1

Particles and continuous materials

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.

These matters were not fully resolved until the seventeenth century when Isaac Newton formulated his three laws of motion, which form the basis of all the mechanics described in this book. The laws refer to the motion of individual particles, which are defined as objects with mass (so that, for example, the earth exerts a gravitational pull on them), but which occupy single points (that is, they have no size). Of course, every real body, even one as small as an atom or an electron, has a finite size, but the laws of particle mechanics can be directly applied both to real bodies in isolation (like the arrow of Zeno's paradox, or the earth in its motion round the sun, or an individual red blood cell) and to extensive regions of continuous matter which can be deformed into different configurations. Examples of such deformable materials include all elastic solids, like steel, rubber, and blood vessel walls, and all fluids, like water, treacle, blood plasma and air. Both liquids and gases are described here as fluids, since the laws of motion are applied in exactly the same way to each.

Newton's laws can be applied to bodies of finite size because it can be proved that a body will move as if all of its mass and all the external forces acting on it were concentrated at one point. This point is called the centre of mass.* [*The centre of mass of a body is the same as its centre of gravity: if in the region of the earth's surface the body is suspended by a string successively attached to various parts of it, there is one point in the body through which the straight line formed by extending the line of the string downwards always passes. This point is the centre of gravity.] Thus the flight of the centre of mass of Zeno's arrow is the same as that of a particle of the same mass, acted on by the same forces of gravity and air-resistance. Similarly the motion through space of the moon, or the earth, or another planet, can be described by particle mechanics. So can the motion of the centre of mass of a blood cell, as long as the forces exerted on it by the surrounding plasma are known. However, the tumbling of a red cell, or the rotation of the earth about its axis, or any other motion of a body relative to its centre of mass, depends on the detailed shape of the body, and cannot be described as if the body were a particle.

The application of Newton's laws to the motion of continuous deformable materials is more difficult to justify. It is bound up with the implicit assumption that the fluids and solids we are interested in are continuous materials. In fact, physicists have long known that all matter is made up of molecules, bound together in various configurations by forces of various strengths,* [*In a solid, the intermolecular forces are very strong, and the molecules vary their relative positions only slightly; the spacing between molecules is comparable to their size. In a liquid, the intermolecular forces are less strong; molecules can move about readily (although their spacing is still comparable to their size), and they undergo frequent collisions. In a gas, the intermolecular forces are weak, and the spacing is large compared with molecular dimensions, although it is still a very small distance (approximately 3 x 10^-9 m (3 nm) for air at normal temperature and pressure).] and consisting of numbers of atoms. These in turn consist of central nuclei, surrounded by clouds of electrons, moving in orbits whose diameters are large compared with those of the nuclei. The motion of electrons round a nucleus is analogous to that of the planets round the sun, and like the solar system, most of an atom (and hence most matter) consists of empty space. Some typical dimensions are given in Table 1.1. It might be supposed that each nucleus, and each electron, or each atom, or even each molecule could be regarded as a particle, and its motion under the influence of the intermolecular forces deduced from Newton's Laws. However, in air at normal temperature and pressure, for example, there are roughly 10²⁰ molecules per cubic centimetre,  and the position of each one would have to be specified precisely. Such a task is virtually impossible. The fact that the spacing between molecules is usually very small compared with the dimensions of the natural or experimental regions of fluid whose motion we wish to describe (see Table 1.1) indicates how we can overcome the difficulty. We may suppose the material to be divided up into a large number of elements whose dimensions are very small compared with those of the region of interest, but which still contain a very large number of molecules. With regard to the experiment, such an element effectively occupies a point, and can therefore be considered as a particle; with regard to molecular motions, however, it is very large, and its overall properties, like its velocity, or the density of the material in it, can be obtained by averaging over all the molecules which comprise it. We are thus able to ignore the random nature of molecular motion and treat materials as continuous. Newton's laws can now be applied to each element of the material (called a fluid element, or fluid particle, when the material is a fluid), and a precise and useful description of the motion as a whole will emerge.

In blood there are some very large molecules (e.g. lipoproteins, diameter about 3-5 x 10^-8 m), and it flows in some very narrow tubes (some capillaries have a diameter as low as 4 x 10^-6 m), but even so the tube diameter is large compared with molecular dimensions. Thus blood plasma, for example, can be treated as a continuous fluid in the manner outlined above. Whole blood, however, cannot always be so treated, since it consists not only of plasma but also of large numbers of cells which amount to about 45 per cent of volume in normal man, and consist primarily of red blood cells (see Chapter 10). It would be convenient if the cells were small and numerous enough for their separate identity to be ignored, and their effect on the motion of whole blood, regarded as a continuous fluid, to be described in an average way. This is the case in large arteries (the diameter of the aorta, for example, is roughly 2000 times that of a red cell), but the diameter of a capillary is comparable to that of a red cell, and a description of flow in such small vessels must treat plasma and cells separately. To sum up, then, whole blood is effectively continuous in large vessels, but is not so in the microcirculation; plasma is continuous in both.

In Part I of this book, we shall develop the fundamental mechanics of continuous fluids and solids, although we must first outline Newton's laws of particle mechanics. The mathematical symbols which appear are used solely as a form of shorthand, facilitating the precise expression of mechanical laws. They are all explained in words wherever they first appear, and a reader who knows some calculus will find much of the notation familiar.

Chapter 2