A multi-scale model for solute transport in a wavy-walled channel

This paper concerns steady flow and solute uptake in a wavy-walled channel, where the wavelength and amplitude of the wall are comparable to each other but are much shorter than the width of the channel. The problem has two primary asymptotic regions: a core region where the walls appear flat at leading order and a wall region where there is full interaction between advection, diffusion and uptake at the wavy wall. For weak wall uptake, the effective uptake from the core is shown to increase with wall waviness in proportion to surface area, whereas for stronger wall uptake, it is found that the uptake from the core can be reduced as the wall amplitude increases. Conditions are identified under which this approximation is uniformly valid in a full channel flow, accounting for inlet conditions, and a comprehensive survey of the asymptotic distributions of solute both along and across the channel is provided. It is also shown how this multiscale approach can readily be extended to account for channel walls with multiple lengthscales of spatial variation.