The term ‘mass transfer’ or ‘mass transport’ encompasses a vast range of processes involving the movement of matter within a system. It is not possible to provide a simple definition of its scope, except to state that we are concerned with the movement of particular molecular species within a system and with the factors which affect the movement. We can introduce the subject by means of two simple examples, although these in no way describe its full breadth.

A puddle of water on the road surface slowly evaporates, the liquid water progressively being transferred as vapour to the air above it. The rate of evaporation depends upon such factors as humidity, the ambient air temperature relative to the ground and the speed of the wind over the surface of the puddle.

If a crystal of copper sulphate is dropped into a beaker of water it slowly dissolves in the water and produces a concentrated solution around the crystal surface. In time this dissolved material diffuses further and further into the surrounding water. The speed with which the crystal dissolves and the rate of transfer of dissolved copper sulphate to the bulk of the water phase can be modified by a number of factors. For example, the process would occur more quickly in hot water than cold, or if the beaker were stirred.

The walls of blood vessels are elastic and can change their size or shape when different forces are applied to them. These forces include both the pressures and shear stresses exerted by the blood, and the constraints imposed by surrounding tissue. In this chapter, therefore, we both outline the basic principles governing the mechanics of deformable solids and show to what extent they are applicable to blood vessel walls, rather than leaving the application to a later chapter. The essentials of solid mechanics are of course contained in Newton's laws of particle motion; a solid material, like a fluid, can be thought of as split up into a large number of small elements, to each of which the laws can be applied. Again, the forces on the elements consist of long-range body forces and short-range stress forces; it is in the relationship between the stresses and the deformations of the material that solid and fluid mechanics differ.

Definitions of elastic properties

We should begin with a few definitions. An elastic material is one which deforms when a force is applied to it, but returns to its original configuration, without any dissipation of energy, when the force is removed. This means that all the elements return to their original positions. The first understanding of elasticity was obtained by Robert Hooke (the English astronomer and physicist) in 1678, from experiments with metal wires.

This chapter is concerned with the mechanical properties of the blood and its constituents. We shall examine in Chapters 12 to 15 the flow of blood in blood vessels and its contact with their walls. The mechanics of fluids, discussed in Chapters 4 and 5, provide a background to the material that follows.

Blood is a suspension of the formed elements (the various blood cells) and some liquid particles (the chylomicrons) in the plasma. Plasma itself is an aqueous solution containing numerous low molecular weight organic and inorganic materials in low concentration, and about 7% by weight of protein (Table 10.1). The mechanical property of blood which is of principal interest to us is its viscosity. In order to understand what determines the viscosity of whole blood we must first consider what governs the viscosity of simple fluids and suspensions, then the mechanical properties of the plasma (p. 155) and the suspended elements (p. 157), and finally whole blood (p. 169).

Viscosity of fluids and suspensions

It was noted in Chapter 1 that the physical features of liquids, gases and solids are directly related to their molecular structure and that both liquids and gases are classed as fluids, because they flow when a shear stress is applied. The property which relates the rate of shearing to the shear stress is the viscosity (p. 37) and we must now consider the factors that determine the viscosity of a fluid.

It helps in understanding the physics of a liquid if at the same time we consider a gas. Gases are much less dense than liquids; therefore, the molecules of a gas are farther apart than those of a liquid.

The science of mechanics comprises the study of motion (or equilibrium) and the forces which cause it. The blood moves in the blood vessels, driven by the pumping action of the heart; the vessel walls, being elastic, also move; the blood and the walls exert forces on each other, which influence their respective motions. Thus, in order to study the mechanics of the circulation, we must first understand the basic principles of the mechanics of fluids (e.g. blood), and of elastic solids (e.g. vessel walls), and the nature of the forces exerted between two moving substances (e.g. blood and vessel walls) in contact.

As well as studying the relatively large-scale behaviour of blood and vessel walls as a whole, we can apply the laws of mechanics to motions right down to the molecular level. Thus, ‘mechanics’ is taken here to include all factors affecting the transport of material, including both diffusion and bulk motion.

The study of mechanics began in the time of the ancient Greeks, with the formulation of ‘laws’ governing the motion of isolated solid bodies. The Greeks believed that, for a body to be in motion, a force of some sort had to be acting upon it all the time; the physical nature of this force, exerted for example on an arrow in flight, was mysterious. The need for such a force was related to one of the paradoxes of the Greek philosopher Zeno: that the arrow occupies a given position during one instant, yet is simultaneously moving to occupy a different position at a subsequent instant.

When I arrived at the Physiological Flow Studies Unit, Imperial College, in 1971, the writing of The Mechanics of the Circulation was already underway. The book had been commissioned by Oxford University Press to be delivered in 1972 and the Tuesday afternoon book meeting was a regular event. From the outset, the purpose of the book was seen as presenting cardiovascular mechanics in a rigorous but accessible way. It was not meant to be a textbook, but an introduction to the subject that would be useful to a wide range of readers from medical students to experts in either mechanics or cardiovascular physiology.

The Mechanics of the Circulation was finally published in 1978 and it was obvious that the authors had succeeded in their purpose. It was a truly interdisciplinary book, its authors having trained in medicine, mathematics and engineering, but there was a continuity of style and content that remains unusual in multidisciplinary, multi-author books. Individual authors wrote the first drafts of the different sections of the book closest to their expertise, but they all had an equal say in the final product which, as evidenced by the time it took to write the book and the heat that was generated in those weekly meetings, was no easy task. The book had an enormous impact on the emerging field of cardiovascular mechanics and, by extension, on the development of the discipline of bioengineering as an essentially multidisciplinary field of study. It was reprinted and published as a paperback.

The book had an enormous impact on the emerging field of cardiovascular mechanics and, by extension, on the development of the discipline of bioengineering as an essentially multidisciplinary field of study. It was reprinted and published as a paperback.

Computational fluid dynamics does not provide an exact solution to all problems, but is in many cases a reliable tool that can provide useful results when it is employed by an experienced user. An inexperienced user, on the other hand, may obtain very nice graphs that are very far from being a prediction of the stated problem. Some of the problems arise from the many default settings in commercial CFD codes, since the user may obtain results without knowing what the code is doing by accepting settings that are not appropriate for the specific problem. The user must make an active decision regarding each setting due to the fact that many problems can arise from a user failing to understand what the proper settings should be. This chapter provides some guidelines that can help a new user to avoid the most common mistakes. Many more recomendations selected by experienced CFD users can be found in the ‘Best Practice Guidelines’ for single-phase flows [20] and for dispersed multiphase flows [21] by the European Research Community on Flow Turbulence and Combustion (ERCOFTAC).

A CFD simulation contains both errors and uncertainties. An error is defined as a recognizable deficiency that is not due to a lack of knowledge, whereas an uncertainty is a potential deficiency that is due to a lack of knowledge.

In this chapter, our primary purpose is to go beyond Stokes flow to tackle the very difficult problem of understanding the influence of fluid inertia on particle-laden flows. Specifically, the issue of interest is the effect of inertia at the particle scale. Following the structure of the preceding two chapters, we consider first the influence of inertia on sedimentation, and then on shear flows of particle-laden fluid, where we will also consider the rheological consequences of inertia. Inclusion of inertia changes the form of the equation of motion, and even weak inertia can have singular effects when large domains are considered; for both sedimentation and shear, we provide a sketch of results obtained using the singular perturbation method of matched asymptotic expansions in the limit of weak inertia, i.e. at small Reynolds number. While Stokes flow is a good approximation near the particle, a pronounced change in symmetry of the disturbance flow caused by the particle is seen if we are far enough away, as the fore–aft symmetry of Stokes flow is completely lost in this “far-field” region.

We can only give an outline of the subject of inertial suspension flow, as most issues are far from completely resolved. In the previous two chapters, the issues which remain unclear are primarily collective, whereas the microhydrodynamic theory is well-established. For inertial suspensions, the level of understanding at the microscopic, i.e. single and pair, level is incomplete. Hence understanding of collective phenomena based on the microscopic physics is not well-developed and may expand rapidly.

Finally in this book, we would like to broaden the discussion to topics where the understanding is less clear. The future of the subject will involve study of these open questions, but we do not intend to suggest that the list of topics that we are discussing is all-inclusive, or even to suggest these topics as priorities. Instead we seek to provide some indication of the scope of activities for which the concepts developed in this book may find future use.

Moving toward open questions While some of the issues discussed in the last chapters are mostly settled (or perhaps will be resolved soon), there remain greater challenges in many areas of suspension flows. We are perhaps touching on the more obvious of issues which come to mind following the exposition in the preceding chapters, and thus we likely miss novel avenues of study. Nonetheless, a list of issues in suspensions where many open questions remain includes:

Dense suspensions: Flow of suspensions approaching the maximum packing limit is often referred to as “dense suspension flow” and this condition raises special issues which we have only noted briefly in this book. In particular, for such mixtures, the particle surfaces are likely to make enduring contacts, and the details of surface roughness and friction coefficient will play a role in the behavior. How such contact forces interact with hydrodynamic lubrication forces in dense suspensions, and the relation of dense suspension flow to dense granular flow in which the interstitial fluid is a gas, are open questions of interest.

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.