Asymptotic heat equation for crossing superconductive thin walls

Abstract

This work deals with the homogenization of the nonstationary heat conduction which takes place in a binary three-dimensional medium consisting of an ambiental phase having conductivity of unity order and a set of highly conductive thin walls crossing orthogonally and periodically. This situation covers in fact three types of microstructures, usually called: box-type (or honeycomb), gridwork and layered. The study is based on the energetic procedure of homogenization associated to a control-zone method, specific to the geometry of the microstructure and to the singularity of the conductivity coefficients. In the present case the main result is the system that governs the asymptotic behaviour of the temperature distribution in this binary medium. It displays a significant increase of the conductivity due to the superconductive thin walls, revealing their seemingly paradoxal behaviour of having an everlasting action on the environment, in spite of an obvious vanishing volume. Moreover, the dependence of this behaviour with respect to the relative thicknesses of the walls can be detailed..

Keywords:

• homogenization
• conduction
• fine-scale
• box-type structure

AMS Subject Classifications::

• 35B27
• 35K57
• 76R50

Acknowledgements

This work was done during the visit of Dan Polişevschi at the I.R.M.A.R.'s Department of Mechanics (University of Rennes 1) whose support is gratefully acknowledged.