A Semi-Implicit Phase Field Model for Droplet Evolution
A semi-implicit phase field method for droplet evolution is proposed. Using the phase field Cahn-Hilliard equation, we are able to track the interface in multiphase flow. The idea of a semi-implicit finite difference scheme is reviewed and employed to solve two nonlinear equations, including the Navier-Stokes and the Cahn-Hilliard equations. The use of a semi-implicit method allows us to have larger time steps compared to explicit schemes. The governing equations are coupled and then solved by a GMRES solver (generalized minimal residual method) using modified Gram-Schmidt orthogonalization. To show the validity of the method, we apply the method to the simulation of a rising droplet, a leaky dielectric drop and the coalescence of drops. The numerical solutions to the phase field model match well with existing solutions over a defined range of variables.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1316474Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 272
 J. W. Cahn and J. E. Hilliard, "Free energy of a nonuniform system. I. Interfacial free energy," The Journal of chemical physics, vol. 28, 1958, pp. 258-267.
 W. Boettinger, J. Warren, C. Beckermann, and A. Karma, "Phase-field simulation of solidification 1," Annual review of materials research, vol. 32, 2002, pp. 163-194.
 L.-Q. Chen, "Phase-field models for microstructure evolution," Annual review of materials research, vol. 32, 2002, pp. 113-140.
 J. Hua, L. K. Lim, and C.-H. Wang, "Numerical simulation of deformation/motion of a drop suspended in viscous liquids under influence of steady electric fields," Physics of Fluids (1994-present), vol. 20, 2008, p. 113302.
 X. Yang, J. J. Feng, C. Liu, and J. Shen, "Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method," Journal of Computational Physics, vol. 218, 2006, pp. 417-428.
 C. Liu and J. Shen, "A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method," Physica D: Nonlinear Phenomena, vol. 179, 2003, pp. 211-228.
 S. Hu and L. Chen, "A phase-field model for evolving microstructures with strong elastic inhomogeneity," Acta materialia, vol. 49, 2001, pp. 1879-1890.
 D. Salac and W. Lu, "A local semi-implicit level-set method for interface motion," Journal of Scientific Computing, vol. 35, 2008, pp. 330-349.
 D. Saville, "Electrohydrodynamics: The Taylor-Melcher leaky dielectric model," Annual review of fluid mechanics, vol. 29, 1997, pp. 27-64.
 T. Inamuro, T. Ogata, S. Tajima, and N. Konishi, "A lattice Boltzmann method for incompressible two-phase flows with large density differences," Journal of Computational Physics, vol. 198, 2004, pp. 628-644.
 G. Taylor, "Studies in electrohydrodynamics. I. The circulation produced in a drop by electrical field," in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1966, pp. 159-166.
 Y. Lin, P. Skjetne, and A. Carlson, "A phase field model for multiphase electro-hydrodynamic flow," International Journal of Multiphase Flow, vol. 45, 2012, pp. 1-11.
 J. Baygents, N. Rivette, and H. Stone, "Electrohydrodynamic deformation and interaction of drop pairs," Journal of Fluid Mechanics, vol. 368, 1998, pp. 359-375.
 Q. Yang, B. Q. Li, and Y. Ding, "3D phase field modeling of electrohydrodynamic multiphase flows," International Journal of Multiphase Flow, vol. 57, 2013, pp. 1-9.