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We extend our previous results characterizing the loading properties of adiffusing passive scalar advected by a laminar shear flow in ducts and channelsto more general cross-sectional shapes, including regular polygons and smoothedcorner ducts originating from deformations of ellipses. For the case of thetriangle, short time skewness is calculated exactly to be positive, whilelong-time asymptotics shows it to be negative. Monte-Carlo simulations confirmthese predictions, and document the time scale for sign change. Interestingly,the equilateral triangle is the only regular polygon with this property, allother polygons possess positive skewness at all times, although this cannotcannot be proved on finite times due to the lack of closed form flow solutionsfor such geometries. Alternatively, closed form flow solutions can beconstructed for smooth deformations of ellipses, and illustrate how thepossibility of multiple sign switching in time is unrelated to domain corners.Exact conditions relating the median and the skewness to the mean are developedwhich guarantee when the sign for the skewness implies front (back) loadingproperties of the evolving tracer distribution along the pipe. Short and longtime asymptotics confirm this condition, and Monte-Carlo simulations verifythis at all times.

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