Numerical simulations of a third-grade fluid flow on a tube through a contraction

Based on a director theory approach related to fluid dynamics we reduce the nonlinear three-dimensional equations governing the axisymmetric unsteady motion of a non-Newtonian incompressible third-grade fluid to a one-dimensional system of ordinary differential equations depending on time and on a single spatial variable. From this new system we obtain the unsteady equation for the mean pressure gradient and the wall shear stress both depending on the volume flow rate, Womersley number and viscoelastic parameters over a finite section of a straight, rigid and impermeable tube with variable circular cross-section. We present some numerical simulations of unsteady flows regimes through a tube with a contraction using a nine-directors theory.