(A) 



(B 



Figure 8. 



curves: E 1 consists of a vertical segment L erected at the origin and of an arc of the 

 free streamline in the cavity behind it, the length of L being such that the endpoint 

 of the arc is at (a, b) ; E 2 is the vertical segment x — a, 5= y ^ b. 



2. In plane or axially symmetric Riabouchinsky flow past a convex obstacle, 

 the length of the cavity and the ratio of length to width are monotonically decreasing 

 functions of the cavitation number. This result is a simple consequence of the above 

 comparison theorem, and for sake of illustration of the technique we carry out the 

 details of proof. 



Let Fig. 7a indicate a typical configuration of two Riabouchinsky flows with 

 the same incident velocity past the obstacle C, in which C 15 C 2 and I\ r 2 are the 

 respective image obstacles and free streamlines. If Y t lies below r 2 then direct applica- 

 tion of the comparison theorem at the separation point P shows that the smaller of 

 the two cavities has the larger cavitation number, as required. On the other hand, 

 if I\ intersects r,, as in Fig. 7a, a suitable similarity contraction takes C + T 1 + C 1 

 into a curve C"+ r' x + C\ lying within the cavity C + T 2 + C 2 and such that r / 1 

 has a point of contact with r 2 (Fig. 7b). Under this contraction the flow speed at 

 corresponding points can be preserved. It follows by the comparison theorem that the 

 speed on r' x , and hence on r l5 exceeds that on r 2 . This proves the monotonicity of 

 the cavity length. The monotonicity of the length-width ratio is proved similarly. Let 

 the cavity with the smaller cavitation number be contracted to have the same length 

 as the other. It suffices now to prove its width smaller. Suppose this were not so; then 

 a typical configuration of the two flows would be as in Fig. 8a. Another contraction 

 of the flow with the smaller cavitation number would bring it into a position where 

 its free streamline lies below the other, and at one point is in contact with it (Fig. 8b) . 

 The comparison theorem is now applicable, and provides an evident contradiction, 

 thereby establishing the required monotonicity property. 



291 



