FLUID RESISTANCE.] MECHANICAL PHILOSOPHY. HYDRODYNAMICS. 



T67 



cular to that direction. Now DC, the force in the 

 direction of the stream, varies as B C sin. D B C ; that is, 

 as B C sin. A : but, as shown above, B C itself varies as 

 sin. 2 A ; therefore, the force in the direction of the stream 

 varies as sin. 3 A. 



This, of course, is on the assumption that the other 

 component of the force of the stream in its own direction 

 namely, the component represented by B A, in the 

 direction of the plane itself is of no effect. 



The force B D, perpendicular to the stream, is as B C 

 cos. DBC=^BC cos. A ; hence the force with which 

 the stream impels an oblique plane upwards, in a 

 direction perpendicular to the stream, varies as sin. 2 A 

 cos. A. 



It must be observed, that in the foregoing propositions 

 the plane is regarded as fixed while receiving the force of 

 the stream, or else the fluid is regarded as quiescent, and 

 the plane as moving through it. If both be regarded as 

 moving in the same or in opposite directions, then the 

 velocity spoken of above must be considered as the re- 

 lative velocity ; that is, it is the difference of the velo- 

 cities if the plane and stream move in the same direction, 

 and the sum of the velocities if they move in the op- 

 posite direction. Thus, if V be the velocity of the 

 stream, and v the velocity of a float-board of an under- 

 shot water-wheel, then the perpendicular pressure on 

 the float-board will vary as (V r) z . 



The student has already been apprised, at the com- 

 mencement of the present brief chapter on the motion of 

 incompressible fluids, that the results of the mathemati- 

 cal theory often require considerable modification to 

 render them accordant with actual experiment. This 

 arises from the difficulty of assigning correct values to 

 all the circumstances attending that motion. We have 

 seen, in treating of fluids spouting from an orifice, that 

 observation has discovered the orifice not to be the nar- 

 rowest channel through which the stream uniformly 

 flows, and consequently that the velocity there, is less 

 than at the vena contracts, which ought therefore to be 

 regarded as the true orifice. 



It was found by Newton that the velocity at the vena 

 contracta is, to the velocity of the orifice, as ^' 2 to 1 ; 

 and since, in falling bodies, the spaces passed through are 

 as the squares of the velocities acquired, it follows that 

 the space to be fallen through to give a body the velocity 

 with which the fluid spouts from the orifice, is only half 

 the space necessary to give the velocity at the vena con- 

 tracta ; that is, the velocity at the orifice would be 

 acquired by a body falling through only half the altitude 

 of the surface above the orifice. 



In like manner, in treating of fluid resistances, the 

 commonest observation shows that a stream impinging 

 upon a plane surface cannot have the full effect which 

 theory assigns. The velocity of the stream is, to a cer- 

 tain degree, impeded by what may be termed the back- 

 water ; moreover, the velocity is not in general the same 

 at all depths. Yet the results of theory may usually be 

 applied with safety in the comparison of different similar 

 cases : for instance, the comparative results as to quan- 

 tity of fluid discharged, time of emptying, <fec., from 

 equal orifices in two different vessels, disregarding the 

 vena contracta, would accord very nearly with experi- 

 ment. And, in like manner, fluid-pressures may be 

 relatively estimated and computed from theory : for in- 

 stance, though theory may assign an erroneous value for 

 the effect of a stream upon the rudder of a ship, yet the 

 position of greatest effect, as deduced from theory, may 

 very well accord with actual observation ; so the effect of 

 a stream of water upon the floats of a water-wheel may 

 be incorrectly assigned by theory, and yet the degree in 

 which the velocity of the wheel should fall short of that 

 of the stream, in order that the greatest effect may be 

 produced, is found to be verified pretty closely by prac- 

 tical experience. 



To determine particular values for the unknown, or 

 arbitrary quantities, which enter into a general expres- 

 sion, so that for those values the expression may be a 

 maximum or a minimum, requires the aid of the differ- 

 ential calculus. But in the two cases mentioned above, 



the principles to be employed are so elementary, and tho 



knowledge required of that science so trifling, that 



although in general we are interdicted from usin.^ the 



| calculus, we shall give the two problems adverted to'as a 



i conclusion to the present portion of our subject. We 



may premise, however, that in order to determine x so 



that any expression of the form 



A + Bx + Cz 2 + Dx 3 + <fcc 



(1) 



may be a maximum, all we have to do is to disregard the 

 i term independent of x that is, the term A to write 

 down all the other terms, first converting each exponent 

 of x into a factor, and writing each exponent a unit 

 smaller ; thus 



B + 2Cx + 3D.c 2 + &c., 



and then to equate this expression to : that is, to solve 

 the equation B + 2Cx + 3Dx 2 + &c. = 0. The value 

 of x, necessary to render (1) the greatest possible if it 

 can be made the greatest possible for any value of x 

 will be found by solving this equation. 



PKOBLEM. To find the angle at which the rudder of a 

 ship must be inclined to the stream, so that the effect, in 

 the direction perpendicular to the stream, may be the 

 greatest possible. 



It has been seen above that the effect varies as 



sin.lA cos. A, or as (1 cos. 2 A) cos. A = cos. A 

 cos. 3 A = x x 3 



where x is put for the cosine of the required inclination. 



Since x x 3 = a maximum 

 .'. 1 3x* = /. x = VJ 



hence the angle whose cosine is ^/J will be the inclina- 

 tion necessary. 



PROBLEM. The velocity V of the stream being given, 

 to determine the velocity v of an undershot-wheel, so that 

 the greatest possible effect may be produced. 



The pressure upon the same area of float-board will 

 vary as (V r) 2 . As this pressure moves with the velo- 

 city v, the effect will be greatest when 



(V ) 2 = V 2 !) 2Vt 2 + u 3 = a maximum ; 

 in order to which, the unknown quantity must satisfy 

 the condition 



The solution of this quadratic gives 

 v = V, and = V 



The first value of renders the proposed expression a 

 minimum : that is, the least quantity of work is per- 

 formed by the wheel when it moves with the same velo- 

 city as the stream which is obvious : the other value of 

 v is that which renders the expression a maximum ; and 

 shows that the greatest amount of work is performed 

 when the wheel moves with a velocity equal to one-third 

 the velocity of the stream. But, for the practical effects 

 of water-power, acting through the medium of water. 

 wheels, and other hydraulic machines, the student is 

 referred to the section on APPLIED MECHANICS, in the 

 present volume. 



[In the preceding chapters, the theories of momentum 

 in dynamics, and the equal pressure of liquids in all 

 directions as a fact in hydrostatics, have been investigated. 

 Presuming the student to have become acquainted with 

 the principles involved in each of these subjects, we shall 

 now illustrate their application in pile-driving and pile- 

 removing. In the construction of bridges, <bc., it is 

 often desirable to obtain a more solid foundation than 

 the earth at the surface affords. This is especially the 

 case when a bridge has to be built over a river, and tho 

 arches of which must be divided so as to be supported 

 by their base on the bed of the stream. By means of 

 the momentum of a mass of metal, piles are driven into 

 the earth, forming the bed, and for this purpose the pile- 

 driving machine is used. The following is a description 

 of one of these machines, as represented in the annexed 

 plate, Figs. 1 and 2. 



