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An alternative approach to the heat equation

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Abstract

The technique of conformal transformations of the metric, widely used in general relativity and in cosmology, is applied to the analysis of heat conduction in an anisotropic medium, in which the thermal conductivity is described by a tensor instead of a scalar. The anisotropic medium is reduced to an effective equivalent one, which is isotropic. The simplification is achieved for a particular conformal factor of the transformation, uniquely determined under physically reasonable assumptions on the thermal conductivity, density, and specific heat of the medium. Another application consists in the formal elimination of source or sink terms from the heat equation by using a suitable conformal transformation.

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Notes

  1. The medium described by eqs. (3.18) and (3.19) is isotropic around the point r = 0, but globally anisotropic.

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Acknowledgements

The author was supported by the NATO Advanced Fellowship Programme through the National Research Council of Italy (CNR).

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Correspondence to Valerio Faraoni.

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Faraoni, V. An alternative approach to the heat equation. Heat Mass Transfer 41, 32–36 (2004). https://doi.org/10.1007/s00231-004-0517-5

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  • DOI: https://doi.org/10.1007/s00231-004-0517-5

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