Melde’s Experiment on a Vibrating Liquid Foam Microchannel

We subject a single Plateau border channel to a transverse harmonic excitation, in an experiment reminiscent of the historical one by Melde on vibrating strings, to study foam stability and wave properties. At low driving amplitudes, the liquid string exhibits regular oscillations. At large ones, a nonlinear regime appears and the acoustic radiation splits the channel into two zones of different cross section area, vibration amplitude, and phase difference with the neighboring soap films. The channel experiences an inertial dilatancy that is accounted for by a new Bernoulli-like relation.

Figure 1

(a) Sketch of the experimental setup. (b) Snapshot of an oscillating 15 cm-long PB (60 Hz). Inset: A 5 cm-long PB (top) and the envelope of the PB motion (bottom). (c) Experimental dispersion relation for $R_i=0.1\mathrm{mm}$ and for several lengths of the PB (2 to 7 cm) and widths of the films (0.5, 1.5, and 3.3 cm). The slope of the straight line is $c=2.0\mathrm{m}/\mathrm{s}$. The dashed blue line shows the result of the model of Derec et al. [13].

Figure 2

Image sequence obtained at increasing driving amplitude. The red arrows emphasize the position of the front. (a) Bright field imaging (60 Hz). (b) Dark field imaging (40 Hz). The red dashed line indicates $z_0$, the position selected to plot the space-time diagram of Fig. 3.

Figure 3

(a) Space-time diagram for the experiment reported in Fig. 2. Labels (1) to (8) indicate the time positions of the numbered images of Fig. 2. The oscillations of both the PB and the film are visible. The passage of the front at $z_0$ occurs for time label (6). (b) Enlargement prior to time label (6). The PB and the envelope of the film are outlined in blue and red, respectively, over one oscillation time period. (c) PB radius $R$ (black circles) and phase difference $\mathrm{\Delta }\varphi =2\pi f\mathrm{\Delta }t$ (light gray circles) as functions of time. The passage of the front occurs for $R_c=0.42\mathrm{mm}$. (d) Same as (b) after time label (6).

Figure 4

(a)–(c) Results on the amplitude $A_{\mathrm{PB}}$ of the PB. The solid lines are given by Eq. (2) with 0.084 as the proportionality factor. (a) $A_{\mathrm{PB}}$ as a function of $R$ for a frequency $f=40\mathrm{Hz}$ and various initial radii $R_i$ [0.32 (square), 0.61 (triangle), and 0.96 (diamond) mm]. The vertical dashed line pinpoints the critical radius $R_c=0.42\mathrm{mm}$. (b) $A_{\mathrm{PB}}$ as a function of $R$ for an initial radius $R_i=0.85\mathrm{mm}$ and various frequencies (30, 40, 60, and 120 Hz). (c) $A_{\mathrm{PB}}^2$ as a function of $(\gamma /\rho f^2)[(1/R)-(1/R_i)]$ (log-log scale). (d)–(f) Results on the amplitude difference $\mathrm{\Delta }A$. (d) $\mathrm{\Delta }A$ as a function of $R$ for the same experiments as in (a). $R_c=0.42\mathrm{mm}$ is represented. The solid lines are guides for the eyes. (e) $\mathrm{\Delta }A/A_{\mathrm{PB}}$ as a function of $R$. Solid lines are parabolic fits, $[R/r(f){]}^2$, where $r$ is a free parameter adjusted for each frequency (30, 40, 60, and 80 Hz). (f) $\mathrm{\Delta }A/A_{\mathrm{PB}}$ as a function of $[R/r(f){]}^2$ (log-log scale).

Figure 5

(a) Sketch of the model. The PB is symbolized by the red empty spot. It is driven by the horizontal soap film, whose maximum in the $xy$ plane is displayed by the blue solid spot. Each film pulls on the PB with a force $2\gamma$ per unit length. (b) Measurements of $R_c$ (square) and $r$ (circle) and their interpolations (dashed lines) given by Eqs. (1) and (4). The model $R_{\mathrm{th}}$ is displayed by the solid line.