Analysis of a Compressible Two-Fluid Stokes System with Constant Viscosity

Basic properties of a reduced viscous compressible gas–liquid two-fluid model are explored. The model is composed of two conservation laws representing mass balance for gas and liquid coupled to two elliptic equations (Stokes system) for the two fluid velocities and obtained by ignoring acceleration terms in the full momentum equations. First, we present a result that shows existence and uniqueness of regular solutions for a fixed time T0 > 0 which depends on the initial data and the constant viscosity coefficients. Moreover, T0 can be large when the viscosity coefficients are large. However, for a fixed set of viscosity coefficients, we conjecture that the smooth solution might blow up, at least, as time tends to infinity. This result is backed up by considering a numerical example for a fixed set of viscosity coefficients demonstrating that for smooth and small initial data with no single-phase regions, the solution may tend to produce both single-phase regions and blow-up of mass gradients as time becomes large.