Axisymmetric motion of a proposed generalized non-Newtonian fluid model with shear-dependent viscoelastic effects

Three-dimensional numerical simulations of non- Newtonian fluid flows are a challenging problem due to the particularities of the involved differential equations leading to a high computational effort in obtaining numerical solutions, which in many relevant situations becomes infeasible. Several models has been developed along the years to simulate the behavior of non-Newtonian fluids together with many different numerical methods. In this work we use a one-dimensional hierarchical approach to a proposed generalized third-grade fluid with shear-dependent viscoelastic effects model. This approach is based on the Cosserat theory related to fluid dynamics and we consider the particular case of flow through a straight and rigid tube with constant circular cross-section. With this approach, we manage to obtain results for the wall shear stress and mean pressure gradient of a real three-dimensional flow by reducing the exact three-dimensional system to an ordinary differential equation. This one-dimensional system is obtained by integrating the linear momentum equation over the constant cross-section of the tube, taking a velocity field approximation provided by the Cosserat theory. From this reduced system, we obtain the unsteady equations for the wall shear stress and mean pressure gradient depending on the volume flow rate, Womersley number, viscoelastic coefficients and the flow index over a finite section of the tube geometry. Attention is focused on some numerical simulations for constant and non-constant mean pressure gradient using a Runge-Kutta method.