Perturbation analysis of Thomas and Windle's model of case II transport

Thomas and Windle's model of Case II transport is analyzed for a semiinfinite medium by a singular perturbation technique. Two adjacent boundary layers separate equilibrated and dry regions. A thin boundary layer of width similar to O(M-(1/2)/ln M), where M (much greater than 1) dictates how rapidly the mixture's viscosity decays with liquid concentration, sits next to the equilibrated enter left region. Here, quasi-steady diffusion balances relaxation. A thicker intermediate layer of width similar to O(M(-1/2)) separates the lefthand boundary layer and the dry outer region on the right, where both relaxation and unsteady diffusion participate in the transport. Matching the solutions at leading order specifies the moving front's speed, v: v similar to M(1/2). The analysis indicates that relaxation significantly affects the nearly dry region just ahead of the moving from. This disagrees with the widely accepted view that ordinary diffusion dominates in the nearly dry righthand region. Approximating that ordinary diffusion dominates in this region leads to a step-exponential concentration profile at the front and a simple analytical solution for the front speed v with the correct M scaling. This approximate result accurately predicts the values of v determined by direct numerical solutions.