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.

arXiv comment21 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:1710.01360

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Cited by in corpus (5)

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• Steady-state ballistic thermal transport associated with transversal motions in a damped graphene lattice subjected to a point heat source
• Thermal equilibration in a one-dimensional damped harmonic crystal
• Non-stationary elastic wave scattering and energy transport in a one-dimensional harmonic chain with an isotopic defect