Publisher Summary

This chapter discusses the ideal fluids or fluids in motions in which thermal conductivity and the viscosity of the fluids are unimportant. Fluid dynamics concerns itself with the study of the motion of fluids, such as liquids and gases. As the phenomena considered in fluid dynamics are macroscopic, a fluid is regarded as a continuous medium. The mathematical description of the state of a moving fluid is effected by means of functions that give the distribution of the fluid velocity and of any two thermodynamic quantities pertaining to the fluid. Such quantities are, in general, functions of the coordinates and of time. The equations of fluid dynamics are simplified in the case of steady flow, which is one where the velocity is constant in time at any point occupied by fluid.

§1 The equation of continuity

Fluid dynamics concerns itself with the study of the motion of fluids (liquids and gases). Since the phenomena considered in fluid dynamics are macroscopic, a fluid is regarded as a continuous medium. This means that any small volume element in the fluid is always supposed so large that it still contains a very great number of molecules. Accordingly, when we speak of infinitely small elements of volume, we shall always mean those which are “physically” infinitely small, i.e. very small compared with the volume of the body under consideration, but large compared with the distances between the molecules. The expressions fluid particle and point in a fluid are to be understood in a similar sense. If, for example, we speak of the displacement of some fluid particle, we mean not the displacement of an individual molecule, but that of a volume element containing many molecules, though still regarded as a point.

The mathematical description of the state of a moving fluid is effected by means of functions which give the distribution of the fluid velocity v = v(x, y, z, t) and of any two thermodynamic quantities pertaining to the fluid, for instance the pressure p(x, y, z, t) and the density ρ(x, y, z, t). All the thermodynamic quantities are determined by the values of any two of them, together with the equation of state; hence, if we are given five quantities, namely the three components of the velocity v, the pressure p and the density ρ, the state of the moving fluid is completely determined.

All these quantities are, in general, functions of the coordinates x, y, z and of the time t. We emphasize that v(x, y, z, t) is the velocity of the fluid at a given point (x, y, z) in space and at a given time t, i.e. it refers to fixed points in space and not to specific particles of the fluid; in the course of time, the latter move about in space. The same remarks apply to ρ and p.

We shall now derive the fundamental equations of fluid dynamics. Let us begin with the equation which expresses the conservation of matter. We consider some volume V0 of space. The mass of fluid in this volume is , where ρ is the fluid density, and the integration is taken over the volume V0. The mass of fluid flowing in unit time through an element df of the surface bounding this volume is ρv · df; the magnitude of the vector df is equal to the area of the surface element, and its direction is along the normal. By convention, we take df along the outward normal. Then ρv · df is positive if the fluid is flowing out of the volume, and negative if the flow is into the volume. The total mass of fluid flowing out of the volume V0 in unit time is therefore

ρv⋅df,

where the integration is taken over the whole of the closed surface surrounding the volume in question.

Next, the decrease per unit time in the mass of fluid in the volume V0 can be written

∂∂t∫ρd V⋅

Equating the two expressions, we have

∂t∫ρd V=−∮ρv⋅df. (1.1)

The surface integral can be transformed by Green’s formula to a volume integral:

ρv⋅df=∫div(ρv)d V.

Thus

[∂ρ∂t+div(ρv)]d V=0.

Since this equation must hold for any volume, the integrand must vanish, i.e.

ρ/∂t+div(ρv)=0. (1.2)

This is the equation of continuity. Expanding the expression div (ρv), we can also write (1.2) as

ρ/∂t+ρ div v+v⋅ grad ρ=0. (1.3)

The vector

=ρv (1.4)

is called the mass flux density. Its direction is that of the motion of the fluid, while its magnitude equals the mass of fluid flowing in unit time through unit area perpendicular to the velocity.

§2 Euler’s equation

Let us consider some volume in the fluid. The total force acting on this volume is equal to the integral

∮p d f

of the pressure, taken over the surface bounding the volume. Transforming it to a volume integral, we have

∮pdf=−∫ grad p d V.

Hence we see that the fluid surrounding any volume element dV exerts on that element a force - d V grad p. In other words, we can say that a force - grad p acts on unit volume of the fluid.

We can now write down the equation of motion of a volume element in the fluid by equating the force - grad p to the product of the mass per unit volume (ρ) and the acceleration dv/dt:

d v/dt=− grad ρ. (2.1)

The derivative dv/dt which appears here denotes not the rate of change of the fluid velocity at a fixed point in space, but the rate of change of the velocity of a given fluid particle as it moves about in space. This derivative has to be expressed in terms of quantities referring to points fixed in space. To do so, we notice that the change dv in the velocity of the given fluid particle during the time dt is composed of two parts, namely the change during dt in the velocity at a point fixed in space, and the difference between the velocities (at the same instant) at two points dr apart, where dr is the distance moved by the given fluid particle during the time dt. The first part is (∂v/∂t)dt, where the derivative ∂v/∂t is taken for constant x, y, z, i.e. at the given point in space. The second part is

x∂v∂x+dy∂v∂y+dz∂v∂z=(dr⋅ grad )v.

Thus

 V=(∂v∂t)dt+(dr⋅ grad )v,

or, dividing both sides by dt,†

vdt=∂v∂t+(v⋅ grad )v. (2.2)

Substituting this in (2.1), we find

v∂t+(v⋅ grad )v=−ρ1 grad ρ. (2.3)

This is the required equation of motion of the fluid; it was first obtained by L. Euler in 1755. It is called Euler’s equation and is one of the fundamental equations of fluid dynamics.

If the fluid is in a gravitational field, an additional force ρg, where g is the acceleration due to gravity, acts on any unit volume. This force must be added to the right-hand side of equation (2.1), so that equation (2.3) takes the form

v∂t+(v⋅ grad )v=− grad ρρ+g. (2.4)

In deriving the equations of motion we have taken no account of processes of energy dissipation, which may occur in a moving fluid in consequence of internal friction (viscosity) in the fluid and heat exchange between different parts of it. The whole of the discussion in this and subsequent sections of this chapter therefore holds good only for motions of fluids in which thermal conductivity and viscosity are unimportant; such fluids are said to be ideal.

The absence of heat exchange between different parts of the fluid (and also, of course, between the fluid and bodies adjoining it) means that the motion is adiabatic throughout the fluid. Thus the motion of an ideal fluid must necessarily be supposed adiabatic.

In adiabatic motion the entropy of any particle of fluid remains constant as that particle moves about in space. Denoting by s the entropy per unit mass, we can express the condition for adiabatic...