Flow of transversely isotropic fluid in curved pipes

Abstract

We study a fully developed creeping flow of nematic liquid crystals in a curved pipe with a circular cross-section. A special limit of zero Deborah number and an infinite Ericksen number limit is considered, so the flow and orientation of the director are described by the transversely isotropic fluid (TIF) model. Analytical solutions are obtained by expanding the governing equation in the limit of small pipe curvature δ, defined as the ratio of the pipe radius to the radius of the bend. We show that the presence of bend curvature alters the orientation of the microstructure and axial velocity profile. The additional non-Newtonian stress generates a secondary flow, which consists of two counter-rotating vortices. Two mechanisms responsible for the secondary motion are identified (1) combination of normal stresses with pipe curvature (2) asymmetry of the stress distribution. The relative strength and direction of those mechanisms in the framework of the transversely isotropic fluid model is quantified through a parameter q. For all liquid crystals considered in this paper, the stress asymmetry dominates, thus controlling the direction of the secondary flow, which is opposite to the secondary flow induced by inertia. Finally, we quantify the contribution of the secondary motion to the viscous dissipation and show that additional shearing increases viscous losses and power consumption without contributing to the throughput.