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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.
We develop a mean-field theory of compressibility effects in turbulent magnetohydrodynamics and passive scalar transport using the quasi-linear approximation and the spectral $tau$-approach. We find that compressibility decreases the $alpha$ effect a
We consider the evolution of a decaying passive scalar in the presence of a gaussian white noise fluctuating linear shear flow known as the Majda Model. We focus on deterministic initial data and establish the symmetry properties of the evolving poin
The advection and mixing of a scalar quantity by fluid flow is an important problem in engineering and natural sciences. If the fluid is turbulent, the statistics of the passive scalar exhibit complex behavior. This paper is concerned with two Lagran
The reduction of dimensionality of physical systems, specially in fluid dynamics, leads in many situations to nonlinear ordinary differential equations which have global invariant manifolds with algebraic expressions containing relevant physical info
In scalar turbulence it is sometimes the case that the scalar diffusivity is smaller than the viscous diffusivity. The thermally-driven turbulent convection in water is a typical example. In such a case the smallest scale in the problem is the Batche