A Cahn–Hilliard–Navier–Stokes system with degenerate mobilities: a finite difference solution

Abstract The Cahn-Hilliard equation is increasingly popular in two-phase flow simulations due to its implicit capture of the interface and its easy extension to higher dimensions. However, it also has its drawbacks, for instance, the mass shrinkage of a small drop in a large computational domain. To that end, a Cahn–Hilliard equation with a degenerate mobility is proposed and coupled with the Navier–Stokes equation. To solve the system equations, a simple and efficient finite difference method is employed. The Laplacian of the chemical potential is discretized using a modified central difference scheme. It is this modification that lends the model to larger time steps. Moreover, the method is fully explicit. The model was tested on a number of cases and compared with the Cahn–Hilliard equation with a constant mobility. It was shown that the new model can conserve mass better, thus sustaining a small drop longer due to the eliminated bulk diffusion. The model was also compared with experimental and analytical outcomes, showing reasonable agreement.