Uncovering bistability phenomena in two-layer Couette flow experiments using non-local evolution equations

This paper investigates the stability of interfacial long waves in two-layer plane Couette flow using a nonlinear, non-local asymptotic model derived from the Navier–Stokes equations and valid for thin upper layers. Non-locality enters through a coupling of the thin and main layers, and crucial inertial effects are retained. The models generically support bistability phenomena observed in experiments where two stable travelling waves, one unimodal and the other bimodal, are recorded at the same lid velocity. In direct comparisons with experiments, the models show remarkable agreement, both qualitatively and quantitatively. The two stable travelling waves are identified and their basins of attraction characterised via large-time computations for different initial conditions. We also identify a new symmetry-breaking travelling-wave branch bifurcating from the bimodal family, compute higher-wavenumber travelling-wave branches and present time-periodic orbits arising via Hopf bifurcation. A symmetry is also presented that links solutions for thin upper layers to those corresponding to thin lower layers. The instability of the two solutions is shown to be identical.