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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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References

[1]Abramowitz, M. & Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover Publications, New York.Google Scholar
[2]Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. (2009) Wetting and spreading. Rev. Mod. Phys. 81, 739–805.CrossRefGoogle Scholar
[3]Chan, T. S., Gueudré, T. & Snoeijer, J. H. (2011) Maximum speed of dewetting on a fiber. Phys. Fluids 23, 112103, 1–9.Google Scholar
[4]Duffy, B. R. & Wilson, S. K. (1997) A third-order differential equation arising in thin-film theory and relevant to Tanner's law. Appl. Math. Lett. 10, 63–68.CrossRefGoogle Scholar
[5]Eggers, J. (2004) Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 93, 094502, 1–4.CrossRefGoogle ScholarPubMed
[6]Eggers, J. (2005) Existence of receding and advancing contact lines. Phys. Fluids 17, 082106, 1–10.CrossRefGoogle Scholar
[7]Janeček, V., Andreotti, B., Pražák, D., Bárta, T. & Nikolayev, V. S. (2013) Moving contact line of a volatile fluid. Phys. Rev. E 88, 060404, 1–5.CrossRefGoogle ScholarPubMed
[8]Karabut, E. A. & Pukhnachev, V. V. (2008) Steady-state conditions of a nonisothermal film with a heat-insulated free boundary. J. App. Mech. Tech. Phys. 49, 568–579.CrossRefGoogle Scholar
[9]Karpitschka, S. & Riegler, H. (2012) Noncoalescence of sessile drops from different but miscible liquids: hydrodynamic analysis of the twin drop contour as a self-stabilizing traveling wave. Phys. Rev. Lett. 109, 066103, 1–5.CrossRefGoogle ScholarPubMed
[10]Limat, L. & Stone, H. A. (2004) Three-dimensional lubrication model of a contact line corner singularity. Europhys. Lett. 65, 365–371.CrossRefGoogle Scholar
[11]Neogi, P. (2010) Bead formation near the contact line in forced spreading. Chem. Eng. Sci. 65, 4572–4578.CrossRefGoogle Scholar
[12]NIST Handbook of Mathematical Functions. Olver, F.W.J., Lozier, D.W., Boisvert, R.F. & Clark, C.W. (editors), U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge (2010). Online version: http://dlmf.nist.govGoogle Scholar
[13]Pukhnachov, V. V. (2002) Model of a viscous layer deformation by thermocapillary forces. Eur. J. Appl. Math. 13, 205–224.CrossRefGoogle Scholar
[14]Pukhnachev, V. V. (2007) Equilibrium of a free nonisothermal liquid film. J. Appl. Mech. Tech. Phys. 48, 310–321.CrossRefGoogle Scholar
[15]Pukhnachev, V. V. & Dubinkina, S. B. (2006) A model of film deformation and rupture under the action of thermocapillary forces. Fluid Dyn. 41, 755–771.CrossRefGoogle Scholar
[16]Snoeijer, J. H. & Andreotti, B. (2013) Moving contact lines: Scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269–292.CrossRefGoogle Scholar
[17]Tuck, E. O. & Schwartz, L. W. (1990) A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev. 32, 453–469.CrossRefGoogle Scholar
[18]Voinov, O. V. (1977) Inclination angles of the boundary in moving liquid layers. Zh. Prikl. Mekh. Tekh. 2, 92–99.Google Scholar