Leidenfrost effect: Accurate drop shape modeling and refined scaling laws

We here present a simple fitting-parameter-free theory of the Leidenfrost effect (droplet levitation above a superheated plate) covering the full range of stable shapes, i.e., from small quasispherical droplets to larger puddles floating on a pocketlike vapor film. The geometry of this film is found to be in excellent quantitative agreement with the interferometric measurements of Burton et al. [Phys. Rev. Lett. 109, 074301 (2012)]. We also obtain new scalings generalizing classical ones derived by Biance et al. [Phys. Fluids 15, 1632 (2003)] as far as the effect of plate superheat is concerned and highlight the relative role of evaporation, gravity, and capillarity in the vapor film. To further substantiate these findings, a treatment of the problem by matched asymptotic expansions is also presented.

Figure 1

Schematic illustration of a Leidenfrost drop. The drop levitates on a vapor cushion of nonuniform thickness profile $h\left(r\right)$, composed by a pocket surrounded by an annular neck. Due to evaporation, an outward radial flow develops in the vapor layer. $R_{\max }$ is the droplet radius as seen from above, while $R_{\mathrm{neck}}$ is the radius of the neck (at its thinnest section). The outer drop region (assumed to be an equilibrium shape) and the inner vapor layer region (described using lubrication theory including evaporation) are solved separately and numerically matched at the point of coordinates ($R_p$, $h_p$). $R_c$ is defined from the outer drop shape as the “contact” radius of a superhydrophobic drop at equilibrium.

Figure 2

Numerically determined shapes of Leidenfrost drops for $R_{\max }=$ 0.1, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, and $\mathcal{E}=1.21\times {10}^{-6}$ ($T_w^{*}=370{}^{\circ }\mathrm{C}$ for water). The lengths are scaled by the capillary length ${\ell }_c^{*}={({\gamma }^{*}/{\rho }_{\ell }^{*}{g}^{*})}^{1/2}$ (i.e., $2.5$ mm for water).

Figure 3

Numerical calculations compared to experimental measurements (Burton et al. [9] and Biance et al. [10]) of the characteristics of the vapor layer under a Leidenfrost water drop. (a) The film thickness at the neck, (b) the depth of the vapor pocket, and (c) the neck radius are plotted as a function of the drop size $R_{\max }^{*}$ (the capillary length is ${\ell }_c^{*}=2.5$ mm).

Figure 4

(a) Radius at the neck $R_{\mathrm{neck}}$ and “contact” radius $R_c$ versus drop size $R_{\max }$ for $\mathcal{E}=1.21\times {10}^{-6}$ ($T_w^{*}=370{}^{\circ }\mathrm{C}$ for water). (b) Smallness parameter $h_{\mathrm{center}}/R_c$ underlying the lubrication approximation versus $R_{\max }$. The inset compares $R_{\mathrm{neck}}$ and $R_c$ for a larger range of sizes than in Fig. 4.

Figure 5

Dimensionless vapor film thicknesses (a) at the neck and (b) at the center as a function of the evaporation number for various drop sizes. [(c) and (d)] The same thicknesses versus the drop size for an evaporation number $\mathcal{E}=1.21\times {10}^{-6}$.

Figure 6

Graph of the auxiliary function defined in Eq. (26).

Figure 7

Various quantities as functions of the maximum horizontal radius of the drop for $\mathcal{E}=1.21\times {10}^{-6}$ (water at $T_w^{*}=370{}^{\circ }\mathrm{C}$) when relevant. (a) Contact radius $R_c$ (solid line) and curvature at the contact line ${\kappa }_c$ (dotted lined) of the associated equilibrium superhydrophobic droplet. (b) Smallness parameters of the problem ($\delta$ the lower curve and $\tilde{\delta }$ the upper curve). (c) $h_{\mathrm{neck}}$ results of the present asymptotic analysis (dashed, green line) and of the computation in Sec. 3 (solid blue line). (d) $h_{\mathrm{center}}$, with the line formats meaning the same as above, while the dot-dashed (red) line is for the semiheuristic result accounting for droplet elevation.

Figure 8

Various quantities as functions of $\mathcal{E}$ for $R_{\max }=3$ [lower curves in (a) and (b) and upper ones in (c) and (d)] and $R_{\max }=1$ (vice versa). The curves nearly coincide in (a). The line format meaning is the same as described in the caption to Fig. 7.