Advection improves homogenized models of continuum diffusion in one-dimensional heterogeneous media

Abstract

We propose a novel homogenization method for one-dimensional continuum diffusion models with spatially variable (heterogeneous) diffusivity. Our method, which extends recent work on stochastic diffusion, assumes the constant-coefficient homogenized equation takes the form of an advectiondiffusion equation with effective (diffusivity and velocity) coefficients. To calculate the effective coefficients, our approach involves solving two uncoupled boundary value problems over the heterogeneous medium and leads to coefficients depending on the spatially-varying diffusivity (as usual) as well as the boundary conditions imposed on the heterogeneous model. Computational experiments comparing our advection-diffusion homogenized model to the standard homogenized model demonstrate that including an advection term in the homogenized equation leads to improved approximations of the solution of the original heterogeneous model.

Exponential and Weibull models for spherical and spherical-shell diffusion-controlled release systems with semi-absorbing boundaries

“…. Equations ( 20)-( 22) are derived by applying the linear differential operator Lϕ := (18) and combining the definition of M (x) (18) with the boundary conditions ( 3)-( 4) and ( 13)-( 14) 20 . In summary, since the boundary value problem ( 20)-( 22) admits a simple closed-form solution, closed-form analytical expressions can be obtained for M (x) (and hence λ) in terms of the full-order model parameters (D, L, a 0 , b 0 , c 0 , a 1 , b 1 and c 1 ).…”