Role of edge effects and fluid depth in azimuthal Faraday waves

Faraday waves are created in experiment by mechanically vibrating a cylindrical container. Frequency scans are performed, and the acceleration threshold above which a surface wave appears is measured using a laser light approach, giving the instability tongue in frequency-acceleration space. The spatial structure of the surface wave, which conforms to the cylindrical container for wavelengths comparable to the container size, is determined using long-exposure-time white light imaging and is defined by the mode number pair $(n,\ell )$. Edge conditions at the container sidewall are controlled to create a (i) pinned or (ii) freely sliding contact-line. The driving frequency with the smallest threshold acceleration is identified as the natural frequency for that particular mode number pair. A theoretical model is developed using a viscous potential approximation to compute the natural oscillations of a viscoelastic material in a cylindrical container with a flat interface for both a pinned and a freely sliding contact-line. A closed-form expression is given for the freely sliding case, and a Rayleigh-Ritz procedure is used for the pinned case. The agreement is good between theoretical predictions and experimental observations for Triton/water mixtures, glycerol/water mixtures, and agarose gels, notwithstanding the large range in parameter space and the very different boundary conditions considered.

Figure 1

Experimental schematic, reproduced with permission from Shao et al. [46].

Figure 2

Illustration of the experimental conditions for (a) a freely sliding and (b) a pinned contact-line.

Figure 3

Experimentally observed modes $(n,\ell )$.

Figure 4

Instability tongue for water with a pinned contact-line and $H=7$ mm plotting acceleration $A$ against driving frequency $f_d$.

Figure 5

Acceleration $A$ against driving frequency $f_d$ for the Triton/water mixture contrasting the free-sliding (S) and pinned (P) contact-line condition for fluid height (a) $H=7$ mm and (b) $H=22$ mm.

Figure 6

Acceleration $A$ against driving frequency $f_d$ for glycerol/water mixtures with height $H=22$ mm and pinned contact-line illustrating (a) the full frequency scan and (b) only the (1,2) mode.

Figure 7

Acceleration $A$ against driving frequency $f_d$ for glycerol/water mixtures with height $H=7$ mm and pinned contact-line for the (1,2) mode.