30 HYDRODYNAMICAL RELATIONS 



grad P = p grad co 



dp ^ 1 dP ^ 1 dP ^ p doi 

 dt ~ /dP\ ' dt ~ c^ dt ~~ d" dt 



\dp)s 



where c^ = {dP)/{dp)s. With these substitutions, the equation of con- 

 tinuity (2.2) and equation of motion (2.4) become, 



(2.21) d'^^ = 4t^ 



c^ dt 



— = — + (vgrad) V = -grad co 



It is convenient here to introduce a function of co and v, which may 

 be called the kinetic enthalpy 12, defined by the relation 



(2.22) 12 = CO + Jvv = CO + Jy^ 



the name arising from the fact that the term (l/2)v2 is the kinetic energy 

 per gram of fluid. It is easily shown that 



grad CO = grad 12 + (v • grad) v + v (curl v) 



In further development, radial flow for which curl v = is assumed, 

 and the equations are profitably examined in terms of a velocity po- 

 tential 4> defined by v = —grad 4>. The second of Eqs. (2.21) becomes 

 12 = d4>/dt and the first may then be written 



If a solution of the form 4> = <J>/r is assumed and the Laplace operator 

 V^ expressed in spherical coordinates, this equation becomes 



?' 

 ar2 



'2 c^' de " c^[_2dr dt^ \ 



and the kinetic enthalpy is given by rl2 = d^/dl = G{r, t). 



The analysis of Kirkwood and Bethe is based on determining 

 G(r, t) from the boundary conditions and equation of state for the 

 fluid. In order to visualize this approach, it is instructive to consider 

 the propagation of the function G{r, t) in two limiting cases. For in- 

 compressible motions c^ 00, and Eq. (2.23) becomes 



