We investigate the transport of a solute past isolated sinks in a bounded domain when advection is dominant over diffusion, evaluating the effectiveness of homogenization approximations when sinks ared istributed uniformly randomly in space. Corrections to such approximations can be non-local, nonsmooth and non-Gaussian, depending on the physical parameters (a Péclet number Pe, assumed large, and a Damköhler number Da) and the compactness of the sinks. In one spatial dimension, solute distributions develop a staircase structure for large Pe, with corrections being better described with credible intervals than with traditional moments. In two and three dimensions, solute distributions are near-singular at each sink (and regularized by sink size), but their moments can be smooth as a result of ensemble averaging over variable sink locations. We approximate corrections to a homogenization approximation using a moment-expansion method, replacing the Green’s function by its free-space form, and test predictions against simulation.We show how, in two or three dimensions, the leading-order impact of disorder can be captured in a homogenization approximation for the ensemble mean concentration through a modification to Da that grows with diminishing sink size.
|Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
|Accepted/In press - 3 May 2022