In Part I, we have presented the basis of microhydrodynamics. In its broad definition, microhydrodynamics represents the theory of viscous fluid flows at small spatial scales. For the purposes of the present book on suspensions, we have specifically considered the single- and pair-body dynamics of small particles immersed in viscous fluid. In Part II, we will describe macroscopic phenomena encountered in flows involving a large number of particles interacting through viscous fluid. We will also provide an introduction to methods developed for understanding and (hopefully) predicting certain macroscopic phenomena in suspensions in terms of the microscopic concepts described in Part I, combined with ideas that fall in the realms of statistical physics and dynamical systems.

In this transitional chapter, we are concerned primarily with introducing statistical techniques and concepts from stochastic processes which we will apply in the following chapters. We will also briefly consider the related issue of chaotic dynamics.

Statistical physics

The theoretical framework for relating microscopic mechanics to macroscopic or bulk properties is statistical physics or statistical mechanics. Understanding systems made up of many interacting particles is far from trivial, and the difficulty involved is not just a mere question of solving the hydrodynamic equations with better, faster computers. The collective interactions between the particles can give rise to quite unexpected qualitative behavior, often much simpler than the microscopic motions seem to suggest.

The purpose of this book is to provide an introduction to suspension dynamics, and so we (the authors) thought it would be good to give some historical (as well as personal) perspective on the study of suspensions. Early development of the subject was largely due to two “schools,” one in England and one in the United States. In England, the subject developed from the fluid mechanical tradition at the University of Cambridge, dating from the work of G. G. Stokes and H. Lamb in the mid- and late-1800s. The subject developed in earnest from the work of George Batchelor and collaborators at Cambridge's Department of Applied Mathematics and Theoretical Physics (DAMTP). In the United States, the development of the discipline took place primarily in chemical engineering departments, largely through the efforts of Andreas Acrivos and a number of his students at the University of California Berkeley, Stanford University, and the City College of New York (CCNY). The authors' approaches to suspensions owe much to these “schools” of suspension dynamics. Élisabeth Guazzelli was introduced to the subject by Bud Homsy at Stanford University and extended interactions with John Hinch of the University of Cambridge. Jeff Morris received his introduction to suspensions as a doctoral student of John Brady (a student of Acrivos) at the California Institute of Technology.

In this chapter, we describe shear flows of suspensions. The goal here is to illustrate the connection between the particle-scale interactions and bulk suspension phenomena. At the microscopic scale, we consider the interactions of discrete particles and the resulting microstructural arrangement, while at the bulk scale the mixture is described as a continuous effective fluid. The connection between the scales is provided by the rheology, i.e. by the stress response of the bulk material. This chapter describes non-Newtonian properties as well as shear-induced diffusivity exhibited by suspensions, and presents an introduction to the relationship between these properties and the flow-induced microstructure. Irreversibility of the bulk motion seen in shear-induced particle migration demonstrates how the interplay of Stokes-flow hydrodynamics, outlined in Part I, with other particle-scale forces leads to some unexpected behavior. As we consider the average material behavior and its relation to the microscopic interactions, it is natural to apply concepts from statistical physics introduced in Chapter 5.

A number of the issues raised in this chapter are topics of active research in rheology and multiphase flow; while we provide a few references as a guide to further information on specific issues, recent reviews by Stickel and Powell (2005), Morris (2009), and Wagner and Brady (2009) provide fuller coverage of the literature.

Suspension viscosity

We have all heard that “blood is thicker than water.” Blood is, in fact, a suspension of red blood cells in a Newtonian plasma.

Mobile particulate systems are encountered in various natural and industrial processes. In the broadest sense, mobile particulate systems include both suspensions and granular media. Suspensions refer to particles dispersed in a liquid or a gas. Familiar examples include aerosols such as sprays, mists, coal dust, and particulate air pollution; biological fluids such as blood; industrial fluids such as paints, ink, or emulsions in food or cosmetics. Suspension flows are also involved in numerous material processing applications, including manufacture of fiber composites and paper, and in natural processes such as sediment transport in rivers and oceans. In common usage, a suspension refers to solid particles as the dispersed state in a liquid, while an emulsion concerns liquid droplets dispersed in another immiscible fluid, and an aerosol is specific to the case of a suspension of fine solid or liquid particles in a gas. We focus on the case of a suspension in this text.

In the flow of suspensions, the viscous fluid between the particles mediates particle interactions, whereas in dry granular media the fluid between the particles is typically assumed to have a minor role, doing no more than providing a resistive drag, and this allows direct contact interactions. Familiar examples of granular media include dry powders, grains, and pills in the food, pharmaceutical, and agricultural industries; sand piles, dredging, and liquefaction of soil in civil engineering; and geophysical phenomena such as landslides, avalanches, and volcanic eruptions.

The sedimentation of particles is one of the basic flows involving suspensions. It is also one of the oldest known separation techniques, e.g. to clarify liquids (or alternatively to recover particles) or to separate particles of different densities or sizes. Sedimentation is also ubiquitous in natural phenomena such as the fall of rain drops and dust particles in the atmosphere, or mud sedimentation in rivers or in estuaries. In this chapter, we focus our attention primarily on the sedimentation of small solid spheres of equal size and density for which the Reynolds number is small. We will, however, also take a brief glance at particles having different size, density, and shape at the end of the chapter. The long-range and many-body nature of the hydrodynamic interactions between the particles that we have introduced in Part I is the key feature in describing a number of interesting and unexpected phenomena in sedimentation. These interactions give rise to complex and collective dynamics which are not completely understood and remain the subject of active research. This chapter is based on the reviews of Davis and Acrivos (1985) and Guazzelli and Hinch (2011), where the reader can find further information.

One, two, three … spheres

When a sphere of radius a and density ρp settles in a quiescent viscous fluid, it generates a disturbance flow which decays very slowly away from the translating particle, i.e. as the inverse of the distance to the sphere for the dominant portion, as shown in Chapters 2 and 3.

Introduction

Chemical engineering is the field of applied science that employs physical, chemical, and biochemical rate processes for the betterment of humanity. This is a sweeping statement, and it contains two essential concepts: rate processes and betterment of humanity. The second is straightforward and is at the heart of all engineering. The engineer designs processes and tangible objects that meet the real or perceived needs of the populace. Some civil engineers design bridges. Some mechanical engineers design engines. Some electrical engineers design power systems. The popular perception of the chemical engineer is someone who designs and operates processes for the production of chemicals and petrochemicals. This is an historically accurate (if incomplete) image, but it describes only a small fraction of the chemical engineers of the early twenty-first century.

Let us turn first to the concept of rate processes, which is the defining paradigm of chemical engineering, and consider an example. Everyone is familiar with the notion that medication taken orally must pass through the digestive system and across membranes into the bloodstream, after which it must be transported to the relevant location in the body (a tumor, a bacterial infection, etc.) where it binds to a receptor or reacts chemically. The residual medication is transported to an organ, where it is metabolized, and the metabolic products are transported across still more membranes and excreted from the body, perhaps in the urine.

Introduction

Biotechnology is a major component of modern chemical engineering, and biotechnology appears to many observers to be a new thrust; yet, as noted in Chapter 1, biochemical engineering has been an essential part of chemical engineering since the development of the modern profession in the early part of the twentieth century. One important aspect of biotechnology, in fact its most traditional component, is the use of microorganisms to effect chemical change. We cited the microbiological production of acetone and penicillin as two classic examples in Chapter 1.

In terms of annual throughput, the activated sludge process for wastewater treatment is by far the most widely used biochemical process in the world, and it provides a useful framework for discussing some interesting features of bioreactor design and performance. The entire process flowsheet is shown schematically in Figure 8.1. The wastewater feed contains organic materials, commonly measured in toto as biological oxygen demand (BOD), that are used as nutrients by microorganisms; the organisms produce water and CO2 as metabolic products. The primary settler is there to remove large objects. The heart of the process is the aeration basin; this is a reactor in which a suspension of microorganisms in porous flocs is brought into contact with the BOD. The microorganisms are aerobic, meaning that they require oxygen for metabolism, so air or enriched air is added to keep the oxygen concentration in the water above a critical level of about 2 g/m3 (2 ppm).