Numerical simulations of a second-order fluid with normal stress coefficients depending on the shear rate

We analyze the unsteady flow of an incompressible generalized second-order fluid in a straight rigid tube, with circular cross-section of constant radius, where the normal stress coefficients depend on the shear rate by using a power law model. The full 3D unsteady model is simplified using a one-dimensional hierarchical approach based on the Cosserat theory related to fluid dynamics, which reduces the exact three-dimensional equations to a system depending only on time and on a single spatial variable. From this new system we obtain the relationship between mean pressure gradient and volume flow rate over a finite section of the tube. Attention is focused on some numerical simulation under constant mean pressure gradient and on the analysis of perturbed flows