Transport of a passive scalar in wide channels with surface topography

TL;DR

This paper extends classical dispersion theory to account for surface topography in wide channels, deriving an anisotropic dispersion tensor that reveals how surface shape influences scalar transport under shear flow.

Contribution

It introduces a generalized asymptotic dispersion model for structured channel surfaces with small roughness, including Fourier-expansible shapes, and quantifies the anisotropic effects on dispersion.

Findings

Dispersion tensor depends on surface wavelength and amplitude.

Tilted surface corrugations enhance dispersion along a principal direction.

Fourier modes contribute linearly independent corrections to classical dispersion.

Abstract

We generalize classical dispersion theory for a passive scalar to derive an asymptotic long-time convection-diffusion equation for a solute suspended in a wide, structured channel and subject to a steady low-Reynolds-number shear flow. Our theory, valid for small roughness amplitudes of the channel, holds for general surface shapes expandable as a Fourier series. We determine an anisotropic dispersion tensor, which depends on the characteristic wavelengths and amplitude of the surface structure. For surfaces whose corrugations are tilted with respect to the applied flow direction, we find that dispersion along the principal direction (i.e., the principal eigenvector of the dispersion tensor) is at an angle to the main flow direction and becomes enhanced relative to classical Taylor dispersion. In contrast, dispersion perpendicular to it can decrease compared to the short-time diffusivity of the particles. Furthermore, for an arbitrary surface shape represented in terms of a Fourier decomposition, we find that each Fourier mode contributes at leading order a linearly-independent correction to the classical Taylor dispersion tensor.