Secondary flows due to axial shearing of a third grade fluid between two eccentrically placed cylinders

In general, purely axial flows of non-Newtonian fluids are not possible in straight pipes of noncircular cross section. The secondary flow pattern for the flow of various non-Newtonian fluids in pipes of non-circular cross section has been studied by many authors. The method which is used is invariably a perturbation technique using either the driving force for the problem or one of the material constants as the parameter for the expansion. The former implies that the solution is a perturbation of the state of rest, while the latter implies that the null solution corresponds to the Newtonian solution. However, in many problems of practical relevance, flows of a particular non-Newtonian fluid, with specific values for the non-Newtonian parameters (not necessarily small), take place under a finite driving force, making such approaches of dubious value. For instance, as a consequence of perturbing in the driving force, the secondary flows appear only at the fourth order. In this paper we use a perturbation technique in which the perturbation parameter is a geometric measure of the departure from the geometry in which rectilinear flow is possible. Such an approach allows one to study perturbation of flows which are not the null state, and this, in turn, leads to secondary flows at first order.