A diffuse-interface immersed-boundary method for two-dimensional simulation of flows with moving contact lines on curved substrates

We propose an approach to simulate flows with moving contact lines (MCLs) on curved substrates on a Cartesian mesh. The approach combines an immersed boundary method with a three-component diffuse-interface model and a characteristic MCL model. The immersed boundary method is able to accurately enforce the no-slip boundary condition at the solid surface, thereby circumventing the penetration of the gas and the liquid into the solid by convection. On the other hand, using the three-component diffuse-interface model can prevent the gas and liquid from infiltrating into the solid substrate through the diffusive fluxes during the interface evolution. A combination of these two methods appears to effectively conserve the mass of the phases in the computation. The characteristic MCL model not only allows the contact lines to move on the curved boundaries, but makes the gas–liquid interface to intersect the solid object at an angle in consistence with the prescribed contact angle, even with the variation of surface tangent at the solid substrate. We examine the performance of the approach through a variety of numerical experiments. The mass conservation and interface shapes at equilibrium were tested through the simulation of drop spreading on a circular cylinder. The dynamic behaviors of moving contact lines were validated by simulating the droplet spreading on a flat substrate, and we compared the numerical results against theoretical predictions and previous experimental observations. The method was also applied to the simulations of flows with curved boundaries and moving contact lines, such as drop impact on a sphere and water entry of a sphere. Finally, we studied the penetration process of a two-dimensional drop into a porous substrate that consists of a cluster of circular cylinders.