which is what the equation (9) becomes when p is constant. To prove the corresponding formula for compressible fluids, we remark that the fluid entering at A now brings with it, in addition to its energies of motion and position, the intrinsic energy or + ρ +fdp, per unit mass. The addition of these terms converts the equation (10) into the equation (9). In most cases of motion of gases, the relation (4) of Art. 7 holds, and (9) then becomes 30. Equations (10) and (11) shew that, in steady motion for points along any one stream-line†, the pressure is, cæteris paribus, greatest where the velocity is least, and vice versa. This statement, though opposed to popular notions, is obvious if we reflect that a particle passing from a place of higher to one of lower pressure must have its motion accelerated, and vice versa. Some interesting practical illustrations and applications of the principle are given by Mr Froude in Nature, Vol. XIII. 1875. It follows that in any case to which the aforesaid equations apply there is a limit which the velocity cannot exceed if the motion be continuous. For instance, let us suppose that we have a liquid flowing from a reservoir where the motion is sensibly zero, and the pressure equal to P, and that we may neglect the P external impressed forces. therefore We have then in (10) C= and ρ Hence if q exceed p 2P actual fluids are unable to support more than a very slight, if any, *Tait's Thermodynamics, Art. 174 (first edition). + This restriction is, by (9), unnecessary when a velocity-potential exists. degree of tension, without rupture. The limiting velocity is, by (12), approximately that with which the fluid would escape from the reservoir into a vacuum. In the case of water at the atmospheric pressure, this velocity is that due to the height of the water-barometer, or roughly, about 45 feet per second. The question as to what takes place when the limiting velocity is reached will be considered in Art. 94. 31. We conclude this chapter with a few simple applications of the equations. Example 1. Steady motion under the action of gravity. A vessel is kept filled up to a constant level with liquid which escapes from a small orifice in its walls. The origin being taken in the upper surface, let the axis of z be vertical, and its positive direction downwards, so that V=— gz. If we suppose the area of the upper surface large compared with that of the orifice, the velocity at the former may be neglected. Hence, determining the constant in (10) so that p= P (the atmospheric pressure), when z = 0, we have At the surface of the issuing jet we have p= P, and therefore i.e. the velocity is that due to the depth below the upper surface. This is Torricelli's Theorem. We cannot however at once apply this result to calculate the rate of efflux of the fluid, for two reasons. In the first place, the of a great number of issuing fluid must be regarded as made up elementary streams converging from all sides towards the orifice. Its motion is not, therefore, throughout the area of the orifice, everywhere perpendicular to this area; but becomes more and more oblique as we pass from the centre of the orifice to the sides. Again, the converging motion of the elementary streams must make the pressure at the orifice somewhat greater in the interior of the jet than at its surface, where it is equal to the atmospheric pressure. The velocity, therefore, in the interior of the jet will be somewhat less than that given by (14). Experiment shews however that the converging motion above spoken of ceases at a short distance beyond the orifice, and that the jet then becomes approximately cylindrical. The ratio of the area of the section S' of the jet at this point (called the 'vena contracta') to the area S of the orifice is called the 'coefficient of contraction.' If the orifice be simply a hole in a thin wall, this coefficient is found to be about 62. If a short cylindrical tube be attached externally, the value of the coefficient is considerably increased; if, on the other hand, there be attached a short tube projecting inwards, the coefficient is about 5. The paths of the particles at the vena contracta being nearly straight, there is little or no variation of pressure as we pass from the axis to the surface of the jet. We may therefore assume the velocity there to be uniform, and to have the value given by (14), where z now denotes the depth of the vena contracta below the surface of the liquid in the vessel. The rate of efflux is therefore √2gz.pS'. 32. The calculation of the form of the issuing jet presents great difficulties, and has only been effected in one or two simple cases. (See Arts. 96, 97, below.) It is, however, easy to shew that the coefficient of contraction cannot (in the absence of friction) fall below the value. For the pressure of the fluid at the walls of the vessel is approximately equal to the statical pressure P+gpz, except near the orifice, where on account of the velocity q becoming sensible, it is, by (13), somewhat less. Assuming it for the moment to be equal to the statical pressure, we see that the total horizontal pressure exerted on the fluid by the vessel is PS+gpff zdS....... .(15), where the integration extends over the area S of the orifice. The horizontal pressure exerted by any one element of the vessel's walls is in fact balanced by that due to an opposite element, except in the case of those elements which are opposite to the orifice. The first term of (15) is balanced by the pressure P of the atmosphere on the portion of fluid external to the vessel; so that the total horizontal force acting on the fluid is gp ffzdS, or gpzS, if z be the depth of the centre of inertia of the orifice. It is this force which produces the momentum with which the fluid leaves the vessel. The mass of fluid which in unit time passes the vena contracta is pqS, and the momentum which this carries away with it is pq S'. Hence, substituting the value of q from (14), we have gpzS=2gpzS'.. ...(16), or, since z, z are nearly equal S': S=1:2, approximately. Since, however, the pressure on the wall is, near the orifice, sensibly less than the statical pressure P+gpz, the total horizontal force acting on the fluid somewhat exceeds the value (15). The left-hand side of (16) is therefore too small, and the ratio S': S is really greater than 1. The above theory is taken from a paper by Mr G. O. Hanlon, in the 3rd volume of the Proceedings of the London Mathematical Society, and from a note appended thereto by Professor Maxwell. In one particular case, viz. where a short cylindrical tube, projecting inwards, is attached to the orifice, the assumption on which (16) was obtained is sensibly exact; and the value of the coefficient of contraction then agrees with experiment. Compare Art. 97. 33. Example 2. A gas flows through a small orifice from a receiver, in which the pressure is p, and the density p1, into an open space where the pressure is p2*. We assume that the motion has become steady. In the receiver, at a distance from the orifice, we have p=p1, q= 0, sensibly. This determines the value of C in equation (11). Neglecting the external forces, we find for the velocity of efflux where P2 is the density of the issuing gas at the vena contracta. If c be the velocity of sound in the gas of the receiver, we have (Chapter VIII.) c2= P1; and therefore, taking account of (4), Art. 7, YP1 : Pi * See Joule and Thomson, On the Thermal Effects of Fluids in Motion, Proc. R. S. May, 1856. Also Rankine, Applied Mechanics, Arts. 637, 637 a. The maximum velocity of efflux occurs when p1 = 0, i.e. when the gas escapes into a vacuum; it is × velocity of sound, or, for atmospheric air at 32° F., about 2413 feet per second. The rate of escape of mass however depends on the value of qP2, or which does not continually increase as p, diminishes, but attains a maximum when or, for atmospheric air at 32° F., about 997 feet per second. The 'reduced velocity,' i.e. the velocity of a current of the density p, of the gas in the receiver which would convey matter at the same rate is got by dividing the expression (18) by p1, and is, when a maximum, about 632 feet per second for air at 32° F. 34. Example 3. A mass of liquid rotates, under the action of gravity only, with constant angular velocity w about the axis of z supposed drawn vertically upwards. also By hypothesis, u = - -wy, v = wx, w = 0; X=0, Y=0, Z=-g. The equation of continuity is identically satisfied, and the dynamical equations of motion become |