Axisymmetric Swirling Motion of Viscoelastic Fluid Flow Inside a Slender Surface of Revolution

Motived by the aim of modelling the behavior of swirling flow motion, we present a 1D hierarchical model for an Rivlin-Ericksen fluid with complexity n = 2, flowing in a circular straight tube with constant and no 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. 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 average 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.