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portion of matter as it moves along. In another method, which is indeed more consistent with the Eulerian notation, we fix our attention on a certain region of space, and investigate the change in its properties produced as well by the flow of matter inwards and outwards across the boundary as by the action of external forces on the included mass*.

Let denote the measure, estimated per unit volume, of any quantity connected with the properties of a fluid, and let us calculate the rate of increase of Q in a rectangular space dady dz having its centre at (x, y, z). This is expressed by

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Now the amount of Q which enters per unit time the specified region

across the yz-face nearest the origin is (Qu —

(Qu

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the amount which leaves the region in the same time by the oppos

d.

ite face is (Qu+dx) dy dz. The two faces together give a

gain of

d. Qu
dx

dx dy dz per

unit time. Calculating in the same

way the effect of the flow across the remaining faces, we have for the total gain of Q due to the flow across the boundary the formula

-(d. Qu+ d. Qv + d. Qw)

dy dz

dx dy dz.

(15).

First, let us consider the change of mass, i. e. we put Q = p, the mass per unit volume. Since the quantity of matter in any region can vary only in consequence of the flow across the boundary, the expressions (14) and (15) must in this case be equal; this gives the equation of continuity in the form (8).

Next, let us take the change of momentum, making Q = pu, the momentum parallel to x per unit mass. The momentum contained in the space dxdy dz is affected not only by the passage of matter carrying its momentum with it across the boundary, but also by the forces acting on the included matter, viz. the pressure and the external impressed forces. The effect of these resolved

* See Maxwell, On the Dynamical Theory of Gases, Phil. Trans. 1867, p. 71. Also, Greenhill, Solutions of Cambridge Problems for 1875, p. 178.

parallel to x is found as in Art. 6, to be

(px-dr) dx dydz..

.(16).

Hence, (14) is now equal to (15) and (16) combined, which

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Performing the differentiations, and simplifying by means of the equation of continuity, we are led again to the first of equations (2), and in like manner the second and third equations may be obtained.

13. Another interesting application of the method of Art. 12 is to make Q = (q2+V+ E)p, the energy per unit mass. Here q denotes the resultant velocity √(u2 + v2+w2), V the potential energy per unit mass with reference to the external im

dV
"
dx

pressed forces (viz. we have X=- &c.),

.), and E the intrinsic

energy. In a liquid we have E=0. If the system of external forces do not change with the time the alteration in the energy contained within the space dx dydz is due to the flow of matter carrying its energy with it, and to the work done on the contained matter by the pressure of the surrounding fluid. The total rate at which this pressure works is

-(d,pr

d.pu, d.pv, d.pw
+
dy dz

+

dx dy dz.........

(18).

The verification of the formula obtained by equating (14) to the sum of (15) and (18) is left as an exercise for the student.

14. To obtain by the same method a proof of the surfacecondition (11) of Art. 10, let in Fig. 1 (Art. 3) P denote a point of the fluid infinitely close to the surface F=0; and let A, B, C be the points in which this surface is met by three straight lines drawn through P parallel to the axes of co-ordinates. Then if PA, PB, PC = a, ß, y respectively, we have

dF
F÷ &c.,
dx'

where F denotes the value of the function Fat P(x, y, z). The rate of flow of matter into the space included between the three

planes meeting in P, and the surface F=0, is ultimately

p(ußy +vya+waß);

and the rate of increase of the mass included in this space is ultimd

ately (paßy). Equating these expressions, substituting for dt

a, B, y their values, and omitting infinitesimals of higher order than the second, we readily find

dF dF dF dF
+ v +w- =
dy dz

dx

which agrees with (11).

dt'

Impulsive Generation of Motion.

15. If at any instant impulsive forces act on the mass of the fluid, or if the boundary conditions suddenly change, a sudden alteration in the motion may take place. The latter case may arise, for instance, when a solid immersed in the fluid is suddenly set in motion.

Let p be the density, u, v, w the component velocities immediately before, u', v', w' those immediately after the impulse, X', Y', Z' the components of the external impulsive forces per unit mass,

the impulsive pressure, at the point (x, y, z). The change of momentum parallel to x of the element defined in Art. 6 is then pdxdydz(u-u); the x-component of the external impulsive forces is pdxdydzX', and the resultant impulsive pressure in the same

απ

direction is dxdy dz. Since an impulse is to be regarded as da an infinitely great force acting for an infinitely short time (7, say), the effects of all finite forces during this interval are neglected. Hence,

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These equations might also have been deduced from (2), by multiplying the latter by dt, integrating between the limits 0 and 7,

putting X'= ['" Xdt, &c., ≈ =

-=[ pdt, and then making ☛ vanish.

T

In a gas an infinite pressure would involve an infinite density; whereas no change of density can occur during the infinitely short time of the impulse. Hence, in applying (19) to the case of a gas we must put = 0, whence

u' - u = X', v' - v=Y', w' - w = Z'.

....

.(20).

In a liquid, on the other hand, an instantaneous change of motion can be produced by the action of impulsive pressures only, even when no impulsive forces act bodily on the mass. In this case we have X', Y', Z' each = 0, so that

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If we differentiate these equations with respect to x, y, z, respectively, and add, and if we further suppose the density to be uniform, we find by (9) that

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The problem then, in any given case, is to determine a value of satisfying this equation and the proper boundary conditions* ; the instantaneous change of motion is then given by (21).

The Lagrangian Forms of the Equations.

16. Let a, b, c be the initial co-ordinates of any particle of fluid, x, y, z its co-ordinates at time t. We here consider x, y, z as functions of the independent variables a, b, c, t; their values in terms of these quantities give the whole history of every particle of the fluid. The velocities parallel to the axes of co-ordinates of

* It will appear in Chapter III. that (save as to additive constants) there is only one value of which does this.

dx dy dz
dt'

the particle (a, b, c) at time t are at at' dt'

accelerations in the same directions are

and the component

d2z

d'x d'y
dt dt dt •

Let p

and p be the pressure and density in the neighbourhood of this particle at time t; X, Y, Z the components of the external impressed forces per unit mass acting there. Considering the motion of the mass of fluid which at time t occupies the differential element of volume dxdydz, we find by the same reasoning as in Art. 6,

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These equations contain differential coefficients with respect to x, y, z, whereas our independent variables are a, b, c, t. To eliminate these differential coefficients, we multiply the above dx dy dz

equations by da'da da respectively, and add; a second time

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These are the Lagrangian forms of the dynamical equations.

We

17. As before, two additional equations are required. have, first, a relation between p and p of the form (3), (4), or (5), as the case may be. To find the form which the equation of continuity assumes in terms of our present variables, we consider the element of fluid which originally occupied a rectangular parallelepiped having the corner nearest the origin at the point (a, b, c), and its edges dr, db, dc parallel to the axes. At the time t the

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