Model for self-organized Leidenfrost rotating polygons as cnoidal waves

The remarkable appearance of self-organized regular and peaked polygonal rotating patterns in shallow Leidenfrost rings is investigated as a balance between surface tension geometry and nonlinear terms of Euler equation. Using the Boussinesq shallow convection approximation and a specialized expansion of the Laplace equation solutions, we derive a nonlinear equation that can be integrated in terms of elliptic functions. The model rigorously accounts for surface tension, the contribution of the poloidal rolling vortex, and the interplay between buoyancy-driven and thermocapillary flows. We obtain cnoidal waves solutions describing the dynamics of the inner free surface of the Leidenfrost ring, to predict polygonal patterns in liquid nitrogen. These predictions are compared with experimental observations.Additionally, we introduce a simplified model based on poloidal averaging of the capillary pressure, leading to a Korteweg-de Vries-type equation. This simplified model not only reproduces the cnoidal wave solutions but also predicts new trigonometric solutions, offering insights into the formation of peaked polygons.