A contact line boundary condition for the second-order phase-field model

The diffuse interface method is a popular tool for simulating multiphase flows, and models that use the conservative 2^nd-order phase field equation have many benefits over higher-order models, including bound preservation and faster computation. An ongoing challenge for such simulations is treatment of contact lines where the phase interface meets solid boundaries. A robust treatment requires attention to the details of coupling between the mass and momentum equations while honoring the added physics due to the diffuse nature of the interface. In this work, we present an asymptotically derived boundary treatment consistent with the diffuse nature of the interface and assess its performance over several canonical test cases.