On scaling of diffuse-interface models

Application of diffuse–interface models (DIM) yields a system of partial differential equations (PDEs) that generally requires a numerical solution. In the analyses of multiphase flows with DIM usually an artificial enlargement of the interface thickness is required for numerical reasons. Replacing the real interface thickness with a numerically acceptable one, while keeping the surface tension constant, can be justified based on the analysis of the equilibrium planar interface, but demands a change in the local part of the free energy. In a non-equilibrium situation, where the interface position and shape evolve with time, we need to know how to change the mobility in order to still model the same physical problem. Here we approach this question by studying the mixing of two immiscible fluids in a lid-driven cavity flow where the interface between the two fluids is stretched roughly linearly with time, before break-up events start. Scaling based on heuristics, where the mobility is taken inversely proportional to the interface thickness, was found to give fairly well results over the period of linear interface stretching for the range of Péclet numbers and viscosity ratios considered when the capillary number is O(10). None of the scalings studied was, however, able to capture the break-up events accurately.