512 



HYDRODYNAMICS. 



Compari- 

 son of the 

 Table with 

 Du Buat's 

 formula. 



On the re- 

 sistance of 

 fluids. 



It will be seen from a comparison of this Table with 

 Du Buat's formula in p. 500, that there is a consider, 

 able agreement between them. For great depths of the 

 wasteboard, Du Buat's formula gives a much greater 

 discharge than Smeaton's Table. At small depths of 

 the wasteboard, Du Buat's formula gives results less 

 than those of Smeaton's Table, while for intermedi- 

 ate depths the results approach very near each other. 

 The 18 inch gauge, for example, with a depth of 18 

 inches, discharges, according to Smeaton, 515.54' cubic 

 feet in a minute ; whereas, according to Buat, it should 

 discharge 554.15 cubic feet The same gauge, at a 

 depth of only one foot, discharges 7-71 cubic feet; where- 

 as, according to Du Buat, it should discharge only 7.254 

 cubic feet. The same notch, with a depth of 8 inches, 

 discharges 164.86 cubic feet; and, according toDu Buat, 

 it should discharge 1 64.20 cubic feet, which is very 

 nearly the same result. 



CHAP. V. 

 ;Ow .THE PERCUSSION AND RESISTANCE OF FLUIDS. 



As the laws of the resistance of fluids can be deter- 

 mined only from experiment, we shall not occupy our 

 pages with theoretical discussions, which are of no prac- 

 tical utility. It will be necessary, however, to make 

 the reader acquainted with the ordinary theory of the 

 resistance of fluids, which may be comprehended in a 

 few propositions. 



SECT. I. On the Theory of the Resistance of Fluids, of Hmds, 



IF a body is moved through a fluid medium, it expe- 

 riences an obstruction in its motion, which is called 

 the resistance of the fluid ; but if the fluid is in motion, 

 and strikes the body at rest; the force sustained by 

 the body is called the percussion of the fluid. The 

 force exerted upon the body is obviously the same in 

 both these cases ; and the percussion and the resistance 

 of fluids follow the same laws. The ordinary theory 

 which we are about to explain, may be used without 

 much risk of error in all cases where the angles of im- 

 pulse is not below 60, which is the case in wheels 

 moved by the force of water. 



PROP. I. 



If a fluid, whose particles have all the same velocity, 

 strikes a plane surface, the resistance will be as the 

 product of the squares of the velocity of the fluid, the 

 density of the fluid, and the area of the plane. 



The resistance must obviously be equal to the force 

 with which each particle strikes the plane, multiplied 

 by the number of particles which strike it in a given 

 time. But the force of each particle is as its velocity, 

 and the number of particles which strike the plane in 

 any given time, must also be as the velocity. Hence the 

 resistance will be as the square of the velocity. It is 

 obvious also that the resistance will be proportional to 

 the density of the fluid, as the number of particles 

 which strike the plane in the same time must be pro- 

 portional to the density; the number of particles 

 which strike the plane must likewise increase with the 

 area of the plane; 'and therefore the whole resistance 

 must be proportional to the square of the velocity of 

 the fluid, the density of the fluid, and the area of the 

 surface .of the plane. 



PROP. II. 



If a fluid in motion strikes a plane surface at rest, 

 inclined to the direction in which the fluid moves, the 

 resistance perpendicular to the plane is proportional to 

 the square of the sine of the angle of inclination. 



Let AB be the plane surface, inclined at an angle Pi ATE 

 ABC to the direction DE, or CB of the motion of the CCCXIX. 

 fluid. Draw AC perpendicular to DE. Then it is Fi S- 9l 

 obvious that the number of particles which strike 

 against the surface AB is proportional to AC, for none 

 of those which are beyond A and C can have any ef- 

 fect upon the plane. Likewise, if we take EF to re- 

 present the velocity of the fluid, and resolve this ve- 

 locity into the two velocities FG, perpendicular to the 

 surface of the plane, and GE parallel to the same sur- 

 face, it is manifest that the part GE has no effect in 

 acting against the plane. Hence the part of the force, 

 which acts perpendicular to the plane is FG, or the 

 sine of the angle GEF := ABC, the inclination of the 

 plane. That is the force which acts perpendicular to 

 the plane is proportional to sin. ABC ; and the number 

 of particles which strike the plane is also proportional 

 to sin. ABC, consequently the resistance must be pro- 

 portional to sin. ABC x sin. ABC = sin. 2 ABC, or 

 the square of the sine of the angle of inclination. 



COR. The resistance which the plane experiences in 

 the direction of its motion, is proportional to the cube 

 of the sine of the angle of inclination. For as the re 

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