Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body

Li, Jingyu; Rosencrans, Steve; Wang, Xuefeng; Zhang, Kaijun

doi:10.1090/S0002-9939-08-09766-9

Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body
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by Jingyu Li, Steve Rosencrans, Xuefeng Wang and Kaijun Zhang PDF

Proc. Amer. Math. Soc. 137 (2009), 1711-1721 Request permission

Abstract:

Of concern is the protection from overheating of an isotropically conducting body by an anisotropically conducting coating which is thin compared to the scale of the body. We assume either that the whole thermal tensor of the coating is small or that it is small in the directions normal to the body (a case we call “optimally aligned coating”). We study the asymptotic behavior of the solution to the heat equation with Dirichlet boundary conditions on the outer surface of the coating, as the thickness of the coating shrinks. We obtain the exact scaling relations between the thermal tensor and the thickness of the coating so that the effective (limiting) condition on the boundary of the body is of Dirichlet, Robin or Neumann type, with the last condition indicating good insulation.

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