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

•An axisymmetric conduction problem with a discontinuous temperature boundary is solved.•The local solution near the discontinuity is represented using a two-dimensional dipole.•Three domain sizes, i.e., infinite, semi-infinite and finite cylinders, are considered.•A simple expression for the local normal wall heat flux near the discontinuity is obtained.•The accuracy is verified in comparison to the heat flux predicted via high order extrapolation.