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

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.

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).

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).

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.