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37361 results in Cambridge Textbooks

Page 1852 of 1869

  • 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.

Page 1852 of 1869