Splitting Schemes for a Navier-Stokes-Cahn-Hilliard Model for Two Fluids with Different Densities

Abstract

In this work, we focus on designing efficient numerical schemes to approximate a thermodynamically consistent Navier-Stokes/Cahn-Hilliard problem given in [3] modeling the mixture of two incompressible fluids with different densities. The model is based on a diffuse-interface phase-field approach that is able to describe topological transitions like droplet coalescense or droplet break-up in a natural way. We present a splitting scheme, decoupling computations of the Navier-Stokes part from the Cahn-Hilliard one, which is unconditionally energy-stable up to the choice of the potential approximation. Some numerical experiments are carried out to validate the correctness and the accuracy of the scheme, and to study the sensitivity of the scheme with respect to different physical parameters.Mathematics subject classification: 35Q35, 65M60, 76D05, 76D45, 76T10.

Comparison of energy stable simulation of moving contact line problems using a thermodynamically consistent Cahn–Hilliard Navier–Stokes model

“…For (1) to (5) and (8) together with no-slip for v and no-flux for ϕ, several thermodynamically consistent discretization schemes were proposed in the last years. Here, we refer to [9,10,13]. Especially in [9] the influence of spatial adaptivity on the fully discrete energy law is discussed.…”

“…In both cases we can decouple the Navier-Stokes equation and the Cahn-Hilliard equation by using an augmented velocity field in (18), see for example [11,13,18,37]. Here we substitute −…”

“…It is valid also for different densities of the involved fluids, but specific contact line dynamics are not included. Recently, several numerical schemes for solving this system have been proposed, see for example, [9][10][11][12][13]. All these schemes are thermodynamically consistent in the sense, that they mimic the energy law from [8] in the time discrete or even in the fully discrete setting.…”

Bernaise: A Flexible Framework for Simulating Two-Phase Electrohydrodynamic Flows in Complex Domains

“…However, in order to simulate larger systems faster, it is preferable to use a splitting scheme to solve for each field sequentially. One such splitting scheme was outlined in [70], based on the energy-stable scheme without electrochemistry as developed by [68,69]. However, that scheme did not take into account that the electric permittivities in the two fluids may differ, and when they do, the phase field and the electrochemistry computations become coupled through the electric field [71].…”

“…Based on the sharp-interface model benchmarks of Hysing et al [66], Aland and Voigt [67] provided benchmarks of bubble dynamics comparing several formulations of phase-field models (without electrodynamics). Energy-stable numerical schemes for the same case were presented and analyzed in [68,69]. Campillo-Funollet et al [62] provided preliminary simulations of the two-phase electrohydrodynamics model in their paper, however with a simplified formulation of the chemical potential of the solutes.…”