2 Overview

Ma & Burton (Reference Ma and Burton2018) present exhaustive experiments with six different fluids and a large range of temperatures and drop volumes. Thanks to the slight curvature of the substrate, they could keep stable large drops ( $R>4l_{c}$ ) that otherwise would have broken under the action of a growing bubble of vapour below (Snoeijer, Brunet & Eggers Reference Snoeijer, Brunet and Eggers2009); hence they observed vibrating drops with wavenumber $n$ from 2 to 13, and $R$ up to 2.4 cm. Furthermore, they measured the pressure underneath the drop, and found that the appearance of the instability goes hand in hand with pressure fluctuations, of typical frequency $f_{p}$ . Also, they used a smart visualisation technique that evidenced capillary waves not only on top and at the edge of the drop, but also underneath. A sketch of their system is depicted in figure 1. One of their most striking results is the independence of $f_{p}$ (and so of the frequency of the drop oscillations $f_{d}$ ) and azimuthal wavelength $\unicode[STIX]{x1D706}_{d}$ on the drop radius. They found that $f_{d}$ is dependent only on $l_{c}$ , which is dependent only on liquid properties. Figure 2(a) shows an oscillating drop from below, where both the external corrugation and capillary waves below are visible. Figure 2(b) is the corresponding power spectrum that evidences the selection of frequency for these waves. The wavenumber $n$ self-adapts and is found to increase linearly with the radius, $n\sim R$ .

Figure 1.

The overall sketch of the drop levitating on its own vapour underneath.

Figure 2.

(a) A star-shaped drop of acetone levitating on its own vapour, viewed from below, and the radial capillary waves underneath. (b) The corresponding power spectrum in frequency and wavenumber.

These results seem in sharp disagreement with previous ones which showed a significant dependence of $f_{d}$ on $R$ (Adachi & Takaki Reference Adachi and Takaki1984; Bouwhuis et al. Reference Bouwhuis, Winkels, Peters, Brunet, van der Meer and Snoeijer2013; Caswell Reference Caswell2014). However, the latter studies examined the situation of smaller droplets ( $R<4l_{c}$ ), for which Ma & Burton explain the dependence on $R$ with an analytical approach.

Ma & Burton’s measurements and analysis of air flow suggest that the selection on $f_{p}$ (and then on $f_{d}$ and $\unicode[STIX]{x1D706}_{d}$ ) originates from gravity–capillary waves provoked by the shear stress exerted on the vapour–liquid interface, which explains why oscillations appear above a threshold in temperature or in air flow. This analysis also reconciles the different measurements obtained for small drops and large puddles. These waves cannot develop under too small drops, as their wavelength $\unicode[STIX]{x1D706}_{c}\simeq 3.03l_{c}$ would then be larger than $R$ . Instead, the profile underneath takes the shape of a bubble with a ‘nip’, i.e. a region of minimal air thickness, where most of the viscous stress responsible for lubrication-induced levitation takes place (Snoeijer et al. Reference Snoeijer, Brunet and Eggers2009). The radius $R$ remains the only length scale in the radial direction, and the selection of $f_{p}$ is set by $R$ . Conversely the air layer under large puddles on curved substrates can exhibit several of these minima (see figure 2 in Ma & Burton Reference Ma and Burton2018), with $\unicode[STIX]{x1D706}_{c}$ being the distance between them. Therefore, $\unicode[STIX]{x1D706}_{c}$ becomes the radial length scale selected by the hydrodynamics, which sets the selection of $f_{p}$ independently of $R$ . It is insightful to revisit previous measurements in light of Ma & Burton’s, as one sees more clearly that the power-law decrease of $f_{d}$ with $R$ holds only for $R$ smaller than $2l_{c}$ (roughly 5 mm for water) (Adachi & Takaki Reference Adachi and Takaki1984; Bouwhuis et al. Reference Bouwhuis, Winkels, Peters, Brunet, van der Meer and Snoeijer2013), while $f_{d}$ seems to reach a plateau for larger  $R$ .

Overall, with systematic and exhaustive measurements, Ma & Burton unveil the complex hydrodynamical mechanism and frequency selection process that lead to vibrating Leidenfrost star drops. Not only do their measurements with large puddles complete the previous ones, but their analysis explains the selection of frequency over the whole range of drop size.

3 Future

Remaining questions first concern the selection of $f_{p}$ and how its mechanism should hold under different conditions. For instance, it would be interesting to check how an undulated substrate, with period ranging between $l_{c}$ and $\unicode[STIX]{x1D706}_{c}$ , could trigger or hinder oscillations and modify the selected $f_{p}$ . Another possibility would be to address these questions under different conditions, like drops sandwiched in a Hele-Shaw cell (Celestini et al. Reference Celestini, Frisch, Cohen, Raufaste, Duchemin and Pomeau2014).

Finally, one can wonder whether or not these oscillations could be converted into audible sound waves ( $f_{p}$ higher than 50 Hz), if higher harmonics appear or even if the fundamental has a high enough frequency, as in Celestini et al. (Reference Celestini, Frisch, Cohen, Raufaste, Duchemin and Pomeau2014). Incidentally, the analogy between these drops and some musical instruments is especially striking. Think of a wind instrument with a vibrating reed, such as a clarinet (Wolfe Reference Wolve2018). Depending on how strongly the performer blows, and on the length of the pipe, either the eigenmode of the reed is excited (a so-called ‘burn note’ generally unsuitable in a musical performance) or the selected sound is due to the vibration of the air column in the pipe (Helmholtz resonator), which couples to the reed vibrations and makes a nice and controlled sound. The levitating drops, depending on their size, are also susceptible to excite vibration modes that are either intrinsic to their size or are tuned by the capillary length intrinsic to the liquid.