The velocity potential and Laplace's equation

(a) Occurrence of irrotational flows

If a closed curve in an in viscid fluid for which ρ = f(p) has at some time no circulation round it, then by Kelvin's theorem there is never any circulation round the curve. Of course there are no such ‘ideal’ fluids around, but we have seen that at high Reynolds number and away from boundaries and other awkward regions, a fluid will behave in a near enough ideal fashion. Hence we expect that there can be large regions of flow which have no circulation round any circuit, and hence no vorticity. Thus it is well worth discussing irrotational flows, which have

∇ × v = 0.

Naturally we must not try to use irrotational flow theory in those regions where we have already seen vorticity to be inevitably developed from the no-slip condition and the diffusive action of viscosity, such as in boundary layers, wakes, eddies and enclosed regions. But away from these regions we can use irrotational flow theory provided that the flow is started, or arrives, with no vorticity. For example, the following flows are closely irrotational.

Basic ideas and equations of state

(a) Functions of state

In our discussions so far we have introduced five functions to describe the motion of a fluid:

An author makes his excuses in a preface, so here are mine. For a number of years at Bristol University we tried to find a suitable text to introduce fluid dynamics to second year mathematics students, and failed. The modern texts with the ‘right’ attitude to the subject were too hard for a first course, the older texts were dominated by potential theory and unrealistic examples. This text has been tried in draft form for several years on our students, and has been judged ‘hard, but interesting’. New work in mathematics is always hard, but I believe that the level chosen here is a suitable one.

I apologise to my colleagues for the gross over-simplification of their work and their subject which is committed in this book. And also for the errors and misapprehensions – students, beware! all texts have mistakes in them. I thank my colleagues for helpful comments and discussions over many years; I also thank a succession of seminar speakers for maintaining my awareness of the full range of fluid dynamics.

Introduction

The single most obvious way in which the relationship between language and context is reflected in the structures of languages themselves, is through the phenomenon of deixis. The term is borrowed from the Greek word for pointing or indicating, and has as prototypical or focal exemplars the use of demonstratives, first and second person pronouns, tense, specific time and place adverbs like now and here, and a variety of other grammatical features tied directly to the circumstances of utterance.

Essentially deixis concerns the ways in which languages encode or grammaticalize features of the context of utterance or speech event, and thus also concerns ways in which the interpretation of utterances depends on the analysis of that context of utterance. Thus the pronoun this does not name or refer to any particular entity on all occasions of use; rather it is a variable or place-holder for some particular entity given by the context (e.g. by a gesture). The facts of deixis should act as a constant reminder to theoretical linguists of the simple but immensely important fact that natural languages are primarily designed, so to speak, for use in face-to-face interaction, and thus there are limits to the extent to which they can be analysed without taking this into account (Lyons, 1977a: 589ff).

The importance of deictic information for the interpretation of utterances is perhaps best illustrated by what happens when such information is lacking (Fillmore, 1975: 38–9).

Introduction

In this Chapter we shall be centrally concerned with the organization of conversation. Definitions will emerge below, but for the present conversation may be taken to be that familiar predominant kind of talk in which two or more participants freely alternate in speaking, which generally occurs outside specific institutional settings like religious services, law courts, classrooms and the like.

It is not hard to see why one should look to conversation for insight into pragmatic phenomena, for conversation is clearly the prototypical kind of language usage, the form in which we are all first exposed to language – the matrix for language acquisition. Various aspects of pragmatic organization can be shown to be centrally organized around usage in conversation, including the aspects of deixis explored in Chapter 2 where it was shown that unmarked usages of grammatical encodings of temporal, spatial, social and discourse parameters are organized around an assumption of co-present conversational participants. Presupposition may also be seen as in some basic ways organized around a conversational setting: the phenomena involve constraints on the way in which information has to be presented if it is to be introduced to particular participants with specific shared assumptions and knowledge about the world. The issues touch closely on the distinction between given and new (see e.g. Clark & Haviland, 1977), and concern constraints on the formulation of information (that is, the choice of just one out of the indefinitely many possible descriptions of some entity – see Schegloff, 1972b), both of which are important issues in conversational organization.