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Leidenfrost levitation of a spherical particle above a liquid bath: Evolution of the vapour-film morphology with particle size

Part of: Basic methods in fluid mechanics Asymptotic theory Incompressible viscous fluids

Published online by Cambridge University Press:  17 February 2022

R. BRANDÃO  and
O. SCHNITZER Open the ORCID record for O. SCHNITZER [Opens in a new window]
R. BRANDÃO
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK emails: [email protected], [email protected]
O. SCHNITZER
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK emails: [email protected], [email protected]
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Abstract

We consider a spherical particle levitating above a liquid bath owing to the Leidenfrost effect, where the vapour of either the bath or sphere forms an insulating film whose pressure supports the sphere’s weight. Starting from a reduced formulation based on a lubrication-type approximation, we use matched asymptotics to describe the morphology of the vapour film assuming that the sphere is small relative to the capillary length (small Bond number) and that the densities of the bath and sphere are comparable. We find that this regime is comprised of two formally infinite sequences of distinguished limits which meet at an accumulation point, the limits being defined by the smallness of an intrinsic evaporation number relative to the Bond number. These sequences of limits reveal a surprisingly intricate evolution of the film morphology with increasing sphere size. Initially, the vapour film transitions from a featureless morphology, where the thickness profile is parabolic, to a neck–bubble morphology, which consists of a uniform pressure bubble bounded by a narrow and much thinner annular neck. Gravity effects then become important in the bubble leading to sequential formation of increasingly smaller neck–bubble pairs near the symmetry axis. This process terminates when the pairs closest to the symmetry axis become indistinguishable and merge. Subsequently, the inner section of that merger transitions into a uniform-thickness film that expands radially, gradually squishing increasingly larger neck–bubble pairs into a region of localised oscillations sandwiched between the uniform film and what remains of the bubble whose radial extent is presently comparable to the uniform film; the neck–bubble pairs farther from the axis remain essentially intact. Ultimately, the uniform film gobbles up the largest outermost bubble, whereby the morphology simplifies to a uniform film bounded by localised oscillations. Overall, the asymptotic analysis describes the continuous evolution of the vapour film from a neck–bubble morphology typical of a Leidenfrost drop levitating above a flat solid substrate to a uniform-film morphology which resembles that in the case of a large liquid drop levitating above a liquid bath.

Keywords

Leidenfrost effectlevitationcapillary wavesasymptotic analysisthin films

MSC classification

Primary: 76D08: Lubrication theory
Secondary: 76M45: Asymptotic methods, singular perturbations 34E10: Perturbations, asymptotics
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 6 , December 2022 , pp. 1117 - 1169
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. (1965) Handbook of Mathematical Functions, 3rd ed., Dover, New York, NY.Google Scholar
Adda-Bedia, M., Kumar, S., Lechenault, F., Moulinet, S., Schillaci, M. & Vella, D. (2016) Inverse leidenfrost effect: levitating drops on liquid nitrogen. Langmuir 32(17), 4179–4188.CrossRefGoogle ScholarPubMed
Benilov, E. S. & Benilov, M. S. (2015) A thin drop sliding down an inclined plate. J. Fluid Mech. 773, 75–102.CrossRefGoogle Scholar
Benilov, E. S., Benilov, M. S. & Kopteva, N. (2008) Steady rimming flows with surface tension. J. Fluid Mech. 597, 91–118.CrossRefGoogle Scholar
Benilov, E. S., Chapman, S. J., McLeod, J. B., Ockendon, J. R. & Zubkov, V. S. (2010) On liquid films on an inclined plate. J. Fluid Mech. 663, 53.CrossRefGoogle Scholar
Biance, A.-L., Clanet, C. & Quéré, D. (2003) Leidenfrost drops. Phys. Fluids 15(6), 1632–1637.CrossRefGoogle Scholar
Bowles, R. I. (1995) Upstream influence and the form of standing hydraulic jumps in liquid-layer flows on favourable slopes. J. Fluid Mech. 284, 63–96.CrossRefGoogle Scholar
Brandão, R. & Schnitzer, O. (2020) Spontaneous dynamics of two-dimensional leidenfrost wheels. Phys. Rev. Fluids 5, 091601.CrossRefGoogle Scholar
Burton, J. C., Sharpe, A. L., Van Der Veen, R. C. A., Franco, A. & Nagel, S. R. (2012) Geometry of the vapor layer under a leidenfrost drop. Phys. Rev. Lett. 109(7), 074301.CrossRefGoogle Scholar
Celestini, F., Frisch, T. & Pomeau, Y. (2012) Take off of small leidenfrost droplets. Phys. Rev. Lett. 109(3), 034501.CrossRefGoogle ScholarPubMed
Cooray, H., Cicuta, P. & Vella, D. (2017) Floating and sinking of a pair of spheres at a liquid–fluid interface. Langmuir 33(6), 1427–1436.CrossRefGoogle Scholar
Cuesta, C. M. & Velázquez, J. J. L. (2012) Analysis of oscillations in a drainage equation. SIAM J. Math. Anal. 44(3), 1588–1616.CrossRefGoogle Scholar
Cuesta, C. M. & Velázquez, J. J. L. (2014) Existence of solutions describing accumulation in a thin-film flow. SIAM J. Appl. Dyn. Syst. 13(1), 47–93.CrossRefGoogle Scholar
Duchemin, L., Lister, J. R. & Lange, U. (2005) Static shapes of levitated viscous drops. J. Fluid Mech. 533, 161–170.CrossRefGoogle Scholar
Galeano-Rios, C. A., Cimpeanu, R., Bauman, I. A., MacEwen, A., Milewski, P. A. & Harris, D. M. (2021) Capillary-scale solid rebounds: experiments, modelling and simulations. J. Fluid Mech. 912, A17.CrossRefGoogle Scholar
Gauthier, A., Diddens, C., Proville, R., Lohse, D. & van der Meer, D. (2019) Self-propulsion of inverse leidenfrost drops on a cryogenic bath. Proc. Natl. Acad. 116(4), 1174–1179.CrossRefGoogle Scholar
Hall, R. S., Board, S. J., Clare, A. J., Duffey, R. B., Playle, T. S. & Poole, D. H. (1969) Inverse leidenfrost phenomenon. Nature 224(5216), 266–267.CrossRefGoogle Scholar
Hewitt, I. J., Balmforth, N. J. & De Bruyn, J. R. (2015) Elastic-plated gravity currents. Eur. J. Appl. Math. 26(01).CrossRefGoogle Scholar
Hinch, E. J. (1991) Perturbation Methods, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Jensen, O. E. (1997) The thin liquid lining of a weakly curved cylindrical tube. J. Fluid Mech. 331, 373–403.CrossRefGoogle Scholar
Jones, A. F. & Wilson, S. D. R. (1978) The film drainage problem in droplet coalescence. J. Fluid Mech. 87(2), 263–288.CrossRefGoogle Scholar
Lagubeau, G., Le Merrer, M., Clanet, C. & Quéré, D. (2011) Leidenfrost on a ratchet. Nat. Phys. 7(5), 395–398.CrossRefGoogle Scholar
Lister, J. R., Thompson, A. B., Perriot, A. & Duchemin, L. (2008) Shape and stability of axisymmetric levitated viscous drops. J. Fluid Mech. 617, 167.CrossRefGoogle Scholar
Maquet, L., Sobac, B., Darbois-Texier, B., Duchesne, A., Brandenbourger, M., Rednikov, A., Colinet, P. & Dorbolo, S. (2016) Leidenfrost drops on a heated liquid pool. Phys. Rev. Fluids 1(5), 053902.CrossRefGoogle Scholar
Matlab. (2020) version 9.90.0 (R2020a), The MathWorks Inc., Natick, Massachusetts.Google Scholar
Pomeau, Y., Le Berre, M., Celestini, F. & Frisch, T. (2012) The leidenfrost effect: from quasi-spherical droplets to puddles. C. R. Mech. 340(11–12), 867–881.CrossRefGoogle Scholar
Quéré, D. (2013) Leidenfrost dynamics. Annu. Rev. Fluid Mech. 45, 197–215.CrossRefGoogle Scholar
Snoeijer, J. H., Brunet, P. & Eggers, J. (2009) Maximum size of drops levitated by an air cushion. Phys. Rev. E 79(3), 036307.CrossRefGoogle ScholarPubMed
Sobac, B., Maquet, L., Duchesne, A., Machrafi, H., Rednikov, A., Dauby, P., Colinet, P. & Dorbolo, S. (2020) Self-induced flows enhance the levitation of leidenfrost drops on liquid baths. Phys. Rev. Fluids 5(6), 062701.CrossRefGoogle Scholar
Sobac, B., Rednikov, A., Dorbolo, S. & Colinet, P. (2014) Leidenfrost effect: accurate drop shape modeling and refined scaling laws. Phys. Rev. E 90(5), 053011.CrossRefGoogle ScholarPubMed
van Limbeek, M. A. J., Sobac, B., Rednikov, A., Colinet, P. & Snoeijer, J. H. (2019) Asymptotic theory for a leidenfrost drop on a liquid pool. J. Fluid Mech. 863, 1157–1189.CrossRefGoogle Scholar
Wilson, S. D. R & Jones, A. F. (1983) The entry of a falling film into a pool and the air-entrainment problem. J. Fluid Mech. 128, 219–230.CrossRefGoogle Scholar
Yiantsios, S. G. & Davis, R. H. (1990) On the buoyancy-driven motion of a drop towards a rigid surface or a deformable interface. J. Fluid Mech. 217, 547–573.CrossRefGoogle Scholar