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

TL;DR

The paper addresses the stability and nonlinear selection of interfacial waves in two-layer Couette flow, focusing on bistability between unimodal ($2L$-periodic) and bimodal ($L$-periodic) travelling waves observed experimentally. It develops a nonlinear, nonlocal evolution equation that retains inertial effects and couples a thin upper layer to a thick lower layer, then computes traveling waves via a Fourier-based Newton solver and analyzes linear stability with an eigenvalue problem. The authors demonstrate a bistability window with coexistence of stable branches, validate the model against Barthelet et al. (1995) data across a range of Reynolds-like parameters, and extend comparisons to interfacial profiles and harmonic amplitudes, achieving remarkable quantitative agreement. They also identify a secondary branch arising near $oldsymbol{\Lambda \,\approx\,0.0358}$ and a second bistability region, offering practical guidance for experiments and future analytical work on bifurcations in nonlocal multilayer flows.

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.

Figures (5)

• Figure 1: Two superposed fluid layers in a channel, driven by the upper-plate shear with speed $U$. Blue and red curves represent unimodal and bimodal respectively. $d = h_2/h_1 = 0.25$ and $U_L = 0.138\,\mathrm{m\,s^{-1}}$ fixed across the cases, while $U/U_L$ varies.
• Figure 2: Bifurcation diagrams, computed travelling waves and their stability for $R = 709$, $\nu = 0.01$, $m = 2.76$. (a) Wave speed $C$ versus $\Lambda$; (b) $L^2$-norm versus $\Lambda$. (c) Wave profiles for branch 1. (d) Wave profiles for branch 2.
• Figure 3: Evolution of the interfacial position: dominated by the fundamental and by the second harmonic. (a) Experimental traces (from BartheletCharruFabre1995, p. 49); (b) Model computation; (c) Basins of attraction; bistability for $\Lambda \in [0.0155, 0.0357]$.
• Figure 4: Interfacial profile for $U/U_L= \{1.06, 1.13,1.33,1.58, 1.77\}$. $U_L = 0.138~\mathrm{m\,s^{-1}}$. (a) Experimental shapes (from BartheletCharruFabre1995, p.36); (b) Model computation. Amplitudes are normalised with the saturated amplitude for $U/U_L=1.77$.
• Figure 5: Harmonic amplitudes for $U/U_L= 1.13,1.33,1.58, 2.25$. (a) Experimental traces (from BartheletCharruFabre1995, §§ 5.4 and 5.5); (b) Model computation.