The Temperature Pattern of a Homogeneous Square Area with Adjacent Sides Moving without Acceleration under Boundary Conditions of the First Kind

Abstract

A square area with homogeneous thermal and physical characteristics, deformed preserving 2-D similarity, is investigated. At the initial moment of time, two adjacent sides start moving respectively towards the abscissa and ordinate axes with constant speed while remaining equidistant to the other two adjacent sides (the fixed and moving sides are kept at different constant temperatures). A nonlinear initial boundary value problem with boundary conditions of the first kind and special coordinates immobilizes the moving boundary of the area into a fixed one with the corresponding transformation of the initial boundary value problem for the fixed boundaries with respect to the multiplicative variable of two unknown functions, which are defined by additional initial boundary values. These were solved by the successive application of integral sine transformations on pseudo-space variables. This enables the solution of the original problem to be notated analytically using special quadratures. The computational experiment proved the correctness of the solution and the absolute fulfillment of the initial conditions. The results also illustrate the adequacy of the qualitative calculations for the heating process of a quadratic area with moving adjacent boundaries. This approach can be applied to the differently directed motion of adjacent boundaries, to uniformly retarded or uniformly accelerated motion. Considering that Fourier's and Fick's laws are mathematically similar, the solution and its generalization are of practical importance in describing mass transfer processes, such as crystallization or dissolution.