Stress-Assisted Diffusion: A Free Boundary Problem
Robert W. Cox · SIAM Journal on Applied Mathematics · 1991
A Stefan problem is formulated for a diffusion equation where the Fickian flux is supplemented by a stress gradient term. The stress in turn obeys a concentration-dependent evolution equation that includes impulsive and relaxation effects. The speed of the free boundary is taken to be proportional to the flux at the front. For short times, the front position is found to be proportional to $t^{1/2} $. Using singular perturbation and numerical methods, three types of “long-time” behavior are found, depending on relationships between the coefficient functions and the parameters in the problem: (i) the front position is proportional to $t^{1/2} $; (ii) the front position is proportional to t; or (iii) the front stops in a finite time and the solution becomes singular (infinite gradients) at that point.