Ogilvie 



Fig. (5-1). Planing Problem In Fig. (5-2). Planing Problem in 

 the Physical Plane the Plane of the 



Complex Potential, 



z = X + iy plane. Let F(z) = <j>(x,y) + ii|j(x,y) be the complex 

 velocity potential for this problem. Then F(z) effects a mapping of 

 the z plane onto an F plane, as shown in Fig. (5-2), in which 

 points are marked to correspond to Fig. (5-i). It is assumed that 

 4> = and ijj = at the stagnation point. Furthermore, we have set 

 4i = Ua on the upstream free- surface streamline, IJ, which implies 

 that a is the thickness of the jet and that Ua is the rate at which 

 fluid leaves in the jet. Of course, F(z) is not known yet. 



We can also consider that the z plane is mapped by the 

 function w(z) = dF/dz. w(z) is the "complex velocity , " that is, 

 w = u - iv, where u and v are the velocity components in the x 

 and y directions, respectively. The entire fluid region Is mapped 

 by w(z) Into the region bounded by a half-circle and Its diameter, 

 as shown In Fig. (5-3). Again, points are marked to correspond to 

 Fig. (5-1). The diameter Is the Image of the planing surface, on 

 which the direction of the velocity vector Is known, and the circle Is 

 the Image of the entire free surface, on which the magnitude of the 

 velocity vector Is known (from the Bernoulli equation) . Again, we 

 note that the mapping function Itself Is not yet known. 



® 



Fig. (5-3). 



Planing Problem In 

 the Plane of the 

 Complex Velocity. 



Fig. (5-4). 



Planing Problem In 

 the Auxiliary {t) 

 Plane, 



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