On two-fluid equations for dispersed incompressible two-phase flow

Abstract

This paper deals with the two-fluid formulation for dispersed two-phase flow. In one space dimension the equations with constant viscosity for both phases are shown to result in a locally in time well-posed periodic problem but without viscosity there are regions in phase space where the problem is non-hyperbolic. It is shown that the viscid problem linearized at constant states in the non-hyperbolic region is terribly unstable for small viscosity coefficients. By numerical experiments for the solution to a model problem it is demonstrated that for smooth initial data in the non-hyperbolic region, discontinuities seem to form. Second order artificial dissipation is added, in an effort to regularize the problem. Numerical examples are given, for different types of initial data. For initial data that is smooth in the non-hyperbolic region, the formation of jumps, or viscid layers, is strongly dependent on the amount of artificial dissipation. No convergence is obtained as the amount of artificial dissipation is diminished. On the other hand, if the initial data is smooth only in the hyperbolic region and with jumps through the non-hyperbolic region, then the jumps or viscid layers that later form, can damp the onset of new layers. In this case convergence in the L2 sense seem to hold, the computed solutions to the regularized problem approach a weak solution as the artificial dissipation is diminished.