AN ASYMPTOTIC LOCAL SOLUTION FOR AXISYMMETRIC HEAT CONDUCTION PROBLEMS WITH TEMPERATURE DISCONTINUITY IN BOUNDARY
An asymptotic local solution to the axisymmetric heat conduction problem with a discontinuous temperature boundary condition in the radial direction is presented. The Laplace equation in cylindrical coordinate (r, z) is expanded near the discontinuity. The leading order term arises from the step-change in the boundary condition and the local solution can be represented using a two-dimensional dipole. The next term reflects the effect of curvature at the position of the temperature discontinuity and is a particular solution to the Poisson equation. Subsequent higher-order terms involve general solutions to the Laplace equation in Cartesian coordinates and depend on conditions on the far end boundaries. The geometric configuration of infinite solid cylinder (0 ≤
r < ∞, 0 ≤ z < ∞) with the temperature discontinuity located at r = 1, z = 0 is considered in this work. Exact solution for temperature is used to determine the coefficients in the higher order terms of the asymptotic solution. As a result, 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.