An analytical solution to the thermal problems with varying boundary conditions around a moving source

•We find an analytical solution to a problem with varying BC and moving source.•The BC varying in both time and space around a moving source in a finite domain.•We use the established solution to simulate underwater friction stir welding.•The analytical results are compared with the numerical results obtained by Comsol.•The analytical results and the numerical results are in good agreement.

This article devises an analytical solution to the practical industrial applications, in which the thermal boundary conditions are varying in both time and space around a cooling or heating source moving along a finite domain. Among the various applications of such a solution, the jet impingement cooling and underwater welding and cutting would be of great importance. Afterwards, a numerical example regarding the temperature field caused by an underwater solid state welding process is provided by using the developed analytical solution. Finally, in order to evaluate the accuracy of the presented analytical model, the analytically obtained results are compared with the results attained by FEM-Comsol commercial software.