Rich McLaughlin (UNC-CH), Colloquium

Tea at 3:30 pm in 330 Phillips Hall

Title:  Tailoring the Tails in Taylor Dispersion

Abstract:  The interaction of the physics of advection and diffusion is a well studied problem in turbulent transport, and gives rise to quantified understandings of things like wind chill effects.  Diffusive effects tend to be strongly dependent upon a contaminants exposed surface area to the fluid flow.  The interaction of fluid flow with diffusion can be quickly understood by thinking about how a pollutant, say a dye, can be stretched by a non-spatially constant fluid flow with to have greater surface area exposed to the fluid over which diffusivity effects can be greatly enhanced.  This is the mechanism first described mathematically by Sir G. I. Taylor in 1953, where he explicitly computed this effect for laminar flow in a straight, circular pipe, computing an effective, “renormalized”, diffusivity which is seen to be tremendously boosted over the usual molecular diffusivity, which typically is only on the order of 10^(-6) cm^2/s.  This phenomena sets in after waiting a sufficient amount of time for molecular diffusion to have time to act, and as such is a long time asymptotic theory.  Very, very interesting effects occur on shorter timescales, which is what we describe in this lecture.

The pure diffusion equation preserves initial symmetries.  But we will show how the interaction of fluid advection with diffusion can break up-stream/downstream initial symmetries, and lead to, for example, chemical deliveries which are front-loaded, arriving with a sharper front, then depart with a much gentler tail, or the reverse of this depending upon the shape of the cross-section as we show in this lecture.   The lowest order statistic which captures these differences is the skewness, which is the centered, normalized third moment of the up-stream/downstream distribution of tracer.  The sign of this quantity, under suitable conditions, distinguishes the front-loaded versus back-loaded distributions.  These different effects are of interest in micro-scale fluidics, and in drug delivery.

We explore the role different geometries (amongst rectangular and elliptical domains of arbitrary aspect ratios) play in controlling emerging up-stream/downstream asymmetries in the cross-sectionally averaged distribution of diffusing passive scalars advected by laminar, pressure driven shear flows.  We show using a combination of rigorous analysis, asymptotic expansions, and Monte-Carlo simulations, that on short time scales relative to the shortest diffusion times, elliptical domains preserve initial uptream/downstream symmetric distributions, while rectangular ducts break this symmetry.  Skinny ducts produce distributions with negative skewness, while fat ducts produce positive skewness for symmetric initial data which is uniformly distributed in the cross-section.  There is a special aspect ratio of approximately 2-1 ratio for which symmetry is preserved.  In turn, long-time (relative to the longest diffusion timescale) exact analysis and simulation shows that all geometries generically break symmetry before ultimately symmetrizing in infinite time. Laboratory experiments are presented which confirm our predictions.

This work is joint with Manuchehr Aminian, Francesca Bernardi, Roberto Camassa, and Dan Harris.