Dancing rivulets in an air-filled Hele-Shaw cell

TL;DR

This work investigates a novel pattern-forming instability of a thin rivulet in an air-filled Hele-Shaw cell under spatially uniform acoustic forcing. By combining experiments with a depth-averaged Navier–Stokes model and a weakly nonlinear multiple-scale analysis, the authors show that two coexisting wave modes—transverse and longitudinal—are parametrically amplified through a triadic resonance mediated by the linear forcing response, yielding a finite-wavelength pattern whose wavelength is selected by the resonance condition. Key contributions include the derivation of coupled amplitude equations, identification of resonance constraints that fix the pattern, and a quantitative account of threshold, phase relations, and nonlinear saturation, all corroborated by Fourier-space data. The findings illuminate a parametric mechanism arising from additive forcing in a dissipative, quasi-1D system and suggest avenues for exploring Faraday-like phenomena and wave turbulence in rivulet dynamics with potential applications to microfluidic and coating processes.

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.

Figures (12)

• Figure 1: (left) Experimental apparatus (not to scale, schematic view). The measuring scale and the pump are not shown. (middle) Typical image, with detection of path and width of the rivulet. The path is defined as the centreline of the bright zone, which corresponds to the interior of the rivulet. The width is the distance between the detected menisci (blue lines in the figure) (right) On top, illustration of the fact that light refracted by the menisci does not reach the camera. On the bottom, definition of the cell spacing $b$ and rivulet width $w$.
• Figure 2: Dispersion relations of transverse (left) and longitudinal waves (right), in the absence of damping. The axis is made dimensionless through appropriate scaling. The horizontal axes correspond to the dimensionless wavevector amplitude $k\, w_0$, while vertical axes correspond to the dimensionless angular frequency $\omega \, w_0/u_0$. The gray lines correspond to pure advection $\omega = k\, u_0$.
• Figure 3: Spatio-temporal representation of the experimental position $z(x,t)$ of the rivulet as a function of time and space. Cell gap $b=0.58 \pm 0.02mm$, flow rate $Q=25.6 \pm 0.9\cubic mm\per s$, excitation frequency $\omega_0/(2\pi)=40Hz$. (bottom left): Position $z$ of the rivulet (color scale) as a function of time $t$ and position $x$. Darker regions delimited by plain lines correspond to the parts where the rivulet is the heaviest, while lighter regions delimited by dashed lines correspond to parts where the rivulet is the thinnest (see fig. \ref{['fig:spatio_w']}). (right): Time-averaged position (over 60ms) of the rivulet as a function of space. Labeled ticks are spaced by $\lambda=2\pi/k = 6.74mm$. (top): Spaced-averaged position (over 17mm) of the rivulet as a function of time. Labeled ticks are spaced by $2\pi/\omega_0 = 25.0ms$.
• Figure 4: Spatio-temporal representation of the experimental width $w(x,t)$ of the rivulet as a function of time and space. Cell gap $b=0.58 \pm 0.02mm$, flow rate $Q=25.6 \pm 0.9\cubic mm\per s$, excitation frequency $\omega_0/(2\pi)=40Hz$. (left): Width $w$ of the rivulet (color scale) as a function of time $t$ and position $x$. (right): Width of the rivulet interpolated along the plain black line represented on the left plot. The plain and dashed lines correspond to the width delimiting the darker and lighter regions on figure \ref{['fig:spatio_z']}, respectively. The abscissa corresponds to a counter-advected position $x - v_w t$, with $v_w=0mm\per s$ the phase speed of longitudinal waves. Labeled ticks are spaced by $\sqrt{\lambda^2 + (v_w T_0)^2}$.
• Figure 5: Relationship between $k$ and $\omega_0$. Symbols correspond to experimental measurements of the wavenumber $k$ (which is the same for transverse and longitudinal waves). Experiments done with cell gap $b=0.6mm$ and flow rate $Q=26 \pm 1\cubic mm\per s$. Lines correspond to $\abs{\omega_w^+ - \omega_z^-} = kv_\text{c} (1 + k w_0)$, i.e. $\varepsilon_z\varepsilon_w = -1$ (purple) and $\abs{\omega_w^- - \omega_z^-} = kv_\text{c} \abs{1 - k w_0}$, i.e. $\varepsilon_z\varepsilon_w = +1$ (ocher). The line curves are computed without fitting, using the experimental values of the parameters. The theoretical prediction corresponds to the modes $\omega_w^+, \omega_z^-$ being unstable (purple curve). Note that the experimental points often fall slightly to the right of the curve, i.e. the wavevectors $k$ are larger than expected: this is because the points were recorded at a finite amplitude for which nonlinear detuning is measurable (see section \ref{['subsec:nonlinear']}).