206] DETERMINATION OF VISCOSITY. 553 



the viscosity of fluids. His verification of the proportionality of the 

 flow to the fourth power of the radius of the tube has been taken 

 as a proof that the liquid does not slide when in contact with a solid. 

 As another example of steady flow let us consider uniplanar 

 cylindrical flow, in which each particle moves in a circle with velocity 

 depending only on the distance from the axis, as in the case of the 

 lubricant between a journal and its bearing. Each cylindrical stratum 

 then revolves like a rigid body, which requires 



246) u = &y, v = ox, 



where co depends only on r "J/# 2 + y 2 . We then find 



du xy do 3u y* do 



= -- -=y ~ = O --- -3? 



cx r dr cy r dr 



' dv . X* dco dv xy dm 



x 



. 

 = <*> H --- 



-5' o~ = ~^- 

 r dr oy r dr 



and, most easily by the application of equation 86), 141, and by 

 the expression of z/co in terms of r, 





d(o\ ~ y 



~ 



- -=- 2- -=-t 

 r drj r dr 



dco\ . rt x dco 



o) . 1 dco\ . rt x 

 r + - -J-) + 2- 

 * ' r dr) ' r dr 



Thus the first two of equations 233) become 



/d 2 a . 3 d&A 1 x dp 



-f vyl-j-j- H -3-1 -- - i 

 \dr* r dr) Q r dr 



249) 



2 ( I ^ \ * 2/ **.P 



\c?r 2 r dr/ Q r dr 



Multiplying the first by y } the second by x and subtracting, 



250) 4^ + - ?- = 0, 



a?* 2 r ar 



a differential equation whose solution is 



251) = + &. 



Determining the constants so that CD = for r = R and o> = & for 

 r = R*, 



252) ,= f-V Jj^-'.-i|. 



1 2 \ J 



Multiplying equations 249) by x and y respectively, and adding, we 

 have to determine PJ 



253) eo 2 r = ^- 



For the stresses we obtain, using equations 230), 



v I xydo)\ (x*-y* 



X n = ( p 2p -^ ^-J cos (nx) H- it I - * -rr I cos 



