A non-scale-invariant form for coarse-grained diffusion-reaction equations

Abstract

The process of mixing and reaction is a challenging problem to understand mathematically. Although there have been successes in describing the effective properties of mixing and reaction under a number of regimes, process descriptions for early times have been challenging for cases where the structure of the initial conditions is highly segregated. In this paper, we use the method of volume averaging to develop a rigorous theory for diffusive mixing with reactions from initial to asymptotic times under highly segregated initial conditions in a bounded domain. One key feature that arises in this development is that the functional form of the averaged differential mass balance equations is not, in general, scale invariant. Upon upscaling, an additional source term arises that helps to account for the initial configuration of the reacting chemical species. In this development, we derive the macroscopic parameters (a macroscale source term and an effectiveness factor modifying the reaction rate) defined in the macroscale diffusion-reaction equation and provide example applications for several initial configurations.

Preasymptotic Taylor dispersion: evolution from the initial condition

“…They also found that the dispersion coefficient was a function of the initial configuration. In a sequence of two papers, Wood (2009, equations (17) and (29)) and Wood & Valdés-Parada (2013) developed theories that explicitly included the influence of the initial condition as a source term in the averaged mass balance equation; this process was then used by Ostvar & Wood (2016) to describe the average diffusion and reaction from an initial (unmixed) configuration. Independently, Balakotaiah & Ratnakar (2010) also developed balance expressions specifically for the Taylor dispersion problem in which source terms arising from the initial conditions were employed; a Kramers-Moyal-like expansion was used, but truncated at second order.…”