An Interface-Capturing Regularization Method for Solving the Equations for Two-Fluid Mixtures (Communications in Computational Physics, 2013)
Many problems in biology involve gels which are mixtures composed ofa polymer network permeated by a fluid solvent (water). The two-fluid model is awidely used approach to described gel mechanics, in which both network and solventcoexist at each point of space and their relative abundance is described by their volumefractions. Each phase is modeled as a continuum with its own velocity and constitutive law. In some biological applications, free boundaries separate regions of gel andregions of pure solvent, resulting in a degenerate network momentum equation wherethe network volume fraction vanishes. To overcome this difficulty, we develop a regularization method to solve the two-phase gel equations when the volume fraction ofone phase goes to zero in part of the computational domain. A small and constantnetwork volume fraction is temporarily added throughout the domain in setting upthe discrete linear equations and the same set of equation is solved everywhere. Theseequations are very poorly conditioned for small values of the regularization parameter, but the multigrid-preconditioned GMRES method we use to solve them is efficientand produces an accurate solution of these equations for the full range of relevant regularization parameter values.