130 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



For the external fluid we have 



*dK ...... (7). 



91. A particular, but very important, case of the problem of 

 the preceding Article is that of the motion of a solid sphere in an 

 infinite mass of liquid which is at rest at infinity. If we take 

 the origin at the centre of the sphere, and the axis of x in the 

 direction of motion, the normal velocity at the surface is 

 u%/r, = u cos 0, where u is the velocity of the centre. Hence 

 the conditions to determine (/&amp;gt; are (1) that we must have V 2 &amp;lt; = 

 everywhere, (2) that the space-derivatives of $ must vanish at 

 infinity, and (3) that at the surface of the sphere (r = a), we must 

 have 



.. ..(1). 



dr 



The form of this suggests at once the zonal harmonic of the first 

 order ; we therefore assume 



, d 1 , cos 6 



$ = ^-T- - = -A r . 

 ax r r- 



The condition (1) gives 2A/a? = u, so that the required solution 

 is $ = Jucos0 ..................... (2)*. 



It appears on comparison with Art. 56 (4) that the motion of 

 the fluid is the same as would be produced by a double-source of 

 strength -Jua 3 , situate at the centre of the sphere. For the forms 

 of the stream-lines see p. 137. 



To find the energy of the fluid motion we have 



T 



cos 2 . 27ra sin d . add 



(3), 



if m = f 7r/oa 3 . It appears, exactly as in Art. 68, that the effect of 

 the fluid pressure is equivalent simply to an addition to the inertia 



* Stokes, &quot; On some cases of Fluid Motion,&quot; Camb. Trans, t. viii. (1843) ; 

 Math, and Pliys. Papers, t. i., p. 41. 



Dirichlet, &quot; Ueber einige Falle in welchen sich die Bewegung eines festen Korpers 

 in einem incompressibeln fliissigen Medium theoretisch bestimmen liisst,&quot; Berl. 

 Monatsber., 1852. 



