Author(s)

F. Carapau & A. Sequeira

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 one dimensional (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. Keywords: Cosserat theory, nine directors, unsteady rectilinear flow, axisymmetric motion, pressure gradient, second order fluid. 1 Introduction A possible simplification to a three-dimensional model for an incompressible viscous fluid inside a domain is to consider the evolution of average flow quantities using simpler one-dimensional models. Usually, in the case of flow in a tube, the classical 1D models are obtained by imposing additional assumptions and integrating both the equations of conservation of linear momentum and mass over the cross section of the tube. Here, we introduce a 1D model for non-Newtonian Rivlin-Ericksen fluids of second order in an axisymmetric tube, based on the nine-

Keywords

Cosserat theory, nine directors, unsteady rectilinear flow, axisymmetric motion, pressure gradient, second order fluid.