CRITICAL VELOCITY 



53 



Log h 



in a parallel pipe, and the existence of a " critical velocity," below which 

 the motion is steady, and above which it is unsteady, Reynolds determined 

 the law of resistance in the two cases and the points at which the change 

 takes places, by measuring the loss of head accompanying different velocities 

 of flow. On plotting a curve showing velocities and losses of head 

 (Fig. 28) it was found that up to a certain velocity, for any given tube, these 

 points lie on a straight line passing through the origin of co-ordinates. 

 Above this velocity, the points lie more or less on a smooth curve, 

 indicating that the loss of head is possibly proportional to v n . 



To test this, and if so to determine the value of n, the logarithms of 

 the loss of head and of 

 the velocity were plotted 

 (Fig. 29), since if 



h="kv n 



log h = log k -f- n log v 

 which is the equation to 

 a straight line, inclined 

 at an angle to the axis 

 of log v (where tan 0=n), 

 and cutting off an inter- 

 cept = log k on the axis 

 of log h. 



It was then found that 

 with velocities increasing 

 between each pair of ex- 

 periments, the plotted 

 points lie on a straight 

 line up to -a certain point 

 A, the value of 6 for this 



line being 45. Up to this point n is unity and h is proportional to v. 

 At A, which marks the higher critical velocity, or the point at which 

 motion, initially steady, becomes sinuous, the law suddenly changes and 

 h increases very rapidly. The relation between h and v, however, follows 

 no definite law, until the point B is reached, where the velocity is about 

 1*3 times that at A. Above this point a perfectly definite law holds, the 

 plotted points from B to C and onwards lying on a straight line. 



The angle of inclination 6 of this line varies with the material and 

 surface of the pipe, but is constant for any one pipe. 



With a lead pipe, tan 6 = 1/722, so that here the law of resistance 

 above the critical point is h a v 1 ' 1 *. 



Log \, 

 FIG. 29. 



