A scrutiny of streamline calculation in analytical solutions of Oseen flow past cylinders and spheres

The problems associated with the calculation of streamlines in the analytical solutions of Oseen's equation for the creeping flow past cylinders and spheres are addressed in this paper. First, the analytical solutions to Oseen flow past cylinders and spheres are presented. Then, the solutions and their computer implementation are validated against existing data. To examine the correctness of the streamlines obtained as contours of the stream function, the true streamlines are calculated by directly integrating the velocity components. The comparison shows that the stream function proposed by past researchers is correct for the cylinder flow whereas it is incorrect for the sphere flow. Thus, the streamline patterns of the sphere flow as predicted by the stream function, both approximate and full formulations, are erroneous. In particular, it overpredicts the separation angle and the size of the recirculation zone. The correctness of the stream function for flow around a cylinder is mathematically proved. Additionally, it is rigorously shown why the stream function for flow around a sphere is incorrect. Specifically, the analysis shows that while the circumferential velocity component derived from the stream function is accurate for the cylinder, it is incorrect for the sphere.