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 nonNewtonian 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.

One-dimensional viscoelastic fluid model where viscosity and normal stress coefficients depend on the shear rate

“…The case, m = 1 and n > 0 on condition (6), have been studied by Carapau et al [27], considering a tube with constant radius. Moreover, the case m = 1 and n = 1 on condition (6), have been studied by Carapau et al [28,30,32,34] under different perspectives, considering a tube with constant and variable radius. Finally, the case m > 0 and n = 1 on condition (6), have been studied by Carapau [31], considering a tube with constant radius.…”

“…Recently, the nine-director theory has been applied to blood flow by Robertson and Sequeira [25] and by Carapau and Sequeira [26]. Also by Carapau et al [27][28][29][30] and by Carapau [31][32][33][34] considering non-Newtonian fluids, under different geometries and perspectives. This theory it was validated on the special case of a uniform tube of constant radius for Newtonian fluids (see [15]), and for non-Newtonian fluids (see [26,28]).…”

“…Also by Carapau et al [27][28][29][30] and by Carapau [31][32][33][34] considering non-Newtonian fluids, under different geometries and perspectives. This theory it was validated on the special case of a uniform tube of constant radius for Newtonian fluids (see [15]), and for non-Newtonian fluids (see [26,28]). Also, this theory it was validated on the case of a linearly tapered tube for non-Newtonian fluids (see [34]).…”

1D dynamics of a second-grade viscous fluid in a constricted tube

“…Recently, this theory has also been applied in haemodynamics to model blood flow in the vascular system, see [2], [6], [20]. The same theory has been developed for different models in the case of a uniform rectilinear tube, see [3], [4], [5].…”