Dispersion in steady and time-oscillatory two-dimensional flows through a parallel-plate channel

Abstract

A multiple-scale perturbation theory is developed to analyze the advection-diffusion transport of a passive solute through a parallel-plate channel. The fluid velocity comprises a steady and a time-oscillatory component, which may vary spatially in the transverse and streamwise directions, and temporally on the fast transverse diffusion timescale. A long-time asymptotic equation governing the evolution of the transverse averaged solute concentration is derived, complemented with Taylor dispersion coefficients and advection speed corrections that are functions of the streamwise coordinate. We demonstrate the theory with a two-dimensional flow in a channel comprising alternating shear-free and no-slip regions. For a steady flow, the dispersion coefficient changes from zero to a finite value when the flow transitions from plug-like in the shear-free section to parabolic in the no-slip region. For an oscillatory flow, the dispersion coefficient due to an oscillatory flow can be negative and two orders of magnitude larger than that due to a steady flow of the same amplitude. This motivates us to quantify the relative magnitude of the steady and oscillatory flow such that there is an overall positive dispersion coefficient necessary for an averaged (macrotransport) equation. We further substitute the transport coefficients into the averaged equation to compute the evolution of the concentration profile, which agrees well with that obtained by solving the full two-dimensional advection-diffusion equation. In a steady flow, we find that while the shear-free section suppresses band broadening, the following no-slip section may lead to a wider band compared with the dispersion driven by the same pressure gradient in an otherwise homogeneously no-slip channel. In an unsteady flow, we demonstrate that a naive implementation of the macrotransport theory with a (localized) negative dispersion coefficient will result in an aphysical finite time singularity (or “blow-up solution”), in contrast to the well-behaved solution of the full advection-diffusion equation.