If a rigid cylinder of any shape rotates at angular velocity co about an axis parallel to 

 its generators, the velocity of its surface at any point distant Z from the axis has a magnitude 

 q = CO CD and is directed at right angles to the radius. Hence, by similar triangles, as illustrated 

 in Figure 166b, 



^n In dS 



q CO M ds 



where dco is the increment of co corresponding to ds. Using this value of q^ and Equation 

 [99a], 



(9i// dco ^ ^ 



dijj = ds — - coTo — ds — cocodco, 

 ds ds 



whence 



iL - — coZ^ = C [99b] 



2 



after integrating. Or, if the origin is taken on the axis, so that co = x + y , 



ip CO {x^ + y^) = C = constant. [99c] 



If any known stream function is inserted for ip in this equation and any chosen value of 

 C, the equation defines a certain curve. A rigid cylinder or shell may be inserted along this 

 curve; then ip and the associated potential cp will represent a possible flow around the cylinder, 

 or inside the shell, when it is rotating at angular velocity co about the axis from which co is 

 measured. 



The velocity of the fluid relative to the cylinder or shell, or relative to axes rotating 

 with it, may be of interest. Let q^, q~, qa denote components of the velocity in the direc- 

 tions of cylindrical coordinates s, To, 6, where the axis of z is drawn along the axis of rotation 

 and the angle 6 is measured around it in the positive direction of rotation; and let q^', q^, q^ 

 denote the corresponding components of the relative velocity. Then any point sharing in the 

 rotation has a velocity coco \n the direction of qn. Hence 



(See Reference 1, Art. 71, 72.) 



CO -^co 



[99d,e,f] 



246 



