Motion of a Sphere through a Viscous Fluid. 117 



Hence the relations w==— U 3 v = 0, w = which are to hold 

 for r = a will be satisfied, provided 



0= | Ua, A = | va, A 1= - \ Ua 3 , . . (29) 



approximately; and it will be noted that the condition for the 

 success of the approximation is that ka, or Vajv, should be small. 



The equations (24), (2,5), (26), (29) agree with Prof. 

 Oseen's solution of the problem, obtained by a different 

 process. 



To find the distribution of velocity in the neighbourhood 

 of the sphere we may use the formulae ('26) and (28), with 

 the values of the constants given in (29). The result is 

 identical with (1) if regard be had to the altered meaning of 

 u. The resistance experienced by the sphere has the same 

 value (6/iaJJ) as on Stokes's theory*. 



In other respects the motion differs widely from that 

 represented by Stokes's formulae ; and the further inter- 

 pretation is very interesting. In the first place, as pointed 

 out by Prof. Oseen, the stream-lines are no longer symmetrical 

 with respect to the plane x = 0, the motion being in fact no 

 longer reversible. Again, the (doubled) angular velocity of 

 the fluid elements is 



and is therefore insensible, on account of the exponential 

 factor alone, except within a region bounded, more or less 

 vaguely, by a paraboloidal surface, having its focus at 0, for 

 which k(r — x) has a moderate constant value. This region 

 may be referred to as the " wake," although it includes a 

 certain space on the upstream side of the sphere. If we 

 superpose a general velocity — U parallel to a, the residual 

 velocity tends, for large values of r and for points outside 

 the wake, to become purely radial, as if due to a simple 

 source of strength 47rA , or 6irva, at the origin. This is 

 compensated by an inward flow in the wake ; thus for points 

 along the axis of the wake, to the right, where x = r, we find 



3Ua 1 Ua" _ 



2 r 2 r 6 v 



This indicates a velocity following the sphere (when the 



* The addition to the pressure (see the footnote on p. 113) is now 

 IpW at the surface of the sphere. This being- uniform, aoes not affect 



the resultant. 



