Thermal State of a Region with a Thermally Insulated Moving Boundary

Abstract

Mathematical model representations of the temperature effect in regions with a thermally insulated moving boundary are developed. The boundary conditions for thermal insulation of a moving boundary are formulated both for locally equilibrium heat transfer processes within the classical Fourier phenomenology and for more complex locally nonequilibrium processes within the Maxwell–Cattaneo–Lykov–Vernott phenomenology, taking into account the finite speed of heat propagation. The applied problem of heat conductance and the theory of thermal shock for a region with a moving thermally insulated boundary, free from external and internal influences, is considered. An exact analytical solution of the formulated mathematical models for equations of the hyperbolic type is obtained. Methods and theorems of operational calculus and Riemann–Mellin contour integrals are used to calculate the originals of complex images with two branch points. A mathematical apparatus for the equivalence of functional structures for the originals of the obtained operational solutions is proposed. It is shown that the presence of a thermally insulated moving boundary leads to the appearance of a temperature gradient in the region and, consequently, to the appearance in the region of a temperature field and corresponding thermoelastic stresses of a wave nature. A numerical experiment is presented and the possibility of transition from one form of analytical solution of the temperature problem to another equivalent form is shown. The described effect manifests itself both for equations of the parabolic type based on classical Fourier phenomenology and for equations of hyperbolic type based on the generalized phenomenology of Maxwell–Cattaneo–Lykov–Vernott.