122 HYDROMECHANICS FIG. 18. Vena Contracts. through which a body will move with a uni- form velocity after it has fallen through a height equal to the vertical distance between the surface of the liquid and the orifice. If this distance is 16'1 ft., the velocity acquired will be 32'2 ft. per second, and therefore the theoretical quantity discharged from an ori- fice 4 in. in diameter, whose centre is IG'l ft. below the surface, would he equal to a cylin- der 4 in. in diameter and 32 '2 ft. long, and containing 4,828-5 cubic inches, or about 21-83 gallons. The actual discharge from a thin ori- fice not furnished with an ajutage is however much less, being only about two thirds of the theoretical amount. The loss is owing partly to fric- tion, but mainly to the interference of converging currents moving within the vessel toward the ori- fice. This interfer- ence may be shown by employing a glass vessel having a per- foration in its bottom, as represented in fig. 18. If particles of some opaque substance having nearly the same specific gravity as wa- ter, so that they will remain suspended in it for a space of time, be mingled with the wa- ter, they will be seen to move in the direc- tion indicated by the lines in the figure, which are nearly direct. If the jet is carefully ob- served, it will be seen that it is not cylin- drical, and that for a distance from the orifice of about half its diameter it resembles a trun- cated cone with the base at the orifice. This contraction of the stream is called the vena contracta, and its smallest diameter is stated to be from 0-6 to 0-8 of that of the orifice. When the stream has a direction downward near- ly vertical, it continues to dimi- nish beyond the vena contracta, in consequence of the increased velocity caused by the force of gravity, the size being in the inverse proportion to the velo- city. The increased velocity at the vena contracta is due to the pressure which forces the par- ticles of water into a narrower channel. As the jet continues to fall, it forms a series of ventral and nodal segments, as shown in fig. 19. The ventral segments are composed of drops elon- gated horizontally, as shown at a a, while the nodal segments are elongated vertically, as seen at J &; and as the segments have fixed positions, it follows that the drops in falling are alternately elongated vertically and hori- zontally. If the orifice is in the side of the FIG. 19. FIG. 20. vessel and discharges horizontally, the size of the stream does not diminish in the same man- ner as when falling vertically, and it is sooner broken. If a cylindrical tube or ajutage whose length is from two to three times its diameter is fitted to the orifice, the rate of efflux may be increased to 80 per cent, of the theoretical amount. The velocity will be somewhat di- minished, but the vena contracta will be larger in proportion. If the inner end of the ajutage has a conical shape with the base toward the interior, the efflux may be further increased to 95 per cent. ; and it has been found that if the outer end of the tube is also enlarged, the efflux may be still further increased to very nearly the theoretical amount, say 98 per cent. When a cylindrical ajutage is used, there will be a partial vacuum formed between the sides of the tube and the contracted vein, as shown in fig. 20. If a pipe ascending from a reservoir of water is let into this part of the ajutage, the water will rise in the pipe; and if the height is not too great, the vessel may be emptied. The re- sistance offered by conduits is a sub- ject of great importance in practical hydro- mechanics, upon which extended experiments have been made. When the length of the aju- tage bears more than a certain proportion to its diameter, the efflux is reduced to about the same amount as when the stream issues through a thin orifice, that is, about 62 per cent, of the theoretical amount. With a pipe of 1J in. in diameter and 30 ft. long, the efflux will he only about half that from a thin orifice, or 31 per cent, of the theoretical amount. This reduc- tion is caused by friction between the liquid and the tube, as well as between its particles, and .is greater with cold than with warm liquids. This resistance to motion, or approach to rigid- ity, which as conferred by cold, is called vis- cosity, and is a principle which has to be taken into account in nearly all very careful hydrau- lic calculations. Resistance of Liquids to the Motion of Solid Bodies. This will depend upon the form and size of the body. The following are two important laws : 1. With the same ve- locity, the resistance is proportional to the ex- tent of surface applied by the solid to the li- quid in the direction of motion. 2. With the same extent of surface, the resistance is pro- portional to the square of the velocity. These laws may be demonstrated experimentally, but their truth will also be apparent from the fol- lowing considerations. In regard to the fir^t law, it will be easily understood that with the same velocity the amount of water displaced will be the measure of resistance, and that a surface of two square feet will displace twice
