Efficient and practical phase-field method for the incompressible multi-component fluids on 3D surfaces with arbitrary shapes

Abstract

The fluid dynamics on a curved surface can be used to approximately describe many physical phenomena in the natural world, such as the atmospherical circulation on a planet. The present work develops a simple, practical, and efficient numerical method for the multi-phase fluid simulations on arbitrarily curved surfaces in 3D space. The multi-component Cahn–Hilliard type phase-field fluid model is adopted to capture the motion of multiple interfaces. The velocity field and pressure are updated by solving the incompressible Navier–Stokes equations. To achieve the decoupled computations of pressure and velocity, the Chorin's projection method with pressure correction is considered. A linear semi-implicit time-marching scheme is designed for the phase-field equations. By adopting the embedding method and appropriate pseudo-Neumann boundary conditions, the direct calculations on the surface can be extended into a 3D narrow band domain including the surface. The standard Laplacian operator can be used to replace the surface Laplace–Beltrami operators and the finite difference method is used to perform the discretization in space. In each time step, all variables can be updated in a step-by-step manner because the proposed method is totally decoupled. Therefore, the whole algorithm is highly efficient and easy to implement. Extensive numerical experiments indicate that the proposed method has good performance for the multi-component simulations on curved surfaces with complex shapes.