INPART I of the present work, the flow field predicted by the steady rotation of a sphere in a viscoelastic fluid is investigated. The equations of motion and continuity are solved together with the rheological equation of state from Giesekus model. By using the slow flow approximation method, the solution up to the second-order approximation, is obtained through the expansion of the dynamical variables in terms of the dimensionless retarded time .

In the present paper, the solution of this problem up to the third-order approximation is obtained. We try to explain some secondary flow phenomena in viscoelastic fluids for which normal stresses occur in steady shear flow. From the obtained stream function it is found that the normal stresses produce a secondary flow towards the sphere in the equatorial plane. After the fluid approaches the sphere, it winds up in close windings to the pole and then runs on a trumpet-like surface symmetric with respect to the pole axis. We introduce the interpretation of this higher order effect due to the third-order solution of the problem. The calculation of the third-order secondary flow fairly well predicts the experimental results qualitatively.