Thermal conductivity in a homogeneous strip with a linear change in thickness under boundary conditions of the first kind

An accurate analytical solution has been obtained in quadratures of the initial boundary value problem for one-dimensional unsteady-state heat-transfer equation with boundary conditions of the first kind for an endless strip, while one of its boundaries is moving at a constant preset speed decreasing the strip thickness. Preliminarily, through the self-similar change of the spatial variable, the initial system of equations has been reduced to a fixed boundary system, to which the method of partitioning of dependent variables has been applied. The requirement that the coefficients before the first-order derivative must be equal to zero for the self-similar derivative and separately included function in a modified equation in partial derivative of parabolic type has allowed to determine the general structure of the solution containing an unknown function. This function is presented as a superposition of two potentials, which are proportionally connected using the self-similar derivative, what has made it possible to simplify the modified equation and to apply the classical Fourier sine integral transformation for its solution. The computation results has shown the dynamics of the local temperature profile along the changing strip thickness at a constant speed, while the kinetics of the average integral temperature shows (unlike with the case of absence of boundary movement) the presence of the maximum that shifts with the growth of the ratio of the boundary movement speed to the heat transfer speed by the conductivity to the fixed boundary. This is explained by the intensive heating up of the strip material in the conditions of the decreasing of its thickness; meanwhile, with the increase in the boundary movement speed (or with the use of material with reduced thermal conductivity), it approaches the fixed boundary. By assuming that the strip thickness is a parameter, the problem in the initial wording is solved using the method of the one-sided Laplace integral time transformation. This solution, when using the linear dependence of parameter on time, correlates with the obtained accurate solution, and therefore it can be used for the preliminary evaluation of the required characteristics of a process under consideration.