24 Mr Molin, An examination of Searle's method 



The angular velocity of the liquid about the axis of the cylinders, 

 at a distance r from the axis, is given by 



_ 277 62 /^ ^ 



^ T ' a^-bAr^~ 



When r = b, the radius of the inner rotating cylinder, 



oj = 1^ = 27r/r, 



and when r = a, the internal radius of the outer fixed cylinder, 

 CO =0. This problem was first treated, not quite accurately, by 

 Newton. The above results were given substantially by Stokes *, 

 and are also given by Lambf and by SearleJ. 



The rate of shearing, rdco/dr, varies somewhat as r increases 

 from b to a, as is shown by the formula 



doi 27r 2a%^ 



r 



dr T ' (a2 _ §2) ^2 • 



We have only taken into account the friction between the 

 coaxal cylindrical layers of the liquid and not the friction between 

 the horizontal layers in proximity to the bottom surface of the 

 movable cylinder, and have not considered the conditions that 

 arise near that surface. In practice, only the lower end of the 

 rotating cylinder is exposed to viscous action ; Dr Searle makes an 

 allowance for this end by writing 



^^^•r+i' (2) 



where I is the length by which the height, /, of the liquid, in the 

 simple theory, must be increased, in order that the increase of 

 couple shall correspond to the viscous action in proximity to the 

 end surface and the edge of the rotating cylinder. 



Dr Searle gives a graphical method of determining k. The 

 values of MT are plotted against I, and he says, "It will be found 

 that the points lie on a straight line, which cuts the axis of I at 

 a distance k from the origin." Dr Searle adds "If the corresponding 

 total load hung from each thread be M grammes, it will be found, 

 on repeating the observation with various loads, that MT is 

 constant for a given level of liquid. This result confirms the 

 fundamental assumption that the viscous stress at each point is 

 proportional to the rate of shearing of the liquid." 



* G. G. Stokes, Brit. Ass. Report, p. 539, 1898. 

 t H. Lamb, Hydrodynamics, Third Ed., p. 546, 1906. 

 loal ^' ^' ^' ®^*^^®' ^°^- ^^f-' P- 602. Compare C. Brodman, Wied. Ann., 45, p. 163, 



