MECHANICAL PHILOSOPHY. UYDRODYNAMICS. [SPOUTING FLUIDS. 



the column f fluid in the assumed unit of time ; and M 

 thin time is indefinitely small, it must be the velocity 

 which the fluid has at the orifice itself, or that with which 

 it actually spouts out. As the vessel is kept constantly 

 full, the initial cucnmitances continue invariable, so that 

 the fluid is discharged with the same uniform velocity. 



Now, if a heavy body fall freely through a height n by 

 the action of gravity it acquires during the time of fall, 

 a velocity = 2A ; that is, a velocity that would carry it 

 through 2k in the same time (DYNAMICS, page 736). 

 Consequently the fluid issues from the orifice A with a 

 constant velocity cuual to that which a heavy body would 

 acquire in falling through the height B A. 



If this height be h feet, the accelerating force of 

 gravity being </ 32-2 feet, we have, for the constant 

 Telocity of the spouting fluid, jfyh, the velocity 

 per second. 



The fluid spouts out with the same velocity whether 

 the small orifice be in the side or in tho bottom of the 

 Teasel, provided only that in both cases the hole be at 

 the same depth below the surface of the fluid, since the 

 pressure of a fluid is the same in all directions at the 

 same depth. 



The stream being, " 1M - 



as it were, thus 

 projected with a 

 uniform velocity r. 

 the curve it will 

 describe in issuing 

 from the side of a 

 vessel (as in the 

 margin) will be a 

 paralivia, of which 

 the directrix is 

 the horizontal line 

 through B (Fig. 

 180), the surface of 

 tin- Iluid (DYNAMICS*, page 739). The hole through which 

 the fluid spouts must be so small as to render the pres- 

 sures at different parts of it insensibly different from 

 that at the centre of the orifice. 



QUANTITY OF FLUID DISCHARGED PER 

 SECOND. As the velocity is constant, the quantity of 

 fluid discharged per second is found by multiplying the 

 area of the orifice by the length J2yh ; thus, if a be the 

 area of the orifice, the discharge per second is a >J'2ijh 

 cubic feet. Let h= 20 inches, and a = 1 square inch, then 

 a J2gK - ^(2 X 322 x 12 X 20) = J 15456 = 124 3 

 cubic inches. 



Since a*/'2yh is the discharge per second, in t seconds 

 a quantity Q will be discharged, such that 



Q-.I^.M-Jt ... .(1) 



This, therefore, expresses the time in which a proposed 

 quantity Q of the fluid will be delivered, Q being ex- 

 pressed in cubic feet. 



THE VENA CONTRACTA. The foregoing con- 

 clusioiis respecting the velocity with which a fluid spouts 

 from a small hole in the vessel containing it, and the 

 quantity of the fluid delivered in a given time, have 

 been compared with many carefully conducted exprri- 

 ments. The most satisfactory way of ascertaining practi- 

 cally the velocity of the discharge, ia by observing the 

 altitude to which the fluid spouts when forced to take au 

 upward direction, as in the accompanying figure. If the 

 uniform velocity of the issuing Fig.187. 



tlui.l be that due to the fall of 

 a heavy body from the sur- 

 face to the orifice, the fluid 

 ought to spout up as high as 

 that surface ; and such is 

 found to be very nearly the 

 case. We might expect that 

 the theoretical result w<mM 

 tint be riqidly fulfilled, but 

 that the height reached would 

 fall a little short of the sur- 

 face of the fluid in the vessel ; 

 friction and the re- 



sistance of the air would act as opposing forces. Hut 

 there is another cause in operation which must now be 

 Dotted 



\\ hen a passage is made in a vessel for the exit of tho 

 fluid contained in it, the equilibrium of the entire mass 

 is disturbed, and motion takes place among all the par- 

 ticles. The instant that the plug is removed, the velocity 

 is due to the direct pressure upon it ; but as lateral and 

 oblique pressures all round the hole, cause the neigh- 

 bouring fluid to flow sideways towards the main stream, 

 when this surrounding fluid reaches the aperture, it does 

 so with a certain velocity transverse to the direction of 

 the main stream, which, in consequence, becomes slightly 

 contracted a little beyond the opening. This contraction 

 of the fluid vein is called by Newton the vena coiitrarta ; 

 and it is through the section of tho vena contracta that 

 the fluid flows, as we have supposed it to flow through 

 the orifice itself. The distance of this section from the 

 orifice is nearly equal to the railing of the orifice, and its 

 area about |ths of that of the orifice. 



If, therefore, the orifice be small as, for instance, a 

 gimlet-hole in a cask the confounding the orifice with 

 the vena contracta can occasion but very little error in 

 the height or distance to which the fluid spouts ; but a 

 considerable departure from the truth will result from 

 calculating the quantity of fluid delivered in any time, 

 on the supposition that the sectional area of the spout- 

 ing stream is A instead of f A ; for instance, in the ex- 

 ample above, the quantity of fluid discharged per second, 

 in-i.'ad of being 124-3 cubic inches, is only 77'7 cubic 

 inches. 



If the vessel be not kept full, and the surface be there- 

 fore allowed to descend, the velocity with which the 

 descent commences will be, to the velocity with which it 

 passes through the vena contracta, as the area of the 

 section at the hitter to the area of the surface (page 7<33). 



Since the curve, which the fluid spouting from a small 

 hole in the side of the vessel assumes, is that of a para- 

 bola the same curve, in fact, that is described by a 

 projectile impelled in the same direction and with the 

 same velocity as the fluid at the vena contracta every- 

 thing connected with the extent of range, time of de- 

 scribing it, greatest altitude, <tc. , may be determined 

 as in the theory of projectiles. The following are a few 

 illustrations : 



If the fluid spout from a point in the middle of die 

 upright side of a full vessel, the horizontal range will bo 

 the greatest ; and from points at equal distances above 

 and below the middle point, the ranges will be equal. 



Let A B (Fig. 188) be the upright side of the vessel 

 perforated at the middle point C, and also at two points 

 D, E, equidistant from C. 



Fig. 198. 



The velocity with which the fluid issues from one of 

 these points, as D, is that which would be acquired in 

 falling through B D. Since gravity acts during thu 

 whole fall of the fluid to 1', a particle arrives at P in 

 the time that a Ixxly would fall through 1) A. 



Hence the particle at D has a horizontal velocity that 

 would carry it forward through a space equal to 2 B D m 

 tlic time that a body would fall from B to D ; conse- 

 qin-nily. we luive only to tind how often this time u 



