1D model of swirling flow motion of a viscous fluid in a circular straight tapered tube

We present a 1D model for a viscous fluid with axial symmetric swirling motion flowing in a circular straight tube with variable 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. The velocity field approximation satisfies both the incompressibility condition and the kinematic boundary condition exactly. From this new system, we derive the equation for the wall shear stress and the relationship between mean pressure gradient, volume flow rate and swirling scalar function over a finite section of the tube. Also, we obtain the corresponding partial differential equation for the swirling scalar function.