Spectral stability of travelling waves in a thin-layer two-fluid Couette flow

Abstract

We consider linear stability of travelling waves in a thin-film model for two-fluid Couette flow when a thin layer of the more viscous fluid resides next to the stationary wall. We prove that in a neighbourhood of a bifurcation point, characterized by a positive integer k b , the principal branch ( k b = 1 ) is spectrally stable while all other branches ( k b > 1 ) are spectrally unstable. For larger amplitude travelling waves, we establish a number of conditional theorems where the conditions were checked with help of computer assist for a set of parameter values. Using these theorems, we rigorously confirm earlier numerical evidence (D. Papageorgiou & S. Tanveer, Proc. R. Soc. A , (doi: 10.1098/rspa.2019.0367 )) on stability and instability of travelling waves over a range of parameters.