In this paper we explore the potential of an explicit constitutive equation in predicting secondary flows in straight, non–circular tubes in steady and fully developed flow. We show that secondary flows appear at first order in k a constitutive parameter in the new constitutive structure when solving the field through an asymptotic approach. The structure of the transversal stream–function predicted by the new structure is the same as that obtained at third order in Weissenberg number with the Modified Phan–Thien–Tanner constitutive equation aside from constitutive constants. We present predictions of the velocity field and a comparison with the corresponding predictions of the MPTT model. These results are obtained using the analytical algorithm based on the shape factor concept coupled with asymptotic expansions developed and used previously by the authors to solve related problems.