Roll waves in two-layer Hele-Shaw flows

Abstract

In this paper we study the emergence and development of roll waves in two-layer fluid flow in a Hele-Shaw cell. We propose the mathematical model of such flow and define the conditions of transition from stable state to instability in the form of the roll waves. We find out the physical parameters of flows at which the roll waves exist. A linear stability analysis and the Whitham criterion of roll waves existence are used for solving the problem and arrive to identical conclusions on depths of upper and lower layers at which violation of flow stability occurs. The numerical calculations for the obtained mathematical model at found ratios of densities, viscosities and depths of layers are performed. They confirm development of the roll waves of finite amplitude from small oscillations of the interface.