(412g) Dispersion in Steady Two-Dimensional Flows through a Parallel-Plate Channel

We conducted a multi-scale perturbation analysis to study the advection-diffusion transport of a passive solute in a two-dimensional flow through a narrow parallel-plate channel, where the fluid velocity is steady and varies in the cross-stream and streamwise directions. A macro-transport equation governing the evolution of the cross-sectionally averaged solute concentration at long times was derived, in which we identified a correction to the advection speed and a Taylor dispersion coefficient that inherit a dependence on the streamwise coordinate from the fluid velocity. While both velocity components contribute to the advection speed correction, only the streamwise velocity is responsible for Taylor dispersion.

We applied the macro-transport equation to examine dispersion in a two-dimensional pressure-driven flow in a heterogeneous channel comprising alternating shear-free and no-slip regions, relevant to electroosmotic flows and flows in superhydrophobic environments. The results therefrom were compared with those obtained from a unidirectional Poiseuille flow in a homogeneously no-slip channel, in which case we recovered the classical constant dispersivity with no zero advection speed correction. In contrast, in the heterogeneous channel 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. The mean velocity faithfully represents the advection speed in each region far away from the shear-free-no-slip transition, at which a localized negative correction is found to account for the decrease in the velocity. We further substituted the transport coefficients into the macro-transport equation to compute the evolution of the concentration profile. The evolution predicted by the macro-transport equation agrees well with that obtained by numerically solving the two-dimensional advection-diffusion equation. We found that while the shear-free section suppresses band broadening, the following no-slip section may lead to a wider band compared to an otherwise homogeneously no-slip channel. This suggests caution being exercised in controlling band broadening in channels with surface heterogeneities, such as encountered in capillary electrophoresis.