Capillary solitons on a levitated medium

A water cylinder deposited on a heated channel levitates on its own generated vapor film owing to the Leidenfrost effect. This experimental setup permits the study of the one-dimensional propagation of surface waves in a free-to-move liquid system. We report the observation of gravity-capillary waves under a dramatic reduction of gravity (up to a factor 30), leading to capillary waves at the centimeter scale. The generated nonlinear structures propagate without deformation and undergo mutual collisions and reflections at the boundaries of the domain. They are identified as Korteweg–de Vries solitons with negative amplitude and subsonic velocity. The typical width and amplitude-dependent velocities are in excellent agreement with theoretical predictions based on a generalized Korteweg–de Vries equation adapted to any substrate geometry. When multiple solitons are present, they interact and form a soliton turbulencelike spectrum.

Figure 1

Sketch of the experimental setup. (a) 100 mL of distilled water is poured onto a 45 cm long quarter-pipe-shaped channel, heated at $280{}^{\circ }\text{C}$, forming a levitated liquid cylinder. A wave maker is used to generate a disturbance $\eta (x,t)$ of the cylinder interface. (b) Side view of the channel: We focus on the propagation of waves along the free right-hand border of the interface. The inclination angle $\alpha$ of the container is tunable. (c) Sketch of the geometry investigated theoretically. (d) Photography of the experimental setup.

Figure 2

Power spectrum $S_{\eta }(k,\omega )$ of the cylinder right interface $\eta (x,t)$ for $\alpha =2^{\circ }$ as a function of $k$ and $\omega$. Color is log scaled. The energy is localized along two branches of propagation. I: Gravity-capillary waves given by Eq. (2) (solid line). The dashed line corresponds to the dispersion relation of pure capillary waves. II: Sloshing modes, i.e., in-phase oscillations of both the left- and right-hand sides of the interface. Inset: Experimental ($\circ$) minimum phase velocity $c_0$ as a function of $\alpha$. $c_0$ is measured from a fit of the dispersion relation in the limit $k\to 0$. Theoretical model $c_0=\sqrt{g^{\prime }{H}^{\prime }}$ [see Eq. (3)] with no fit parameter (dashed line) and with an adjusted geometrical factor $H_{\mathrm{eff}}=0.65H^{\prime }$ (solid line).

Figure 3

Top: $\eta (x,t)$ of a capillary soliton as a function of time (solid line) with a time step $0.04$ s between two neighboring profiles ($\alpha =4^{\circ }$). Left inset: Experimental data ($\mathrm{\Delta }$) and theoretical shape of a negative amplitude soliton given by Eq. (6) (solid red line). Right inset: Rescaling of the solitonic shape for various soliton amplitudes using the expression of $\mathrm{\Delta }$ given by Eq. (7) ($\times$). Bottom: Normalized velocity $c/c_0$ as a function of the normalized amplitude $A_0/3H_{\mathrm{eff}}$ for $\alpha =3^{\circ }$ and $4^{\circ }$ and data from Falcon et al. [15] (gray symbols). Linear relations correspond to Eq. (7) for three theoretical values of ${\mu }_1$ [flat channel ${\mu }_1=3/2$, triangular geometry in Fig. 1 ${\mu }_1=5/2$], and best experimental fit (${\mu }_1=3\pm 0.4$). Typical error bars are shown on four data points.

Figure 4

$S_{\eta }\left(\omega \right)$, calculated for each $x$ location and then spatially averaged. A ${\omega }^{-1.5}$ power law (red dashed line) is observed at low $\omega$, with a cutoff at $\omega =18$ rad/s corresponding to the sloshing mode. This low frequency regime can be interpreted as a solitary wave turbulence regime. Inset: Normalized PDF of $\eta$. High amplitude negative events lead to a departure from the Gaussian statistics (red dashed line).