Hence, if these equations hold, the line x = a may form part of a rotating boundary. Since it is 

 evident from Equation [103c, d] that everything repeats when 6 is increased by 120 deg or 240 

 deg, two other similar lines must exist. These three lines will enclose an equilateral triangle 

 centered at the origin. 



Substituting for A in Equation [103c, d], 



= — r sin 30, i/i = - — r cos 3 6; 

 6a 6a 



[103e,f] 



whence 



q = r sin ^ d, qn = - ;• cos 3 0, q 



[I03g,h,i] 



These formulas represent the flow inside a vessel in the form of an equilateral triangular 

 prism, rotating with angular velocity about an axis parallel to its length and passing through 

 the center of its section. The vertices are at r = 2a and = 60 deg, 180 deg, and 300 deg; 

 the sides are of length s = 2 (2a cos 30°) or s = 2^f^a. The instantaneous streamlines are 

 illustrated in Figure 172. 



The pressure p, when oj is constant, is 



given by Equation [lie] with w = r or, using 



Equations [103h,i], 



f = (r + 4 ar cos 3 + p^). 



8a^ 



Figure 172 — Absolute streamlines for 

 fluid within a triangular prism 

 rotating about its axis. 



[103j] 



The kinetic energy of the fluid per unit 

 length is 



r, = — p {\q dxdy = _ 



1 2 ^ ^ 80v'3 



o4 2 

 -=. pS CO . 



[103k] 



(For notation and method; see Section 34; 

 Reference 1, Article 72; Reference 2, Section 

 9.72.) 



254 



