Introduction to nonlinear theory

Introductory remarks

Nonlinear theories are of three more or less distinct kinds. In one, properties of arbitrarily-large disturbances are deduced directly from the full Navier–Stokes equations. Consideration of integral inequalities yields bounds on flow quantities, such as the energy of disturbances, which give stability criteria in the form of necessary or sufficient conditions for growth or decay with time. An admirable account of such theories is given by Joseph (1976). They have the advantage of supplying mathematically rigorous results while incorporating very few assumptions regarding the size or nature of the disturbances. Sometimes, these criteria correspond quite closely to observed stability boundaries. The bounds for onset of thermal (Bénard) instability and centrifugal (Rayleigh–Taylor) instability in concentric rotating cylinders are particularly notable successes. Often, however, the bounds are rather weak: this is especially so for shear-flow instabilities, where local details of the flow typically play an important rôle which cannot be (or, at least, has not been) incorporated into the global theory.

The second class of theories relies on the idea that linearized equations provide a satisfactory first approximation for those finite-amplitude disturbances which are, in some sense, sufficiently small. Successive approximations may then be developed by expansion in ascending powers of a characteristic dimensionless wave amplitude. These are known as weakly nonlinear theories, and they have proved successful in revealing many important physical processes.

Introduction

Waves occur throughout Nature in an astonishing diversity of physical, chemical and biological systems. During the late nineteenth and the early twentieth century, the linear theory of wave motion was developed to a high degree of sophistication, particularly in acoustics, elasticity and hydrodynamics. Much of this ‘classical’ theory is expounded in the famous treatises of Rayleigh (1896), Love (1927) and Lamb (1932).

The classical theory concerns situations which, under suitable simplifying assumptions, reduce to linear partial differential equations, usually the wave equation or Laplace's equation, together with linear boundary conditions. Then, the principle of superposition of solutions permits fruitful employment of Fourier-series and integral-transform techniques; also, for Laplace's equation, the added power of complex-variable methods is available.

Since the governing equations and boundary conditions of mechanical systems are rarely strictly linear and those of fluid mechanics and elasticity almost never so, the linearized approximation restricts attention to sufficiently small displacements from some known state of equilibrium or steady motion. Precisely how small these displacements must be depends on circumstances. Gravity waves in deep water need only have wave-slopes small compared with unity; but shallow-water waves and waves in shear flows must meet other, more stringent, requirements. Violation of these requirements forces abandonment of the powerful and attractive mathematical machinery of linear analysis, which has reaped such rich harvests. Yet, even during the nineteenth century, considerable progress was made in understanding aspects of weakly-nonlinear wave propagation, the most notable theoretical accomplishments being those of Rayleigh in acoustics and Stokes for water waves.

When, over four years ago, I began writing on nonlinear wave interactions and stability, I envisaged a work encompassing a wider variety of physical systems than those treated here. Many ideas and phenomena recur in such apparently diverse fields as rigid-body and fluid mechanics, plasma physics, optics and population dynamics. But it soon became plain that full justice could not be done to all these areas – certainly by me and perhaps by anyone.

Accordingly, I chose to restrict attention to incompressible fluid mechanics, the field that I know best; but I hope that this work will be of interest to those in other disciplines, where similar mathematical problems and analogous physical processes arise.

I owe thanks to many. Philip Drazin and Michael McIntyre showed me partial drafts of their own monographs prior to publication, so enabling me to avoid undue overlap with their work. My colleague Alan Cairns has instructed me in related matters in plasma physics, which have influenced my views. General advice and encouragement were gratefully received from Brooke Benjamin and the series Editor, George Batchelor.

Various people kindly supplied photographs and drawings and freely gave permission to use their work: all are acknowledged in the text. Other illustrations were prepared by Mr Peter Adamson and colleagues of St Andrews University Photographic Unit and by Mr Robin Gibb, University Cartographer. The bulk of the typing, from pencil manuscript of dubious legibility, was impeccably carried out by Miss Sheila Wilson, with assistance from Miss Pat Dunne.

The next three chapters, which form the core of this monograph, have two main purposes. One is to give a theoretical explanation for some of the arterial velocity profiles described in § 1.2.4. The other, of greater potential importance in the analysis of arterial disease, is to make predictions of the detailed distribution of wall shear stress in arteries, which is related to the rate of mass transport across artery walls and hence (presumably) to atherogenesis (§ 1.2.6). The second purpose is particularly important because no method has yet been devised to measure wall shear stress accurately as a function of time in vivo. This is rather surprising, considering the probable importance of wall shear, and the first section of this chapter is devoted to an explanation of why it is so difficult to measure. The second section begins the analysis of viscous flow in arteries with a discussion of unsteady entry flow (with flow reversal) in a straight tube. In chapters 4 and 5 respectively, curved and branched tubes are considered, and chapter 5 concludes with a discussion of flow instability in arteries.

The difficulty of measuring wall shear stress

The need for a good frequency response

Since the mechanism by which the wall shear stress influences mass transport across the artery wall is unknown, with the consequence that the relevant features of the wall shear distribution cannot be identified, it is important to understand as many features as possible.

It is the propagation of the pulse that determines the pressure gradient during the flow at every location in the arterial tree, so it is important to begin the mathematical analysis of arterial fluid mechanics with a description of this propagation. The most concise and easily comprehensible outline of the subject is that by Lighthill (1975, chapter 12), and I shall frequently refer to his account in what follows.

One-dimensional theory, uniform tube, inviscid fluid

Basic theory

It is necessary, as in most branches of applied mathematics, to analyse a simple model before introducing the many modifying features present in reality. We therefore start by considering the propagation of pressure waves in a straight, uniform, elastic tube, whose undisturbed cross-sectional area and elastic properties are independent of the longitudinal coordinate, x. We also take the blood to be inviscid, as well as being homogeneous and incompressible (density ρ); the last two assumptions are made throughout this book. The neglect of viscosity is based on the observation (§§ 1.2, 1.3) that the velocity profiles in large arteries are approximately flat, suggesting that the effect of viscosity is confined to thin boundary layers on the walls; this is confirmed mathematically below. We further suppose that the wavelengths of all disturbances of interest are long compared with the tube diameter, so that the velocity profile will remain flat at all times, and the motion of the blood can be represented by the longitudinal velocity component u(x, t), where t is the time.

Introduction

All the velocity profiles measured in arteries (and reported in chapter 1), almost all the profiles measured in models or casts of arterial junctions (chapter 5) and all direct measurements of wall shear-rates in models have been obtained by the use of a hot-film anemometer (or its close relation, an electrochemical shear probe). Therefore it is important to understand how such a device operates, particularly since the main justification for the detailed theoretical analysis of flow in bends and bifurcations (chapters 3 to 5) rests on the claim that hot-film anemometry is not at present capable of the accurate measurement of unsteady wall shear in arteries.

A constant-temperature hot-film anemometer consists of a thin metallic (usually gold) film mounted flush with the surface of an insulated solid probe, which is inserted into the fluid whose velocity is to be measured. The temperature of the film is maintained by an electronic feedback circuit at a fixed value, T1, slightly higher than the temperature of the fluid, T0, which is also assumed to be constant. The power required to maintain it is proportional to the rate at which heat is lost to the fluid, which is in turn related to the velocity of the fluid flowing past the probe. In steady flow, this latter relation is obtained by calibration in known flows, after which the probe can, in principle, be used in any steady flow of the same fluid.

The overall arrangement of the mammalian cardiovascular system can be summarised briefly as follows. The heart is composed of four chambers arranged in two pairs. The thin-walled atrium on each side is connected through a valved orifice to a thick-walled muscular ventricle; each ventricle connects in turn to a major distributing artery, the mouth of which is again guarded by a valve. The left ventricle is the thicker and leads to the aorta (diameter about 2.5 cm in man), through which oxygenated blood is distributed to the tissues of the body. Large arteries branch off the aorta, smaller ones branch off them, and so on for many subdivisions; the number of branchings along any pathway depends on the particular organ being supplied. The final subdivisions of the arterial tree are the arterioles, which have very muscular walls and internal diameters in the range 30–100 μm. These vessels give rise to the capillaries (diameters down to 4 or 5 μm) across the walls of which the principal exchange of fluids and metabolites between blood and the tissues takes place. The blood passes from the capillaries into the smallest veins (venules) and thence into a converging system of increasingly larger veins, finally merging into the superior and inferior venae cavae which join directly to the right atrium of the heart. (An exception to this pattern is the circulation in the heart muscle itself, which drains directly into the right atrium.)

Some knowledge of fluid mechanics is required before the circulation of the blood can be understood. Indeed, the single fact that above all others convinced William Harvey (1578–1657) that the blood does circulate was the presence in the veins of valves, whose function is a passive, fluid mechanical process. He saw that these could be effective only if the blood in the veins flowed towards the heart, not away from it as proposed by Galen (129–199) and believed by the European medical establishment until Harvey's time. Harvey was also the first to make a quantitative estimate of the output of blood from the human heart and this, although a gross underestimate (36 oz, i.e. about 1 litre, per minute instead of about 5 litres per minute) was largely responsible for convincing the sceptics that the arterial blood could not be continuously created in the liver and, hence, that it must circulate.

The earliest quantitative measurements of mechanical phenomena in the circulation were made by Stephen Hales (1677–1761) who measured arterial and venous blood pressure, the volume of individual chambers of the heart and the rate of outflow of blood from severed veins and arteries, thereby demonstrating that most of the resistance to blood flow arises in the microcirculation. He also realised that the elasticity of the arteries was responsible for blood flow in veins being more or less steady, not pulsatile as in arteries.

Flow in symmetric bifurcations

Most branchings in the cardiovascular system are asymmetric, the only major exception in man being the bifurcation where the aorta divides to form the iliac arteries. This is in contrast to the bifurcating airways of the lung, for which the assumption of symmetry is more appropriate, and which have been the subject of extensive research (Pedley, 1977). Furthermore, the precise definition of an asymmetric bifurcation requires the specification of several more parameters than that of a symmetric one (e.g. the ratios of the flow-rates in, and the diameters of, the two daughter tubes, as well as the different angles of branching). There is, therefore, considerably more fluid mechanical information available on the subject of symmetric bifurcations, and this chapter begins with a survey of it (taken largely from the review by Pedley, 1977). It should be said at the start, however, that the problem is still very complicated, and most of the data have been obtained experimentally not theoretically, with steady rather than unsteady flow. Clearly much work remains to be done.

In all the investigations described in this section and the next, the geometry of the bifurcations is taken to be fully three-dimensional. There has been relatively extensive theoretical and experimental work on two-dimensional bifurcations, but since that geometry rules out all secondary motions it is unlikely to have much relevance to the cardiovascular system.

We now turn to the second main feature of the thoracic aorta: its curvature. The aim is to describe flow near the entrance of a curved tube in the same way that the previous section described flow near the entrance of a straight tube. However, we immediately come up against the major difficulty that the fully developed flow to which the entry flow tends, and which in a straight tube is Poiseuille flow (the mean flow) plus an easily calculated oscillatory component, is very complicated, and even the steady component is not yet completely understood. In the next three sections, therefore, we concentrate on fully developed flow in curved tubes, leaving a discussion of entry flow to §§ 4.4 and 4.5.

The reason why the flow in a curved tube is difficult to calculate lies in the fact that the motion cannot be everywhere parallel to the curved axis of the tube, but transverse (or secondary) components of velocity must be present. This follows because in order for a fluid particle to travel in a curved path of radius R with speed w it must be acted on by a lateral force (provided by the pressure gradients in the fluid) to give it a lateral acceleration w2/R. Now the pressure gradient acting on all particles will be approximately uniform, but the velocity of those particles near the wall will be much lower than that of particles in the core, as a result of the no-slip condition.

In this chapter we will consider the processes of formation of mixed layers by externally driven turbulence, and entrainment of fluid across the interfaces bounding such layers. Mixing across a density interface has so far been treated only in the special context of double-diffusive convection (§8.3), but more general stirring mechanisms must now be discussed. The problems of interest here may be identified with cases (d) or (e) of fig. 4.19, in which stirring at one level is used to produce mixing across an interface located some distance below or above the source of turbulent energy.

Various laboratory experiments which have shed light on the mechanism of entrainment at a density interface will be described first. Mixing can be generated by mechanical stirring, or by the production of a mean turbulent flow in the surface layer, and both methods have been used. The observed structure of the surface layers of the ocean and atmosphere discussed in §10.1 suggest that mixing in those regions is dominated by the boundary processes. The laboratory results can immediately be applied to these geophysical examples, and, with the addition of results from the earlier chapters on convection, they can be extended to take account of convective as well as mechanical mixing. It is assumed throughout this chapter that the mixing processes under consideration can be treated as one-dimensional in depth, implying that mixing is uniform (in the mean) over a large horizontal area, with no significant contribution from large scale lateral convection.

The mechanisms responsible for mixing in the interior of a stratified fluid are even less well understood than those described in chapter 9, since the sources of energy are not so obvious, and several different processes must be taken into account simultaneously. The important ideas have already been introduced in earlier sections, but it seems appropriate in this final chapter to take a broader and less detailed view, and to consider together the whole array of mixing phenomena which can be relevant in large natural bodies of stratified fluid. Geophysical examples have figured prominently in this book, and indeed the range of subject matter has been chosen with this final synthesis in mind. The basic facts requiring explanation are somewhat scattered, however, and in order to collect them together and to define the problems to be treated here, a brief summary will now be given of the observed structure of the ocean and atmosphere.

This field is developing rapidly, and the interpretation given here must necessarily be a somewhat tentative and personal one. Nevertheless, it seems useful to sketch how our present knowledge of the separate components can be fitted into a self-consistent picture, at the same time extending some of the earlier arguments so that they can be applied in this wider context.

The observational data

Routine density profiles made using reversing bottles and thermometers show that the ocean is everywhere stably stratified, except for limited regions where bottom water forms intermittently as the water column becomes convectively unstable.

A comparatively recent development in the field of convection has been the study of fluids in which there are gradients of two (or more) properties with different molecular diffusivities. When the concentration gradients have opposing effects on the vertical density distribution, a number of surprising things can happen, and these are the subject of the present chapter. The phenomena were first studied with an application to the ocean in mind (see §8.2.4), and because heat and salt (or some other dissolved substance) are then the relevant properties, the process has been called ‘thermohaline’ or ‘thermosolutal’ convection. Related effects have now been observed in the laboratory using a pair of solutes, and in solidifying metal alloys, and the name ‘double-diffusive convection’ has been chosen to encompass this wider range of phenomena.

The stability problem will first be reviewed, somewhat more fully than was done in previous chapters because of the comparative novelty of the double-diffusive phenomena. It will then be shown that when two components contribute to the vertical density gradient, a series of steps tends to form, with well-mixed layers separated by sharper density interfaces. The detailed structure of these interfaces and measurements of the coupled fluxes across them will also be described.

The stability problem

The mechanism, of instability

In such a system with opposing gradients, the existence of a net density distribution which decreases upwards is not a guarantee of stability.

Buoyancy forces arise as a result of variations of density in a fluid subject to gravity, and produce a wide range of phenomena of importance in many branches of fluid mechanics. Progress in this field has been made largely through the desire to solve very practical problems, arising for instance in meteorology or in hydraulic engineering. This emphasis on particular applications has meant that parallel developments have often been made in different disciplines without much cross reference to related work, and some results, well understood in one context, are less familiar in another where they might be used to advantage. In this book I have attempted to write a coherent account of the various fluid motions which can be driven or influenced by the presence of small density differences. It is intended as a general introduction to the subject and its literature, in which the physical understanding of the phenomena is emphasized, rather than the applications on the one hand or detailed mathematical theory on the other.

The selection of subject matter must always be a personal one, however, and my own research interests have certainly influenced the topics chosen and the amount of space given to each of them. I have worked with laboratory models of small scale processes in the ocean and atmosphere, and so laboratory and geophysical examples come most readily to mind, but comparisons have also been made with results from various fields of engineering where possible.