Effects of rough boundary on the heat transfer in a thin-film flow

Igor Pažanin; Francisco Javier Suárez-Grau

doi:10.1016/j.crme.2013.05.001

Effects of rough boundary on the heat transfer in a thin-film flow

Igor Pažanin ¹ ; Francisco Javier Suárez-Grau ²

¹ Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
² Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/ Tarfia s/n, 41012 Sevilla, Spain

Comptes Rendus. Mécanique, Volume 341 (2013) no. 8, pp. 646-652.

Abstract
Résumé

In this Note, a heat flow through a rough thin domain filled with fluid (lubricant) is studied. The domainʼs thickness is considered as the small parameter ε, while the roughness is defined by a periodical function with a period of order $ε^{2}$ . We assume that the lubricant is cooled by the exterior medium and we describe the heat exchange on the rough part of the boundary by Newtonʼs cooling law. Depending on the magnitude of the heat transfer coefficient with respect to ε, we obtain three different macroscopic models via formal asymptotic analysis. We identify the critical case explicitly acknowledging both roughness-induced effects and the effects of the surrounding medium on heat transfer at main order. We illustrate the obtained results by some numerical simulations.

Dans cette Note, on étudie un flux de chaleur dans un domaine rugueux de faible épaisseur rempli de liquide (lubrifiant). On considère lʼépaisseur du domaine comme le petit paramètre ε, tandis que la rugosité est définie par une fonction périodique de période dʼordre $ε^{2}$ . On suppose que le lubrifiant est refroidi par le milieu extérieur et que lʼéchange de chaleur est décrit sur la partie rugueuse de la frontière par la loi de refroidissement de Newton. En fonction de la valeur du coefficient de transfert de chaleur par rapport à ε, on obtient trois différents modèles macroscopiques via une analyse asymptotique formelle. On identifie le cas critique, reconnaissant explicitement les effets induits par la rugosité et les effets du milieu environnant sur le transfert de chaleur à lʼordre principal. Les résultats obtenus sont illustrés à lʼaide de simulations numériques.

Received: 2013-04-01
Accepted: 2013-05-22
Published online: 2013-06-13

DOI: 10.1016/j.crme.2013.05.001

Keywords: Thin-film lubrication, Heat conduction, Rough boundaries, Asymptotic analysis
Mots-clés : Film fluide mince, Conduction de chaleur, Surfaces rugueuses, Analyse asymptotique

Author's affiliations:

Igor Pažanin ¹; Francisco Javier Suárez-Grau ²

¹ Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia
² Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/ Tarfia s/n, 41012 Sevilla, Spain

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     author = {Igor Pa\v{z}anin and Francisco Javier Su\'arez-Grau},
     title = {Effects of rough boundary on the heat transfer in a thin-film flow},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {646--652},
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     language = {en},
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Igor Pažanin; Francisco Javier Suárez-Grau. Effects of rough boundary on the heat transfer in a thin-film flow. Comptes Rendus. Mécanique, Volume 341 (2013) no. 8, pp. 646-652. doi : 10.1016/j.crme.2013.05.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.05.001/

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