On the development of non-Newtonian secondary flows in tubes of non-circular cross-section

Abstract

The role that the normal stresses play in the generation of non-Newtonian secondary flows in straight tubes of non-circular cross-section, is examined from a theoretical standpoint. It is proven independent of any constitutive assumptions that in order for these secondary flows to occur in the absence of body forces, the axial velocity must give rise to some non-zero difference in the transverse normal stresses. This necessary condition is utilized to demonstrate that the development of secondary flows in certain non-Newtonian fluids can be influenced by the precise structure of the time derivatives in the constitutive equations. In particular, it is shown that certain frame-indifferent convected time rates give rise to secondary flows whereas others do not.