Upscaling of Mixing-Limited Bimolecular Chemical Reactions in Poiseuille Flow

We consider the fast irreversible bimolecular chemical reaction A+B -> C in the Poiseuille flow through a channel, in which A displaces B. This system allows to systematically study the impact of the interaction of interface deformation and diffusion on mixing and reactive transport. At early times, the reaction is diffusion controlled. With increasing time, advection begins to dominate and we find enhanced reaction efficiency due to the deformation of the interface between the two reactants. For times larger than the characteristic diffusion time across the channel, mixing and reaction are quantified by the Taylor dispersion coefficient. Predictions based on Taylor dispersion may significantly overestimate the reaction efficiency at preasymptotic times, when the system is characterized by incomplete mixing. This type of behaviors of incomplete mixing and reaction have been observed in heterogeneous systems across different scales. Channel flow allows to study them in detail for a well-controlled system. We propose a dispersive lamella approach based on the concept of effective dispersion which accurately predicts the full evolution of the product mass. Specifically, this approach captures the impact of interface deformation and diffusive coalescence, which marks the transition to the Taylor regime. It gives insight into the mechanism of incomplete mixing and its consequences for reactive transport in more general porous media flows.