A physical picture of the inverse Leidenfrost effect holding in the limit of vanishing crispation number $Cr$

Assuming axisymmetry and zero gravity, computation and asymptotic analysis are used to find the maximum value of the force $F$ with which a heated non--evaporating sphere (radius $b$) can be pushed against the surface of a volatile liquid. Mass evaporated beneath the sphere flows to the atmosphere as a thin film of vapour, and the pool surface is deformed by the pressure field driving that flow. For $f=F/(2\pi\gamma b)\ll 1$ (surface tension $\gamma$), film thickness $h$ increases monotonically with angle $\theta$ (measured from the sphere bottom). Once $f$ exceeds a critical value, $h(\theta)$ changes form; a maximum $h_0$ occurs at $\theta=0$, and a minimum $h_1$ at $\theta=\theta_1$. With increasing $f$, the ratio $h_0/h_1$ increases, causing an apparent contact line to form at $\theta_1$. For $\theta<\theta_1$, $p(\theta)$ is asymptotically uniform and the pool surface is a spherical cap; for $\theta>\theta_1$, $p$ is atmospheric and the pool surface is the minimal surface tangent to the sphere at $\theta_1$. $p(\theta)$ falls from $p_0$ to atmospheric across a narrow barrier rim within which $h=O(h_1)$. From this picture, it follows that $F=2\pi\gamma b \sin^2\theta_1$, and that the maximum force is $2\pi\gamma b$. A formula for the evaporation rate is also provided.