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.

Rationale and organization

Models for the turbulent stresses and scalar fluxes have been in widespread use since the 1960s, incorporated within CFD codes of a wide range of types and capabilities. Over this period the vast majority of computations have been made using turbulence models simpler than second-moment closure. Quite clearly, such simpler models must deliver satisfactory predictions of some of the flows of interest – for otherwise they would be discarded. This chapter is devoted to such reduced models. The position adopted is that, of course, such simplification makes sense, provided it is made with an appreciation of what has been lost in the process.

This truism applies as much to the numerical solver as to the physical model of turbulence employed, for one would surely never use a three-dimensional, elliptic, compressible-flow solver if one's interests were simply in computing a range of axisymmetric, unseparating boundary layers in liquids. But, if we proceed in the reverse direction, while it is not usually possible to apply a simple numerical solver to flows well beyond the solver's capability, it is all too easy to assume that a turbulence model that functioned very satisfactorily in computing simple shear flows, will perform equally as well in computing complex strains or in the presence of strong external force fields. That is why it is seen as important that simple (or simpler) turbulence models should be arrived at by a rational simplification of the full second-moment closure (having regard for the particular features of the flow to be computed) rather than by adopting some constitutive equation as an article of faith.

Early proposals

The label wall functions was first applied by Patankar and Spalding (1967) as the collective name for the set of algebraic relations linking the values of the effective wall-normal gradients of dependent variables between the wall and the wall-adjacent node (in a numerical solver) to the shear stress, heat or mass flux at the wall.

The underlying purpose of wall functions, as originally proposed, was to allow computations to escape the need to model the very complex flow dynamics associated with the low-Re region that formed the subject of Chapter 6. It may seem absurd that in the region which, from a physical point of view, contains the most complex viscous and turbulent interactions, one resorts to algebraic rather than differential relations to resolve the flow. We note, however, that in Chapter 7 the power of using very simple eddy-viscosity models of turbulence to handle the sublayer has been demonstrated. Wall functions may just be seen as an extrapolation of that simplification strategy; that is, an even cheaper approach to capturing the essentials of the viscosity-affected layer, by exploiting the fact that gradients of dependent variables normal to the wall are dominant and that transport effects are relatively uninfluential. The present chapter first summarizes conventional wall functions and then introduces four more powerful approaches that the authors and their colleagues have developed more recently.

The fact of turbulent flow

Man has evolved within a world where air and water are, by far, the most common fluids encountered. The scales of the environment around him and of the machines and artefacts his ingenuity has created mean that, given their relatively low kinematic viscosities, the relevant global Reynolds number, Re, associated with the motion of both fluids is, in most cases, sufficiently high that the resultant flow is of the continually time-varying, spatially irregular kind we call turbulent.

If, however, our Reynolds number is chosen not by the overall physical dimension of the body of interest – an aircraft wing, say – and the fluid velocity past it but by the smallest distance over which the velocity found within a turbulent eddy changes appreciably and the time over which such a velocity change will occur, its value then turns out to be of order unity. Indeed, one might observe that if this last Reynolds number, traditionally called the micro-scale Reynolds number, Reη, were significantly greater than unity, the rate at which the turbulent kinetic energy is destroyed by viscous dissipation could not balance the overall rate at which turbulence ‘captures’ kinetic energy from the mean flow.

The nature of viscous and wall effects: options for modelling

The turbulence models considered in earlier chapters were based on the assumption that the turbulent Reynolds numbers were high enough everywhere to permit the neglect of viscous effects. Thus, they are not applicable to flows with a low bulk Reynolds number (where the effects of viscosity may permeate the whole flow) or to the viscosity-affected regions adjacent to solid walls (commonly referred to as the viscous sublayer and buffer regions but which we shall normally collectively refer to as the viscous region) which always exist on a smooth wall irrespective of how high the bulk Reynolds number may be. In other words, while at high Reynolds number, viscous effects on the energy-containing turbulent motions are indeed negligible throughout most of the flow, the condition of no-slip at solid interfaces always ensures that, in the immediate vicinity of a wall, viscous contributions will be influential, perhaps dominant. Figure 6.1 shows the typical ‘layered’ composition for a near-wall turbulent flow (though with an expanded scale for the near-wall region) as found in a constant-pressure boundary layer, channel or pipe flow. Although the thickness of this viscosity-affected zone is usually two or more orders of magnitude less than the overall width of the flow (and decreases as the Reynolds number increases), its effects extend over the whole flow field since, typically, half of the velocity change from the wall to the free stream occurs in this region.

Because viscosity dampens velocity fluctuations equally in all directions, one may argue that viscosity has a ‘scalar’ effect. However, turbulence in the proximity of a solid wall or a phase interface is also subjected to non-viscous damping arising from the impermeability of the wall and the consequent reflection of pressure fluctuations. This ‘wall-blocking’ effect, which is also felt outside the viscous layer well into the fully turbulent wall region, directly dampens the velocity fluctuations in the wall-normal direction and thus it has a ‘vector’ character. A good illustration of this effect is the reduction of the surface-normal velocity fluctuations that has been observed in flow regions close to a phase interface, where there are no viscous effects, for example the DNS of Perot and Moin (1995).