(416g) A Model for Solute Macro-Transport in Spatio-Temporally Inhomogeneous Flows: Application to Dispersion in Slip-Stick Channels and Eccentric Annulus

Controlling solute transport in a spatio-temporally inhomogeneous fluid flow is critical in natural and synthetic systems. Here, we conduct a multiple-scale analysis to derive a macrotransport, averaged equation for predicting the long-time solute transport in which the flow may be three-dimensional and oscillate periodically in time. We demonstrate the theory with two flow configurations: the first being a two-dimensional flow in a channel comprising alternating shear-free and no-slip regions, mimicking electroosmotic flows or flows over superhydrophobic surfaces. In contrast to a well-known positive dispersivity (enhanced diffusivity), we show that the dispersivity due to a time-oscillatory flow over inhomogeneous surfaces can be locally negative and two orders of magnitude larger than that due to a steady flow. Thus, a naïve implementation of the macrotransport theory with a localized negative dispersivity will result in an aphysical finite time singularity. We resolve the presence of such a “blow-up solution” by quantifying the relative magnitude of the steady and oscillatory flow such that there is an overall positive dispersivity necessary for an averaged equation. Our second demonstration presents a giant dispersion of solute achievable by flows through an eccentric annulus, relevant to primary cementing of oil wells and cerebrospinal fluid flow inside the spinal cavity. We find that increasing the eccentricity and annulus size gives rise to stronger dispersion in a steady flow driven by a fixed pressure gradient. This relationship holds when the flow becomes unsteady. In the limit of slow oscillation, dispersion due to an oscillatory flow asymptotes to one-half of that by a steady flow. Increasing the oscillation frequency leads to a two-step decay of the dispersivity. The maximum dispersion in an oscillatory flow can be achieved in the limit of slow oscillation and large eccentricity, where dispersion can be O(10³) times larger than that in an otherwise concentric annulus. The present macrotransport theory is readily generalizable to incorporate chemically-reactive and dynamic boundaries, suitable for designing and investigating a wide class of biochemical applications.