Numerical study of incompressible binary fluids on 3D curved surfaces based on the conservative Allen–Cahn–Navier–Stokes model

In this article, we propose a practical and highly efficient finite difference approach for two-phase fluid simulations on three-dimensional (3D) surfaces. The hydrodynamically coupled interfacial motion is captured by using the conservative Allen–Cahn–Navier–Stokes (CACNS) equations. By adopting the closest point method and the pseudo-Neumann boundary condition, the direct computations on curved surfaces are transferred to the 3D simulations in a narrow band domain embedding the surface. The projection method with pressure correction is used to decouple the computations of velocity and pressure. The operator splitting method is used to split the calculation of conservative Allen–Cahn equation into subproblems and the nonlinear part can be analytically solved. Therefore, the whole computation in each time iteration is highly efficient and easy to implement. The numerical experiments on various 3D curved surfaces are investigated to show the good performance of the proposed method.