Tl=^P^(^''s-^^c) 



[17e] 



If the cylinder and shell are in motion, it is only necessary to superpose upon the 

 circulatory flow F2 as just described another flow F^ having a single-valued potential <;&, such 

 as is caused by the motion of the boundaries when F = 0. The kinetic energies associated 

 with these two flows are simply additive, giving a total of 



^Pj'^l^n 



ds+lp^{d,,-^/r,J 



[17f] 



where the first integral extends around both shell and cylinder and i//„ is the stream function 

 for F2 alone. For, if at any point in the fluid the particle velocity due to 0. has a component 

 qj^ normal to the direction of the velocity q^ due to F2, and a component q^ parallel to 5',, 

 then 



?^ = Sin + (?lp + ?2)^ = ?1^ + ?2^ + ^^ip ?2 

 If, now, p g. q2 S A is integrated along any tube of F-, just as p q^/2 was in Equation [17b], 

 pg,^^ i^ again constant, and \^,nds = - (change in 0^) = 0. Thus the product term q^p^^ 

 contributes nothing on the whole to the kinetic energy. 



If several cylinders are present inside the shell, the flow can be resolved into F^ and 

 a number of circulatory flows, in each of which there is the same circulation about all paths 

 encircling a certain one of the cylinders once and zero circulation about all paths not en- 

 circling it. Then the argument can be extended to prove that the total kinetic energy is 

 simply the sum of the energies associated with each of these component flows. 



In the axisymmetric case, the element of area dS may take the form of a ring cut out of 

 the bounding surface by two neighboring planes perpendicular to the axis; see Figure 17. The 

 width of the ring is the length ds of the arc 



Boundary 



that is cut out by the planes from the curve 

 representing, the boundary on a typical plane 

 through the axis, and its perimeter is 277^ 

 where oT denotes distance from the axis; hence 

 its area is dS = 2n7ods. Thus, from Equation 

 [17c], 



T = npfc^ q^dS = npfcf) dip [17g] 



after substituting dS = 2n'a)ds, q^ = (l/SJ) 

 dxjj/dn from Equation [16c], dip/dn = di///ds, 

 and(dip/ds)ds=d4,. . Figure 17 - Illustrating the kinetic energy of 



the fluid for an axisymmetric surface. 

 These formulas can all be shown to 

 hold also for an infinite mass of fluid surrounding a moving internal boundary, provided the 

 velocity vanishes at infinity and provided there is no circulation. Here the integral is taken 

 only over the internal boundary. 



31 



