﻿Motion of a Sphere in a Viscous Fluid. 545 



the boundary conditions are not seriously altered by a con- 

 siderable movement of the sphere. Now we have seen that 

 the motion given by the sphere in sphere solution of (6) also 

 represents very closely the motion in an elongated rectan- 

 gular vessel, and a second order approximation based on this 

 solution may be expected to show the way in which the 

 motion in this case alters with increasing velocity, even if 

 it does not give an exact representation of it. 



There is, however, another cause which limits the validity 

 of Stokes's solution at a great distance from the sphere, 

 which has been pointed out by Oseen and Lamb *. To get 

 really steady motion it is necessary to consider the sphere 

 as at rest in an infinite stream of fluid, or to consider the 

 origin as moving with the sphere. in either case the 

 velocity of the fluid relative to the origin has a fixed con- 

 stant value at infinity. Now the validity of Stokes's solu- 

 tion depends on the possibility of neglecting terms of the 



form w^— compared with terms of the form v^—%* In this 



OX QX 



case <— and ^— ^ diminish indefinitely as the distance from 

 ox ox 1 J 



the origin increases while u remains constant, hence how- 

 ever small u may be, it is impossible to neglect w— when 

 r is great. ° x 



Oseen has obtained a solution in which this fact is taken 

 into account, but as it only differs from the ordinary solu- 

 tion at considerable distances from the sphere, it is not easy 

 to test it experimentally, nor does it seem possible to base a 

 second approximation on it. 



If, instead of referring the motion to an origin that moves 

 with the sphere, we refer it to a fixed origin, coinciding at 

 any instant with the centre of the sphere, the velocity of the 

 fluid far from the origin will be zero, and hence the term 



u^ can be neglected over the whole field. On the other 

 ox 



hand, the motion is not strictly steady except when quan- 

 tities of the second order are neglected. For let the sphere 

 be moving with velocity V and let f be the distance of its 

 centre from a fixed point, then if yjr be the stream function, 



dt t> of 



* Lamb, Phil. Mag-, xxi. p. 112 (1911). 

 Phil. Mag. S. 6. Vol. 29. No. 172. April 1915. 2 N 



