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We consider the equations governing the motion of incompressible second grade fluids in a bounded two-dimensional domain with Navier-slip boundary conditions. We first prove that the corresponding solutions are uniformly bounded with respect to the normal stress modulus α in the L∞-H1 and the L2-H2 time–space norms. Next, we study their asymptotic behavior when α tends to zero, prove that they converge to regular solutions of the Navier–Stokes equations and give the rate of convergence in terms of α.

This paper is concerned with steady, fully developed motion of a Navier--Stokes fluid with shear-dependent viscosity in a curved pipe under a given axial pressure gradient. We establish existence and uniqueness results, derive appropriate estimates, and prove a characterization of the secondary flows. The approximation, with respect to the curvature ratio, of the full governing systems by some Dean-like equation is studied.