Modified diffuse interface fluid model and its consistent energy-stable computation in arbitrary domains

Binary fluid flows in irregular domains are common phenomena in natural and industrial fields. To describe these physical processes, we herein present a novel fluid flow-coupled two-material model reflecting different wetting conditions on an arbitrary fluid-solid interface. A modified ternary Cahn–Hilliard (CH) diffuse interface model is adopted to capture the fluid-fluid interface. One of three components is fixed for all time to represent the profile of complex domains. By considering the contact angle condition on solid-fluid interface and equilibrium interface assumption, an extra wetting term is derived. To treat the fluid flow in domains with solid obstacles, we consider the entire system as a porous medium with variable permeability and add a large penalty term into the incompressible Navier–Stokes (NS) equations. It can be proven that the proposed modified Cahn–Hilliard–Navier–Stokes (CHNS) system satisfies the energy dissipation law (energy stability). Based on the scalar auxiliary variable (SAV) approach with appropriate correction techniques, the linear and consistent energy-stable scheme is developed. We introduce the numerical implementation and estimate the discrete version of energy stability in detail. The proposed method can be implemented on regular Cartesian grids with the absence of explicit boundary treatment. Moreover, the calculations are totally decoupled in each time step. Extensive numerical experiments not only indicate the expected accuracy and stability but also show the superior performance in arbitrary domains with different wetting conditions.