Both latitude and l.o.d. observations show seasonal oscillations that rise well above the noise level of the astronomical spectra (figures 1.1 and 1.2). The principal seasonal oscillation in the wobble is the annual term which has generally been attributed to a geographical redistribution of mass associated with meteorological causes. Jeffreys, in 1916, first attempted a detailed quantitative evaluation of this excitation function by considering the contributions from atmospheric and oceanic motion, of precipitation, of vegetation and of polar ice. Jeffreys concluded that these factors explain the observed annual polar motion, a conclusion that is still valid today, although the quantitative comparisons between the observed and computed annual components of the pole path are still not satisfactory. These discrepancies may be a consequence of (i) inadequate data for evaluating the known excitation functions, (ii) the neglect of additional excitation functions, (iii) systematic errors in the astronomical data, or (iv) year-to-year variability in the annual excitation functions. The semi-annual term in the wobble is much smaller than the annual term, and the astronomical evidence for it is not compelling. This could be expected from the nature of the solution (4.3.12) of the polar motion for a sinusoidally varying excitation: for equal-magnitude excitation functions at the annual and semi-annual frequencies and Q ≃ 100, the annual pole shift will be some eight times larger than the semi-annual pole shift simply because it is closer to the Chandler resonance.

Observations of the Earth's rotation

The astronomical observations

Precise measurements of the time elapsed between two consecutive transits of a star across a meridian determine the l.o.d. with respect to some uniform time scale. Measurement of l.o.d. therefore involves two processes: the astronomical observation of the star transits, and the establishment of a reference time. Changes in the l.o.d. are small, of the order of 10−8, and observations of numerous stars from a number of observatories, over several nights, are required in order that the signal rises above the noise of the measuring process. Thus, what is observed by astronomers is the integrated amount by which the Earth is in advance or behind after a number of days, compared with the uniform time scale. Time kept by the Earth is referred to as universal time (UT)). Strictly speaking, the time interval between successive star transits defines sidereal time, whereas UT is a mean solar time. The relation between these two systems is quite complex and is discussed in detail in most textbooks on spherical astronomy (see, for example, Smart 1962; Woolard & Clemence 1966). For geophysical purposes, the observed quantity can be considered to be the universal time. Star transits are observed with respect to an Earth-fixed meridian, defined by the station coordinates and the body-fixed x-axes.

Introduction

Tidal dissipation and its consequences on the lunar orbit and Earth's rotation have become a classic problem, yet there is probably no other subject in geophysics that has had as long a history of frustration and still attracts very considerable attention from geophysicists, astronomers and oceanographers. That this is so is as much a reflection of the fascination of the subject as an indication of a problem of some importance in understanding the origin and dynamical evolution of the Moon. In his Harold Jeffreys lecture, entitled ‘Once Again – Tidal Friction’, Walter Munk introduced the subject by saying that in 1920 it appeared Jeffreys had solved the problem of tidal dissipation but that we have gone backwards ever since (Munk 1968). Now, some 10 yr later, we have gone full circle, for once again there is agreement between observations and theory. Future new developments may mean that we have to go through the cycle of agreement and disagreement once more, before we can finally conclude that the subject is closed. But if these new results, such as those that may come from lunar laser-ranging analysis, disagree with our present knowledge we can always use Jeffreys' dictum ‘[The analysis] covers only a short interval of time and will probably be improved’ (Jeffreys 1973).

A discussion on the Earth's rotation is conveniently separated into three parts: (i) precession and nutation, (ii) polar motion and (iii) changes in length-of-day (l.o.d.). Precession and nutation describes the rotational motion of the Earth in space and is a consequence of the lunar and solar gravitational attraction on the Earth's equatorial bulge. Polar motion, or wobble, is the motion of the rotation axis with respect to the Earth's crust. Changes in the l.o.d. are a measure of a variable speed of rotation about the instantaneous pole. We are primarily concerned here with the last two components of the motion.

The standard treatment of precession and nutation for a rigid Earth is that by Woolard (1953), but a more comprehensive treatment is by Kinoshita (1977). Observational evidence is discussed by Federov (1963). Further discussions are found in the symposium proceedings edited by Federov, Smith & Bender (1977). The main discrepancies between the observed and theoretical nutations are consequences of the presence of the liquid core. The problem of the precession and nutation of a shell with a liquid-filled spheroidal cavity continues to draw the attention of mathematicians and geophysicists (see, for example, Roberts & Stewartson 1965; Busse 1968; Toomre 1966, 1974). It is touched upon briefly in chapter 3.

Perturbations in the rotation from the rigid body state are caused by motions and deformations of the Earth by a variety of forces. Chapter 2 discusses some general aspects of the deformations of the solid part of the Earth.

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.