PDE/Numerical Analyses and Computations of Some Diffuse Interface Models of Viscous Two-Phase Flows

Diffuse interface methods approximate the separating boundary between two fluid phases using an order parameter u that continuously, and usually monotonically, varies from one value in phase A, say u=+1, to another value in phase B, say u=¬-1, in a boundary layer of small, but finite, thickness. This contrasts a sharp interface formulation employing a characteristic (or indicator) function description. In the diffuse interface approximation, the "location" of the interface can be identified, though somewhat arbitrarily, as the level surface u=0. In many two-phase fluids, such a diffuse description of an interface may be physically justified. In fact, van der Waals had argued that this was the case in the late 19th century, well before the computational significance could be appreciated. Today, however, the diffuse description is typically used as a mathematical formalism (an approximation), and the thickness of the boundary layer is significantly larger than is physically realistic. The smaller the interface thickness is, the smaller the associated approximation error is. I will describe 2 rather simple two-phase flow models, one of which was motivated by my interest in porous media flow and tumor growth, and another of which was motivated by problems in micro-fluidics, lava lamps, bio-films, and vesicle motion. Both are close relatives to the well-known Cahn-Hilliard-Navier-Stokes equation. I will discuss some PDE theory, numerical analysis, and computational issues related to both models and show 2 and 3d computational results.