The role of the theory of flow of an inviscid fluid

We have completed a study of the general effects of the viscosity of the fluid, and are now in a position to take advantage of the fact that the viscosities of the common fluids air and water are quite small. The Reynolds number ρLU/µ (in the notation of §4.7) is usually a measure of the ratio of the representative magnitude of inertia forces to that of viscous forces; and, when this Reynolds number is large compared with unity, viscous forces frequently play a negligible part in the equation of motion over nearly all the flow field. In many cases in which separation of the boundary layer from a rigid boundary does not occur, the flow field tends to the form appropriate to an inviscid fluid, as ρLU/µ → ∞, over the whole of the region occupied by the fluid, and the fact that viscous forces remain significant in certain thin layers in the fluids however large the Reynolds number may be, is of little consequence for many purposes. However, in cases in which the boundary layer does separate from a rigid boundary, the limit is a singular one and, although the region of fluid in which viscous forces are significant may decrease in size to zero as ρLU/µ ∞, the limiting form of the flow field is not the same as that appropriate to a completely inviscid fluid.

While teaching fluid dynamics to students preparing for the various Parts of the Mathematical Tripos at Cambridge I have found difficulty over the choice of textbooks to accompany the lectures. There appear to be many books intended for use by a student approaching fluid dynamics with a view to its application in various fields of engineering, but relatively few which cater for a student coming to the subject as an applied mathematician and none which in my view does so satisfactorily. The trouble is that the great strides made in our understanding of many aspects of fluid dynamics during the last 50 years or so have not yet been absorbed into the educational texts for students of applied mathematics. A teacher is therefore obliged to do without textbooks for large parts of his course, or to tailor his lectures to the existing books. This latter alternative tends to emphasize unduly the classical analytical aspects of the subject, and the mathematical theory of irrotational flow in particular, with the probable consequence that the students remain unaware of the vitally important physical aspects of fluid dynamics. Students, and teachers too, are apt to derive their ideas of the content of a subject from the topics treated in the textbooks they can lay hands on, and it is undesirable that so many of the books on fluid dynamics for applied mathematicians should be about problems which are mathematically solvable but not necessarily related to what happens in real fluids.

Specification of the flow field

The continuum hypothesis enables us to use the simple concept of local velocity of the fluid, and we must now consider how the whole field of flow may be specified as an aggregate of such local velocities. Two distinct alternative kinds of specification are possible. The first, usually called the Eulerian type, is like the specification of an electromagnetic field in that the flow quantities are defined as functions of position in space (x) and time (t), The primary flow quantity is the (vector) velocity of the fluid, which is thus written as u(x, t). This Eulerian specification can be thought of as providing a picture of the spatial distribution of fluid velocity (and of other flow quantities such as density and pressure) at each instant during the motion.

The second, or Lagrangian type of specification, makes use of the fact that, as in particle mechanics, some of the dynamical or physical quantities refer not only to certain positions in space but also (and more fundamentally) to identifiable pieces of matter. The flow quantities are here defined as functions of time and of the choice of a material element of fluid, and describe the dynamical history of this selected fluid element.

Solidss liquids and gases

The defining property of fluids, embracing both liquids and gases, lies in the ease with which they may be deformed. A piece of solid material has a definite shape, and that shape changes only when there is a change in the external conditions. A portion of fluid, on the other hand, does not have a preferred shape, and different elements of a homogeneous fluid may be rearranged freely without affecting the macroscopic properties of the portion of fluid. The fact that relative motion of different elements of a portion of fluid can, and in general does, occur when forces act on the fluid gives rise to the science of fluid dynamics.

The distinction between solids and fluids is not a sharp one, since there are many materials which in some respects behave like a solid and in other respects like a fluid. A ‘simple’ solid might be regarded as a material of which the shape, and the relative positions of the constituent elements, change by a small amount only, when there is a small change in the forces acting on it. Correspondingly, a ‘simple’ fluid (there is no one term in general use) might be defined as a material such that the relative positions of the elements of the material change by an amount which is not small when suitably chosen forces, however small in magnitude, are applied to the material.

Introduction

In this chapter the discussion of the flow of a viscous incompressible fluid of uniform density will be continued.

The values of the kinematic viscosity for air and water are so small that the Reynolds numbers for most of the flow systems of importance, whether in nature or in technology or in the laboratory, are very much larger than unity. A Reynolds number of 103 is attained in air at 20°C when UL has the very modest value 150 cm2/sec, and in water when UL is only 10 cm2/sec, where U and L are representative values for the velocity variations and the distances over which they occur in the flow system concerned. Such small values of the product UL are so readily and so often exceeded that flow at large Reynolds number must be regarded as the standard case.

The largeness of R = UL/v has its implications for the relative importance of the various terms in the equation of motion, as was seen in §4.7. Provided the non-dimensional quantities |Du′/Dt′| and |∇2u′| are both of order unity over most of the flow field (which would of course exclude some simple flow fields, such as steady unidirectional flow in a tube, in which the fluid acceleration is zero everywhere), R is a measure of the ratio of the magnitudes of inertia and viscous forces acting on the fluid; and a flow field for which R ≫ 1 is presumably one in which inertia forces are much greater than viscous forces over most of the field.

Observations and methods

In Chapters 6–8 we have seen how travel time curves depend on the characteristics of the media through which the seismic waves propagate and that the velocity's distribution with depth can be deduced from them. In this chapter we will apply these results to observations regarding the Earth and discuss the results concerning its internal structure obtained. For short distances (less than 1000 km) we can use the flat-Earth approximation and plane geometry. Seismic waves for that range of distances give information on depths of about 100 km, that is, on the crust and part of the upper mantle. For this range of distances we can apply the theory derived in Chapters 6 and 7. For greater distances the spherical shape of the Earth must be considered, so the results of Chapter 8 must be applied. The effects due to the deviations of the form of the Earth from a sphere, that is, mainly its flatness, can be taken into account by using corrections to the spherical model. In seismology these effects are not very important.

The first seismic waves used for the study of the Earth's structure were those produced by earthquakes. Even today this is the main source of information, especially for the deep interior. Among the first tables and curves of travel times of seismic waves were those of Oldham, who in 1906 deduced the existence of the Earth's core. These tables were completed by Zöppritz and Turner and later, in 1914, by Gutenberg.

Parameters and observations

The determination of the source mechanism of earthquakes consists in finding the parameters of the model used in its representation, whose number depends on its complexity. The localization of the source is given by four parameters (φ, λ, h, and t0), the coordinates of the epicenter, depth of focus, and origin time (section 15.2). Generally, in determinations of focal mechanisms, these parameters are assumed to be known, although in some methods some (the focal depth) or all (the centroid) are determined anew. The size given by the magnitude M is independently determined, but, in many methods, the seismic moment M0 is evaluated as part of the mechanism.

The simplest models of source mechanism are those with point sources. For a point shear dislocation, the orientation of the mechanism is given by the angles φ, δ, and λ (section 15.1). For the equivalent double-couple force model, the orientation is given by those of the X and Y axes, that is, by the angles ⊗x, φy, and φy, or by the P and T axes, angles ⊗P, φP, and φT (section 16.2). If the source is not a shear fracture, it can be represented by five components of the moment tensor Mij (for a deviatoric source) or six if there are changes in volume (Chapter 17). Including its size, a shear-fracture point source or DC source is given by four parameters (M0, φ, δ, and λ) plus four for its location, that is a total of eight parameters.

The historical development

The term seismology is derived from two Greek words, seismos, shaking, and logos, science or treatise. Earthquakes were called seismos tes ges in Greek, literally shaking of the Earth; the Latin term is terrae motus, and from the equivalents of these two terms come the words used in occidental languages. Seismology means, then, the science of the shaking of the Earth or the science of earthquakes. The term seismology and similar ones in other occidental languages (séismologie, sismologia, sismología, Seismologie, etc.) started to be used around the middle of the nineteenth century. Information about the main historical developments of seismology can be found in each chapter; a very short overview is given in the following paragraphs.

In antiquity, the first rational explanations of earthquakes, beyond mythical stories, are from Greek natural philosophers. Aristotle (in the fourth century BC) discussed the nature and origin of earthquakes in the second book of his treatise on meteors (Meteorologicorum libri IV). The term meteors was used by the ancient Greeks for a variety of phenomena believed to take place somewhere above the Earth's surface, such as rain, wind, thunder, lightning, comets, and also earthquakes and volcanic eruptions. The term meteorology derives from this word, but in modern use it refers only to atmospheric phenomena. Aristotle, following other Greek authors, such as Anaxagoras, Empedocles, and Democritus, proposed that the cause of earthquakes consists in the shaking of the Earth due to dry heated vapors underground or winds trapped in its interior and trying to leave toward the exterior.