Pi'essttre around Spheres in a Viscous Fluid, 427 



The expression for the resistance of a penchiluni mo\dno; in 

 a viscous fluid is, according to the same author ^, 



F= - ->paV« v/=l(l + ,-i + ,i V-'l"'' • (l*^) 



which, \vhen the conditions for steady motion are applied, 

 becomes 



-F = 67r/x>V, (11) 



for the resultant force on a sphere parallel to the direction of 

 motion. 



These results are obtained on the assumption that there is 

 no slip at the surface, and that the inertia term 



may be neglected in comparison ^Yith the viscous term 



The general form of the results obtained by Stokes t from 

 theory has been recently verified by Mr. H. S. Allen J. Mr. 

 Allen allowed air-bubbles of various size to escape from a 

 small opening under water. The size of the bubble was 

 varied until the velocity with which the bubbles rose in the 

 water or other fluid became constant. The force on the 

 sphere due to its motion in the viscous fluid could then be 

 measured in terms of gravity. Mr. Allen also allowed 

 bicycle bearing-balls to fall through viscous fluids, varying 

 the diameter until constant velocity was obtained. From 

 results thus obtained Mr. Allen concludes that for very small 

 velocities the motion agrees with that deduced theoretically 

 by Stokes. 



When, however, the velocity is greater than a certain 

 definite velocitv o-iven bv the formula 



Y = 4fja — r. , o / ^ \^^) 



^ '^ fJL ^a-\- ZfJi' ' 



the resistance is proportional to the radius to the three- 

 halves power, and when the velocities are considerably 

 greater than the critical velocity the resistance follows the 

 law deduced by Sir Isaac Newton: 



^ = kpar\' (13) 



* Math, and Physical Papers, vol. iii. p. 33. 



t L. c. p. 4. 



X Phil. Mag. [5] vol. 1. pp. 338, 519 (1900). 



