295] 



STEADY MOTION OF A SPHERE. 



531 



The components of stress across the surface of a sphere of radius r are, 

 by Art. 283, 



If we substitute the values of p xxt p xy , p xt , ..., from Art. 284, we find 



d 



-j- (xu +yv + zw\ 



d 



d \ d 



-r--i)w+n-r, 



In the present case we have 



We thus obtain, for the component tractions on the sphere r = a, 



If 8S denote an element of the surface, we find 



The resultant force on the sphere is therefore parallel to .r, arid equal to 



The character of the motion may be most concisely expressed by means of 

 the stream-function of Art. 93. If we put x=r cos 0, the flux (27r\^) through a 

 circle with Ox as axis, whose radius subtends an angle 6 at is given by 



as is evident at once from (v). 



If we impress on everything a velocity - u in the direction of x, we get 

 the case of a sphere moving steadily through a viscous fluid which is at rest 

 at infinity. The stream-function is then 



2\ 



(xiii)* 



The diagram on p. 532, shews the stream-lines ^ = const., in this case, for a 

 series of equidistant values of &amp;gt;//&amp;gt;. The contrast with the case of a Motionless 

 liquid, depicted on p. 137, is remarkable, but it must be remembered that the 



* This problem was first solved by Stokes, in terms of the stream-function, 

 I.e. ante p. 518. 



342 



