1D Viscoelastic Flow in a Circular Straight Tube with Variable Radius

The aim of this paper, is to analyze the axisymmetric unsteady flow of an incompressible second order fluid in a slender body of revolution with circular cross section with variable radius. To make that, we use the Cosserat theory ({\it also called director theory}) related with the fluid dynamics which reduces the exact three-dimensional equations to a system which depends only on a single spatial and time variables. From this 1D system, we obtain for a flow in a rigid and impermeable tube, without swirling motion, the relationship between mean pressure gradient and volume flow rate over a finite section of the tube. Also, we obtain the correspondent equation to the wall shear stress. Moreover, we use the three-dimensional flow {\it theorem of Giesekus} to obtain the three-dimensional average pressure gradient solution for steady creeping flow in a linearly tapered tube. Therefore, we compare this 3D solution for steady creeping flow with the respective solution for average pressure gradient obtained by director theory with nine directors. Numerical simulations are performed for different nondimensional numbers. Flow properties as existence, uniqueness and stability of steady solutions are also given