Thus, when a velocity potential exists, 9S and tA are associated solutions of the two equations 



_: + — =0, -i-(_I — +^Li_] = o [i6h, i] 



A surface over which t// is constant, or a stream surface, is necessarily a surface of 

 revolution. A streamline may follow the axis up to a stagnation point, at which it divides into 

 a sheaf of streamlines which then diverge and form a stream surface. The distance between 

 two given stream surfaces for slightly different values of i!i varies as l/{coq), as is evident 

 from Equation [16c]. 



The older convention as to the signs of ^ and (//, mentioned in Sections 6 and 13, has to 

 be recognized again in the present connection. According to it, the signs before the derivatives 

 would be reversed in Equations [16a, b] and [16d, e], but not in Equations [16f, g], and the 

 direction for d\h/dn would also be reversed. 



17. KINETIC ENERGY OF THE FLUID 



A useful formula in terms of the potential can be obtained for the kinetic energy of the 



fluid. The following simple deduction may be of interest; a more rigorous proof is given in 



Milne-Thomson's book.^ 



Let the fluid be homogeneous and incompressible, and let it be moving with zero 



circulation about all closed curves. Suppose, first, that the region is enclosed within a moving 



finite boundary. Then the entire region can be divided up into slender tubes of flow, such 



that the boundary of each tube consists of streamlines. As illustrated in Figure 16, each tube 



must start and end on the boundary, for the reason stated in Section 8. 



The kinetic energy of the fluid in unit volume is {l/2)pq^\ hence the energy in a single 



tube can be written 



8T = ^lpq^ {8A) ds [17a] 



Figure 16 — Illustrating the kinetic energy of the fluid. 



29 



