Dancing rivulets in an air-filled Hele-Shaw cell

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.

💡 Research Summary

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This paper investigates a novel instability that arises when a thin liquid filament (a rivulet) confined in an air‑filled Hele‑Shaw cell is subjected to a spatially uniform, temporally harmonic acoustic forcing. The authors combine high‑resolution spatiotemporal experiments with a depth‑averaged Navier‑Stokes model and a multiple‑scale analytical framework to reveal that, above a well‑defined forcing amplitude, the rivulet spontaneously develops a regular pattern composed of coupled transverse (center‑line) and longitudinal (width) waves.

The experimental set‑up consists of two parallel glass plates separated by a gap of 0.5–0.6 mm. PTFE oil is injected at the top, forming a vertical, gravity‑driven rivulet that spans the full 1 m height of the cell. Two anti‑phase loudspeakers generate a uniform pressure gradient across the cell, imposing a forcing term (F(t)=A\sin(\omega_f t)) with frequencies ranging from 10 Hz to 2 kHz. High‑speed imaging, together with a custom image‑processing pipeline, extracts the rivulet centreline (z(x,t)) and its width (w(x,t)) at sub‑pixel accuracy.

In the linear regime (forcing amplitude below a critical value (A_c)), both (z) and (w) respond weakly at the forcing frequency and are rapidly damped by viscous dissipation. When (A>A_c), a finite‑wavelength pattern emerges: (z) and (w) oscillate sinusoidally with the same spatial wavelength but with a fixed phase shift of approximately (\pi/2). The wavelength is independent of the forcing frequency, while the threshold amplitude decreases as the forcing frequency increases, indicating an inverse frequency dependence.

The theoretical description starts from the full Navier‑Stokes equations, depth‑averaged across the gap, and assumes a parabolic Poiseuille profile (Re≈150). The resulting one‑dimensional equations for the rivulet width (w(x,t)) and centreline (z(x,t)) contain inertia, gravity, viscous drag (Darcy‑type term (\mu=12\nu/b^2)), capillary restoring forces (with (\Gamma=\pi\gamma/(2\rho))), curvature‑induced pressure gradients, and a contact‑line friction term. Linearisation yields two damped wave branches: a transverse capillary wave governed by the curvature of the centreline and a longitudinal capillary wave associated with variations of the width.

A multiple‑scale expansion separates the fast acoustic time scale (T_f=2\pi/\omega_f) from the slow growth/ saturation time scale. At first order the system exhibits only the zero‑wavenumber response forced directly by the acoustic pressure. At second order, quadratic nonlinearities generate two finite‑wavenumber modes whose frequencies are half the forcing frequency. The resonance conditions

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