The solution of the nonaxisymmetric stationary problem of heat conduction for the polar-orthotropic annular plate of variable thickness

V. V. Korolevich ; D. G. Medvedev

Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2018), pp. 77-87

Résumé

In the work is given the solution of nonaxisymmetric stationary heat conduction problem for profiled polar-orthotropic annular plates with thermally insulated bases. The dependence of the thermophysical characteristics of the plate material of the temperature is taken into account. Temperature values are set on the contours of the annular plate: temperature $T_{0}^{*}$ is constant on the internal contour, and on the outer contour on several arcs with length $l_{i}(i=\overline{1,k})$ – temperature is $T_{1}^{*}(T_{1}^{*}>T_{0}^{*})$. The temperature distribution in such a plate is nonaxisymmetric. It is assumed that the radial $\lambda_{r}$, and tangential $\lambda_{0}$ heat conduction coefficients are linearly dependent on the temperature $T(r,\Theta)$: $\lambda_{r}(T)=\lambda_{r}^{(0)}(1-\gamma T(r,\Theta)), \lambda_{\Theta}(T)=\lambda_{\Theta}^{(0)}(1-\gamma T(r,\Theta))$, here the parameter $\gamma>1$; the constants $\lambda_{r}^{(0)}, \lambda_{r}^{(\Theta)}$ are determined experimentally at the primary temperature $T_{0}$. The primary nonlinear differential heat equation is reduced to a linear differential equation of the $2^{nd}$ kind in partial derivatives when a new function $Z(r, \Theta)=[T(r, \Theta)-\frac{\gamma}{2}T^{2}(r, \Theta)]$ is introduced in consideration.

Détail

Keywords: composite material; temperature; polar-orthotropic annular plate; stationary heat conduction equation; differential equation; Volterra integral equation of the $2^{nd}$ kind; resolvent; quadratic equation; plate of power profile; conical plate; plate of exponential profile.