where q is the velocity, 8A is the cross-section of the tube and ds is an element of distance 

 along it, taken in the direction of q, so that 8 Ads represents an element of volume. But pq8A 

 represents the mass of fluid that passes a given cross section of the tube per second and is 

 constant along the tube, since no fluid crosses its sides. Hence p qSA can be put in front of 

 the integral sign. Furthermore, qds = -d(f) in terms of the velocity potential c^, by Equation 

 [6f]. Hence . ^ 



ST =lpq8AJq ds=-lpq SAjdch =lpq8A(cf>p -(f,(^) [17b] 



where <^p and <f)Q denote values of cf) at the ends of the tube. Now let SS denote the element 

 of area on the bounding surface that is enclosed by the tube, and let q^ be the component of 

 the velocity normal to 8S, taken positive toward the fluid. Then, at the end P where q^ is 

 positive, either q^SS or q 8A represents the rate at which fluid is flowing away from the 

 instantaneous position of 8S; hence at this end q8 A = q^ 8S. At the other end, where q^ is 

 nega.tive, q^8S = - q8 A. Thus 



ST =lp[(cf.q^dS)p +(4^q^SS)^] 



Summation of this expression for all tubes gives for the total kinetic energy 



defy 



T=lpJ4>q„dS^-lpj</.^^dS, [17c] 



in which dS stands for an element of area on the bounding surface, whereas q^ represents the 

 component of velocity normal to the boundary, taken positive toward the fluid, and equals 

 -d(f)/dn by Equation [6f], where ds is replaced by dn, representing an elementary displacement 

 away from the boundary and along the normal. The integral extends over the entire boundary. 

 For two-dimensional motion, let T^ denote the kinetic energy of the fluid between two 

 planes drawn parallel to the planes in which the particles move, and unit distance apart. The 

 integral in Equation [17c] may then be taken only over the included part of the boundary; and, 

 since the motion is the same in all planes, dS may be given the form of a strip of unit length 

 and width ds, where ds is an element of distance along the curve representing the boundary in 

 a typical plane. Thus, provided there is no circulation. 



Here the curve is assumed to be traversed with the fluid lying on the left; d4>/dn is the space 

 rate of change of <^ toward the fluid along the normal to the boundary, and the last expression 

 results from Equation [13c] and the relation (dili/ds)ds = dxlj. 



If circulation is present, the formula for T, must be modified. In the case of a station- 

 ary cylinder inside a stationary cylindrical shell, only circulatory flow is possible, and the 

 tubes of flow are all closed on themselves. Here, in Equation [17b], P and Q coincide and 

 (f) p - (f)^ = r, the circulation, which is the same for all tubes; also, "^qSA, summed for all 

 tubes between two planes unit distance apart, equals i//^ - ifj ^ where xjj ^ is the value of the 

 stream function on the shell and lA its value on the enclosed cylinder. Hence in this case 



30 



