212 REPORT— 1869. 



througli the Torquay -water-main, under small variations in smoothness of internal 

 surface, suggest the necessity of a revision of the theory of " Huid friction," which 

 in its commonly accepted form, ignores the etl'oct of " quality of surface " entirely, 

 and in which, even as amended by later able writers, less importance is attached 

 to it than the variations referred to prove it to possess. 



The theory appears defective in two other important respects. 



(1) In assigning to pipes of different diameters, imder the same hydraulic gra- 

 dient, a How proportioned to the power (|) of the diameter, it proceeds on the 

 assumption that the ratio of the mean velocity to the maximum velocitj' is the same 

 in pipes of all diameters. This is equi-valent to assuming, either that the velocity 

 of the central particles of a flowing column does not exceed that of the circumfer- 

 ential particles more in a 12-inch than in a G-inch pipe ; or that particles of water 

 can glide more freely past the semi-rough surface of the pipe, than they can glide 

 past each other ; whereas, since the latter alternative seems absurd, we ought to 

 expect that within the 12-inch pipe, the particles occupying the central inches 

 must possess, in addition to the velocity of those immediately surrounding them, 

 the whole mean velocity which they would have possessed if flowing indepen- 

 dently in a 6-inch pipe ; an expectation irreconcileable with the law that the deli- 

 very with a given hydraulic gradient is as the power (-|) of the diameter. 



(2) The commonly received theory assigns to every square foot of an extended 

 plane, drawn edgeways tlirough undisturbed fluid, one and the same intensity of 

 I'rictional resistance ; and this is expressed in terms of the velocity of the surface, 

 calculated with reference to the undisturbed part of the fluid, and is supposed to 

 create an equal resistance per square foot throughout the plane ; whereas it is cer- 

 tain that the anterior portions of the surface, in rubbing against the particles which 

 it passes, and experiencing resistance from them, 7iuist impress on them equivalent 

 force in the direction of its motion, and must impart to them some velocity in that 

 direction. Thus, though it may be in some sense asserted that the anterior portions 

 of the plane rub against the contiguous particles with the entire velocity of the plane, 

 since these particles are undisturbed, this cannot be truly asserted of the posterior 

 portions of the plane, since the particles against wliich these rub have already re- 

 ceived a velocity conformable to that of the phtue ; and a " state of motion" will be 

 thus produced in the contiguous particles involving a widening body of fluid, and 

 with increasing velocity imparted to it, as we recede foot by foot sternward along 

 the plane ; forming, in fact, a '' cui-rent," created and left behind, by the transit of the 

 plane, such that if we could integrate the volume of current created in each unit of 

 time, and the exact velocity possessed by each of its particles, the aggregate mo- 

 mentum must be precisely tliat which is due to the frictional resistance of the en- 

 tire plane acting during that unit of time. Obviously the stemward portions of the 

 plane moving forward in such a favouring current, must experience a less intense 

 frictional resistance than the anterior portions. 



A consideration of these objections to the received theory, the latter especially, 

 suggests the question, how the velocity of the surface relatively to the fluid is 

 properly to be estimated, as relevant to, or as governing the intensity of frictional 

 resistance. And in attempting to arrive at this, it is plainly necessary to take ac- 

 count of the manner in which, and the distance to which, the velocity which the 

 rubbing surface imparts to the particles, spreads into the fluid. 



With a view to this, the following propositions seem relevant and admissible. 



( 1) A surface free from such roughnesses and prominences as to produce eddies, 

 if of such quality that the fluid thorouglily zeds it, will experience the same resis- 

 tance in moving past the particles which resist it, as if it had itself consisted of 

 particles of the fluid. 



(2) No particles of the contiguous fluid can be strictly regarded as sliding past 

 the surface, or, vice versa, in the sense in which a solid slides past a solid ; but be- 

 tween the complete motion of the surface, and the complete quiescence of the fluid 

 where it is yet undisturbed, a graduated state of motion must exist, the mamier 

 of the graduation govei-ning the intensity of the force transmitted from particle to 

 particle. 



(3) When a plane slides edgeways through tmdisturbed fluid, part of the force 

 which it transmits into the fluid around it is employed in giving motion to the par- 



