Diffusion cloud around and downstream of active sphere immersed in granular bed through which fluid flows

Abstract

The concentration distribution in the wake of (and around) a soluble or fast reacting sphere, immersed in a granular bed of inert particles through which liquid or gas flows, with "uniform velocity", has been obtained numerically, for solute transport by both advection and diffusion. Fluid flow in the granular bed around the sphere was assumed to follow Darcy's law and the elliptic PDE equation, resulting from a differential material balance on the solute in an elementary control volume, was solved numerically over the "whole range" of values of the Peclet number. The solution gives the steady state concentration contour plots around the sphere and, for each concentration level, the downstream extent and approximate maximum width of the corresponding contour surface were determined. General expressions are presented (as Eqs. (20) and (21)) to relate those variables with the Peclet number and this provides simple formulae to compute the approximate size of the diffusion wake downstream of the active sphere.