Title: Mass distribution and skewness for passive scalar transport in pipes with polygonal and smooth cross-sections,
Abstract: 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, 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 time scale for sign change. Interestingly,
the equilateral triangle is the only regular polygon with this property, all
other polygons possess positive skewness at all times, although this cannot
cannot be proved on finite times due to the lack of closed form flow solutions
for such geometries. Alternatively, closed form flow solutions can be
constructed for smooth deformations of ellipses, and illustrate how 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 (back) 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.