Oscillating and star-shaped drops levitated by an airflow

We investigate the spontaneous oscillations of drops levitated above an air cushion, eventually inducing a breaking of axisymmetry and the appearance of “star drops”. This is strongly reminiscent of the Leidenfrost stars that are observed for drops floating above a hot substrate. The key advantage of this work is that we inject the airflow at a constant rate below the drop, thus eliminating thermal effects and allowing for a better control of the flow rate. We perform experiments with drops of different viscosities and observe stable states, oscillations, and chimney instabilities. We find that for a given drop size the instability appears above a critical flow rate, where the latter is largest for small drops. All these observations are reproduced by numerical simulations, where we treat the drop using potential flow and the gas as a viscous lubrication layer. Qualitatively, the onset of instability agrees with the experimental results, although the typical flow rates are too large by a factor 10. Our results demonstrate that thermal effects are not important for the formation of star drops and strongly suggest a purely hydrodynamic mechanism for the formation of Leidenfrost stars.

Figure 1

Star drops levitated by a steady (i.e., nonpulsating) airflow. Top: mode $n=3$; bottom: mode $n=4$. Figure from Ref. [12]. Reproduced with kind permission of the European Physical Journal (EPJ).

Figure 2

Numerical implementation of the drop levitated by an airflow with uniform upward flow velocity $U_g$. The numerics consist of a coupling between the Boundary Integral method for the inviscid drop and the lubrication approximation for the airflow beneath the drop. The flow inside the drop is assumed to be a potential flow; the flow at the bottom of the drop is a viscous flow, in which inertial effects are neglected.

Figure 3

Sketch of the experimental setup. Illumination and camera view are obtained using a beam splitter. A flow rate $Q$ is prescribed through a porous medium. Since the levitated drop is very mobile, it is held in position by a needle, which also supplies the liquid.

Figure 4

Measured instability threshold $Q_m$, for levitated drops. The upper axis gives the gas velocity, estimated by dividing the total flow rate by the area of the porous medium. Data represent all data points for water and water-glycerine drops, in circles ( and ) and squares ( and ), respectively. Since for the smallest flow rate the drop size could not be measured (it detaches from the needle), $R$ is measured in levitated state instead of sessile state. These points are therefore indicated by a solid symbol ( and ). Note that point corresponds to the chimney instability from Fig. 7b. The theoretical prediction of the critical radius for chimney instability is indicated by the blue dashed line, and red dotted line for the used water and water-glycerine mixture, respectively.

Figure 5

Examples of levitated drop instabilities. All images show water drops, except for (b), which is a water-glycerine drop. (a) Sessile water drop. (b) Levitating water-glycerine drop. (c) Chaotic mode water drop oscillation. (d) Water drop, mode $n=2$ ($R=4.1$ mm, $f=13.8$ Hz). (e) Water drop, mode $n=3$ ($R=6.1$ mm, $f=14.2$ Hz). (f) Water drop, mode $n=4$ ($R=5.2$ mm, $f=17.8$ Hz). (g) Water drop, mode $n=5$ ($R=8.6$ mm, $f=14.3$ Hz). (h) Water drop, mode $n=6$ ($R=6.1$ mm, $f=30.9$ Hz).

Figure 6

The frequency measured for faceted drops as shown in the images of Fig. 5. Each data point $\circ$, corresponds to one water drop measurement. The red solid line is the prediction from the corresponding eigenmode for a puddle, given by Eq. (2).

Figure 7

(a) Top row: an image sequence of the breathing mode oscillation of a large-viscosity drop (water glycerine, $60\mathrm{mPas}$). As the oscillation amplitude is rather small, a space-time diagram is shown as well, which is built from slices similar to the white boxes indicated in the images. (b) For larger drop sizes we observe the formation of a chimney.

Figure 8

Measured oscillation frequency at threshold for high-viscosity drops (see Fig. 4) (blue dots with error bars), combined with numerical results. For the numerics, the measured oscillation frequency (excitation frequency) as a function of the drop top view radius with airflow velocity 1 and 5 m/s, at three different liquid viscosities is shown. In the numerics, the frequency appears to be independent of liquid viscosity, decreases with increasing drop radius, and decreases with increasing airflow velocity.

Figure 9

Two different initial drop shapes (one spherical, one elliptical) of equal volume, converging to the same steady end shape. $\mathrm{Bo}=1$, $\mathrm{Ca}=2.5\times {10}^{-4}$, $\Lambda =11\times {10}^3$. See Fig. 10 for the corresponding pressure profile.

Figure 10

Pressure profile ($P_g$) at the bottom of a steady levitated drop. The largest pressure gradient is typically at the neck, $r=r_n$, such that the profile resembles a plateau. $\mathrm{Bo}=1$, $\mathrm{Ca}=2.5\times {10}^{-4}$, $\Lambda =11\times {10}^3$.

Figure 11

Chimney instability. Shown is the evolution of two drops, with almost equal volumes. The largest drop apparently has a radius just above the chimney threshold, which appears to be about 2.7${\ell }_c$, or 7.3 mm, while the smallest has a radius just below. The large drop does not remain stable, due to the gas pocket breaking through; the small drop keeps its steady shape. $\mathrm{Bo}=2.25$ and 2.5, $\mathrm{Ca}=2.5\times {10}^{-5}$, $\Lambda =11\times {10}^3$.

Figure 12

Time sequence from the evolution of the oscillatory instability of a levitated drop. The simulation is initiated by a spherical drop, released from small height (0.27 mm) (top-left). The top panel row shows the process from the spherical drop shape to an intermediate steady shape. The bottom panel row shows the oscillatory behavior of the drop at a later point in time. $\mathrm{Bo}=1$, $\mathrm{Ca}=2.5\times {10}^{-4}$, $\Lambda =8.2\times {10}^3$.

Figure 13

Drop contours during the final stages of the simulation in an unstable scenario (see Fig. 12). Blue contours are the two extremes, red lines are intermediate. The bottom two plots show the velocity profile inside the drop for the extremes. Note that the liquid velocities, and therefore the oscillations as well, are mainly in the vertical direction. $\mathrm{Bo}=1$, $\mathrm{Ca}=2.5\times {10}^{-4}$, $\Lambda =8.2\times {10}^3$.

Figure 14

Top view radius as a function of time for (a) a stable drop, (b) a case around the transition, and (c) an unstable drop. In the first part of each plot (up to about $100\mathrm{ms}$) the initial, spherical shape of the drop stabilizes towards the “Leidenfrost” state. After this stabilization the oscillations become visible which typically have a much larger frequency (see insets). (a) $\mathrm{Bo}=0.75$, $\mathrm{Ca}=2.5\times {10}^{-5}$; (b) $\mathrm{Bo}=0.80$, $\mathrm{Ca}=5\times {10}^{-5}$; (c) $\mathrm{Bo}=0.80$, $\mathrm{Ca}=5\times {10}^{-4}$. $\Lambda =5.5\times {10}^3$ in all three cases.

Figure 15

Top view radius $R$ versus airflow velocity $U_g$ stability diagram for $\Lambda =11\times {10}^3$. Black dots point out stable configurations: the drop has no tendency to oscillate; white squares indicate oscillating drops.