316 



MOTION OF THE GAS SPHERE 



pulsating sphere can maintain the required flow distributions at a rigid 

 wall, and it is plausible that the sphere should move in this way as if the 

 wall exerted a force on it directly. The addition of such a displacement 

 is not, however, a true solution of the boundary condition at the sphere 

 except in the limit of negligible size, but the required flow at this bound- 

 ary can be restored by placing additional sources at suitable points in- 

 side the sphere. As this region is in fact free of fluid, fictitious sources 





\ \ \ \ \_\ \ 



\ 



\. \ / 



/ 



. \i.'/ 



\ 



\\. 



II / 

 I/. 



/ 





Fig. 8.13 Streamlines of flow for source and image of equal strength. 



of this kind are permissible if the corresponding velocity potentials 

 satisfy Laplace's equation at all points in the fluid. 



The required sources to correct the flow at the sphere can be shown 

 to be a point source and a continuous uniform distribution of negative 

 sources known as a line sink, the disposition of which in the sphere is 

 shown in Fig. 8.14. The point C at which the source is located and the 

 line sink from the center ends is called the inverse point to B and is 

 defined geometrically by the relation OB X OC = a^, where a is the 

 radius of "the sphere. The required strength of the source C is {a/OB)M, 

 if M is the strength of the original images at B, and the distributed line 

 source has a strength M/a per unit length.^ It will be observed that 



° A proof that this distribution restores the required boundary condition at the 

 sphere will be found in the reference of footnote 8, and only a sketch of the method 

 is given here. The proof is most easily made in terms of the Stokes stream function 

 "^j which must be constant over the sphere. The value of ^ at a point P for a point 

 source M at point O is easily siiown to b(^ ^ = M cos 6, where d is the angle between 



