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

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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. App. 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.
LanguageEnglish
Pages721-741
Number of pages21
JournalEuropean Journal of Applied Mathematics
Volume26
Issue number5
Early online date23 Mar 2015
DOIs
Publication statusPublished - Oct 2015

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Closed-form Solution
Heat flux
Heat Flux
Parametric Solutions
Thin films
Unique Solution
Fluids
Critical value
Thin Films
Closed-form
Fluid
Model

Keywords

  • free film
  • non-isothermal flow
  • thermocapillarity
  • Pukhnachev

Cite this

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title = "Closed-form solution of a thermocapillary free-film problem due to Pukhnachev",
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. App. 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 film, non-isothermal flow, thermocapillarity, Pukhnachev",
author = "Duffy, {Brian R.} and Matthias Langer and Wilson, {Stephen K.}",
year = "2015",
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T1 - Closed-form solution of a thermocapillary free-film problem due to Pukhnachev

AU - Duffy, Brian R.

AU - Langer, Matthias

AU - Wilson, Stephen K.

PY - 2015/10

Y1 - 2015/10

N2 - 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. App. 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.

AB - 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. App. 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.

KW - free film

KW - non-isothermal flow

KW - thermocapillarity

KW - Pukhnachev

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