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7864 results in Fluid dynamics and solid mechanics

Page 387 of 394

  • XIII - Water waves

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 289-324

  • Summary

    Background

    It is part of everyone's experience that waves and oscillations occur on the surface of water, and they indeed form a wide range of phenomena for any mathematical theory to deal with. Ocean waves from the tsunami – the ‘tidal wave’ which is generated by an earthquake and which may travel at high speed for huge distances – and the tides themselves, down to small ripples in a calm sea. Other types of disturbance occur on rivers, such as the tidal bore which travels up the River Severn, and the surface waves which can be caused by an irregularity on the bottom. A lake may have a ‘seiche’ in it, which is an oscillation to-and-fro of the surface of the fluid (very much like the same phenomenon in a bath). On a wet day you may see waves on thin sheets of water flowing down a road.

    Real waves have awkward properties: they may break, or have their tops blown off by the wind; they may be suppressed by a small amount of oil on their surfaces; or be running over water of non-uniform depth. We model the waves that you see on water in rather a drastic fashion, to give approachable mathematics, and get some very interesting results; but we cannot cope with too much reality, and so will not predict every visible effect.

    This chapter deals with the easy aspects of waves in oceans, lakes and smaller containers. We cannot deal here with such interesting topics as:

  • (i) the generation of waves by the wind;

  • (ii) the interactions of waves which meet;

  • (iii) the changes in waves as they approach the shore;

  • (iv) waves due to ships, or interaction of waves and ships;

  • (v) energy generation from waves.

    • Some aspects of waves on rivers will be covered in Chapter XV, so we assume in this chapter that the water is at rest apart from near its disturbed surface.

  • XI - Potential theory

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 205-247

  • Summary

    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.

  • (i) Flow of air round a streamlined aeroplane wing or body. The aircraft flies into air that is effectively at rest, and the boundary layers and wake are thin enough to be neglected at a first approximation. Such flows will be discussed in detail in Chapter XVII.

  • (ii) Waves on the surface of reasonably deep water. The boundary condition at the surface does not bring in a noticeable boundary layer because the air is so much less dense than the water. Water waves are dealt with in Chapter XIII.

  • (iii) Sound waves in air (or water) are of such short time scale that diffusive effects have no time to act, and hence irrotational flow is an appropriate model in many cases. Sound waves are considered in Chapter XII.

  • (iv) In certain cases irrotational flow theory is useful for some regions of the flow of a uniform stream past a blunt body.

  • VII - Thermodynamics

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 111-126

  • Summary

    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:

  • (i) density ρ;

  • (ii) velocity v, three components;

  • (iii) pressure p.

    • So far there are only four equations in sight for these functions, the equation of mass-conservation, and the equation of momentum change (three components). We clearly need something like an energy equation, which must include energy of compression of a gas; and it is well known that compressing a gas heats it (try pumping a bicycle tyre), so that temperature will come in as well, as a related quantity to heat energy. In order to discuss these ideas clearly, we must set up some, but not too much, of the theory of thermodynamics. By the end of the chapter we will have three standard mathematical models to work with, one for liquids and two for gases under reasonable conditions; but we should also have a good idea of when and why these models are adequate – a model that is not understood is a model that will be used in the wrong way.

    A new area of physical theory will have new observables, and the mathematical theory will bring in new functions which are not observable and yet which are the best ones for framing the theory. Our early stages in fluid dynamics are helped by the fact that density and velocity are common measurable concepts in other forms of mechanics; and the idea of a stress tensor, which though not directly observable is extremely useful, is not too unlikely a generalisation from a force vector. But thermodynamics brings in ideas which may be quite new to those without much background in physics, so the early stage of thermodynamics needs careful attention, to note where axioms based on experiments are being brought in, and where new definitions are being made.

  • Preface

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp ix-x

  • Summary

    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.

  • Preface

  • P. G. Drazin
  • Book:
    Solitons
    Published online:
    25 March 2010
    Print publication:
    21 July 1983, pp vii-viii

  • Summary

    The theory of solitons is attractive; it is wide and deep, and it is intrinsically beautiful. It is related to even more areas of mathematics and has even more applications to the physical sciences than the many which are indicated in this book. It has an interesting history and a promising future. Indeed, the work of Kruskal and his associates which gave us the inverse scattering transform is a major achievement of twentieth-century mathematics. Their work was stimulated by a physical problem and is also a classic example of how computational results may lead to the development of new mathematics, just as observational and experimental results have done since the time of Archimedes.

    This book originated from lectures given to classes of mathematics honours students at the University of Bristol in their final year. The aim was to make the essence of the method of inverse scattering understandable as easily as possible, rather than to expound the analysis rigorously or to describe the applications in detail. The present version of my lecture notes has a similar aim. It is intended for senior students and for graduate students, phyicists, chemists and engineers as well as mathematicians. The book will also help specialists in these and other subjects who wish to become acquainted with the theory of solitons, but does not go as far as the rapidly advancing frontier of research. The fundamentals are introduced from the point of view of a course of advanced calculus or the mathematical methods of physics.

Solitons
  • P. G. Drazin
  • Published online:
    25 March 2010
    Print publication:
    21 July 1983
  • A 'soliton' is a localized nonlinear wave of permanent form which may interact strongly with other solitons so that when they separate after the interaction they regain their original forms. This textbook is an account of the theory of solitons and of the diverse applications of the theory to nonlinear systems arising in the physical sciences. The essence of the book is an introduction to the method of inverse scattering. Solitary waves, cnoidal waves, conservation laws, the initial-value problem for the Korteweg-de Vries equation, the Lax method, the sine-Gordon equation and Backlund transformations are treated. The book will be useful for research workers who wish to learn about solitons as well as graduate students in mathematics, physics and engineering.

  • Chapter 1 - THE KORTEWEG–DE VRIES EQUATION

  • P. G. Drazin
  • Book:
    Solitons
    Published online:
    25 March 2010
    Print publication:
    21 July 1983, pp 1-12

  • Summary

    The discovery of solitary waves

    This book is an introduction to the theory of solitons and to the applications of the theory. Solitons are a special kind of localized wave, an essentially nonlinear kind. We shall define them at the end of this chapter, describing their discovery by Zabusky & Kruskal (1965). A solitary wave is the first and most celebrated example of a soliton to have been discovered, although more than 150 years elapsed after the discovery before a solitary wave was recognized as an example of a soliton. To lead to the definition of a soliton, it is helpful to study solitary waves on shallow water. We shall describe briefly in this section the properties of these waves, and then revise the elements of the theory of linear and nonlinear waves in order to build a foundation of the theory of solitons. Let us begin at the beginning, and relate a little history.

    The solitary wave, or great wave of translation, was first observed on the Edinburgh to Glasgow canal in 1834 by J. Scott Russell. Russell reported his discovery to the British Association in 1844 as follows:

    I believe I shall best introduce this ph ænomenon by describing the circumstances of my own first acquaintance with it.

Page 387 of 394