Imperial College London
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ProfessorDarrenCrowdy

Faculty of Natural Sciences, Department of Mathematics

Professor in Applied Mathematics
 
 
 
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Contact

 

+44 (0)20 7594 8587d.crowdy Website

 
 
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Location

 

735Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Yariv:2019:10.1017/jfm.2019.362,
author = {Yariv, E and Crowdy, D},
doi = {10.1017/jfm.2019.362},
journal = {Journal of Fluid Mechanics},
pages = {775--798},
title = {Thermocapillary  flow between grooved superhydrophobic surfaces: transverse temperature gradients},
url = {http://dx.doi.org/10.1017/jfm.2019.362},
volume = {871},
year = {2019}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - 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.
AU  - Yariv,E
AU  - Crowdy,D
DO  - 10.1017/jfm.2019.362
EP  - 798
PY  - 2019///
SN  - 0022-1120
SP  - 775
TI  - Thermocapillary  flow between grooved superhydrophobic surfaces: transverse temperature gradients
T2  - Journal of Fluid Mechanics
UR  - http://dx.doi.org/10.1017/jfm.2019.362
UR  - http://hdl.handle.net/10044/1/70192
VL  - 871
ER  -
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