The Solution of Two-Dimensional Oseen Flow Problems Using Integral Conditions

Abstract

A general method of solving Oseen's linearized equations for two-dimensional steady flow of a viscous incompressible fluid past a cylinder in an unbounded field is developed. The analysis is developed in terms of the scalar vorticity and stream function and it is shown that the vorticity for Oseen flow problems can be obtained separately from the stream function. The determination of the vorticity can be effected using conditions of an integral character deduced from the no-slip condition at the cylinder surface together with the conditions at large distances. The independent determination of the vorticity seems to be a new step in Oseen theory. The method enables one to obtain many properties of the flow in terms ofthe Reynolds number by using only the vorticity without the necessity of finding the stream function. The use of integral conditions makes the detailed calculations straightforward, systematic, and elementary. The method is tested by applying it to the case of uniform flow past an elliptic cylinder at an arbitrary angle of incidence and also to cases of symmetrical and asymmetrical flows past circular cylinders. The leading approximation for small Reynolds number is obtained where possible. In the case of flow past a rotating cylinder, the only possible solution is the Oseen solution for the nonrotating case with the addition of a potential vortex.