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

  • Serge N. GavrilovEmail author
  • Anton M. Krivtsov
Original Article


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.


Ballistic heat transfer 2D harmonic scalar lattice Kinetic temperature 


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


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Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering RASSt. PetersburgRussia
  2. 2.Peter the Great St. Petersburg Polytechnic University (SPbPU)St. PetersburgRussia

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