Construction of seamless immersed boundary phase-field method

In this paper, we try to construct the seamless immersed boundary phase-field method for simulating the two-phase flows. The seamless immersed boundary method is one of the Cartesian grid approaches, in which the forcing term is added to the incompressible Navier-Stokes equations in order to satisfy the velocity condition on the boundary. In the seamless immersed boundary method, the forcing term is added not only on the grid points near the boundary but also on the grid points inside the boundary. This method is applied to the phase-field equation, i.e., the Cahn-Hilliard equation for the two-phase flow analysis. By using the Taylor series expansion in multi-variable, the correction term for satisfying the Neumann boundary condition is estimated. The phase separation in a rotated square cavity is considered, in order to validate the present approach. It is found that the rotated solutions obtained on the Cartesian coordinates are the same as the original solution at any time. Then, it is concluded that the present seamless immersed boundary phase-field method is very fruitful for simulating the complicated two-phase flows.