Incomplete mixing and reactions with fractional dispersion

A common barrier to accurately predicting the fate of reactive contaminants is accurately describing the role of incomplete mixing. In this paper we develop a stochastic analytical framework for an irreversible kinetic bimolecular reaction in a system with anomalous transport, governed by the fractional advection–dispersion equation (fADE). The classical well-mixed (thermodynamic) solution dictates that the concentration of reactants after an initial transient decreases proportional to t−1. As the system becomes less and less well-mixed, the rate of reaction decreases relative to the thermodynamic solution, at late times scaling with t−1/(2α) instead of t−1, where 1 < α ⩽ 2 is the fractional order of the dispersion term in the fADE. The time at which this transition takes place is derived, giving an indication of the range of validity of the classical (well-mixed) equation. We verify these analytic results using particle-based simulations of random walks and reactions.

► Incomplete mixing reduces the rate of reaction relative to well-mixed systems.
► As incomplete mixing dominates the late time scaling for reaction goes like t-1/(2α)t-1/(2α) where αα is the fractional order of the dispersion.
► The method of moments can predict the full time solution, including the transition from early to late time scalings.