Maximum size of drops levitated by an air cushion

Liquid drops can be kept from touching a plane solid surface by a gas stream entering from underneath, as it is observed for water drops on a heated plate, kept aloft by a stream of water vapor. We investigate the limit of small flow rates, for which the size of the gap between the drop and the substrate becomes very small, to obtain a full analytical description of stationary drop states and their stability. Above a critical drop radius no stationary drops can exist, below the critical radius two solutions coexist. However, only the solution with the smaller gap width is stable, the other is unstable. We compare to experimental data and use boundary integral simulations to show that unstable drops develop a gas “chimney” that breaks the drop in its middle.

Figure 1

Definitions and sketch of the matching regions. The liquid-gas interface will be denoted by $z=h$. In the “gas pocket” region below the drop, we can write $h$ as a function of the radial coordinate $r$. Due to the overhang exterior of the neck, $h\left(r\right)$ is multivalued in the “drop” region.

Figure 2

Solid line: the function $f$ relates the slope $S_{-}$ to curvature $K_{-}$ of the inner solution [Eq. (23) with $K_{+}=2.17$]. Dotted lines: the function $g$ provides the matching condition between inner region and gas pocket region [Eq. (51) with $\chi ={10}^{-7}$]. The three curves correspond to values $r_n=3.55$ (below critical), $r_n=3.62$ (critical), and $r_n=3.65$ (above critical).

Figure 3

The outer solution of the “drop” region corresponds to a perfectly nonwetting sessile drop. The size of the drop, characterized by $r_n$, sets the curvature $K_{+}$ for the inner solution.

Figure 4

The outer solution fixes the value of $K_{+}$ as a function of the neck radius (which controls the drop volume). The maximal radius $r_0=3.38317\cdots$ gives $K_{+}=2.17\cdots$.

Figure 5

Outer solutions for the gas pocket region (amplitudes normalized to unity). Thin solid lines correspond to $r_n=1,2$; the dashed line, corresponding to $r_n=3$, illustrates that $h$ would have to become negative to realize a neck radius larger than $r_0$. The heavy line shows the maximum possible $r_n=r_0$, corresponding to the first minimum of $J_0\left(r\right)$.

Figure 6

The inner solution $H\left(\xi \right)$ obtained from numerical integration of Eq. (17), with $K_{+}=2.17$ and $K_{-}=0$. It will follow that these values correspond to the critical solution. The minimum value $H_n\approx 0.931$ determines the thickness of the neck (49).

Figure 7

The bifurcation diagram $(K_{-},\tilde{r})$, derived from Eq. (54). Curves correspond to different values of the flux, $\chi ={10}^{-5},{10}^{-7},{10}^{-10}$, revealing the weak dependence on $\chi$. The dashed lines represent perturbations $\delta \tilde{r},\delta K_{-}=-\delta \tilde{r}∕c$, which are tangent to the solution curve. They represent marginal perturbations, separating stable from unstable solutions.

Figure 8

The bifurcation diagram $(\mathrm{Bo},r_n)$, for different values of the flux, $\chi ={10}^{-5},{10}^{-7},{10}^{-10}$. The maximum drop volume is attained slightly before the maximum radius $r_c$. This maximum coincides with the stability boundary shown as the dashed lines in Fig. 7. The dotted lines indicate the asymptotic limit for $\chi \to 0$.

Figure 9

Boundary integral simulation of a drop with parameters $\Gamma =0.02$, $\mathrm{Bo}=4.2$, and $\lambda =100$. The drop relaxes toward a stable state, which is drawn as the heavy line.

Figure 10

The same as Fig. 9, but with a slightly larger $\mathrm{Bo}=4.4$. The air bubble under the center of the drop lifts up to form a chimney. The time interval between the profiles is $\Delta t=3000$, in units of ${\mathcal{l}}_c{\eta }_{\text{gas}}∕\gamma$.

Figure 11

Bifurcation diagram $h_0$ vs $r_n$ for $\chi ={10}^{-4}$ and ${10}^{-7}$. Smaller $\chi$ yield larger radii. Solid lines were obtained from numerical solution of the lubrication equation (82). Dashed lines correspond to asymptotic theory (54).

Figure 12

Critical radius $r_c$ as a function of the flux $\chi$. The numerical values obtained from the lubrication equation (dots) indeed approach the theoretical prediction (55) (solid line). The dashed line indicates the asymptotic value $r_0=3.8317\dots$.