Abstract
We construct a fully-discrete finite element numerical scheme for the Cahn–Hilliard phase-field model of the two-phase incompressible flow system with variable density and viscosity. The scheme is linear, decoupled, and unconditionally energy stable. Its key idea is to combine the penalty method of the Navier–Stokes equations with the Strang operator splitting method, and introduce several nonlocal variables and their ordinary differential equations to process coupled nonlinear terms. The scheme is highly efficient and it only needs to solve a series of completely independent linear elliptic equations at each time step, in which the Cahn–Hilliard equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability and solvability of the scheme and carry out numerous accuracy/stability examples and various benchmark numerical simulations in 2D and 3D, including the Rayleigh–Taylor instability and rising/coalescence dynamics of bubbles to demonstrate the effectiveness of the scheme, numerically.
Highlights
When the Cahn–Hilliard–Navier–Stokes model (CH-NS, for short) of a two-phase incompressible flow system is coupled with variable density and viscosity, the existence of the variable density will cause many complicated terms to appear in the coupled system consisting of the Cahn–Hilliard equation and the incompressible Navier– Stokes equation
We note that many nonlinear coupling items satisfy the so-called “zero-energy-contribution” feature. By combining this particular property with the operator splitting method, we introduce several nonlocal variables and design some special ordinary differential equations (ODE)
Following the error analysis carried out for the different scheme of the Navier–Stokes coupled Cahn–Hilliard phase-field model for the constant density case, cf. [6], the error analysis associated with the developed scheme in this article will be performed in our future work
Summary
When the Cahn–Hilliard–Navier–Stokes model (CH-NS, for short) of a two-phase incompressible flow system is coupled with variable density and viscosity, the existence of the variable density will cause many complicated terms to appear in the coupled system consisting of the Cahn–Hilliard equation and the incompressible Navier– Stokes equation This model can simulate many interesting fluid interface phenomena very effectively, cf [1, 13, 16, 22, 25, 32], how to construct an effective fully discrete numerical scheme for such a highly complex coupled nonlinear system, making it linear, fully decoupled, and capable of owning unconditional energy stability has always been a very challenging problem. We expect that the resulting scheme can solve as many constant coefficient systems as possible, so as to reduce the computational cost as much as possible while maintaining linearity, and owning a decoupling structure and unconditional energy stability To this end, we note that many nonlinear coupling items satisfy the so-called “zero-energy-contribution” feature (see Rem. 2.1).
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