The kinetic energy of the fluid per unit length of the cylinder is, from Equations 

 [106v,w] and Equation [17d], 



2 ^"' 



T, = -^^ (a?- - b^f f sin^ 2ridn = — (a^ - b^)^ o? . [106z] 



^16 / 16 



For a lamina (6 = 0) 



T, = — no 0)^ a". [106a'] 



^ 16 



Circulation V about the cylinder or lamina can be included by adding the same terms 

 containing F as in Section 84. 



Combined Translation and Rotation, or Rotation about Other Axes. 



Any two-dimensional motion of the cylinder or lamina perpendicular to its generators 

 can be resolved into a motion of translation and a rotation about its axis. Such a motion can 

 also be regarded as a pure motion of rotation about some other axis. The corresponding 

 expressions for (^ and i/' can be constructed by adding those for the component motions, with 

 or without circulation; and the velocity can be found by adding the two velocities vectorially. 



In the case F = 0, when the kinetic energy is calculated by substituting the combined 

 (^ and ^A, as obtained from Equations [106v,w] and from Equations [84b, c] with ^- ^ , in 

 Equation [17d] or in f^ = - (p/2) ^(fidxp, the same terms in V'^ and oj^ are obtained as before, 

 and in addition a product term in coU. The latter contains, besides constants, the integral 



J ^ \-2{b cos a cos t] + a sin a sin -q) sin 2 7/ + cos 2 r]{- b cos a sin rj + a sin a cos tj)! d-q. 

 



This integral, however, equals zero. Hence the kinetic energies of translation and of rotation 

 are simply additive; their sum is the total energy. 



As an example, if a lamina of width 2a is rotating at angular velocity cd about an axis 

 lying in its plane, parallel to its edges but displaced a distance j8a from its median line, the 

 translation to be added is one perpendicular to its plane at velocity (J = j8aw. Hence, from 

 Equation [841] with a = 7r/2 and Equation [106z], the total kinetic energy of the fluid per unit 

 length of the lamina is 



1 + 8B^ 



r - !_ 



1 



16 

 Streamlines for this case are shown in Figure 175 



261 



np a'* u?. [106b'] 



