466 HYDROMECHANICS [HYDRAULICS. In so far as the vibrations may be considered isochronous, the distance between consecutive corresponding points of the recurrent figure, or, as it may be termed, the wave length of the figure, is di rectly proportional to the velocity of the jet, that is, to the square root of the head of water. For low he ids the measurements confirm this law. For higher heads there is an increase of the wave lengths in a higher ratio than the velocity of the jet. This points to a de parture from isochronous vibration, the nature of which is investi gated in Lord Kayleigh s paper. IY. THEORY OF THE STEADY MOTION OF FLUIDS. 25. The general equation of the steady motion of a fluid given under Hydrodynamics furnishes immediately three results as to the distribution of pressure in a stream which may here be assumed. (a. ) If the motion is rectilinear and uniform, the variation of pressure is the same as in a fluid at rest. In a stream flowing in an open channel, for instance, when the effect of eddies produced by the roughness of the sides is neglected, the pressure at each point is simply the hydrostatic pressure due to the depth below the free surface. (b. ) If the velocity of the fluid is very small, the distribution of pressure is approximately the same as in a fluid at rest. (c.) If the fluid molecules take precisely the accelerations which they would have if independent and submitted only to the external forces, the pressure is uniform. Thus in a jet falling freely in the air the pressure throughout any cross section is uniform and equal to the atmospheric pressure. (d.) In any bounded plane section traversed normally by streams which are rectilinear for a certain distance on either side of the sec tion, the distribution of pressure is the same as in a fluid at rest. DISTRIBUTION OF ENERGY IN INCOMPRESSIBLE FLUIDS. 26. Application of the Principle of the Conservation of Energy to Cases of Stream Line Motion, The external and internal work done on a mass is equal to the change of kinetic energy produced. In many hydraulic questions this principle is difficult to apply, be cause from the complicated nature of the motion produced it is difficult to estimate the total kinetic energy generated, and because in some cases the internal work done in overcoming frictional or viscous resistances cannot be ascertained ; but in the case of stream line motion it furnishes a simple and important result known as Bernoulli! s theorem. Let AB (fig. 30) be any one elementary stream, in a steadily moving fluid mass. Then, from the steadiness of the motion, AB is a fixed O o B _j_ i I NN V A 4 I t A i f^ B "" 1 Z X Fig. 30. path in space through which a stream of fluid is constantly flowing. Let 00 be the free surface and XX any horizontal datum line. Let o> be the area of a normal cross section, v the velocity, p the intensity of pressure, and z the elevation above XX, of the elementary stream AB at A, and w lt j, v lt z 1 the same quantities at B. Suppose that in a short time t the mass of fluid initially occupying AB comes to A B . Then AA , BB are equal to vt, vj, and the volumes of fluid A A , BB are the equal inflow and outflow = Qt = a>vt = (t> l v i t > in the given time. If we suppose the filament AB surrounded by other filaments moving with not very different velocities, the fric tional or viscous resistance on its surface will be small enough to be neglected, and if the fluid is incompressible no internal work is done in change of volume. Then the work done by external forces will be equal to the kinetic energy produced in the time con sidered. The normal pressures on the surface of the mass (excluding the ends A, B) are at each point normal to the direction of motion, and do no work. Hence the only external forces to be reckoned are gravity and the pressures on the ends of the stream. The work of gravity when AB falls to A B is the same as that of transferring A A to BB ; that is, GQ< (z-zj. The work of the pressures on the ends, reckoning that at B negative, because it is opposite to the direction of motion, is (pcaxvt) - (p l ca l xv } t) = Q (p ~ Pi). The change of kinetic energy in the time t is the differ ence of the kinetic energy originally possessed by AA and that finally acquired by BB , for in the intermediate part A B there is no change of kinetic energy, in consequence of the steadiness of the r< motion. But the mass of AA and BB is Qt, and the change of 9 kinetic energy is therefore Qq ^L - JL. j . Equating this to the g 2 2 / work done on the mass AB, GQt(z- Dividing by GQt and rearranging the terms, , P i * I i P i 2 /"I . or, as A and B are any two points, 1> 2 n I- -~- + s =- constant = H .... (2). Now is the head due to the velocity v, P~ is the head equiva- %c/ G lent to the pressure, and z is the elevation above the datum (see 15). Hence the terms on the left are the total head due to velocity, pressure, and elevation at a given cross section of the filament. z is easily seen to be the work in foot-pounds which would be done by 1 lb of fluid falling to the datum line, and similarly J and -- are the quantities of work which would be done by 1 lb of fluid due to the pressure p and velocity v. The expression on the left of the equation is, therefore, the total energy of the stream at the sec tion considered, per lb of fluid, estimated with reference to the datum line XX. Hence we see that in stream line motion, under the restrictions named above, the total energy per lb of fluid is uni formly distributed along the stream line. If the free surface of the fluid 00 is taken as the datum, and - h, - h are the depths of A and B measured down from the free surface, the equation takes the form 2/ii 2 2g G 2y G or generally 1- J - h = constant (3) 27. Second Form of the Theorem of Bernoulli. Suppose at the two sections A, B (fig. 31) of an elementary stream small vertical pipes are introduced, which may be termed pressure columns A" B" T t T" ct i 4, i A. HF=-James500 (talk) TV: "~-- ___ 1"
- _._
5 B C f - H ~ 1
- -
i
^
! __ i i - _
-,--- c c XI 1 1 1 Y i -X. Fig. 31. ( 8), having their lower ends accurately parallel to the direction of flow. In such tubes the water will rise to heights corresponding to the pressures at A and B. Hence h = and b ^-^~ r* r* jr IJT Consequently the tops of the pressure columns A and B will be at total heights b + c = -P + z and l + c = ^- + z 1 above the datum G G line XX. The difference of level of the pressure column tops, or the fall of free surface level between A and B, is therefore and this by equation (1), 26, is That is, the fall of free surface level between two sections is equal to the difference of the heights due to the velocities at the sections. The line A B is sometimes called the line of hydraulic gradient, though this term is also used in cases where friction needs to be
taken into account. It is the line the height of which above datum
