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36569 results in Cambridge Textbooks

Page 1813 of 1829

  • II - Physical preliminaries

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 25-31

  • Summary

    Background knowledge

    No book like this can go right back to the very beginning, so we must expect you to have some general ideas from earlier study of applied mathematics or physics. In particular it will help considerably if you have some acquaintance with the following.

  • (i) Newtonian dynamics, including energy, momentum and angular momentum (moment of momentum).

  • (ii) The physics of any sort of wave motion, including the ideas of standing and travelling waves and reflection at barriers.

  • (iii) The atomic nature of matter, and especially that a gas is made up of well-separated rapidly moving molecules.

    • Naturally it will be a help if you have further background, but there will be quite a lot of explanation for everything that is essential. If you find there is not enough description in some areas, you must go to other texts.

    Mathematical modelling

    The activity of modelling (in a mathematical sense) is one that causes unprepared students some trouble. The theories and examples that we deal with in a course like this seem too unreal to be worth the trouble, and unless the reasons behind the theories and the choice of examples are explained, applied mathematics gets a bad name. This happened many years ago in elementary statics and dynamics with interminable questions about ladders leaning against smooth walls and particles sliding down rough wedges with smooth bases.

    The problems of the real subject of fluid dynamics are of great complexity, involving many physical effects and a considerable set of non-linear partial differential equations. These problems cannot be solved either by advanced techniques or by ‘putting them on the computer’: the techniques do not exist and the machines are neither powerful enough nor sophisticated enough (to reject spurious solutions). The problems can only be approached by omitting, after much careful thought and perhaps some experiments, a large number of the physical effects; then perhaps a special case can be dealt with analytically, and this will show what sort of calculation the machine must be programmed for in the more general case.

  • VIII - The equation of motion

  • A. R. Paterson
  • Book:
    A First Course in Fluid Dynamics
    Published online:
    28 May 2018
    Print publication:
    10 November 1983, pp 127-139

  • Summary

    The fundamental form

    We use the same method to derive the equation for momentum as we used for the continuity (or mass) equation. The rate of increase of momentum inside any volume V, as sketched in fig. VIII. 1, is equal to the sum of the

  • (i) rate of momentum inflow through S,

  • (ii) total forces acting inside V,

  • (iii) total forces acting on S.

    • Now flow rates through S are all controlled by v · dS, a volume per time, which is proportional to local area and to local outward velocity; since the ith component of momentum per volume is just ρvi, we must have a first term

    where the minus converts it to an inflow.

    The second term arises from the body force, and is therefore

    where Fi is the (ith component of) the body force per mass.

    The interaction across a fluid surface is given by the stress tensor, and since we want the force on V due to fluid outside, and have the normal direction taken as out of V, this is

    Thus the equation of motion for the fluid inside S is

    As before we put all terms as volume integrals, by using the (generalised) divergence theorem, and noting that

    This leads us to

    where we have used the fact that V is time independent in the first term. Finally, because V is arbitrary and the integrand is assumed to be smooth, we have

    This is the equation of motion. We may rearrange it by using the mass equation

    into the form

    Note the occurrence of the term ρv· ∇v, and that the left side is just

    i.e. density times acceleration following the particle.

    Stress and rate of strain

    (a) Isotropic and deviatoric parts of the stress tensor

    This equation of motion is of no help at all until we have a form for σij.

  • 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.

Page 1813 of 1829