Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method

Shokrpour Roudbari, M.; Şimşek, G.; Brummelen, E.H. van; Van Der Zee, K. G.

Abstract

© 2018 World Scientific Publishing Company. While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier-Stokes-Cahn-Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman-Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with non-solenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and breakup dynamics of droplets including pinching due to gravity.

Citation

Shokrpour Roudbari, M., Şimşek, G., Brummelen, E. V., & Van Der Zee, K. G. (2018). Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method. Mathematical Models and Methods in Applied Sciences, 28(4), 733-770. https://doi.org/10.1142/S0218202518500197

• Journal Article Type: Article
• Acceptance Date: Dec 7, 2017
• Online Publication Date: Mar 20, 2018
• Publication Date: Apr 1, 2018
• Deposit Date: Dec 11, 2017
• Publicly Available Date: Mar 21, 2019
• Journal: Mathematical Models and Methods in Applied Sciences
• Print ISSN: 0218-2025
• Electronic ISSN: 1793-6314
• Publisher: World Scientific
• Peer Reviewed: Peer Reviewed
• Volume: 28
• Issue: 4
• Pages: 733-770
• DOI: https://doi.org/10.1142/S0218202518500197
• Keywords: Navier-Stokes Cahn-Hilliard; Quasi-incompressible two-phase-flow; Mixture theory; Thermodynamic consistency; Diffuse interface; Energy-stable scheme
• Public URL: https://nottingham-repository.worktribe.com/output/930053
• Publisher URL: https://www.worldscientific.com/doi/abs/10.1142/S0218202518500197
• Additional Information: Electronic version of an article published as Mathematical Models and Methods in Applied Sciences [Volume, Issue, Year, Pages] doi:10.1142/S0218202518500197 ©copyright World Scientific Publishing Company https://www.worldscientific.com/doi/abs/10.1142/S0218202518500197
• Contract Date: Dec 11, 2017