Asymptotic transport theory for solutes in channels with patterned surface topography.
Our goal was to determine how surface geometry changes long-time transport in low-Reynolds-number channel flows. Classical Taylor dispersion explains how shear flow enhances spreading in a smooth channel. In (Roggeveen et al., 2023), we extended this theory to wide channels with small-amplitude surface topography.
Problem setup: passive tracers move through a wide channel with a structured lower surface and an imposed shear flow. At long times, cross-channel diffusion averages the dynamics into an effective planar transport problem.
The result is an effective long-time convection-diffusion equation with an anisotropic dispersion tensor. By decomposing the surface into Fourier modes, we could compute how each mode modifies the classical dispersion tensor, allowing general patterned surfaces to be treated systematically.
One useful consequence is that tilted surface corrugations can rotate the principal direction of dispersion away from the mean flow direction. Transport can be enhanced along one direction while suppressed along another, giving a controlled way to reason about mixing, spreading, and anisotropic transport in structured channels.
Single-mode topography
Because the asymptotic correction is mode-by-mode, a single sinusoidal surface gives the cleanest visual example of the mechanism. For a single sinusoidal mode, the theory predicts both drift and an anisotropic correction to dispersion. The paired animations below show how the surface pattern reorganizes spreading and drift in a shear flow.
Baseline Taylor dispersion near a flat surface.Surface-induced drift for a related single-mode topography. The surface mode is arranged at 45 degrees with respect to the direction of travel.
Gallery of Fluid Motion
This video was prepared for an APS Division of Fluid Dynamics Gallery of Fluid Motion submission with collaborators. It is a visual companion to the asymptotic theory, showing how surface structure can reorganize solute spreading in a shear flow.
References
2023
Transport of a passive scalar in wide channels with surface topography: An asymptotic theory
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 asymptotic theory relies on a domain perturbation approach for small roughness amplitudes of the channel and 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 diffusion tensor.
