U = = work done per unit of time on the liquid which enters the pipe by the pressure of that in the reservoir, U1 work carried away per unit of time by the liquid which flows from the extremity of the pipe, U work expended on the various resistances which are opposed to the descent of the liquid in the reservoir and to its passage from the reservoir through its aperture into the pipe, work expended on the resistance of the internal surface of the pipe to the flow of the liquid along it, internal work of the resistance of the films to the flowing of each film over the surface of the next in succession; then, by the principle of virtual velocities, Letv % U=U1+U2+U2+U4• • velocity of any film, velocity of the filament which coincides with the axis of the pipe, V= velocity of the film which is in contact with the surface of the pipe, R internal radius of pipe, r= radius of the film whose velocity is v, p resistance per unit of surface to the sliding of the film 7= length of the pipe, w= weight of cubic unit of liquid. Let the unit of length in all the above measurements be the French metre, and the unit of weight the French kilogramme. The weight of the liquid which flows out of the pipe per unit of time is represented by R wv (2πrdr). This weight of liquid is therefore that which descends through the heighth in the reservoir per unit of time; .. U=h wv (2πrdr)=2πwh Ο R vrdr.. (3) Also the work U, which the liquid flowing out of the pipe carries away with it per unit of time, is represented by half its vis viva. But the weight of liquid which flows per unit of time between two films whose radii are r and r+dr is w(2πrdr)v. Half the The work U, expended per unit of time on the resistance of the internal surface of the pipe to the flow of the liquid is equal to the entire resistance of the surface multiplied by the velocity V of the liquid in contact with it, since V is the distance through which that resistance is overcome per unit of time. But the resistance of the pipe per unit of surface is To determine U4, which represents the aggregate internal work of the mutual resistances of the successive films of liquid, let it be observed that, as v represents the velocity of the film dv whose radius is r, v- dr represents that of the film whose radius is r+dr, the negative sign being taken because as r increases v diminishes. The distance by which one film slips over the next in the unit of time is therefore represented by — (dv)dr. But the resistance opposed to this slipping is 2πrlp, R R dv .. U1 = − ( TM 2πrlp(dv) dr = — 2πl (pr(d) dr. To determine the unit of resistance p which is opposed by the film whose radius is r+dr to the motion over it of that whose radius is r, let the velocity v—(d)dr of the former film be sup posed to be communicated in an opposite direction to both. The resistance of the one film opposed to the motion over it of the other will not thus be changed, but the former will be brought to rest, and the other will move over it with the velocity dv dr. The case will thus become the same with that of the dr film which moves in contact with that fixed to the internal surface of the pipe, except that the constants μ, and λ, will have different values. Let these values be and λ, then R dv 2 dr ; dr λ . . U1 = − 2 ml [ " {μ + x [ (d2 ) dr]®}r (dv)dr, in which the second term may be neglected as of infinitely small or considering separately the case of a portion of the liquid bounded by a film whose radius is r, in which equation the work of the bounding film of the portion of the liquid which is considered separately from the rest, is included in the last integral. Differentiating the above equation, considering U, constant, and reducing, 2glu (d) พ dr (7) If there were no resistance to the flow of the films over one another, or to the flow of the liquid over the internal surface of the pipe, the whole of the work done by the weight of the liquid in the reservoir on that in the pipe per unit of time would be accumulated in the liquid discharged per unit of time. Let u be the velocity the liquid would under these circumstances acquire. Then R2v WπR2v represents the discharge per unit of time, and v represents the work accumulated in it. Also hwTR2v represents the work done upon it per unit of time by the pressure of the liquid in the reservoir, WTR2v v2=hwπR2v; 2g 2g .. v2=2gh. (8) The same result is arrived at, as it ought to be, by making μ-0 in equation (7), and substituting v for v. By equation (7), 2glμ (de), v (v2 —v2) dr (d); 1 == 1 v (v2-v2) ; บ-ข h Let =i. Then integrating between the limits O and r, and In certain of the experiments of M. Darcy, the velocities of films of water were determined at three different distances (r) from the axes of pipes of diameters varying from 0.188 to 0.500 metre, under heads of water (h) varying from 0.202 to 13.427 metres, the values of v corresponding to which varied from 1.18 to 16:56 metres. Throughout this range of experiments the values of were for the same values of r approxi mately the same. ( This will be seen by inspecting the following Table I., in the two central columns of which are given the values of the ratio in respect of the first, second, and fourth films, as determined by the experiments of M. Darcy in pipes of different diameters, and of different degrees of roughness internally, and under different heads of water. These values of are calculated in respect of all the experiments of M. Darcy in which he has determined the values of v and vo; and in all, the value of that fraction is approximately the same for the same value of r. |