Enhanced diffusivity and skewness of a diffusing tracer in the presence of an oscillating wall

TL;DR

This paper develops a theoretical framework for understanding how oscillating walls influence the enhanced diffusivity and skewness of a diffusing tracer in fluid flows, validated by experiments with particle tracking velocimetry.

Contribution

The paper introduces a new formalism for calculating effective diffusivity and skewness in oscillating wall flows, including a novel series formula and experimental validation.

Findings

Effective diffusivity diverges at high frequencies for finite viscosities.

Skewness can be controlled via oscillation phase in nonlinear flows.

Theory matches experimental measurements of tracer distribution.

Abstract

We develop a theory of enhanced diffusivity and skewness of the longitudinal distribution of a diffusing tracer advected by a periodic time-varying shear flow in a straight channel. Although applicable to general fluid flow, we restrict the examples of our theory to the tracer advected by flows that are induced by a periodically oscillating wall in a Newtonian fluid between two infinite parallel plates as well as flow in an infinitely long duct. We first derived the formula of the flow produced by the wall motions. Second, we calculate the second Aris moment for all time and its long-time limiting effective diffusivity as a function of the geometrical parameters, frequency, viscosity, and diffusivity. Using a new formalism based upon the Helmholtz operator we establish a new single series formula for the variance. We show that the viscous-dominated limit results in a linear shear layer for which the effective diffusivity is bounded. For finite viscosities, the enhanced diffusion diverges in the high-frequency limit. We present a study of the effective diffusivity surface as a function of the non-dimensional parameters which shows how a maximum can exist for various parameter sweeps. Physical experiments are performed in water using particle tracking velocimetry to quantitatively measure the fluid flow. Using fluorescein dye as the passive tracer, we document that the theory is quantitatively accurate. Specifically, image analysis suggests that the distribution variance be measured using the full width at half maximum is robust to noise. Further, we show that the scalar skewness is zero for linear shear flows at all times, whereas for the nonlinear Stokes layer, the skewness sign can be controlled through the oscillating phase. Last, for single-frequency wall motion, the long-time skewness decays at the faster rate as compared with the case with steady flow.