357 



cavity (1); see Figure 2. The dimensionless quantity 

 x/rc is on the horizontal axis. In x > the 

 numerically computed function 6rc(x) is plotted 

 and also the asymptotic expression (102) is given. 

 It appears that the asymptotic expression is a good 

 approximation for Sr^ix) also for rather small 

 values of x/tq. 



The Figures 4(a) through (e) correspond to 

 decreasing values of a. This is equivalent to 

 increasing values of the speed U with constant p, 

 r, and p„ - Pc- The length of the waves on the 

 cavity is an increasing function of a, as was stated 

 in Section 4. Further we observe from these figures 

 the following: 



i) An increase of the speed U induces an increase 

 of the amplitude of the waves on the cavity, 

 ii) When U is relatively large, the point of 

 separation is near the point where 6rjj(x) attains 

 the value 6rjj(-«') . When U is small, the point of 

 separation is near the point where 6rj^(x) = 0. 



The latter phenomenon is easily understood, since 

 we can imagine two reasons for which the fluid may 

 separate from the hub: First, the radius of curva- 

 ture of the hub may be so small that the fluid 

 particles are unable to keep contact with the hub. 

 This effect dominates in the case of a relatively 

 high speed U. Second, the value of 6rh(x) may 

 become negative. Then the centrifugal force makes 



the fluid particles leave the hub. This effect is 

 important in the case of a low speed. 



In Figure 5 we have plotted the same functions 

 for a different shape of the hub. Here 6r. (x) 

 consists of a straight line, a part of a parabola, 

 and another straight line. It appears that quali- 

 tatively the same effects occur. 



In Figure 5 we have only one value of the param- 

 eter, a, (a = 1) , but we have plotted a family of 

 functions 6rj^(x). The plot of 6r(,(x) is omitted, and 

 we have indicated the point of separation with a 

 dot. The amplitudes of the waves in 6r^(x) at x = ■» 

 are denoted in a table underneath Figure 6 . These 

 numbers are the amplitudes divided by 6rj^(-'°). 



From this figure we observe the following: 

 iii) If 6rj^(x) decreases abruptly as a function of 

 X, then the amplitude of the waves on the cavity is 

 relatively large. 



iv) The sign of (Srj^ (x) at the point of separation 

 can be positive or negative, depending on the 

 function 6rj^ (x) . If 5rh(x) decreases more and more 

 slowly as a function of x, then the va],ue of Srj^ (x) 

 at the point of separation approaches zero from 

 below. 



We will compare the effects on the cavity by 

 changing a, or U with constant p and p - p , and 

 of the function 6r (x) . In order to give a rough 



description of the dependence on 6r, (x) , we use the 



h 



'/I I I / I 1 1 1 1 



'/////// / / / 



'//////// 'rj~ 



10.00 



////////// /// 



//'///////'///// 



10.00 



FIGURE 5. The functions Srj, [ (x+s) /r^,] 

 {hatched curve), i5r|,(x/rj,) and the asymptotic 

 expression (102) (a.e.). The values of a are 

 a)4, b)2, c)l, d)0.5, e)0.25. The point of 

 separation is at x=0. 5rj^ is given by 

 6rjj(x/r^)=l for x < 0, = l-(x/rj,)2/2 for 

 < X < r^, and =1.5-x/r for x > r^. The values 

 of s/r^ are a)1.683, b)1.759, c)1.805, d)0.361, 

 e)0.052. 



