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Thermocapillary flow between grooved superhydrophobic surfaces: transverse temperature gradients

Published online by Cambridge University Press:  24 May 2019

Ehud Yariv Open the ORCID record for Ehud Yariv [Opens in a new window]  and
Darren Crowdy Open the ORCID record for Darren Crowdy [Opens in a new window]
Ehud Yariv*
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
Darren Crowdy
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
*
†Email address for correspondence: udi@technion.ac.il
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Abstract

We consider the thermocapillary motion of a liquid layer which is bounded between two superhydrophobic surfaces, each made up of a periodic array of highly conducting solid slats, with flat bubbles trapped in the grooves between them. Following the recent analysis of the longitudinal problem (Yariv, J. Fluid Mech., vol. 855, 2018, pp. 574–594), we address here the transverse problem, where the macroscopic temperature gradient that drives the flow is applied perpendicular to the grooves, with the goal of calculating the volumetric flux between the two surfaces. We focus upon the situation where the slats separating the grooves are long relative to the groove-array period, for which case the temperature in the solid portions of the superhydrophobic plane is piecewise uniform. This scenario, which was investigated numerically by Baier et al. (Phys. Rev. E, vol. 82 (3), 2010, 037301), allows for a surprising analogy between the harmonic conjugate of the temperature field in the present problem and the unidirectional velocity in a comparable longitudinal pressure-driven flow problem over an interchanged boundary. The main body of the paper is concerned with the limit of deep channels, where the problem reduces to the calculation of the heat transport and flow about a single surface and the associated ‘slip’ velocity at large distance from that surface. Making use of Lorentz’s reciprocity, we obtain that velocity as a simple quadrature, providing the analogue to the expression obtained by Baier et al. (2010) in the comparable longitudinal problem. The rest of the paper is devoted to the diametric limit of shallow channels, which is analysed using a Hele-Shaw approximation, and the singular limit of small solid fractions, where we find a logarithmic scaling of the flux with the solid fraction. The latter two limits do not commute.

JFM classification

Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 871 , 25 July 2019 , pp. 775 - 798
Copyright
© 2019 Cambridge University Press 

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References

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