Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

Abstract

We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system of stochastic ordinary differential equations. We perform a continualization procedure and derive an infinite system of linear partial differential equations for covariance variables. The most important results of the paper are the deterministic differential–difference equation describing non-stationary heat propagation in the lattice and the analytical formula in the integral form for its steady-state solution describing kinetic temperature distribution caused by a point heat source of a constant intensity. The comparison between numerical solution of stochastic equations and obtained analytical solution demonstrates a very good agreement everywhere except for the main diagonals of the lattice (with respect to the point source position), where the analytical solution is singular.

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Acknowledgements

The authors are grateful to V.A. Kuzkin, A.S. Murachev, and E.V. Shishkina for useful and stimulating discussions.

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Correspondence to Serge N. Gavrilov.

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This work is supported by Russian Science Foundation (Grant No. 18-11-00201).

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Gavrilov, S.N., Krivtsov, A.M. Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source. Continuum Mech. Thermodyn. 32, 41–61 (2020). https://doi.org/10.1007/s00161-019-00782-2

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Keywords

  • Ballistic heat transfer
  • 2D harmonic scalar lattice
  • Kinetic temperature