Conical potential flow about bodies of revolution

The method of Green's functions and a formulation in terms of conal functions are used to solve the axisymmetric Laplace equation in a domain bounded externally or internally by conical boundaries. The particular application of interest is to determine the pressure and the velocity distribution about slender bodies lying on the axis of a conical tunnel in an incompressible and irrotational flow.Expressions are derived for the velocity potential and the stream function of both an isolated source and a ring source in the interior of a conical domain. These basic potential functions are used to formulate Fredholm integral equations of the first and second kinds for the source distribution generating a prescribed slender body in a conical tunnel. The integral equation for the axial source distribution is solved by expanding the solution in terms of a convergent Legendre series. An integral equation which renders directly the surface velocity distribution, by using a surface of a prolate spheroid inside the cone. Both examples involve a radial undisturbed flow. Finally, new expressions for the conal functions, which are more suitable for numerical computations, are also presented together with some numerical results.