On Mixing and Segregation: From Fluids and Maps to Granular Solids and Advection–Diffusion Systems

Abstract

In fluids, diffusive transport and fluid parameters are determined by the thermally driven movement of molecules, whereas in granular media, thermal effects play no role in transport. Despite this fundamental difference, mixing in fluids and mixing and segregation in granular solids share many similarities. The advection−diffusion equation formalism unites the two systems and can be used in both cases to predict and understand how and when mixing occurs, through the use of strategic simplifications. We illustrate this connection with a fluid mixing example (the rotated potential mixer) and a granular segregation example (the segregation of a bidisperse mixture flowing in a chute). ■ INTRODUCTIONMixing in fluids and granular materials shares similarities, but also crucial differences. Every problem involving fluid motion, diffusion, and even reaction, can, in principle, be understood in terms of the Navier−Stokes equations and the advection− diffusion equation. However, it is rare that analytic solutions and their accompanying insights exist; instead, the coupled partial differential equations (PDEs) are solved numerically for one set of parameters at a time. In contrast to fluids, there is not yet a universal continuum description that fully captures flow and mixing in granular systems. Instead, discrete element method (DEM) simulations provide a powerful computational microscope that can reveal nearly all details for flows involving millions of particles. These details on their own, although useful for a particular set of parameters, lack the power to provide a deeper understanding of the principles governing granular flow and mixing. To advance our knowledge on both frontsfluid flow and granular flowother approaches are needed that are both accurate and sufficiently elegant to reveal the fundamental nature of the mechanisms driving mixing, and, in the case of granular solids, segregation as well. We present two examples, representing extremes, and show how they connect.The foundation of understanding fluid mixing rests on maps. Repeated applications of the right types of maps in certain mixing systems leads to stretching and folding and chaos. 1 The map route, a purely kinematical approach to mixing, was crucial over the last two decades to understanding fluid mixing via chaotic advection. Only lately have computational algorithms advanced to the point that directly solving the Navier−Stokes and advection−diffusion equations has become possible and practical. Here, we present an example that sits midway between the two approaches: simple enough that the structure created by the repeated applications of maps is evident, but rich enough, when one adds diffusion, that it serves as a test bed for the application of recent algorithms based on the advection− diffusion formalism. 2 In fact, the repeated application of a map to solve the advection−diffusion equation has provided a methodology to solve problems in which chaotic advection is accompanied by diffusion. 3−7 The method uses operator splitting, which allows...

Diffusion, mixing, and segregation in confined granular flows

“…Note that Eq. predicts that the steady‐state degree of segregation, as characterized, for example, by both the thickness of the pure layers ( c i near 0 and near 1) and the sharpness of the change in concentration between the layers, is insensitive to the shear rate, since both D and w p , i are linear in . Note also that the steady‐state degree of segregation predicted by Eq.…”

“…are zero, and the right‐hand side depends only on the second derivative in y . The resulting ordinary differential equation has a solution for the predicted depthwise concentration profiles with the simple form: where is a nondimensional ratio of segregation to diffusive time scales, , C D = 0.042 is the leading coefficient for diffusion described in Figure , S is the segregation coefficient in the segregation velocity equation, h is the height of the shear cell, and A = e 1/(2 λ ) is an integration constant for equal volume mixtures that satisfies the condition . This analytic solution is valid for constant diffusion and segregation coefficients, but Eq.…”

Simulation and modeling of segregating rods in quasi‐2D bounded heap flow

“…To do so, we apply a form of a continuum model that has been successfully applied to quantitatively predict bidisperse size segregation of spherical particles in different geometries [45][46][47][48] as well as bidisperse density segregation 49 and multiand polydisperse size segregation. 50 This continuum segregation model has the general form…”

“…To do so, we apply a form of a continuum model that has been successfully applied to quantitatively predict bidisperse size segregation of spherical particles in different geometries as well as bidisperse density segregation and multi‐ and polydisperse size segregation . This continuum segregation model has the general form where c i is the concentration of species i ( l for long and s for short) such that c i = f i / f , where f i is the local volume fraction of species i , and f is the total volume fraction for all species.…”

Measuring segregation characteristics of industrially relevant granular mixtures: Part I – A continuum model approach

“…An alternative is to use continuum models to describe the segregation process. Previous research has shown that a continuum model approach can accurately predict segregation in flows of glass-like, frictional particles in multiple geometries, including bounded heaps [5,12,19,20], chutes [31,32,33], cylindrical tumblers [32,34], and planar shear cells [35].…”