Time dependence and local structure of tracer dispersion in oscillating liquid Hele-Shaw flows

Abstract

Passive tracer dispersion in oscillating Poiseuille liquid flows of zero net velocity is studied experimentally in a Hele-Shaw cell and numerically by 2D simulations: this study is particularly focused on the time dependence and local properties of the dispersion. The dispersion mechanism is found to be controlled by the ratio τm/T of the molecular diffusion time across the gap and the oscillation period (when molecular diffusion parallel to the flow is negligible). The 2D numerical simulations complement the experiments by providing the local concentration c(x, z, t) at a given distance z from the cell walls (instead of only the average over z). Above a time lapse scaling like τm, the variation of c with the distance x along the flow becomes a Gaussian of width constant with z while the mean distance x̄ may depend both on z and t. For τm/T ≲ 2, the front spreads through Taylor-like dispersion and the normalized dispersivity scales as τm/T. The front oscillates parallel to the flow with an amplitude constant across the gap; its width increases monotonically at a rate modulated at twice the flow frequency, due to variations of the instantaneous dispersivity. For τm/T ≳ 20, the molecular diffusion distance during a period of the flow is smaller than the gap and the normalized dispersivity scales as (τm/T)−1. The oscillations of the different points of the front follow the local fluid velocity: this produces a reversible modulation of the global front width at twice the flow frequency and in quadrature with that in the Taylor-like regime.