Basic model of heat transfer process in a radiation-transparent solid body with absorbing spherical layer

Theoretical studies on the problem of laser initiation of explosive decomposition in heterogeneous energetic materials place significant importance on the mathematical model of the heat transfer process in an isotropic solid body containing an absorbing inclusion in the form of a spherical layer. Difficulties are known to arise when seeking an analytical solution to the corresponding problem of non-stationary heat conduction, both through the application of Laplace’s integral transformation with respect to the time variable and singular integral transformation with respect to the spatial variable. This explains the absence of an analytical solution for the considered problem using current methods.

The task of determining the temperature field in an isotropic solid body with an absorbing penetrating radiation inclusion in the form of a spherical layer is formulated. The implemented mathematical model of the heat transfer process represents a mixed problem for a system of three second-order partial differential equations of the parabolic type in the presence of a non-stationary heat source in the system. An analytical method for solving the problem is proposed, which consists of two main stages.

The first stage involves finding the solution to the problem in the space of the Laplace integral transformation image, followed by its asymptotic expansion. The procedure used allows for the estimation of the extent and radius of the boundary of the thermal disturbance zone for large values of the Fourier number. The second stage, based on the application of the developed finite integral transformation with respect to the spatial variable for a three-layer region, completes the procedure of constructing an analytically closed solution to the original problem of non-stationary heat conduction.