324 Mr. H. S. Allen on the Motion of 



when the effective weight is equal to the resistance consequent 

 on the viscosity of the fluid. In the case of very slow motion 

 this resistance is 



/3a + dp, 



where V is the constant velocity, 



a is the radius of the sphere, 



fju is the coefficient of viscosity of the liquid, 



/3 is the coefficient of sliding friction. 

 Equating this to the effective weight §7rg(<T—p)a s , we obtain 

 for the limiting velocity 



where a is the density of the sphere, and p is the density of 

 the liquid. 



This equation only holds good when the velocity is so small 

 that squares may be neglected. Those terms in the equations 

 of motion of the fluid which represent its inertia are ne- 

 glected in comparison with those due to its viscosity *. Con- 

 sideration of these terms shows that Ypa must be small com- 

 pared with p. Or if we write p = vp, so that v is the kinematic 

 coefficient of viscosity, Va must be small compared with v. 



We may call that value of a which makes Ya = v the 

 "critical radius/'' Denoting it by a, we find 



- 8 _ V j3a+ 'dp 

 a ~ *9PK°—P) 0a + 2p 

 Both the terminal velocity and the critical radius involve 

 the coefficient of sliding friction fi. The value of the fraction 



r\ i o 



a .J 1 nes between 1 and §, as /3 varies from infinity (no 



slipping) to zero (infinite slip). Whethamf has shown that 

 for steady flow through a capillary tube no slipping occurs at 

 the surface of separation of solid and liquid. 



In the case in which yS is infinite, to which therefore the 

 greatest interest attaches, 



d 9 p 7 



9/x 



2 9P[<r-p) 



* Lamb, ' Hydrodynamics,' p. 533 : Lord Itayleigh, Phil. Mag'. (5) 

 xxx vi. pp. 364-365, 1893 (2). 



f Phil. Trans, clxxxi. (A) pp. 559-582 (1890). 



