A linear mass and energy conserving numerical scheme for two-phase flows with thermocapillary effects

In this paper, a thermodynamically consistent phase-field model is employed to simulate the thermocapillary migration of a droplet. The model equations consist of a general Navier-Stokes equation for the two-phase flows, a Cahn-Hilliard equation for the diffuse interface, and a heat equation, and meanwhile satisfy the balance laws of mass, energy and entropy. In particular, the total energy of the system includes kinetic energy, potential energy and internal energy, which leads to a highly coupled and nonlinear equation system. We therefore develop a linear mass and energy conserving, semi-decoupled numerical method for the numerical simulations. As the model contains a heat (energy) equation, a simple error term introduced by the temporal discretization of the momentum equation can be absorbed into the heat equation, such that the numerical solutions satisfy the conservation laws of mass and energy exactly at the temporal discrete level. Several numerical tests are carried out to validate our numerical method.