Model for self-organized Leidenfrost rotating polygons as cnoidal waves

TL;DR

The study tackles the formation of self-organized rotating polygonal patterns in shallow Leidenfrost rings by deriving a nonlinear contour evolution equation for the inner free surface under the Euler equations in a Boussinesq shallow-convection regime. The model incorporates a rolling poloidal vortex and detailed capillary pressure via mean curvature, yielding $cn$-type cnoidal wave solutions in a co-rotating frame, and also presents a reduced KdV-type description through pressure averaging. Two complementary routes are used: a full nonlinear equation that fits experimental polygons (4–8 edges) and a KdV-like reduced model that explains both standard and peaked polygons, with exact and approximate cnoidal-wave forms. Experimental data from liquid nitrogen Leidenfrost rings validate the theoretical predictions, showing good agreement and revealing the crucial roles of rolling vortices and surface tension, while identifying avenues for 3D thermo-convective extensions and stability analyses.

Abstract

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.

Figures (10)

• Figure 1: Typical liquid nitrogen Leidenfrost ring formation inside a rigid cylinder. The liquid layer maintained outside the cylinder creates a regime which favors formation of rotating hollow polygons inside. $R$ is the inner cylinder radius, and $\vec{r}_{\Gamma}(\theta,t)$ is the parametrization of the $\Gamma$ directrix describing the inner liquid boundary.
• Figure 2: A normal cross section in the Leidenfrost ring of height $h$. $R_0$ is the radius corresponding to the static equilibrium ($\gamma=0$, $\Gamma$ circular) and $R$ is the radius of the external solid boundary. The inner boundary of the liquid is the normal channel surface $\Sigma$ approximated by a circular arc of radius $a$ (thick black curve) and center $C$ which is not necessary placed on the radius $R_0$. $\Sigma$ is parameterized by the contour $\vec{r}_{\Gamma}(\theta,\varphi,t)$. Liquid surface deformations at the proximity of the wall and bottom are not shown here, and the figure is not drawn to scale. In this example $\gamma<0$
• Figure 3: Example of an inner liquid boundary $\Sigma$ generated by a hexagonal shape $\gamma$. The mean curvature $H_{\Sigma}$, Eq. (\ref{['eqp']}) is represented in colored density plot, with legend. The corners of the contour $\Gamma$ have large positive curvature $\kappa_{\Gamma}$ and can be compensate the large half-cylinder negative curvature $-1/a$ of the tubular surface, resulting in $H_{\Sigma}\simeq 0$ around these critical points (red regions).
• Figure 4: Examples of cnoidal waves solutions Eq. (\ref{['el']}) representing hollow polygons in the co-rotating frame, with $4,5$ and $6$ edges. All geometric parameters occurring in Eq. (\ref{['eq5']}) and the amplitude of the contour deformation $\zeta$ are shown. The polygons with $n=4,5,6$ edges are also presented in Figs. \ref{['figs6543']} left, right and Fig. \ref{['figs6543b']} left, respectively.
• Figure 5: Matching ratios in Eq. (\ref{['203']}) with experimental parameters for hexagon $\zeta=0.053$cm, $m=0.216, R_0=1.18$ cm, $p=3.44$, see Table I, for the whole range . The red curves represent the left-hand sides of the terms in Eq. (\ref{['203']}) as functions of the poloidal angle $\phi$. The black lines represent the right-hand side terms of the same equation, obtained from experiments. The theory-experiment match is acceptable for the large part of the range of the poloidal angle.