGHAUS, S. Electrostatic stabilization of fluid microstructures. Appl. Phys. Lett. 81 (2002), 2303.
28. NAVIER, C. L. M. H. Mémoire sur les lois du movement des fluides. Mem. Acad. R. Sci. Inst. France 6 (1823), 389-440.
29. ONSAGER, L. Reciprocal relations in irreversible processes I. Phys. Rev. 37 (1931), 405-426. Zbl 0001.09501
30. PRINS, M. W. J., WELTERS, W. J. J., & WEEKAMP, J. W. Fluid control in multichannel structures by electrocapillary pressure. Science 291 (2001), 277-280.
31. QIAN, T., WANG, X. P., & SHENG, P. A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564 (2006), 333-360. Zbl pre05068409 MR 2261865 (Pubitemid 44420003)
32. RACKE, R., & ZHENG, S. The Cahn-Hilliard equation with dynamic boundary conditions. Adv. Differential Equations 8 (2003), 83-110. Zbl 1035.35050 MR 1946559
33. SAAD, Y. Iterative Methods for Sparse Linear Systems. 2nd ed., SIAM (2003). Zbl 1031.65046 MR 1990645
34. SAVILLE, D. A. Electrohydrodynamics: The Taylor-Melcher leaky dielectric model. Ann. Rev. Fluid Mech. 29 (1997), 27-64. MR 1435033
35. STRUWE, M. Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer, Berlin (1996). Zbl 0864.49001 MR 1411681
36. VALLET, M., VALLADE, M., & BERGE, B. Limiting phenomena for the spreading of water on polymer films by electrowetting. Eur. Phys. J. B 11 (1999), 583-591.
37. WALKER, S. W., & SHAPIRO, B. Modeling the fluid dynamics of electrowetting on dielectric(EWOD). J. Microelectromech. Systems 15 (2006), 986-1000.
38. YOUNG, T. An essay on the cohesion of fluids. Philos. Trans. Roy. Soc. London 95 (1805), 65-87