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

TL;DR

This study models the stability and bistability of interfacial waves in two-layer Couette flow using a nonlinear nonlocal equation, revealing stable wave states, bifurcations, and strong agreement with experiments.

Contribution

It introduces a novel nonlocal asymptotic model capturing bistability phenomena and complex wave bifurcations in two-layer Couette flow, validated against experimental data.

Findings

Identification of two stable travelling wave solutions

Discovery of a new symmetry-breaking wave branch

Observation of time-periodic orbits via Hopf bifurcation

Abstract

This paper investigates the stability of interfacial long waves in two-layer plane Couette flow using a nonlinear, nonlocal asymptotic model derived from the Navier-Stokes equations and valid for thin upper layers. Nonlocality 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.