Asymptotics of the circulation around a three-dimensional thin wing

We are interested in finding The velocity distribution at the wings of an aeroplane. Within the scope of a three-dimensional linear theory we analyse a model which is formulated as a mixed screen boundary value problem for the Helmholtz equation (Delta + k(2))Phi = 0 in R-3\(S) over bar, where Phi denotes the perturbation velocity potential, induced by the presence of the wings and (S) over bar:= (L) over bar boolean OR (W) over bar with the projection L of the wings onto the (y(1), y(2))-plane and the wake W. Not all Cauchy data are given explicitly on L, respectively W. These missing Cauchy data depend on the wing circulation Gamma. Gamma has to be fixed by the Kutta-Joukovskii condition: del Phi should be finite near the trailing edge x(t) of L. We reduce here this screen problem to an equivalent mixed boundary value problem in R-+(3). The main problem is in both cases the calculation of Gamma. In order to find Phi we use the method of matched asymptotics for some small geometrical parameter epsilon and the ansatz Gamma = Gamma(0) + epsilon Gamma(1)+ ... which makes it possible to split the problem into a sequence of problems for Gamma(0), Gamma(1),.... Concretely, we calculate Gamma(0) and Gamma(1) explicitly by the demand of vanishing intensity factors of the solutions of the corresponding mixed problems at the borderline between L and W. Especially, we point out that Gamma(0) can be obtained by solving a two-dimensional problem for every cross-section of L while Gamma(1) indicates the interaction of these cross-sections.