Decoupled finite element scheme of the variable-density and viscosity phase-field model of a two-phase incompressible fluid flow system using the volume-conserved Allen–Cahn dynamics

We aim to develop a fully-discrete version numerical scheme in this article for solving a variable density and viscosity phase-field model involving a nonlinear coupling between the conserved Allen–Cahn equation and the incompressible fluid dynamics. The scheme uses the finite element method for spatial discretization and the linearly stabilized-explicit method for time discretization. The fully-decoupled structure is achieved by applying the “zero-energy-contribution” feature satisfied by coupled nonlinear terms that include the advection and the surface tension, where two nonlocal auxiliary variables are used. These nonlocal auxiliary variables, combined with the operator Strang-splitting method, play as the keys to obtaining an efficient numerical algorithm. At each time step, only a series of full decoupling elliptic equations need to be solved. We rigorously demonstrate the solvability and unconditional energy stability of the scheme, and verify its effectiveness through carrying out various numerical examples including 3D droplets rising simulations.