Binary thermal fluids computation over arbitrary surfaces with second-order accuracy and unconditional energy stability based on phase-field model

In this paper, we propose an efficient surface computational system with second-order spatial and temporal accuracy to solve multiple physical field coupling problems over arbitrary surfaces. The computational system is coupled with heat transfer equation and incompressible Navier–Stokes equation based on a phase-field model. Due to the discretization of the triangular grids, we define the discretized gradient operator, divergence operator and the Laplace–Beltrami operator with second-order spatial accuracy. The Crank–Nicolson-type scheme is used to confirm the temporal accuracy. The Navier–Stokes equation is solved by the projection method. We use the biconjugate gradient stabilized method to solve the involved equations. The discrete system is provable to be unconditionally stable and the mass conservation law is satisfied during the computation, which implies that the proposed method is not limited by the temporal step. Several computational tests are conducted to show the efficiency, robustness and accuracy of the proposed scheme.