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.

在本文中，针对任意曲面上的二元流体流动模型，提出了一种二阶时空精确、无条件能量稳定的方案。我们构建了一种新颖的表面离散有限体积方法，用于具有二阶空间精度的表面计算。可以基于由三角形网格组成的表面网格获得离散化. 为了获得二阶时间精度，我们在投影框架下将 Crank-Nicolson 型方法应用于 Cahn-Hilliard-Navier-Stokes 系统。所得系统通过 Jacobi 型迭代法和双共轭梯度稳定法求解。所提出的方案被证明是无条件能量稳定的，这意味着可以使用更大的时间步长。此外，我们的方案已被证明满足质量守恒性质。提出了各种数值实验来证明所提出方法的效率和鲁棒性。