Swirling motion of a second-order viscous fluid in a straight tube

We consider a second-order viscous fluid with swirling motion flowing in a circular straight tube with constant radius. 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 depending only on time and on a single spatial variable. From this 1D model, we derive the relationship between mean pressure gradient and volume flow rate over a finite section of the tube as well the wall shear stress