Impurity transport in fractal media in the presence of a degrading diffusion barrier

Abstract

We have analyzed the transport regimes and the asymptotic forms of the impurity concentration in a randomly inhomogeneous fractal medium in the case when an impurity source is surrounded by a weakly permeable degrading barrier. The systematization of transport regimes depends on the relation between the time t₀ of emergence of impurity from the barrier and time t_* corresponding to the beginning of degradation. For t₀ < t_*, degradation processes are immaterial. In the opposite situation, when t₀ > t_*, the results on time intervals t < t_* can be formally reduced to the problem with a stationary barrier. The characteristics of regimes with t_* < t < t₀ depend on the scenario of barrier degradation. For an exponentially fast scenario, the interval t_* < t < t₀ is very narrow, and the transport regime occurring over time intervals t < t_* passes almost jumpwise to the regime of the problem without a barrier. In the slow power-law scenario, the transport over long time interval t_* < t < t₀ occurs in a new regime, which is faster as compared to the problem with a stationary barrier, but slower than in the problem without a barrier. The asymptotic form of the concentration at large distances from the source over time intervals t < t₀ has two steps, while for t > t₀, it has only one step. The more remote step for t < t₀ and the single step for t > t₀ coincide with the asymptotic form in the problem without a barrier.