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In this work, we propose a fully discrete energy stable scheme for thephase-field moving contact line model with variable densities and viscosities.The mathematical model consists of a Cahn-Hilliard equation, a Navier-Stokesequation and the generalized Navier boundary condition for the moving contactline. A scalar auxiliary variable is adopted to transform the governing systeminto an equivalent form, allowing the double well potential to be treatedsemi-explicitly. A stabilization term is added to balance the explicitnonlinear term originating from the surface energy at fluid-solid interface. Apressure stabilization method is used to decouple the computation of velocityand pressure. Some subtle implicit-explicit treatments are adopted to deal withconvention and stress terms. We establish a rigorous proof of energy stabilityfor the proposed time-marching scheme. Then a finite difference method onstaggered grids is used to spatially discretize the constructed time-marchingscheme. We further prove that the fully discrete scheme also satisfies thediscrete energy dissipation law. Numerical results demonstrate accuracy andenergy stability of the proposed scheme. Using our numerical scheme, we analyzethe contact line dynamics through a shear flow driven droplet sliding case.Three-dimensional droplet spreading is also investigated on a chemicallypatterned surface. Our numerical simulation accurately predicts the expectedenergy evolutions and it successfully reproduces expected phenomena that an oildroplet contracts inwards on a hydrophobic zone and spreads outwards quickly ona hydrophilic zone.

A fully discrete energy stable scheme for a phase-field moving contact line model with variable densities and viscosities