286 MOTION OF THE GAS SPHERE 



chosen that it gives correctly any prescribed values of velocity at bound- 

 ary surfaces of the fluid. At rigid boundaries, the component of veloc- 

 ity normal to the surface must be equal to that of the surface, and in an 

 infinite medium, the velocity must vanish at infinity at least as rapidly 

 as l/r^ if the total kinetic energy of flow is to remain finite. 



In the cases to be considered, the fluid is subjected to external 

 forces. If the force per unit mass of fluid be denoted by F, the equa- 

 tion of motion becomes 



(8.17) Po^+ ho grad v^ = p,F - grad P 



ot 



The only forces which need be considered are conservative and hence 

 derivable from a potential function 12 defined by F = —grad 12. Intro- 

 ducing the potentials ip and 12 in Eq. (8.17) gives 



— po — grad ip + \po grad v^ = — po grad 12 — grad P 

 dt 



Space and time differentiations are interchangeable and this equation 

 can therefore be integrated with respect to the position variables to give 

 the generalized Bernoulli's equation 



(8.18) £ + = ^ _ iv^ + fXO 



po ot 



where F{t) is a function of time only. If the motion is steady (does not 

 change in character with time), d(p/dt = and F(t) = constant, and 

 hence 



p 



h Iv^ + 12 = constant 



Po 



which is the familiar form of Bernoulli's theorem for steady flow. 



For the motion of the gas sphere, the only external force is gravity 

 which has the potential — gz, where g is the acceleration of gravity, and 

 the depth z is measured as a positive distance below the point in the 

 liquid at which the hydrostatic pressure would be zero. If the surface 

 of the water is at atmospheric pressure the origin of z is therefore thirty- 

 four feet above the surface, this distance being the length of the column 

 of water which exerts a pressure of one atmosphere. From this equa- 

 tion, prescri})cd conditions on the pressure at boundary surfaces fur- 

 ther define the allowable solutions for (p. For example, at the free sur- 

 face of the liquid the pressure must be the same everywhere (ordinarily 



