Axisymmetric motion of a second order viscous fluid in a circular straight tube under pressure gradients varying exponentially with time

Abstract

The aim of this paper is to analyze the axisymmetric unsteady flow of a non-Newtonian incompressible second order fluid in a straight rigid and impermeable tube with circular cross-section of constant radius. To study this problem, we use the 1D nine directors Cosserat theory approach which reduces the exact three-dimensional equations to a system depending only on time and on a single spatial variable. From this system we obtain the relationship between mean pressure gradient and volume flow rate over a finite section of the tube. Assuming that the pressure gradient rises and falls exponentially with time, the 3D exact solution for unsteady volume flow rate is compared with the corresponding 1D solution obtained by the Cosserat theory using nine directors.