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Modeling of heat conduction via fractional derivatives

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Abstract

The modeling of heat conduction is considered by letting the time derivative, in the Cattaneo–Maxwell equation, be replaced by a derivative of fractional order. The purpose of this new approach is to overcome some drawbacks of the Cattaneo–Maxwell equation, for instance possible fluctuations which violate the non-negativity of the absolute temperature. Consistency with thermodynamics is shown to hold for a suitable free energy potential, that is in fact a functional of the summed history of the heat flux, subject to a suitable restriction on the set of admissible histories. Compatibility with wave propagation at a finite speed is investigated in connection with temperature-rate waves. It follows that though, as expected, this is the case for the Cattaneo–Maxwell equation, the model involving the fractional derivative does not allow the propagation at a finite speed. Nevertheless, this new model provides a good description of wave-like profiles in thermal propagation phenomena, whereas Fourier’s law does not.

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Notes

  1. Named also Cattaneo–Vernotte equation or Cattaneo–Maxwell–Vernotte equation.

Abbreviations

\({\mathbf{x}},x\) :

Space variables

t, s :

Time variables

\(\tau ,\tau _\alpha \) :

Relaxation times

\(\theta \) :

Absolute temperature

\({\mathbf{g}}\) :

Absolute temperature gradient

\({\mathbf{q}}\) :

Heat flux vector

\(k,\kappa _1\) :

Heat conductivity

\(\kappa _0\) :

Heat conductivity per unit of time

c :

Heat capacity

r :

External heat supply

\(\varepsilon ,\hat{\varepsilon }\) :

Internal energy densities

\(\eta \) :

Entropy density

\(\psi ,\hat{\psi }\) :

Helmholtz free energy densities

\(\Theta \) :

Thermal displacement

\(\xi \) :

Internal dissipation rate

\(\overline{\theta }^t(\cdot )\) :

Summed history of the absolute temperature

\( {{\mathbf{g}}}^t(\cdot )\) :

Past history of the temperature gradient

\( \overline{{\mathbf{g}}}^t(\cdot )\) :

Summed history of the temperature gradient

\( {{\varvec{\zeta }}}^t(\cdot )\) :

Relative history of the heat flux

\( \overline{{\mathbf{q}}}(t,\cdot )\) :

Summed history of the heat flux

\({\mathfrak {K}}, \beta ,\gamma , {\mathfrak {h}}, g, j\) :

Memory kernels

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Acknowledgements

The research leading to this work has been developed under the auspices of INDAM-GNFM, Italy.

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Authors and Affiliations

  1. Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40127, Bologna, Italy

    Mauro Fabrizio

  2. DICATAM, Via Valotti 9, 25133, Brescia, Italy

    Claudio Giorgi

  3. DIBRIS, Via Opera Pia 13, 16145, Genoa, Italy

    Angelo Morro

Authors
  1. Mauro Fabrizio
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  2. Claudio Giorgi
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  3. Angelo Morro
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Correspondence to Claudio Giorgi.

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Fabrizio, M., Giorgi, C. & Morro, A. Modeling of heat conduction via fractional derivatives. Heat Mass Transfer 53, 2785–2797 (2017). https://doi.org/10.1007/s00231-017-1985-8

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