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Closed-form solution of a thermocapillary free-film problem due to Pukhnachev

Published online by Cambridge University Press:  23 March 2015

BRIAN R. DUFFY ,
MATTHIAS LANGER  and
STEPHEN K. WILSON
BRIAN R. DUFFY
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK email: b.r.duffy@strath.ac.uk, m.langer@strath.ac.uk, s.k.wilson@strath.ac.uk
MATTHIAS LANGER
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK email: b.r.duffy@strath.ac.uk, m.langer@strath.ac.uk, s.k.wilson@strath.ac.uk
STEPHEN K. WILSON
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK email: b.r.duffy@strath.ac.uk, m.langer@strath.ac.uk, s.k.wilson@strath.ac.uk
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Abstract

We consider the steady two-dimensional thin-film version of a problem concerning a weightless non-isothermal free fluid film subject to thermocapillarity, proposed and analysed by Pukhnachev and co-workers. Specifically, we extend and correct the paper by Karabut and Pukhnachev (J. Appl. Mech. Tech. Phys. 49, 568–579, 2008), in which the problem is solved numerically, and in which it is claimed that there exists a unique solution for any value of a prescribed heat-flux parameter in the model. We present a closed-form (parametric) solution of the problem, and from this show that, on the contrary, solutions exist only when the heat-flux parameter is less than a critical value found numerically by Karabut and Pukhnachev, and that when this condition is satisfied there are in fact two solutions, one of which recovers that obtained numerically by Karabut and Pukhnachev, the other being new.

Keywords

free filmnon-isothermal flowthermocapillarityPukhnachev
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Special Anniversay Issue 5: Celebrating 75 years , October 2015 , pp. 721 - 741
Copyright
Copyright © Cambridge University Press 2015 

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