A nonlocal Fourier's law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices

Noël Challamel; Cécile Grazide; Vincent Picandet; Arnaud Perrot; Yingyan Zhang

doi:10.1016/j.crme.2016.01.001

A nonlocal Fourier's law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices

Noël Challamel ¹ ; Cécile Grazide ¹ ; Vincent Picandet ¹ ; Arnaud Perrot ¹ ; Yingyan Zhang ²

¹ Université de Bretagne Sud, UBS – Institut Dupuy de Lôme, Centre de Recherche, Rue de Saint Maudé, BP92116, 56321 Lorient Cedex, France
² School of Computing, Engineering & Mathematics, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia

Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 388-401.

Résumé

This study focuses on heat conduction in unidimensional lattices also known as microstructured rods. The lattice thermal properties can be representative of concentrated thermal interface phases in one-dimensional segmented rods. The exact solution of the linear time-dependent spatial difference equation associated with the lattice problem is presented for some given initial and boundary conditions. This exact solution is compared to the quasicontinuum approximation built by continualization of the lattice equations. A rational-based asymptotic expansion of the pseudo-differential problem leads to an equivalent nonlocal-type Fourier's law. The differential nonlocal Fourier's law is analysed with respect to thermodynamic models available in the literature, such as the Guyer–Krumhansl-type equation. The length scale of the nonlocal heat law is calibrated with respect to the lattice spacing. An error analysis is conducted for quantifying the efficiency of the nonlocal model to capture the lattice evolution problem, as compared to the local model. The propagation of error with the nonlocal model is much slower than that in its local counterpart. A two-dimensional thermal lattice is also considered and approximated by a two-dimensional nonlocal heat problem. It is shown that nonlocal and continualized heat equations both approximate efficiently the two-dimensional thermal lattice response. These extended continuous heat models are shown to be good candidates for approximating the heat transfer behaviour of microstructured rods or membranes.

Reçu le : 2015-10-23
Accepté le : 2016-01-02
Publié le : 2016-03-17

DOI : 10.1016/j.crme.2016.01.001

Mots clés : Heat equation, Lattice, Fourier's law, Nonlocal thermodynamics, Gradient Fourier's law, Nonlocality, Diffusion equation

Affiliations des auteurs :

Noël Challamel ¹ ; Cécile Grazide ¹ ; Vincent Picandet ¹ ; Arnaud Perrot ¹ ; Yingyan Zhang ²

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     author = {No\"el Challamel and C\'ecile Grazide and Vincent Picandet and Arnaud Perrot and Yingyan Zhang},
     title = {A nonlocal {Fourier's} law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {388--401},
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     number = {6},
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     doi = {10.1016/j.crme.2016.01.001},
     language = {en},
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Noël Challamel; Cécile Grazide; Vincent Picandet; Arnaud Perrot; Yingyan Zhang. A nonlocal Fourier's law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices. Comptes Rendus. Mécanique, Volume 344 (2016) no. 6, pp. 388-401. doi : 10.1016/j.crme.2016.01.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.01.001/

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