PIPE FLOW 195 



addition we express p\ p$ as a difference h of head in feet of water, 

 equation (2) becomes 



, flv" -f'lv" , 8) 



- 7 - ~ 



The analogy between the two cases is, however, not exact, in that, 

 while with a solid moving through a large body of water any 

 disturbance set up at the surface may be propagated over any unknown 

 distance, becoming less marked as the distance from the solid increases 

 and finally dying out altogether, any such disturbance in a pipe has a 

 strictly limited range of extension, but is in general communicated to the 

 whole mass of water in motion. 



Where the motion through the pipe is everywhere steady, it is entirely 

 governed by the law of simple viscous resistance and the conditions are 

 accurately stated by the formulae of p. 69. Thus Poiseuille (1845), 

 experimenting on tubes of very fine bore (between *02 and '10 millimetres), 

 found the resistance to motion to be directly proportional to the velocity, 

 and to the pipe length, and inversely to the square of the diameter, and 

 deduced the law (see p. 69) : 



82 /x Iv 

 loss of pressure = ^ 



Many experimental researches have been carried out from time to 

 time to determine the law of resistance with sinuous or unsteady motion, 

 and it is now proposed to consider the results of these somewhat in 

 detail. 



Probably the most complete series of experiments is that of Darcy, 

 who in 1857, experimenting on cast-iron pipes having diameters ranging 

 from *5 inch to 20 inches, and lengths of 110 yards, concluded that the 

 resistance is proportional to the length and to the square of the velocity 

 and is inversely proportional to the diameter. Thus expressed, the law 

 becomes 



c being a constant for any one type of surface. 



It was found, however, that this formula was not strictly applicable to 

 all diameters of pipe, since, as d increases, the resistance diminishes 

 according to a slightly higher power of d than the first. Darcy founu 

 that, within the range of his experiments, keeping c a constant, this 



d 



exponential value of d might be replaced by ) i L and thus 



h Wd) 



o 2 



