349 



the fluid. Another approximation is that our theory 

 will be linear. In order for this theory to be 

 valid it is necessary that there be no abrupt changes 

 in radius of the hub and cavity. In real fluids the 

 viscosity can have an important influence on the 

 point of separation [Wu (1972)], however, this 

 effect is too complicated to be treated by our 

 method. We will not take into account the dependence 

 of the local vapor pressure on the curvature of the 

 interface between vapor and liquid. Surface tension 

 is included in the general formulation of the prob- 

 lem. The more detailed considerations, as well as 

 the numerical calculations, will be confined to 

 zero surface tension. 



One of the unknowns of the problem is the value 

 of the axial coordinate of the point of separation. 

 This value can be determined by demanding that there 

 is no place at the wetted area where the pressure 

 is lower than the prescribed pressure in the cavity 

 and by demanding that the flow cannot penetrate the 

 surface of the hub. 



The problem is very similar to the shrink fit 

 problem, in the theory of elasticity, of an unbounded 

 elastic medium with a circular two-sided infinite 

 hole [Sparenberg (1958)]. This hole is occupied by 

 a half infinite axially symmetric rigid body and 

 the problem is to calculate the contact pressure 

 between the body and surrounding medium when for 

 instance shear stresses are supposed to be zero. 

 Also, in this case, the edge of the region of con- 

 tact has to be determined. 



The way in which we solve our problem is 

 analogous to the way in which the aforementioned 

 elastic problem can be solved. First we determine 

 a Green function. This is, in our case, the 

 deformation of the two-sided infinite cavity with 

 swirl when a rotationally symmetric pressure of a 

 Dirac 6 function type is applied at the circular 

 cylindrical wall. By using this Green function as 

 a kernel we can write down a Wiener-Hopf integral 

 equation for the unknown contact pressure causing 

 the fluid flow along the hub. This integral 

 equation is solved ni:mierically by the finite element 

 method. 



2. EQUATIONS OF MOTION AND BOUNDARY CONDITIONS 



where u, v, and w are the velocity components in the 

 X, r, and 9 direction, p is the pressure, and T is 

 2tt times the circulation around the axis. From 

 Bernoulli's equation it follows that 



p^(r) = p^-pT^/2r^ 



(2) 



Poa is the ambient pressure in the fluid and p (r) 

 ■* p^ for r ->• ". On the wall of the cavity for 



r + r we have 

 c 



P (r ) = p - pr^/2r •^ 

 o c ■» c 



Pc - P'^/'^c 



(3) 



where p is the pressure inside the cavity and pa 

 is the surface tension of the fluid. In the 

 following we assume 



p > 

 c 



(4) 



hence, the ambient pressure at infinity is larger 

 than the pressure in the cavity. From (3) it 

 follows 



- pa + /p^a^ + 2 pr^ (p - p ), 



CO C 



2 (P 



(5) 



We had to choose the positive root under the 

 assiamption (4) . For (p - p ) < we would have 

 chosen the negative root, however, this would 

 yield an unstable situation. In the case of zero 

 surface tension (5) simplifies to 



r^ = r/p/2(p_^-p^) 



(6) 



The equations of motion for a time dependent fluid 

 flow are 



3u - 1" _ 1 3p 

 p 8x 



3u - du 



dt dx 3r 



3v 



, 3v , 3v w^ 1 3p 



r-+u— +v- = — - 



dt 3x 3r r p 3r 



(7) 



(8) 



First we consider a two-sided infinite circular 

 undisturbed cavity of radius r^,, with swirl in an 

 inviscid and imcompressible fluid of density ~p. 



FIGURE 1. Undisturbed cavity flow. 



The undisturbed velocity field and pressure field are 



r 



3w _ 3w _ 3w vw 



--+U--+V-— + =0 



dt dx 3r r 



Also, we have to satisfy 



,--.-, 3u , 3v V 

 div (u, V, w)= -— + -— + —= 0. 

 3x 3r r 



r 



w = — 

 r 



(9) 



(10) 



For a disturbed motion which satisfies (7)... (10) 

 it remains true that (1) 



(11) 



u = U, V = 0, w 



Po(r) , r >rc. 



(1) 



otherwise a circular contour floating with the 

 fluid would change its circulation which is im- 

 possible when external force fields inside the 

 fluid are absent. This follows also from (9) which 

 is satisfied by (11) . Hence substituting (11) into 



(7), (10) we are left with the following three 



equations for the three unknown functions u,v, and 

 P- 



