356 



the hub, we must have B < for a hub with a smooth 

 surface . We found above that B > and hence B 

 must vanish. For our numerical approximation this 

 implies that the coefficient, Cj , must vanish. Now 

 the value of the shift, s, is determined by iteration 

 so that Cj vanishes. 



When f (x) has been computed we can compute the 

 shape of the cavity in x > with a numerical 

 integration of (73) for x > 0. Using (40) with 

 a =0, we can derive an expression for 6r^(x) for 

 x -> ". The derivation is similar to the derivation 

 of (55) and, therefore we give the result only: 



2C, 



&r (x) 



where : 



o 



(102) 

 {Ai sin(? x/r ) + A2COS(C x/r )}, 



7. NU^4ERICAL RESULTS 



In this section we give computermade plots of the 

 shape of the cavity 6r|_,(x) for a number of shapes 

 of the hub 6rj^ (x) and for a number of values of the 

 dimensionless parameter a. We consider the case of 

 zero surface tension, hence a is given by (87) or 

 by (43) with a = 0: 



2(P„ 



Pc' 



(104) 



pU^ 



It follows from (88) that 6r[^(x) depends on r^ for 

 a fixed hub. The value of r^ is given by (6) : 



(f)^^r(p„ 



Pc> 



(105) 



Aj = lim I 



E-l-O -" 

 o 

 A2 = lim I 



E + -<» 



e cos(S x/r ) f{x) dx, (103a) 

 c 



e sin(5x/r ) f{x) dx. (103b) 

 2 c 



However we can vary a without changing 6rjj(x) by 

 varying U and keeping p, T and p„ - p^, constant. 



In the Figure 4 the function 6rh(x) is plotted; 

 it consists of a straight horizontal line and part 

 of a parabola. The x-axis is chosen so that x = 

 at the point of separation. No scale-unit is given 

 in the vertical direction, since 6rj,(x) and 6rc(x) 

 are the linearized perturbations of the undisturbed 



'/iiiui I III n 1 1 1 n I D 



Tri 1 1 1 1 1 1 1 1 n 1 1 



'/ 1 ! I I I / I I I I / I 



-I.OO -2.00 0.0 



'// / / 1 / 1 / / / / / / / / 1 / 



FIGURE 4. The functions Srj, [(x+s)/r^) 

 (hatched curve), '5rj,(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. Sr^, is given by 

 5rh(x/rj;) = 1 for x < and = 1- (x/r^,) ^ for 

 X > 0. The values of s/r^, are a) 1.092, 

 b)1.017, c)0.558, d)0.070, e)0.014. 



V / / / / / ) / I / / / / I / 



<V; 



