Dancing rivulets in an air-filled Hele-Shaw cell

TL;DR

This study investigates how an externally forced thin fluid rivulet in a Hele-Shaw cell exhibits nonlinear resonant patterns, combining experiments and a depth-averaged Navier-Stokes model to understand wave interactions and pattern formation.

Contribution

It introduces a combined experimental and theoretical analysis of nonlinear wave resonance in a forced rivulet system, revealing mode selection and pattern dynamics.

Findings

Resonant three-wave interactions determine pattern wavelength.

The model predicts the instability threshold frequency.

Experimental observations confirm the phase-locking of waves.

Abstract

We study the behaviour of a thin fluid filament (a rivulet) flowing in an air-filled Hele-Shaw cell. Transverse and longitudinal deformations can propagate on this rivulet, although both are linearly attenuated in the parameter range we use. On this seemingly simple system, we impose an external acoustic forcing, homogeneous in space and harmonic in time. When the forcing amplitude exceeds a given threshold, the rivulet responds nonlinearly, adopting a peculiar pattern. We investigate the dance of the rivulet both experimentally using spatiotemporal measurements, and theoretically using a model based on depth-averaged Navier-Stokes equations. The instability is due to a three-wave resonant interaction between waves along the rivulet, the resonance condition fixing the pattern wavelength. Although the forcing is additive, the amplification of transverse and longitudinal waves is effectively parametric, being mediated by the linear response of the system to the homogeneous forcing. Our model successfully explains the mode selection and phase-locking between the waves, it notably allows us to predict the frequency dependence of the instability threshold. The dominant spatiotemporal features of the generated pattern are understood through a multiple-scale analysis.