A Variational Calculation of Diffusive Flux in a Mixed Boundary Value Problem

William Pons; Stanley Pons

doi:10.5802/crmeca.198

A Variational Calculation of Diffusive Flux in a Mixed Boundary Value Problem

William Pons ¹; Stanley Pons ^2,³

¹ Department of Chemistry, Clemson University, Clemson SC 29634, USA
² Department of Chemistry, University of Utah, Salt Lake City, UT 84112 (Prior affiliation where work was conceived)
³ Pons(EI), 14, r. Eugene Giraud, 06560 Valbonne, France

Comptes Rendus. Mécanique, Volume 351 (2023), pp. 247-263.

Abstract
Résumé

A variational solution to the transient heat flow measure above a closed conductive region of arbitrary perimeter aspect, held at constant temperature, and which is embedded in an otherwise insulating boundary plane, is presented. Upon developing the general variational formulation, a full range solution for the circular disk conductor is considered as an example when implementing a two term trial function that comprises a general form based on known physical solutions to the problem at long and short times. The particular combination of the Lagrange density for the total transient diffusive flux and of the form of the trial functions are evidently effective in dealing with the difficulties often experienced when dealing with mixed boundary value heat transfer problems of the parabolic type which have a non-periodic time variable.

Une solution variationnelle à la mesure du flux thermique transitoire au-dessus d’une région conductrice fermée d’aspect de périmètre arbitraire, maintenue à température constante, et qui est encastrée dans un plan limite autrement isolant, est présentée. Après avoir développé la formulation variationnelle générale, une solution complète pour le conducteur en forme de disque circulaire est considérée comme un exemple lors de la mise en œuvre d’une fonction d’essai à deux termes qui comprend une forme générale basée sur des solutions physiques connues du problème à des temps longs et courts. La combinaison particulière de la densité de Lagrange pour le flux diffusif transitoire total et de la forme des fonctions d’essai est évidemment efficace pour traiter les difficultés souvent rencontrées lors du traitement des problèmes de transfert de chaleur à valeur limite mixte de type parabolique qui ont une variable temporelle non périodique.

Received: 2023-01-31
Accepted: 2023-05-04
Published online: 2023-06-22

DOI: 10.5802/crmeca.198

Keywords: Variational solution, Embedded conductor, Mixed boundary condition, Heat equation, Circular Disk
Mots-clés : Solution variationnelle, conducteur incorporé, condition limite mixte, équation de la chaleur, disque circulaire

Author's affiliations:

William Pons ¹; Stanley Pons ^{2, 3}

¹ Department of Chemistry, Clemson University, Clemson SC 29634, USA
² Department of Chemistry, University of Utah, Salt Lake City, UT 84112 (Prior affiliation where work was conceived)
³ Pons(EI), 14, r. Eugene Giraud, 06560 Valbonne, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {William Pons and Stanley Pons},
     title = {A {Variational} {Calculation} of {Diffusive} {Flux} in a {Mixed} {Boundary} {Value} {Problem}},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {247--263},
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     year = {2023},
     doi = {10.5802/crmeca.198},
     language = {en},
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William Pons; Stanley Pons. A Variational Calculation of Diffusive Flux in a Mixed Boundary Value Problem. Comptes Rendus. Mécanique, Volume 351 (2023), pp. 247-263. doi : 10.5802/crmeca.198. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.198/

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