A one-dimensional model for unsteady axisymmetric swirling motion of a viscous fluid in a variable radius straight circular tube Fernando Carapau a,⇑ , João Janela b a Universidade de Évora, Dept. Matemática e CIMAUE, Rua Romão Ramalho 59, 7001-671 Évora, Portugal b Universidade Técnica de Lisboa, ISEG, Dept. Matemática e CEMAPRE, Rua do Quelhas 6, 1200-781 Lisboa, Portugal article info Article history: Keywords: One-dimensional model Swirling motion Unsteady flow Hierarchical theory abstract A one-dimensional model for the flow of a viscous fluid with axisymmetric swirling motion is derived in the particular case of a straight tube of variable circular cross-section. The model is obtained by integrating the Navier–Stokes equations over cross section the tube, taking a velocity field approximation provided by the Cosserat theory. This procedure yields a one-dimensional system, depending only on time and a single spatial variable. The velocity field approximation satisfies exactly both the incompressibility condition and the kinematic boundary condition. From this reduced system, we derive unsteady equations for the wall shear stress and mean pressure gradient depending on the volume flow rate, the Womersley number, the Rossby number and the swirling scalar function over a finite section of the tube geometry. Moreover, we obtain the corresponding partial differ- ential equation for the scalar swirling function. Ó 2013 Published by Elsevier Ltd. 1. Introduction In this paper we present a one-dimensional model for the swirling motion of a viscous fluid, based on the Cosserat theory – also called director theory. The swirling features in flow fields are commonly called vortices. For most purposes (see e.g., Lugt, 1972; Kitoh, 2006; Nissan and Bressan, 1961; Moene, 2003), a vortex is characterized by a swirling motion of fluid around a central region. The swirling flow through a straight tube of variable circular cross-section is a complex turbulent flow and it is still challenging to predict and it is computationally demanding to simulate the full three-dimensional equa- tions for swirling flows, which makes the direct 3D numerical simulation infeasible in many relevant situations. In recent years, the computational dynamics of two-dimensional swirling flows has been studied extensively with the purpose of bet- ter understanding the underlying physical phenomena and getting insight on important applications like the study of hur- ricanes and tornadoes (see e.g., Guinn and Shubert, 1993; Lewellen, 1993). Here we apply the Cosserat theory (see Caulk and Naghdi (1987)) to reduce the full three-dimensional system of fluid equations to a one-dimensional system of partial differ- ential equations, which depend only on time and on a single spatial variable. The basis of this theory (see Duhem (1893) and Cosserat and Cosserat (1908)) is to consider an additional structure of deformable vectors (called directors) assigned to each point on a spatial curve (the Cosserat curve). The use of directors in continuum mechanics goes back to Duhem (1893), who regarded a body as a collection of points, together with associated directions. Theories based on such models of an oriented medium were further developed by Cosserat and Cosserat (1908). This theory has also been used by several authors in 0020-7225/$ - see front matter Ó 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.ijengsci.2013.06.010 ⇑ Corresponding author. Tel.: +351 266745370; fax: +351 266745393. E-mail addresses: flc@uevora.pt (F. Carapau), jjanela@iseg.utl.pt (J. Janela). International Journal of Engineering Science 72 (2013) 107–116 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci