Mathematical Modeling of Heat Transfer in Anisotropic Half-Space Based on the Generalized Parabolic Wave Heat Transfer Equation
В. Ф. Формалев, B. A. Garibyan, А. А. Орехов
Abstract
Based on the well-known Vernott–Cattaneau–Lykov law of heat transfer in thermodynamic nonequilibrium space, a new law of wave heat transfer is proposed in a form coinciding with the classical Fourier law, but with a lagged argument in time equal to the relaxation time characterizing the lag time of the heat flux from the temperature gradient. To justify the new law, the terms in the form of heat flux and its time derivative with a factor equal to the relaxation time are treated as the first two terms of the McLaren series expansion in powers of the relaxation time of the heat flux function, depending on the spatial point and time with a lagged argument equal to the relaxation time. Based on the new law, a wave equation of parabolic type is obtained, which is similar in form to the classical heat conduction equation, but with a lagging argument in time. Using this equation the problem of heat transfer in a transversally anisotropic half-space in relation to the plane bounding the half-space under the action of a point source of thermal energy of exponential type is formulated. By applying the double Fourier transform of two spatial variables and Laplace transform with respect to time, the problem is reduced to the second initial-edge problem of heat transfer in a semi-infinite rod along the third spatial variable. Results obtained with new analytical solution of the whole problem showed the wave nature of heat transfer with discontinuities of the first kind of isotherms along the spatial variables. The isotherms in the half-space are second-order surfaces — ellipsoids, paraboloids, and single-band hyperboloids.