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7931 results in Fluid dynamics and solid mechanics

Page 389 of 397

The Theory of Classical Dynamics
  • J. B. Griffiths
  • Published online:
    25 October 2011
    Print publication:
    30 May 1985
  • This textbook describes in detail the classical theory of dynamics, a subject fundamental to the physical sciences, which has a large number of important applications. The author's aim is to describe the essential content of the theory, the general way in which it is used, and the basic concepts that are involved. No deep understanding can be obtained simply by examining theoretical considerations, so Dr Griffiths has included throughout many examples and exercises. This then is an ideal textbook for an undergraduate course for physicists or mathematicians who are familiar with vector analysis.

  • 9 - Analytical dynamics

  • J. B. Griffiths, Loughborough University
  • Book:
    The Theory of Classical Dynamics
    Published online:
    25 October 2011
    Print publication:
    30 May 1985, pp 236-275

  • Summary

    The approach to classical dynamics proposed so far has been a more or less direct application of Newton's laws of motion. In such an approach the motion of a body or particle can be predicted on the assumption of a given set of external forces which act on it, simply by integrating the equations of motion. However, for complex systems of particles or rigid bodies, it is not always easy to determine appropriate equations for each component, let alone perform the required integration. In practice, using this approach, it is found that each individual type of problem requires its own particular insights and techniques.

    In this chapter the Lagrangian approach to classical dynamics is developed. This approach is based upon two scalar properties of a system, its kinetic energy and work. It leads to a powerful and general method for the solution of dynamical problems which is found to be particularly useful in the analysis of mechanical systems which contain a number of rigid bodies that are connected in some way, but which may move relative to each other. In the traditional approach each component would have to be treated separately in terms of the forces acting on it. However, the Lagrangian approach enables such a system to be considered as a whole.

    The aim here is to develop a general approach which may be applied to any dynamical system. It is found that the equations of motion can be presented in a standard and convenient form.

  • 1 - The Newtonian method

  • J. B. Griffiths, Loughborough University
  • Book:
    The Theory of Classical Dynamics
    Published online:
    25 October 2011
    Print publication:
    30 May 1985, pp 1-15

  • Summary

    As with other scientific theories, theories of dynamics aim to contribute to an understanding and explanation of phenomena that occur in the real world. In the theories of dynamics, it is the general subject of motion that comes under investigation. The aim is to describe how objects move, and to suggest physical reasons as to why they move. In particular they should provide methods for analysing or predicting the motion of specific bodies, and also possibly suggest techniques for controlling the motion of some objects.

    The theory known as classical or Newtonian dynamics, the subject of this book, is one such theory. Before describing it in detail, however, it is convenient to describe in general terms the way in which the theory is used. This is the purpose of this first chapter.

    The technique of mathematical modelling

    The basic method by which any theory of dynamics is applied can be described in terms of three distinct phases. The first phase consists essentially of constructing a simplified model. This is an idealised imaginary representation of some physical situation in the real world. In phase two this theoretical representation is analysed mathematically and its consequences are deduced on the basis of some assumed theory. Finally, in phase three, the theoretical results of phase two are interpreted and compared with observations of the real physical situation. This whole process can of course be repeated many times using different initial representations or different basic theories.

  • Introduction

  • J. B. Griffiths, Loughborough University
  • Book:
    The Theory of Classical Dynamics
    Published online:
    25 October 2011
    Print publication:
    30 May 1985, pp xi-xiv

  • Summary

    The classical theory of dynamics is that scientific theory which was developed through the seventeenth, eighteenth and nineteenth centuries to describe the motion of physical bodies. The theory was originally developed in two distinct parts, one dealing with terrestrial bodies such as a projectile or a pendulum, and the other dealing with celestial bodies, in particular the planets. These two apparently distinct subjects were first brought together by Isaac Newton whose book, known as the Principia, is rightly acknowledged as the first complete formulation of the theory that is now referred to as classical dynamics.

    Since those early days the theory has been thoroughly developed and extended so that it can now be applied to a very wide range of physical situations. Some of the early concepts have been clarified, and others have been added. But, although the theory has now been formulated in many different ways, it is still essentially the same as that originally proposed by Newton. The most significant advances have in fact been those associated with the development of new mathematical techniques, which have subsequently enabled the theory to be applied to situations which previously had proved too difficult to analyse.

    Now, as is well known, a revolution occurred in scientific thinking in the first part of the present century. Classical theories were disproved, and exciting new theories were put forward. The theory of relativity was suggested in order to explain the results obtained when the speed of light was measured.

  • 10 - Variational principles

  • J. B. Griffiths, Loughborough University
  • Book:
    The Theory of Classical Dynamics
    Published online:
    25 October 2011
    Print publication:
    30 May 1985, pp 276-285

  • Summary

    The approaches to the subject of classical mechanics considered so far have relied heavily on the mathematical techniques associated with the study of differential equations. Both in the vectorial approach to Newtonian mechanics, and in the analytic approach to Lagrangian dynamics, the motion of a system is ultimately described in a mathematical model in terms of a set of differential equations. Historically, however, the study of differential equations has proceeded in parallel with the study of the calculus of variations. It was thus natural in the development of the subject that the techniques associated with the calculus of variations should also be applied to the problems of classical dynamics. The variational principles of dynamics obtained in this way have in fact always been considered to be of great importance, and they certainly include a number of very interesting results.

    The advantage of the variational approach is basically that it considers some property of a system over its entire motion. The aim is to find some integral, taken over the whole motion, which has a stationary value with respect to a certain class of permissible variations. Such a principle enables the motion of a system to be stated in a most economical way without reference to any particular coordinate system. It also enables motion to be considered in a more metaphysical way, and thus facilitates the development of alternative physical theories.

  • 5 - Newtonian gravitation

  • J. B. Griffiths, Loughborough University
  • Book:
    The Theory of Classical Dynamics
    Published online:
    25 October 2011
    Print publication:
    30 May 1985, pp 86-102

  • Summary

    The theory of gravitation really starts from a consideration of the motion of bodies near the earth's surface. It is observed that all bodies have a tendency to accelerate vertically downwards unless they are prevented from doing so by impressed forces acting on them. This acceleration is explained in classical dynamics as the effect of a hypothetical gravitational field which exerts a force on any body that is placed in the field. The general theory of gravitation has, however, been deduced largely from a consideration of the motion of heavenly bodies. This is the approach that is followed here.

    Kepler's laws

    The motion of the planets seems to have fascinated many past civilisations. Over many centuries the wanderings of the planets across the sky have been recorded. However, as far as the subject of this book is concerned it is appropriate to start with the work of Johannes Kepler. Kepler had at his disposal the mass of observations recorded by previous generations. In particular, he had access to the detailed observations of Tycho Brahe. Tycho's work was remarkable in the degree of accuracy to which he attained, though his observations like those of his predecessors were all made with the naked eye. It is interesting to notice that Kepler published his first two laws in 1609, the same year as Galileo developed the telescope.

    Kepler's work is significant in that he introduced a new theory relating to planetary motion.

  • 3 - Force, mass and the law of motion

  • J. B. Griffiths, Loughborough University
  • Book:
    The Theory of Classical Dynamics
    Published online:
    25 October 2011
    Print publication:
    30 May 1985, pp 35-66

  • Summary

    We come now to consider the more familiar subject matter of the theory of classical dynamics as we turn to the concepts of force and mass, and the way in which such concepts are used to analyse and predict the motion of a physical body. However, it is necessary first to introduce the concept of a particle.

    The concept of a particle

    In classical dynamics we consider the motion of real physical objects. These are sometimes referred to as bodies. The aim is to develop a method whereby the motion of such bodies can not only be analysed, but also predicted. Unfortunately, the physical bodies whose motions are to be considered are usually fairly complex objects. It is therefore necessary to start by making some simplifying assumptions about such bodies in order that they may easily be represented in a theoretical model.

    In some circumstances it turns out to be comparatively simple to build a theoretical representation of physical bodies. For example, when a stone is thrown into the air, or when considering the motion of a planet in the solar system, the main interest is usually in the general position and velocity of the object, or its general linear motion, rather than its orientation and angular motion. In such situations the information given or required about its motion can usually be stated in terms of the position at different times of some representative point of the body, such as some kind of centre.

  • 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).

Page 389 of 397