130 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



For the external fluid we have 



*dv ...... (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 

 ux/r, = u cos 6, where u is the velocity of the centre. Hence 

 the conditions to determine </> are (1) that we must have V 2 < = 

 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 



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

 order ; we therefore assume 



< d 1 . cos 6 



<b = A-j -- = - A- -. 



UCC T r 



The condition (1) gives 2A/a 3 = u, so that the required solution 

 is </> = iu^cos0 ..................... (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 ^ua 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 



cos 2 . 2ira sin 6 . adO 



(3), 



if m' = f Trpa 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, " On some cases of Fluid Motion," Gamb. Trans, t. viii. (1843) ; 

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



Dirichlet, " Ueber einige Falle in welchen sich die Bewegung ernes festen Korpers 

 in einem incompressibeln fliissigen Medium theoretisch bestimmen lasst," Berl. 

 Monatsber., 1852. 



