Page:Encyclopædia Britannica, Ninth Edition, v. 3.djvu/56

continuous in space and time, and if we follow any portion of it as it moves, the mass of that portion remains invariable. These properties it shares with all material sub stances. In the next place, it is incompressible.

The form of a given portion of the fluid may change, but its volume remains invariable ; in other words, the density of the fluid remains the same during its motion. Besides this, the fluid is homogeneous, or the density of all parts of the fluid is the same. It is also continuous, so that the mass of the fluid contained within any closed surface is always exactly proportional to the volume contained within that surface. This is equivalent to asserting that the fluid is not made up of molecules ; for, if it were, the mass would vary in a discontinuous manner as the volume increases continuously, because first one and then another molecule would be included within the closed surface. Lastly, it is a perfect fluid, or, in other words, the stress between one portion and a contiguous portion is always normal to the surface which separates these portions, and this whether the fluid is at rest or in motion.

We have seen that in a molecular fluid the interdiffusion of the molecules causes an interdiffusion of motion of different parts of the fluid, so that the action between contiguous parts is no longer normal but in a direction tending to diminish their relative motion. Hence the perfect fluid cannot be molecular.

All that is necessary in order to form a correct mathematical theory of a material system is that its properties shall be clearly defined and shall be consistent with each other. This is essential ; but whether a substance having such properties actually exists is a question which comes to be considered only when we propose to make some practical application of the results of the mathematical theory. The properties of our perfect liquid are clearly defined and consistent with each other, and from the mathematical theory we can deduce remarkable results, some of which may be illustrated in a rough way by means of fluids which are by no means perfect in the sense of not being viscous, such, for instance, as air and water.

The motion of a fluid is said to be irrotational when it is such that if a spherical portion of the fluid were suddenly solidified, the solid sphere so formed would not be rotating about any axis. When the motion of the fluid is rotational the axis and angular velocity of the rotation of any small part of the fluid are those of a small spherical portion suddenly solidified.

The mathematical expression of these definitions is as follows: Let u, v, w be the components of the velocity of the fluid at the point (x, y, z), and let

$\alpha ={\frac {dv}{dz}}-{\frac {dw}{dy}}$, $\beta ={\frac {dw}{dx}}-{\frac {du}{dz}}$, $\gamma ={\frac {du}{dy}}-{\frac {dv}{dx}}$ (1),

then α, β, γ are the components of the velocity of rotation of the fluid at the point (x, y, z). The axis of rotation is in the direction of the resultant of α, β, and γ, and the velocity of rotation, ω, is measured by this resultant.

A line drawn in the fluid, so that at every point of the line

${\frac {1}{\alpha }}{\frac {dx}{ds}}={\frac {1}{\beta }}{\frac {dy}{ds}}={\frac {1}{\gamma }}{\frac {dz}{ds}}={\frac {1}{\omega }}$ (2),

where's is the length of the line up to the point x, y, z, is called a vortex line. Its direction coincides at every point with that of the axis of rotation of the fluid.

We may now prove the theorem of Helmholtz, that the points of the fluid which at any instant lie in the same vortex line continue to lie in the same vortex line during the whole motion of the fluid.

The equations of motion of a fluid are of the form

$\rho {\frac {\delta u}{\delta t}}+{\frac {dp}{dx}}+\rho {\frac {dV}{dx}}=0$ (3),

when ρ is the density, which in the case of our homogeneous incompressible fluid we may assume to be unity, the operator ρ/(ρt), represents the rate of variation of the symbol to which it is prefixed at a point which is carried forward with the fluid, so that

${\frac {\delta u}{\delta t}}={\frac {du}{dt}}+u{\frac {du}{dx}}+v{\frac {du}{dy}}+w{\frac {du}{dz}}$ (4),

p is the pressure, and V is the potential of external forces. There are two other equations of similar form in y and z. Differentiating the equation in y with respect to z, and that in z with respect to y, and subtracting the second from the first, we find

${\frac {d}{dz}}{\frac {\delta v}{\delta t}}-{\frac {d}{dy}}{\frac {\delta w}{\delta t}}$ (5).

Performing the differentiations and remembering equations (1) and also the condition of incompressibility,

${\frac {du}{dx}}-{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$ (6),

we find

${\frac {\delta a}{\delta t}}=\alpha {\frac {du}{dx}}+\beta {\frac {du}{dy}}+\gamma {\frac {du}{dz}}$ (7).

Now, let us suppose a vortex line drawn in the fluid so as always to begin at the same particle of the fluid. The components of the velocity of this point are u, v, w. Let us find those of a point on the moving vortex line at a distance ds from this point where

$ds=\omega d\sigma$ (8).

The co-ordinates of this point are

$x+\alpha d\sigma ,y+\beta d\sigma ,z+\gamma d\sigma$ (9),

and the components of its velocity are

$u+{\frac {\delta \alpha }{\delta t}}d\sigma ,v+{\frac {\delta \beta }{\delta t}}d\sigma ,w+{\frac {\delta \gamma }{\delta t}}d\sigma$ (10).

Consider the first of these components. In virtue of equation (7) we may write it

$u+{\frac {du}{dx}}\alpha d\sigma +{\frac {du}{dy}}\beta d\sigma +{\frac {du}{dz}}\gamma d\sigma$ (11),

or

$u+{\frac {du}{dx}}{\frac {dx}{d\sigma }}d\sigma +{\frac {du}{dy}}{\frac {dy}{d\sigma }}d\sigma +{\frac {du}{dz}}{\frac {dz}{d\sigma }}d\sigma$ (12),

or

$u+{\frac {du}{d\sigma }}d\sigma$ (13).

But this represents the value of the component u of the velocity of the fluid itself at the same point, and the same thing may be proved of the other components.

Hence the velocity of the second point on the vortex line is identical with that of the fluid at that point. In other words, the vortex line swims along with the fluid, and is always formed of the same row of fluid particles. The vortex line is therefore no mere mathematical symbol, but has a physical existence continuous in time and space.

By differentiating equations (1) with respect to x, y, and z respectively, and adding the results, we obtain the equation

${\frac {d\alpha }{dx}}+{\frac {d\beta }{dy}}+{\frac {d\gamma }{dz}}=0$ (14).

This is an equation of the same form with (6), which expresses the condition of flow of a fluid of invariable density. Hence, if we imagine a fluid, quite independent of the original fluid, whose components of velocity are α, β, γ, this imaginary fluid will flow without altering its density.

Now, consider a closed curve in space, and let vortex