Spontaneous dynamics of two-dimensional Leidenfrost wheels

It has been recently discovered that liquid Leidenfrost drops levitated by their vapor above a flat hot surface undergo symmetry breaking leading to spontaneous motion (A. Bouillant et al., Nature Physics, 14 1188-1192, 2018). Motivated by these experiments, we theoretically investigate the spontaneous dynamics of Leidenfrost drops on the basis of a simplified two-dimensional model, focusing on near-circular drops small in comparison to the capillary length. The model couples the equations of motion of the drop, which flows as a rigid wheel, and a thin-film model governing the vapor flow, the profile of the deformable vapor-liquid interface and thus the hydrodynamic forces and torques on the drop. We find that the symmetric Leidenfrost state is unstable above a critical drop radius: $R_1$ for a free drop and $R_2>R_1$ for an immobilized drop. In these respective cases, symmetry breaking is manifested in supercritical-pitchfork bifurcations into steady states of pure rolling and constant angular-velocity. In qualitative agreement with the experiments, when an immobilized drop is suddenly released it initially moves at constant acceleration $\alpha g$, where $\alpha$ is an angle characterizing the slope of the liquid-vapor profile and $g$ is the gravitational acceleration; furthermore, $\alpha$ exhibits a maximum with respect to the drop radius, at a radius increasing with the temperature difference between the surface and the drop. At long times, the translation and rotational velocities become comparable and the drop tends to the steady pure-rolling state.