To simulate the two-phase flow of conducting fluids, we propose a coupled model of the Cahn-Hilliard equations and the inductionless and incompressible magnetohydrodynamic (MHD) equations. The model describes the dynamic behavior of conducting fluid under the influence of magnetic field. Based on the “invariant energy quadratization” method, we propose two fully discrete time-marching schemes which are linear, decoupled, unconditionally energy stable, and of second order. The well-posedness and energy stability of the discrete problems are proven. By extensive numerical experiments, we verify the second-order convergence of the numerical methods and demonstrate the capability of the coupled model for simulating two-phase flows.

为了模拟导电流体的两相流，我们提出了Cahn-Hilliard方程与无感应和不可压缩磁流体动力学（MHD）方程的耦合模型。该模型描述了在磁场作用下导电流体的动态行为。基于“不变能量正交化”方法，我们提出了两种完全离散的时间行进方案，它们是线性，解耦，无条件能量稳定且是二阶的。离散问题的适定性和能量稳定性得到了证明。通过大量的数值实验，我们验证了数值方法的二阶收敛性，并证明了耦合模型模拟两相流的能力。