Levitation of a non--volatile drop by an evaporating pool: the inverse Leidenfrost effect

Assuming axisymmetry, zero gravity, and uniform surface tension $\gamma$ and vapour properties (viscosity $\eta$, conductivity $k$ and density $\rho$), we determine the maximum value of the force $F$ with which a heated sphere (radius $b$) can be pressed against the pool surface without rupturing it. The Laplace--Young and Reynolds equations form a coupled system of ODEs determining, in particular, film thickness $h_0/b$ at the sphere bottom as a function of $F/(2\pi \gamma b)$ with $\varepsilon= \frac{\eta k \Delta T}{\gamma b\rho H_{lv}}$, as a parameter (latent heat $H_{lv}$). Numerical solutions for fixed small $\varepsilon$ show that as $F/(2\pi \gamma b)$ is increased from zero, $h_0/b$ first decreases to a minimum. With further increase in $F$, $h_0$ increases until a turning point is reached. There, the slope ${\rm d} h_0/{\rm d} F\to \infty$, and the response curve doubles back on itself to form an upper branch. Near the turning point, the interface shows an apparent contact line with apparent contact angle $\pi$ (on the liquid side). The turning point corresponds to the contact line moving from the lower hemisphere to the upper; during this process, $F/(2\pi \gamma b)$ reaches its maximum (unity). This result is consistent with work by Adda--Bedia et al.(2016).