Mass distribution and skewness for passive scalar transport in pipes with polygonal and smooth cross sections

We extend our previous results characterizing the loading properties of a diffusing passive scalar advected by a laminar shear flow in ducts and channels to more general cross-sectional shapes, including regular polygons and smoothed corner ducts originating from deformations of ellipses. For the case of the triangle and localized, cross-wise uniform initial distributions, short-time skewness is calculated exactly to be positive, while long-time asymptotics shows it to be negative. Monte Carlo simulations confirm these predictions, and document the timescale for sign change. The equilateral triangle appears to be the only regular polygon with this propertyall others possess positive skewness at all times. Alternatively, closed-form flow solutions can be constructed for smooth deformations of ellipses, and illustrate how both nonzero short-time skewness and the possibility of multiple sign switching in time is unrelated to domain corners. Exact conditions relating the median and the skewness to the mean are developed which guarantee when the sign for the skewness implies front (more mass to the right of the mean) or back (more mass to the left of the mean) loading properties of the evolving tracer distribution along the pipe. Short- and long-time asymptotics confirm this condition, and Monte Carlo simulations verify this at all times. The simulations are also used to examine the role of corners and boundaries on the distribution for short-time evolution of point source, as opposed to cross-wise uniform, initial data.