Darrozes 
we must give an a priori geometric description of the separated flowl?], 
This is the reason why we will assume later on, that no separation 
occurs. With this condition, the Kutta-Joukowski theorem leads toa 
physically correct description, but, even with this requirement, one 
does not know the starting shape of the vortex sheet in a non-symme- 
tric flow. So, the solution still depends upon an additive assumptionl9] 
One can go further, assuming that the body is very thin, be- 
cause the flow is undisturbed in a first approximation, and the velocity 
potential @ is formulated in the form of an asymptotic expansion. 
Unfortunately, new difficulties arise when the classical solution of this 
simpler problem is evaluated through a numerical analysis. The flow 
velocity is found to be infinite on the leading edge, the trailing edge 
and the vortex sheet edges. Up till now, theoretical investigations allow 
us to know the singular behaviour of the asymptotic expansion of the 
functions @, in the vicinity of the afore-mentioned lines, but they do 
not suffice to solve complety the problem] . The classical method is 
to consider an inner region in the neighbourhood of the singular lines 
in which we look for the velocity potential in the form of a new asymp- 
totic expansion called the inner approximation. The technique of match- 
ed asymptotic expansions, which is the properway of investigation, has 
been successfully employed by M.J. Lighthilll4]and M. Van Dyke] , 
to solve the two-dimensional problem of a flow past a thin hydrofoil. 
In this paper we apply the same technique for the three- 
dimensional case and it will be seen that the results depend strongly 
upon the wing geometry. 
II FORMULATION OF THE CLASSICAL OUTER PROBLEM 
II. 1 - Basic equations 
Figure 1 shows the body shape with the following assumptions 
and notations. The body is very closed to the plane z = 0, and the inci- 
dent flow is supposed to have a uniform velocity at infinity, parallel to 
the plane xoy, in the x-direction. The planform (S) is the projection of 
the body on this plane, in the z-direction and its limiting curve ()) 
is the projection of a curve ( \) drawn on the body. The curve ()), 
which parametric equations are x(s), y(s) and € z(s) separates the 
body surface into two parts. 
- The upward surface S+ has a given analytic expression : 
Zo sane£to(acpy ) 
- The other part S~ is known in the same way : 
Zameke= (Way yo) 
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