•li Pien and Strom-Tejsen 



where ^ is a vorticity vector, it follows that 



B 



1 1 „ 1 



Bt " Pf ^"^ 2 

 For an incompressible fluid, p^ being a constant, we have 



(A3) 



^■^ = -V(f + ^qM + (q X ^ + F) . (A4) 



3t \Pf 2 



Applying the continuity equation, i.e., Vq = 0, Eq, (A4) yields 



+ Iq^U V(qx^+F) . (A5) 



We introduce a function o and a generalized force vector k such that 



d. = f + i q^ (7) 



and 



Equation (A5) becomes 



^ + F. (A6) 



V^O = Vk. (A7) 



This is a general governing equation for an incompressible inviscid fluid flow 

 subjected to an external force field. 



Now we shall attempt to discuss the physical significance of this equation 

 and its solution as applied to a propeller problem. 



Like an airfoil or wing, a propeller blade is considered to be a lifting sur- 

 face on which forces are distributed, and this surface distribution of forces may 

 be considered to be a limiting case of volume distribution by reducing one of the 

 dimensions of the volume to zero while increasing the force intensity so that the 

 total force is the same. It can be shown that the action of such external forces 

 upon a fluid will produce vortex motion, and specifically, the curl of the force 

 vector represents the time rate of change of the vorticity generated. It can 

 therefore be said that a lifting surface such as a propeller blade is a vorticity 

 generator. When the blade advances in the fluid, it imparts vorticity to the fluid 

 particles along its path as it passes, i.e., vorticity is left in its wake. 



Hence, the entire fluid field may be conveniently divided into three regions, 

 the lifting surface, the wake, and the remaining field, and each region may be 

 described by an appropriate equation based on the previously described general 

 governing equation as follows: 



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