Critical Analysis of Two-Dimensional Heat-Balance Equations for Composite Plates Obtained According to the Variational Principles of the Theory of Heat Conduction. II. Model Problem

We obtain analytic solutions of the model problem of stationary heat conduction for a composite plate. These solutions are constructed with the help of two methods aimed at reducing dimensionality, namely, the method of weighted residuals (the generalized Galerkin method) and the variational method. We consider a homogeneous rectangular elongated plate made of an anisotropic material in the general form. The front faces of the structure are thermally insulated; temperature is specified on one longitudinal end face of the plate, whereas a heat flux is given on the other end face. It is assumed that the input data of the problem and its solution are independent of the longitudinal coordinate. The temperature field is approximated by a second-order polynomial in the transverse coordinate. We also take into account boundary conditions imposed on the front faces. It is shown that, in the case of application of the method of weighted residuals, the resolving differential equation of the problem has the second order. At the same time, in the variational method, it has the fourth order. In the case of solution obtained by the method of weighted residuals, the integral heat flux in the tangential direction is constant and equal to the true value of this quantity. At the same time, in the solution obtained by using the variational method, the indicated integral heat flux oscillates in the tangential direction orthogonal to the longitudinal direction of the plate. The frequency and amplitude of these oscillations depend on the relative thickness of the structure. Moreover, the amplitude of oscillations can be higher than the true value of the integral heat flux by several orders of magnitude. The temperature field in the plate computed by using this method of reducing dimensionality of the problem of heat conduction has a similar oscillating character.