Asymptotic Behavior of a Viscous Liquid-Gas Model with Mass-Dependent Viscosity and Vacuum

TL;DR

This paper studies the long-term behavior of a viscous liquid-gas model with mass-dependent viscosity, analyzing how the mass functions decay over time in vacuum conditions, extending previous Navier-Stokes results.

Contribution

It provides new insights into the asymptotic behavior and decay rates of mass functions in a two-phase viscous flow with vacuum, considering mass-dependent viscosity.

Findings

Derived decay rates for mass functions in vacuum conditions

Extended asymptotic analysis to models with mass-dependent viscosity

Improved understanding of free boundary problems in viscous flows

Abstract

In this paper, we consider two classes of free boundary value problems of a viscous two-phase liquid-gas model relevant to the flow in wells and pipelines with mass-dependent viscosity coefficient. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. We obtain the asymptotic behavior and decay rates of the mass functions $n(x,t)$,\$m(x,t)$ when the initial masses are assumed to be connected to vacuum both discontinuously and continuously, which improves the corresponding result about Navier-Stokes equations in \cite{Zhu}.