MOTION OF THE GAS SPHERE 28B 



Substituting in Eq. (8.20) to determine A and B, we obtain 



(8-21) ^ = 7UJ + 2 7^^'=°'' 



Differentiation to determine radial and tangential components of veloc- 

 ity gives the results 



d(p a^ da , a^ jj ^ 

 Ur= - -T- = -2:^ + -3 ^ cos (9 

 dr r^ at n 



\d(p la^ jj . - 



Ua = = (7 sm ^ 



^ r dd 2r^ 



Both Ur and Uq vanish as 1/r^ and are admissible solutions. The 

 kinetic energy of the fluid is, from Eq. (8.19), given by 



2 I I ^ dn 



the integral being carried out over the bounding sphere of radius a and 

 a second outer boundary which can be considered a sphere of radius R 

 allowed to recede to infinity. The contribution from the large sphere 

 vanishes, while the value for the inner surface is 



(8.22) T = ^ r a ("^ + it/ cos A ("^ + f/ cos ^^ 27r2 sin ddd 



o 



The first term, giving the energy of radial motion, is the same as the 

 value found before (Eq. (8.5)). The second term, giving the transla- 

 tional energy of flow, shows that this energy is the same as if a mass 

 27rpoa^/3, equal to one half the mass of water displaced by the gas sphere, 

 were given the velocity U. The equation of energy is, therefore, ex- 

 pressed by the relation 



(8.23) 27rp,a3 (^ + jpoa'U'^ + "^ poa'gz =Y- E{a) 



where, as in section 8.2, E{a) is the internal energy and the hydrostatic 

 pressure Po = PoQz. In this equation U = —dz/dt, and there are there- 



