The Rayleigh-Taylor instability for the Verigin problem with and without phase transition

TL;DR

This paper investigates the Rayleigh-Taylor instability in two-phase flows with and without phase transition, demonstrating well-posedness and stability properties of equilibrium states in a thermodynamically consistent framework.

Contribution

It introduces a thermodynamically consistent model for two-phase flows with gravity, analyzing stability and well-posedness of equilibria with flat interfaces.

Findings

Equilibria with flat interfaces are identified.

The systems are well-posed in an $L_p$-setting.

Stability analysis of the Rayleigh-Taylor instability is conducted.

Abstract

Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria with flat interface are identified. It is shown that the problems are well-posed in an $L_p$-setting and generate local semiflows in the proper state manifolds. The main result concerns the stability of equilibria with flat interface, i.e. the Rayleigh-Taylor instability.