Microwave joining of two long hollow tubes: an asymptotic theory and numerical simulations

A nonlinear heat equation which models the microwave assisted joining of tw o large SIC tubes is analyzed. By exploiting the small fineness ratio of th e structure and disparate time scales an asymptotic theory for this problem is systematically deduced. Specifically, a one-dimensional nonlinear heat equation is described which governs the temperature in the outer region. Th is is a numerically well posed problem and it is efficiently solved using s tandard methods. This solution is not valid in the inner region which inclu des the microwave source. An inner asymptotic approximation is derived to d escribe the temperature in this region. This approximation yields two unkno wn functions which are determined from matching to the outer solution. The results of the asymptotic theory are compared to calculations done on the f ull problem. Since the full problem is numerically ill conditioned, the asy mptotic theory yields enormous savings in computational time and effort.