One-dimensional viscoelastic fluid model where viscosity and normal stress coefficients depend on the shear rate

We study the unsteady motion of a viscoelastic fluid modeled by a second-order fluid where normal stress coefficients and viscosity depend on the shear rate by using a power-law model. To study this problem, we use the one-dimensional nine-director Cosserat theory approach which reduces the exact three-dimensional equations to a system depending only on time and on a single spatial variable. Integrating the equation of conservation of linear momentum over the tube cross-section, with the velocity field approximated by the Cosserat theory, we obtain a one-dimensional system. The velocity field approximation satisfies both the incompressibility condition and the kinematic boundary condition exactly. From this one-dimensional system we obtain the relationship between average pressure and volume flow rate over a finite section of the tube with constant and variable radius. Also, we obtain the correspondent equation for the wall shear stress which enters directly in the formulation as a dependent variable. Attention is focused on some numerical simulation of unsteady/steady flows for average pressure, wall shear stress and on the analysis of perturbed flows.