Introduction

Unlike solid particles or liquid droplets, gas/vapor bubbles can grow or collapse in a flow and in doing so manifest a host of phenomena with technological importance. We devote this chapter to the fundamental dynamics of a growing or collapsing bubble in an infinite domain of liquid that is at rest far from the bubble. Although the assumption of spherical symmetry is violated in several important processes, it is necessary to first develop this baseline. The dynamics of clouds of bubbles or of bubbly flows are treated in later chapters.

Bubble Growth and Collapse

Rayleigh–Plesset Equation

Consider a spherical bubble of radius, R(t) (where t is time), in an infinite domain of liquid whose temperature and pressure far from the bubble are T∞ and p∞(t) respectively. The temperature, T∞, is assumed to be a simple constant because temperature gradients are not considered. Conversely, the pressure, p∞(t), is assumed to be a known (and perhaps controlled) input that regulates the growth or collapse of the bubble.

Though compressibility of the liquid can be important in the context of bubble collapse, it will, for the present, be assumed that the liquid density, ρL, is a constant. Furthermore, the dynamic viscosity, µL, is assumed constant and uniform. It will also be assumed that the contents of the bubble are homogeneous and that the temperature, TB(t), and pressure, pB(t), within the bubble are always uniform. These assumptions may not be justified in circumstances that will be identified as the analysis proceeds.

Introduction

Dense fluid-particle flows in which the direct particle/particle interactions are a dominant feature encompass a diverse range of industrial and geophysical contexts (Jaeger et al. 1996), including, for example, slurry pipelines (Shook and Roco 1991), fluidized beds (Davidson and Harrison 1971), mining and milling operations, ploughing (Weighardt 1975), abrasive water jet machining, food processing, debris flows (Iverson 1997), avalanches (Hutter 1993), landslides, sediment transport, and earthquake-induced soil liquefaction. In many of these applications, stress is transmitted both by shear stresses in the fluid and by momentum exchange during direct particle/particle interactions. Many of the other chapters in this book analyze flow in which the particle concentration is sufficiently low that the particle-particle momentum exchange is negligible.

In this chapter we address those circumstances, usually at high particle concentrations, in which the direct particle/particle interactions play an important role in determining the flow properties. When those interactions dominate the mechanics, the motions are called granular flows and the flow patterns can be quite different from those of conventional fluids. An example is included as Figure 13.1, which shows the downward flow of sand around a circular cylinder. Note the upstream wake of stagnant material in front of the cylinder and the empty cavity behind it.

Within the domain of granular flows, there are, as we shall see, several very different types of flow distinguished by the fraction of time for which particles are in contact. For most slow flows, the particles are in contact most of the time.

Introduction

In Chapter 2 it was assumed that the particles were rigid and therefore not deformed, fissioned, or otherwise modified by the flow. However, there are many instances in which the particles that comprise the disperse phase are radically modified by the forces imposed by the continuous phase. Sometimes those modifications are radical enough to, in turn, affect the flow of the continuous phase. For example, the shear rates in the continuous phase may be sufficient to cause fission of the particles and this, in turn, may reduce the relative motion and therefore alter the global extent of phase separation in the flow.

The purpose of this chapter is to identify additional phenomena and issues that arise when the translating disperse phase consists of deformable particles, namely bubbles, droplets, or fissionable solid grains.

Deformation due to Translation

Dimensional Analysis

Because the fluid stresses due to translation may deform the bubbles, drops, or deformable solid particles that make up the disperse phase, we should consider not only the parameters governing the deformation but also the consequences in terms of the translation velocity and the shape. We concentrate here on bubbles and drops in which surface tension, S, acts as the force restraining deformation. However, the reader will realize that there would exist a similar analysis for deformable elastic particles. Furthermore, the discussion is limited to the case of steady translation, caused by gravity, g. Clearly the results could be extended to cover translation due to fluid acceleration by using an effective value of g as indicated in Section 2.4.2.

Introduction

In Chapter 9, the analyses were predicated on the existence of an effective barotropic relation for the homogeneous mixture. Indeed, the construction of the sonic speed in Sections 9.3.1 and 9.3.3 assumes that all the phases are in dynamic equilibrium at all times. For example, in the case of bubbles in liquids, it is assumed that the response of the bubbles to the change in pressure, δp, is an essentially instantaneous change in their volume. In practice this would be the case only if the typical frequencies experienced by the bubbles in the flow are very much smaller than the natural frequencies of the bubbles themselves (see Section 4.4.1). Under these circumstances the bubbles would behave quasistatically and the mixture would be barotropic. However, there are a number of important contexts in which the bubbles are not in equilibrium and in which the nonequilibrium effects have important consequences. One example is the response of a bubbly multiphase mixture to high-frequency excitation. Another is a bubbly cavitating flow where the nonequilibrium bubble dynamics lead to shock waves with substantial noise and damage potential.

In this chapter we therefore examine some flows in which the dynamics of the individual bubbles play an important role. These effects are included by incorporating the Rayleigh–Plesset equation (Rayleigh 1917, Knapp et al. 1970, Brennen 1995) into the global conservation equations for the multiphase flow. Consequently the mixture no longer behaves barotropically.

Introduction

One of the most common requirements of a multiphase flow analysis is the prediction of the energy gains and losses as the flow proceeds through the pipes, valves, pumps, and other components that make up an internal flow system. In this chapter we attempt to provide a few insights into the physical processes that influence these energy conversion processes in a multiphase flow. The literature contains a plethora of engineering correlations for pipe friction and some data for other components such as pumps. This chapter provides an overview and some references to illustrative material but does not pretend to survey these empirical methodologies.

As might be expected, frictional losses in straight uniform pipe flows have been the most widely studied of these energy conversion processes and so we begin with a discussion of that subject, focusing first on disperse or nearly disperse flows and then on separated flows. In the last part of the chapter, we consider multiphase flows in pumps, in part because of the ubiquity of these devices and in part because they provide a second example of the multiphase flow effects in internal flows.

Frictional Loss in Disperse Flow

Horizontal Flow

We begin with a discussion of disperse horizontal flow. There exists a substantial body of data relating to the frictional losses or pressure gradient, (−dp/ds), in a straight pipe of circular cross section (the coordinate s is measured along the axis of the pipe).

Introduction

Sprays are an important constituent of many natural and technological processes and range in scale from the very large dimensions of the global air/sea interaction and the dynamics of spillways and plunge pools to the smaller dimensions of fuel injection and ink-jet systems. In this chapter we first examine the processes by which sprays are formed and some of the resulting features of those sprays. Then, because the combustion of liquid fuels in droplet form constitute such an important component of our industrialized society, we focus on the evaporation and combustion of single droplets and follow that with an examination of the features involved in the combustion of sprays.

Types of Spray Formation

In general, sprays are formed when the interface between a liquid and a gas becomes deformed and droplets of liquid are generated. These then migrate out into the body of the gas. Sometimes the gas plays a negligible role in the kinematics and dynamics of the droplet formation process; this simplifies the analyses of the phenomena. In other circumstances the gas dynamic forces generated can play an important role. This tends to occur when the relative velocity between the gas and the liquid becomes large as is the case, for example, with hurricane-generated ocean spray.

Several prototypical flow geometries are characteristic of the natural and technological circumstances in which spray formation is important. The first prototypical geometry is the flow of a gas over a liquid surface.

Introduction

Scope

In the context of this book, the term multiphase flow is used to refer to any fluid flow consisting of more than one phase or component. For brevity and because they are covered in other texts, we exclude those circumstances in which the components are well mixed above the molecular level. Consequently, the flows considered here have some level of phase or component separation at a scale well above the molecular level. This still leaves an enormous spectrum of different multiphase flows. One could classify them according to the state of the different phases or components and therefore refer to gas/solids flows or liquid/solids flows or gas/particle flows or bubbly flows and so on; many texts exist that limit their attention in this way. Some treatises are defined in terms of a specific type of fluid flow and deal with low-Reynolds-number suspension flows, dusty gas dynamics, and so on. Others focus attention on a specific application such as slurry flows, cavitating flows, aerosols, debris flows, fluidized beds, and so on; again, there are many such texts. In this book we attempt to identify the basic fluid mechanical phenomena and to illustrate those phenomena with examples from a broad range of applications and types of flow.

Parenthetically, it is valuable to reflect on the diverse and ubiquitous challenges of multiphase flow. Virtually every processing technology must deal with multiphase flow, from cavitating pumps and turbines to electrophotographic processes to papermaking to the pellet form of almost all raw plastics.

Introduction

This chapter briefly reviews the issues and problems involved in constructing the equations of motion for individual particles, drops, or bubbles moving through a fluid. For convenience we use the generic name particle to refer to the finite pieces of the disperse phase or component. The analyses are implicitly confined to those circumstances in which the interactions between neighboring particles are negligible. In very dilute multiphase flows in which the particles are very small compared with the global dimensions of the flow and are very far apart compared with the particle size, it is often sufficient to solve for the velocity and pressure, ui (xi, t) and p(xi, t), of the continuous suspending fluid while ignoring the particles or disperse phase. Given this solution one could then solve an equation of motion for the particle to determine its trajectory. This chapter focuses on the construction of such a particle or bubble equation of motion.

The body of fluid mechanical literature on the subject of flows around particles or bodies is very large indeed. Here we present a summary that focuses on a spherical particle of radius R and employs the following common notation. The components of the translational velocity of the center of the particle is denoted by Vi(t). The velocity that the fluid would have had at the location of the particle center in the absence of the particle is denoted by Ui(t).

The genesis of solid friction

The laws of friction

Friction is the resistance encountered when one body moves tangentially over another with which it is in contact. The work expended against friction is often redundant, that is, it makes no useful contribution to the overall operation of the device of which the bodies are part, and ultimately must be dissipated as waste heat. Consequently, in most tribological designs our aim is to keep these frictional forces as small as possible. Of course there are exceptions to this general rule, occasions when sufficient friction is essential to continued progress and there are many practical devices which rely on the frictional transmission of power: automobile tyres on a roadway, vehicle brakes and clutches, as well as several of the variable-speed transmission systems now finding wider application. When two objects are to be held together, the only alternative to methods which rely on friction is the formation of some sort of chemical or metallurgical bond between them. The development of this sort of technique–adhesives and ‘superglues’, and even welding and brazing–are relatively recent; ‘traditional’ forms of fixing rely almost exclusively on friction. A nail hammered into a piece of wood is held in place by frictional effects along its length; if the frictional interaction were substantially reduced, the nail would be squeezed out. Similarly, the grip between a nut and a bolt depends on adequate friction between them.

Introduction

With a few important exceptions, engineering devices which involve the contact of loaded, sliding surfaces will only operate satisfactorily, that is, without giving rise to unacceptable amounts of surface damage or wear, when they are provided with adequate lubrication. The lubricant can act in two distinct, but not necessarily mutually exclusive, ways. The first of its functions may be to physically separate the surfaces by interposing between them a coherent, viscous film which is relatively thick (i.e. significantly larger than the size of likely surface asperities). In hydrostatic bearings this film is provided by an external pump and so its presence depends on the continuous operation of an external source of energy. In hydrodynamic bearings its generation relies only on the geometry and motion of the surfaces (hence the term dynamic) together with the viscous nature of the fluid. The second role of the lubricant may be to generate an additional thin, protective coating on one or both of the solid surfaces, preventing, or at least limiting, the formation of strong, adhesive and so potentially damaging friction junctions between the underlying solids at locations of particularly acute loading. If this protective coating has a comparatively low shear strength then the ultimate tangential force of friction can be much reduced: this mechanism of friction limitation is generally known as boundary lubrication. Such boundary films are generally very thin, perhaps only a few (albeit very large) molecules thick, and their formation and survival depends very much on the physical and chemical interactions between components of the lubricant and the solid surfaces.

Introduction

All real materials possess some microstructure which becomes apparent when a representative specimen is viewed with a sufficient degree of magnification. However, engineers usually work at a comparatively large scale and so use macroscopic models of material behaviour: these are likely to involve a very large number of the individual microstructural units which contribute to the overall material response. Engineering properties of the bulk represent average or integrated values, and when dealing with either solids or liquids the material is often treated as a homogeneous continuum. In the case of the linear elastic theory of solids what is required is Hooke's law and the observation that most metals undergo only a very small strain before yielding. Similarly, a Newtonian compressible fluid is described by the linear relationships between the applied shear stress and the resultant shear strain rate, and between density and the applied hydrostatic pressure. Many liquid lubricants, including hydrocarbon mineral oils, are effectively incompressible at moderate pressures so that their density remains constant, and this simplifies the analysis still further.

There is no fundamental reason why the lubricant in a hydrodynamic bearing should not be a vapour or a gas (rather than a liquid such as oil or water); indeed considered as a potential lubricant, air has a number of advantages–such as cleanliness and ease of supply–and it has been used in aerostatic and aerodynamic bearings for several decades.