Rotation Combined with Outflow 



With reference to applications in turbine theory, it is of interest to imagine a line 

 source to exist along the apex of the rotating angle. By the principal of superposition, the 

 source adds a radial component of velocity A/t where 4 is a constant, so that, in place of 

 Equation [101c] or Equation [lOlf], 



sin 2 6» 



A 



?r = ?r = '^'' 



[101k] 



Qq and qa' are unchanged. A volume 2a A of fluid flows outward through the angle, per 

 second and per unit of length perpendicular to the planes of motion. 



A stagnation line now occurs on one face of the angle, on the rear face if co and A 

 have the same sign, at r = r. = (A/oj tan 2a )^^'^. Where r < r , q^'has everywhere the same 

 sign, but beyond this point reversal of the radial velocity occurs from one side of the angle 

 to the other, as it does in the absence of outflow. As r increases, the outflowing fluid that 

 has come from the source becomes crowded more and more against the leading wall of the 

 angle. U A < 0, there is a line sink on the axis, and the fluid that is destined to be absorbed 

 by it is crowded against the trailing edge. The streamlines are illustrated in Figure 169. 



(For notation and method; see Section 34; Reference 10, page 94) 



Figure 169 — Streamlines for the motion of fluid relative to the walls 

 in a rotating angle containing a line source at its apex, 



i.e., on the axis of rotation. i 



(Copied from Reference 10.) 



102. FLUID WITHIN A ROTATING SECTOR 



Consider the fluid within a vessel whose section has the form of a circular sector of 

 radius a and aperture 2a ; let it rotate at angular velocity co about its edge, or about the apex 

 of the sector; see Figure 170. 



This problem differs from the last in the additional boundary condition that ^ = at 

 r = a, where r denotes distance from the axis. The solution can be constructed by adding to 



250 



