In the rotating coordinate system, the vorticity vector, O), can be expressed 

 as the sum of two terms: 



CO = oj +0) (21) 



— — r -^ 



where oj is the vorticity due to the rotation of the coordinate system and O) is 

 — r -^ ^ -w 



the vorticity in the inertial reference frame. From the definition of vorticity, 

 it can be shown that: 



0)^ = V X (rf2e^) = -2g (22) 



If we let £ = xi^ + r e^ (9), |fi x r_\ term on the left-hand side of Equation (17) 

 will be: 



|fl x r| = rn (23) 



Substituting Equations (21) to (23) into Equation (17) with the assumption 

 of the steady flow, we have: 



V <4- V«V + -2 i- (r^)^ + gy V = V x OJ (24) 



'2 p 2 oj— — w 



By integrating Equation (24) along a path in the flow between two arbitrary 

 points, A and B, we obtain the Bernoulli equation: 



/ 



»B - »A = / (V X 0)^) . dr (25) 



where H(£,t) is sometimes called the Bernoulli head and is defined by: 



H(r,f) = i- VV + -E- - i- (r^)^ + gy (26) 



— I o I o 



If we take the integral path dj: along a streamline or a vortex line, i.e., 

 parallel to V or oj , respectively, the integral in Equation (25) vanishes since the 

 dot product in Equation (25) is equal to zero. It then follows that the Bernoulli 

 head is constant along a streamline or a vortex line. 



By taking a reference point. A, as a point along the streamline far upstream 

 of the propeller where the propeller perturbation velocity, V , and the other 



19 



