Reactive strip method for mixing and reaction in two dimensions

Abstract

A numerical method to efficiently solve for mixing and reaction of scalars in a twodimensional flow field at large Péclet numbers but otherwise arbitrary Damköhler numbers is reported. Flow disorder often leads to the formation of lamellar structures for the reactants, thus altering the observed reaction rates. We consider a strip of one reactant in a pool of another reactant, both of which are advected with a known velocity field. We first establish that the conditions under which the system may be described by a locally one-dimensional reaction-diffusion problem is when the strip thickness is smaller than the local radius of curvature and also when the strip thickness is smaller than the distance between adjacent strips. In such a scenario, typical of many mixing systems, the system of advection-diffusion-reaction equation in two-dimensions is thus reduced to a local onedimensional reaction-diffusion equation in the Lagrangian frame attached to the advected strip in strip in such a manner that the effect of advection is systematically decoupled from the diffusion and reaction processes. We first demonstrate the method for the transport of a conservative scalar (the limiting case of zero Damköhler number) under a linear shear flow, point vortex and a chaotic sine flow. We then proceed to consider the situation of a simple bimolecular reaction between two reactants yielding a single product. In essence, the reduction of dimensionality of the problem, which renders the 2D problem 1D, allows one to efficiently model reactive transport under high Péclet numbers which are otherwise prohibitively difficult to resolve from classical finite difference or finite element based methods.