Understanding of the Ill-posed two-fluid model

Multifield models are central to the modeling and simulation of transport processes in multiphase, homogenized systems. The approach is based on an interpenetrating continua description, in which conservation laws are applied to each phase as a separate continuum (field) and constitutive laws are provided to represent interfield interactions. The resulting model is illposed and mathematically complex, in the sense that the equation system is nonhyperbolic, nonlinear and nonconservative. These features are thought to be at the root of difficulties in the numerical solution of the multifield model equation system. Consequently, the fidelity of the numerical solution is confounded by the interplay between uncertainty in physical closure relationships (constitutive laws) and numerical errors due to numerical diffusion and unphysical oscillations. In this paper, we provide a synthesis toward understanding the illposed twofluid model. Issues related to nonhyperbolicity are revisited broadly and brought under the perspective of their physical and mathematical origins. We emphasize the connection between the model's mathematical properties and approaches of mathematical and/or numerical regularization. Lessons learned from past experiences are highlighted. The synthesis leads to principal questions of mathematical, physical and numerical nature that need to be addressed if further progress is to be made. A new approach to mathematical regularization through Virtual Spacetime Relaxation (VSR) is described, and numerical examples that clarify the roles of hyperbolicity and conservatism are offered