An asymptotic local solution for axi-symmetric heat conduction problems with temperature discontinuity on the boundary

Abstract

An asymptotic local solution to the axisymmetric conduction problem with a discontinuous temperature boundary condition in the radial direction is presented. The Laplace's equation in cylindrical coordinates (r, z) is expanded near the discontinuity at r = 1, z = 0. The leading order term arises from the step-change in the boundary condition. The next term reflects the effect of curvature at the position of the temperature discontinuity and is a particular solution to the Poisson's equation. Subsequent higher-order terms involve general solutions to the Laplace's equation in Cartesian coordinates and depend on conditions on the far end boundaries. Three domain sizes are thus considered: (i) infinite solid cylinder (0 ≤ r < ∞, 0 ≤ z < ∞); (ii) semi-infinite solid cylinder with finite radius (0 ≤ r < a, 0 ≤ z < ∞); and (iii) finite solid cylinder (0 ≤ r ≤ a, 0 ≤ z ≤ a). Exact solutions for temperature in those three cases are used to determine the coefficients in the higher order terms of the asymptotic solution. As a result, a simple expression for local normal wall heat flux near the discontinuity is obtained and the accuracy is confirmed by comparing with the heat flux obtained through high order extrapolation of the exact temperature field near the wall.