▪ Abstract  Granular materials segregate. Small differences in either size or density lead to flow-induced segregation, a complex phenomenon without parallel in fluids. Modeling of mixing and segregation processes requires the confluence of several tools, including continuum and discrete descriptions (particle dynamics, Monte Carlo simulations, cellular automata computations) and, often, considerable geometrical insight. None of these viewpoints, however, is wholly satisfactory by itself. Moreover, continuum and discrete descriptions of granular flows are regime dependent, and this fact may require adopting different subviewpoints. This review organizes a body of knowledge that forms—albeit imperfectly—the beginnings of an expandable continuum framework for the description of mixing and segregation of granular materials. We focus primarily on noncohesive particles, possibly differing in size, density, shape, etc. We present segregation mechanisms and models for size and density segregation and introduce chaotic advection, which appears in noncircular tumblers. Chaotic advection interacts in nontrivial ways with segregation in granular materials and leads to unique equilibrium structures that serve as a prototype for systems displaying organization in the midst of disorder.

Predicting segregation of granular materials composed of different-sized particles is a challenging problem. In this paper, we develop and implement a theoretical model that captures the interplay between advection, segregation, and diffusion in size bidisperse granular materials. The fluxes associated with these three driving factors depend on the underlying kinematics, whose characteristics play key roles in determining particle segregation configurations. Unlike previous models for segregation, our model uses parameters based on kinematic measures from discrete element method simulations instead of arbitrarily adjustable fitting parameters, and it achieves excellent quantitative agreement with both experimental and simulation results when applied to quasi-two-dimensional bounded heaps. The model yields two dimensionless control parameters, both of which are only functions of physically control parameters (feed rate, particle sizes, and system size) and kinematic parameters (diffusion coefficient, flowing layer depth, and percolation velocity). The Péclet number, P e, captures the interplay of advection and diffusion, and the second dimensionless parameter, Λ, describes the interplay between segregation and advection. A parametric study of Λ and P e demonstrates how the particle segregation configuration depends on the interplay of advection, segregation, and diffusion. The model can be readily adapted to other flow geometries.Key words: Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfm-keywords.pdf for the full list) †

No abstract

In this paper we briefly review the main issues associated with mixing at the microscale and introduce the papers comprising the Theme Issue.

The simplest mixing problem corresponds to the mixing of a fluid with itself; this case provides a foundation on which the subject rests. The objective here is to study mixing independently of the mechanisms used to create the motion and review elements of theory focusing mostly on mathematical foundations and minimal models. The flows under consideration will be of two types: two-dimensional (2D) 'blinking flows', or three-dimensional (3D) duct flows. Given that mixing in continuous 3D duct flows depends critically on cross-sectional mixing, and that many microfluidic applications involve continuous flows, we focus on the essential aspects of mixing in 2D flows, as they provide a foundation from which to base our understanding of more complex cases. The baker's transformation is taken as the centrepiece for describing the dynamical systems framework. In particular, a hierarchy of characterizations of mixing exist, Bernoulli --> mixing --> ergodic, ordered according to the quality of mixing (the strongest first). Most importantly for the design process, we show how the so-called linked twist maps function as a minimal picture of mixing, provide a mathematical structure for understanding the type of 2D flows that arise in many micromixers already built, and give conditions guaranteeing the best quality mixing. Extensions of these concepts lead to first-principle-based designs without resorting to lengthy computations.