A numerical method for the quasi-incompressible Cahn–Hilliard–Navier–Stokes equations for variable density flows with a discrete energy law

Abstract

In this paper, we investigate numerically a diffuse interface model for the NavierStokes equation with fluid-fluid interface when the fluids have different densities [45]. Under minor reformulation of the system, we show that there is a continuous energy law underlying the system, assuming that all variables have reasonable regularities. It is shown in the literature that an energy law preserving method will perform better for multiphase problems. Thus for the reformulated system, we design a C 0 finite element method and a special temporal scheme where the energy law is preserved at the discrete level. Such a discrete energy law (almost the same as the continuous energy law) for this variable density two-phase flow model has never been established before with C 0 finite element. A Newton's method is introduced to linearise the highly non-linear system of our discretization scheme. Some numerical experiments are carried out using the adaptive mesh to investigate the scenario of coalescing and rising drops with differing density ratio. The snapshots for the evolution of the interface together with the adaptive mesh at different times are presented to show that the evolution, including the break-up/pinch-off of the drop, can be handled smoothly by our numerical scheme. The discrete energy functional for the system is examined to show that the energy law at the discrete level is preserved by our scheme.

Numerical simulation for multi-phase fluids considering different densities with continuous finite element method

“…In this section, we simulate the buoyancy-driven evolution of a rising bubble with equation (17)- (20). A lighter bubble is set in a two phase fluids distributed at the top and bottom of the pipe and the bubble rises from the bottom of this two phase media to the top of the computing domain.…”

“…Phase-field model [16] has become one of the major tools to deal with many dynamical processes in biological morphology, which can be traced back to Cahn et al [17,18] [19,20] developed the model and its advantage is the well-posed nonlinear partial differential equations can satisfy thermodynamics-consistent energy dissipation laws. In [21], a massconserved diffuse interface method is proposed for simulating incompressible flows of binary fluids with large density ratio.…”

Fully Discrete Second-Order Linear Schemes for Hydrodynamic Phase Field Models of Binary Viscous Fluid Flows with Variable Densities

“…The time discretization in this scheme is the analogue of the energy dissipation preserving time discretization used in [30]. Downloaded 04/13/18 to 129.123.124.101.…”

“…However, the numerical schemes developed so far for model (1.3) or model (1.4) in the literature are either linear, first order in time, or, even if they can reach secondorder accuracy in time, they are nonlinear so that a highly nonlinear system has to be solved iteratively [22,28,29,30,31]. The nonlinear systems usually require sophisticated implementation and a smaller time step to ensure convergence of the numerical solver.…”

Energy stable numerical scheme for the viscous Cahn–Hilliard–Navier–Stokes equations with moving contact line

“…For the numerical studies, the Cahn-Hilliard-Navier-Stokes model was highly studied and different kinds of discretizations were proposed, we cite here only few of the existing works: [11,12,[29][30][31][32][33][34][35][36][37][38][39].…”