On scaling of diffuse interface models

A diffuse interface method applied to multiphase flow problems yields a system of partial differentialequations which seldom has an analytical solution and hence, a numerical solution is usuallysought for. For most problems of practical interest, numerical implementation, which is generally ona fixed grid,requires an artificial enlargement of the interface thickness. This replacement of the realinterface thickness with a numerically acceptable interface thickness, while keeping surface tensionthe same, is justified based on an analysis of an equilibrium planar interface, but demands a changeof the local part of the free energy.In a non-equilibrium situation, where the interface position and/or shape evolves with time, afurther understanding on how to change mobility upon replacing the real interface thickness with anumerical approximation is required. We have attempted to answer this question using mixing oftwo immiscible fluids in a lid-driven cavity flow as a case study. Scaling based on heuristics, wheremobility is inversely proportional to the interface thickness,was found to give fairly well results overthe range of capillary numbers, Peclet numbers and viscosity ratios considered.