Oseen Flow past a Flat Plate Inclined to the Uniform Stream

Abstract

The steady two-dimensional flow of an incompressible viscous fluid past a flat plate inclined to the uniform stream is discussed on the basis of Oseen's approximation. An integral equation is obtained for determining the distribution of fundamental singularities along the surface of an arbitrary cylinder which should represent the effect of the cylinder upon the stream. In the case of a flat plate whose angle of inclination α is arbitrary, this equation is solved for small values of the Reynolds number R and it is found that the expressions for the drag coefficient C D and the lift coefficient C L are in accord with the existing ones, respectively. For infinitely large R , we show that this integral equation corresponds to that in Oseen's asymptotic theory of fluid resistance, and the expressions for C D and C L are obtained for arbitrary α. In the case of medium R , we find the solution only for α≪1 by use of a perturbation method, assuming that α 2 R is small compared with unity. Under this condition, we find that \(C_{D}{=}(8/\sqrt{\pi R})[1+{=}\alpha^{2}R/12+\text{O}(\alpha^{4}\text{R}^{2})]\) and C L =πα[1+O(α 4 R 2 )]. The above various results are shown graphically in the whole range of the Reynolds number.