The Nusselt number of a hot sphere levitated by a volatile pool

When placed at the surface of a volatile liquid, a sphere of hot dense non-volatile material remains suspended until it cools sufficiently. The duration of this ‘inverse Leidenfrost’ phenomenon depends on the Nusselt number $Nu$ of the sphere, itself determined by flow in the film of vapour separating particle and liquid. It is shown that provided the Nusselt number is large, it can be calculated numerically using only the Laplace relation and the equations governing the thin film; patching to a solution for the outer thick film is not necessary. This method is demonstrated by using it to determine $Nu$ for a sphere sufficiently small that in the governing equations, the acceleration due to gravity is negligible except where multiplied by the density of the sphere. Numerical results giving $Nu$ as a function of a dimensionless measure of sphere weight are supplemented with analysis showing that, when the weight is of the order of the maximum supportable by surface tension alone, the film consists of a spherical bubble cap bounded by its contact rim. The solutions for these regions are coupled: although the apparent contact angle $\chi$ for the cap is determined within the rim, its value depends on the flow rate arriving from the cap as well as on the additional evaporation from the rim. The latter acts to reduce $\chi$ from the value it would otherwise have, thereby reducing the thickness of the entire cap. For the example treated here, the value of $Nu$ is doubled by this mechanism.