Shock-wave solutions in two-layer channel flow. II. Linear and nonlinear stability

Abstract

We investigate the flow of two immiscible fluids in an inclined channel, building on the work presented in Part I of this study. In this paper, we examine the stability of the flow to spanwise perturbations in both the linear and nonlinear regimes. The evolution equation governing the interfacial dynamics, derived using lubrication theory in Part I, is linearised to study the effect of system parameters on the linear stability of the interface. A transient growth analysis of the linearised equation is carried out with no-flux conditions in the spanwise direction. The results of this analysis reveal that increasing the density and/or the viscosity of the upper layer, and/or increasing the counter-current nature of the flow configuration exerts a stabilising influence. Inspection of the flow profiles indicates that single Lax-shocks and the trailing Lax-shocks in Lax-undercompressive double-shocks are unstable to finger formation; undercompressive shocks and rarefaction waves are stable. In unstably stratified cases, increasing the channel inclination away from verticality, such that a denser upper layer overhangs a less dense lower one, is found to be destabilising. These results are used to guide our transient numerical simulations aimed at studying the nonlinear development of fingering phenomena.