Optimizing the Source Distribution in Fluid Mixing

A passive scalar is advected by a velocity field, with a nonuniform spatial source that maintains concentration inhomogeneities. For example, the scalar could be temperature with a source consisting of hot and cold spots, such that the mean temperature is constant. Which source distributions are best mixed by this velocity field? This question has a straightforward yet rich answer that is relevant to real mixing problems. We use a multiscale measure of steady-state enhancement to mixing and optimize it by a variational approach. We then solve the resulting Euler--Lagrange equation for a perturbed uniform flow and for simple cellular flows. The optimal source distributions have many broad features that are as expected: they avoid stagnation points, favor regions of fast flow, and their contours are aligned such that the flow blows hot spots onto cold and vice versa. However, the detailed structure varies widely with diffusivity and other problem parameters. Though these are model problems, the optimization procedure is simple enough to be adapted to more complex situations.Comment: 19 pages, 23 figures. RevTeX4 with psfrag macro

Reduced models of chemical reaction in chaotic flows

A. Vikhansky and S. M. Co

Conditional moment closure for chemical reactions in laminar chaotic flows

We implement conditional moment closure (CMC) for simulation of chemical reactions in laminar chaotic flows. The CMC approach predicts the expected concentration of reactive species, conditional upon the concentration of a corresponding nonreactive scalar. Closure is obtained by neglecting the difference between the local concentration of the reactive scalar and its conditional average. We first use a Monte Carlo method to calculate the evolution of the moments of a conserved scalar; we then reconstruct the corresponding probability density function and dissipation rate. Finally, the concentrations of the reactive scalars are determined. The results are compared (and show excellent agreement) with full numerical simulations of the reaction processes in a chaotic laminar flow. This is a preprint of an article published in AlChE Journal copyright (2007) American Institute of Chemical Engineers: http://www3.interscience.wiley.com

Granular flow in a rotating cylindrical drum


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Granular flow in a rotating cylindrical drum

Flow of granular material in a partially filled rotating cylinder is studied using a boundary layer approach. Simple equations are derived to describe granular dynamics in a transverse plane. A small parameter is recognized, and the derived equations are solved analytically in the first-order approximation of the small parameter. Cascading layer thickness, mean velocity along the layer and profile of the free surface are determined in a closed analytical form

Kinematics of the mixing of granular material in slowly rotating containers

The mixing of granular material in a two-dimensional slowly rotating container is studied using a kinematic approach. A simple model is proposed to estimate the mixing capacity of a mixer when the particles diffusivity is small. Different geometries of the container are examined numerically. The mixing rate index for the case of a circular drum is derived analytically. The optimum shape of the container and the optimum filling level are discussed

Granular flow in a rotating cylindrical drum

Abstract. { Flow of granular material in a partially lled rotating cylinder is studied using a boundary layer approach. Simple equations are derived to describe granular dynamics in a transverse plane. A small parameter is recognized, and the derived equations are solved analytically in the rst-order approximation of the small parameter. Cascading layer thickness, mean velocity along the layer and prole of the free surface are determined in a closed analytical form. Dynamics, mixing and separation of granular materials in a partially lled rotating cylin-drical drum have been the subject of numerous experimental and theoretical investigations (see, e.g., [1]-[3]). However only a few studies attempted to describe a continuous granular flow in a transverse plane of a rotating drum [1], [3]. In the present work a boundary value approximation is used to describe a two-dimensional granular flow. In the developed approach the transition from solid to liquid-like behavior is considered as a key process for specifying the granular dynamics. Let us consider a cylindrical drum of radius R which rotates with a constant angular velocity! ( g. 1). The drum is partially lled with a granular material of constant bulk density , and the length of the free surface is 2L. The free surface is inclined at an angle of internal frictio