Faraday Instability in a Surface-Frozen Liquid

Faraday surface instability measurements of the critical acceleration, $a_{\mathrm{c}}$, and wave number, $k_{\mathrm{c}}$, for standing surface waves on a tetracosanol (${\mathrm{C}}_{24}{\mathrm{H}}_{50}$) melt exhibit abrupt changes at $T_s=54°\mathrm{C}$, $\sim 4°\mathrm{C}$ above the bulk freezing temperature. The measured variations of $a_{\mathrm{c}}$ and $k_{\mathrm{c}}$ vs temperature and driving frequency are accounted for quantitatively by a hydrodynamic model, revealing a change from a free-slip surface flow, generic for a free liquid surface ($T>T_s$), to a surface-pinned, no-slip flow, characteristic of a flow near a wetted solid wall ($T<T_s$). The change at $T_s$ is traced to the onset of surface freezing, where the steep velocity gradient in the surface-pinned flow significantly increases the viscous dissipation near the surface.

Figure 1

Faraday instability parameters versus temperature for a fixed driving frequency $f=150\mathrm{Hz}$. Open (closed) symbols represent data measured on cooling (heating). (a) Critical wave number, $k_{\mathrm{c}}$. (b) Critical acceleration, $a_{\mathrm{c}}$. Inset of (a): Standing wave pattern at the surface at $T=54°\mathrm{C}$. Solid and dashed lines represent the KM-model calculations assuming free-slip and no-slip boundary conditions, respectively. Arrows mark the discontinuities in the Faraday parameters.

Figure 2

Faraday instability parameters measured at $52°\mathrm{C}$ (triangles, surface-frozen state) and $58°\mathrm{C}$ (circles;, non-surface-frozen state) vs the driving frequency, $f$. (a) Critical wavelength, $k_{\mathrm{c}}$. (b) Critical acceleration, $a_{\mathrm{c}}$. (c) Dispersion relation plot, discussed in the text. The solid and dashed lines are calculated from the KM model assuming free-slip and no-slip boundary conditions, respectively.

Figure 3

Static surface tension ${\gamma }_{\mathrm{s}}$ measured by the Wilhelmy plate method (line) and dynamic surface tension ${\gamma }_{\mathrm{d}}$ (solid triangles) as determined from the surface waves’ dispersion relation for a fixed $f=150\mathrm{Hz}$.