A second-order accurate, unconditionally energy stable numerical scheme for binary fluid flows on arbitrarily curved surfaces

In this paper, a second-order temporal and spatial accurate, unconditionally energy stable scheme for the binary fluid flows model on arbitrarily curved surfaces is proposed. We construct a novel surface discrete finite volume method for the surface computation with second-order spatial accuracy. The discretization can be obtained based on the surface mesh consisting of triangular grids. In order to obtain second order temporal accuracy, we apply a Crank–Nicolson-type method to the Cahn–Hilliard–Navier–Stokes system under the projection framework. The resulting system is solved by the Jacobi-type iteration method and bi-conjugate gradient stabilized method. The proposed scheme is proved to be unconditionally energy stable, which implies that a larger time step can be used. Additionally, our scheme has been proved to satisfy mass conservation property. Various numerical experiments are presented to demonstrate the efficiency and robustness of the proposed method.