Anomalous mixing and reaction induced by superdiffusive nonlocal transport

Spatially nonlocal transport describes the evolution of solute concentration due to mass transfer over long ranges. Such long-range mass transfer, present in many flow situations, changes the character of mixing and consequent chemical reactions. We study mixing in terms of the scalar dissipation and reaction rates for mixing-limited equilibrium reactions, using the space-fractional advection-dispersion equation (fADE) to model long range mass transfer. The scalar dissipation and global reaction rates decay as power-laws at late time. As opposed to the Fickian (local) transport model, local reaction rates are not zero where the concentration has zero gradient. As alpha, the fractional derivative exponent, decreases from two in the fADE, the reaction rate grows larger at the position of zero gradient, due to long-range transfer of reactants from distances larger than Fick's law allows. The reaction rates are also greater far from the reactant source for non-Fickian transport; however, the globally integrated reaction rate decreases with smaller alpha. This behavior may provide a method to investigate spatial nonlocality as a proper model of upscaling: the reaction products would be found in places precluded by Fickian dispersion, and overall reaction rates are suppressed.