Uniform Asymptotic Solutions for the Two-Dimensional Potential Field About a Slender Body

The two-dimensional potential field about an arbitrary slender body, with either a Dirichlet or Neumann boundary condition on the body, is studied. In particular, the two-dimensional irrotational flow of an incompressible inviscid fluid past a slender cylindrical airfoil and the two-dimensional electrostatic field in the exterior of a slender cylindrical conductor are considered. The complete uniform asymptotic expansion of the potential field is obtained with respect to the parameter ε, the slenderness ratio of the profile curve of the cylinder. The camber line of the profile curve is completely arbitrary and hence these results extend the work of Geer and Keller (1968). The potential due to the presence of the body is represented as the superposition of potentials of point sources and point vortices distributed along an arc inside the body. The source and vortex strength distributions satisfy a pair of coupled linear integral equations. The kernels in both of the equations become singular as ε approaches zero. The complete uniform asymptotic expansion of the solution to these equations is found by modifying and adapting the method of Handelsman and Keller (1967). Complete expansions of the virtual mass, the polarizability, and the lift of the cylinder are obtained.