Shocks in a Model for Stress-Driven Diffusion

Diffusion of a solvent into a polymer slab is sometimes characterized by a sharp front that moves into the medium and lasts for a long time. The behavior of such a front cannot be explained by a standard diffusion equation. Model equations are proposed that include solvent flux due to stress gradients in the polymer in addition to the Fickian flux. The stress in turn obeys a concentration-dependent evolution equation. In the limit of small diffusivity, the solution to a penetration problem is shown to have a steep front that progresses into the medium. The equations that govern the front are derived. A method for approximating the front is presented that uses a piece of the long time similarity solution to represent the concentration profile behind the front. The addition of a convective term to the equations is shown to raise the possibility of a traveling wave solution.