Numerical solution of coupled Cahn–Hilliard Navier–Stokes equations for two-phase flows having variable density and viscosity

This work is concerned with the implementation of a decoupled diffuse interface approach for the numerical solution of two-phase flows with a moving interface. The underlying two-phase flow model consists of a mass-preserving Cahn–Hilliard (CH) equation coupled with the incompressible Navier–Stokes equations (NSEs) through surface tension. Due to the higher order nonlinearity and stiff nature of the CH equation, its numerical solution is very challenging. We have used the decoupled pressure projection method for the numerical simulation of the governing equations. The spatial variables are discretized using the finite difference scheme on the staggered grids, while the explicit Euler method is used for time discretization. Four examples include the coalescence of inline rising bubbles, the obliques coalescence of two rising bubbles, and the deformation of single and two different-sized bubbles in a shear flow field is numerically simulated. It is observed that the scheme is capable of tracking the interface accurately and respecting mass conservation and energy dissipation. Moreover, the scheme is efficient, easily implementable, energy stable, and fully decoupled.