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7987 results in Fluid dynamics and solid mechanics

Page 393 of 400

  • I - Mathematical preliminaries

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 7-24

  • Summary

    Background knowledge

    Quite a lot of mathematics is needed for this text, and most of it must be assumed to have been met elsewhere. However, it would be unfair to assume that you have already learned every mathematical technique, so some new methods will be introduced in sufficient detail as they become necessary. If you need to revise any of the topics mentioned below, do it now, before your lack of knowledge interferes with the fluid dynamics that is being expounded.

    (a) Vectors

    Fluid dynamics is about the motion of fluids, so velocities must come in; that is, the whole subject will be full of work with vectors. You must be confident in your use of the two products

    A · B and A × B,

    and in the use of components and unit vectors. These components may be those appropriate to cartesian axes or to polar directions; the important material about polar coordinates and directions is summarised below.

    (b) Functions of several variables

    The velocities in fluid dynamics will in general depend on position and time. e.g.

    v(x, y, z, t).

    For an example of this you can think of weather maps, which usually show wind velocities at the surface. These velocities are different for different parts of the country and change from day to day. So we shall certainly need to consider functions of several variables; whenever possible we shall reduce from the generality of four variables down to special cases involving only two, but we cannot often use only one variable.

    (c) Vector calculus

    Since the velocities change in space and time, we shall need vector calculus. For example, the divergence of v

    ∇·v or div v

    and the curl of v

    ∇ × v or curl v

    are most important. This whole area is vital, and so it is summarised below with special results that are needed in fluid dynamics.

  • Introduction

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 1-6

  • Summary

    Fluid dynamics

    There are two main reasons for studying fluid dynamics. Firstly, understanding (though in some places still only partial) can be gained of a great range of phenomena, many of which are of considerable complexity. And secondly, predictions can be made in many areas of practical importance which involve fluids.

    As we shall see later, a ‘fluid’ is a way of looking at a large collection of particles, so as to avoid dealing with each particle separately. One of the largest examples of such a collection of ‘particles’ is a galaxy, composed of a vast number of individual stars. A more obvious fluid composes the sun: the particles here are largely electrons and nucleons, and the fluid dynamics here is complicated by electromagnetic forces, nuclear reactions and radiation effects. Astrophysics provides another example of fluid motion in the solar wind, the outflow of (isolated) particles from the sun: this is a fluid in which the particles are very thinly spread, but it is a fluid which interacts importantly with the Earth's magnetic field and the upper layers of the atmosphere. Both atmosphere and magnetic field are much studied examples of fluid dynamics; climate predictions, weather forecasts and studies of local climate are of obvious interest, while the origin of the Earth's magnetic field in the inner motions of the Earth's material is rather less obvious, but no less interesting. You may list for yourself some of the physical phenomena associated with the existence of oceans, rivers, lakes and underground water.

    At this sort of scale one can start to see practical considerations coming to the fore. The engineer who designs a hydroelectric system must have a good knowledge of how water will behave and what forces it will exert. As a further example, an aeroplane (or a ship) is built by an interaction of past experience, mathematical calculation and testing of models in wind tunnels (or wave tanks).

  • Hints for exercises

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 483-507

  • Summary

    Chapter I

    1. Resolve in fig. I.1.

    2. Sketches need to be three-dimensional versions of fig. 1.2: you need either a length dz or a length corresponding to dλ and ds will no longer be in the plane of r and θ.

    Otherwise you need to find formulae corresponding to

    dx = dr cos θ − r sin θ dθ,

    and calculate ds2 from

    ds2 = dx2 + dy2 + dz2.

    3. The formulae for are in §2(c) You must differentiate these, and then express the result in terms of these unit vectors.

    4. Taylor's theorem in one-dimension is

    when (φ is smooth enough (twice differentiable with continuous second derivative is certainly enough), and where 0 < θ < 1. The last term on the right is less than some constant times h2 if φ″ is continuous on (x, x + h), because φ″ is then bounded in that interval.

    Remember that h2 = hihi = hkhk.

    5. (i) ∇ × (φA) has ith component

    (ii) When is ∂2u/∂x∂t = ∂2u/∂t∂x?

    (iii) ∇ · A = ∂Ai/∂xi, and here A = φ∇ψ, and ∇ψ has ith component ∂ψ/∂xi.

    (iv), (v) ε123 = + 1, ε213 = −1: look for terms which have these ε in them. Then use ∂2/∂x1∂x2 = ∂2/∂x2∂x1 for smooth functions.

    (vi) This is and use a theorem in §4(c).

    6. ∇ × ∇ = 0, and write what remains as

    ∇ × (A × B),

    which works out rather like Q5(vi). Finally put the proper value back for B.

    7. Use Q5(iii), and the divergence theorem from §5(b).

    8. For ∇ · A proceed as in §6(a), and use formulae for derivatives of from §2(c) and Q3. For ∇ × A evaluate the determinant in §6(c).

  • XII - Sound waves in fluids

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 248-288

  • Summary

    Background

    The basic facts of the propagation and generation of sound are well known. Sound is a pressure disturbance in air (or water) which travels at a rather high speed. Sound is generated in regions of fluid motion; sometimes a motion of solid boundaries is involved, as in a drum; sometimes not, as in a flute. Sound is received in our ears (or by a microphone) where the pressure oscillations are converted into electrical signals.

    The speed of sound is typically around 330–40m s−1 in still air near ground level, and around 1400–50ms−1 in water. Sound also travels through solid materials in the form of elastic waves; we cannot do the theory for such waves here, though it is of the same general type as what we shall do. Often the speed of sound in solids is much higher than the speed of sound in air.

    The pressure fluctuations in sound in air are typically between 10−4 and 1 N m−2, so that the ratio of pressure fluctuations to ambient pressure in the atmosphere lies between

    10−9 and 10−5.

    The smallness of this quantity is one of the basic facts in the simple theory of sound. It is quite possible to get much higher pressure fluctuations, in explosions or in jet engines, so that the ratio of pressures is not small: the simple theory of sound cannot apply to such cases.

    The frequencies of audible sound waves lie between about 20 Hz and 2 × 104 Hz. Below about 20 Hz, oscillations are felt by the body rather than heard by the ear, and so may not be classed as sound waves, though they are still described by the same mathematics – sometimes the word infrasound is used. Similarly, oscillations above 2 × 104 Hz cannot be heard by everyone; again, the mathematics is mainly the same, and the word ultrasound is used.

  • 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.

Page 393 of 400