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

A nonlinear heat equation which models the microwave assisted joining of two large SiC tubes is analyzed. By exploiting the small fineness ratio of the 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. This is a numerically well posed problem and it is efficiently solved using standard methods. This solution is not valid in the inner region which includes the microwave source. An inner asymptotic approximation is derived to describe the temperature in this region. This approximation yields two unknown functions which are determined from matching to the outer solution. The results of the asymptotic theory are compared to calculations done on the full problem. Since the full problem is numerically ill conditioned, the asymptotic theory yields enormous savings in computational time and effort.