Instability investigation of creeping viscoelastic flow in a curved duct with rectangular cross-section M. Norouzi a , M.R.H. Nobari b,n , M.H. Kayhani a , F. Talebi c a Mechanical Engineering Department, Shahrood University of Technology, Shahrood, Iran b Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran c Faculty of Engineering, Semnan University, Semnan, Iran article info Article history: Received 24 August 2010 Received in revised form 16 August 2011 Accepted 17 August 2011 Available online 22 August 2011 Keywords: Instability Curved duct Rectangular cross-section Viscoelstic CEF model Normal stress differences abstract In this paper, instability in the creeping viscoelastic flow inside a curved rectangular duct is investigated numerically for the first time. Using the Criminale–Eriksen–Filbey (CEF) model as the constitutive equation, the governing equations are solved by a second order of finite difference method based on the artificial compressibility algorithm in a staggered mesh. The effects of normal stress differences on the flow stability are investigated. The numerical results obtained indicate that the increase of the negative second normal stress difference of viscoelastic fluid causes stability in the creeping flow in curved ducts, however, the increase of the first normal stress difference intensifies the instability. Furthermore, at the special value of C 2 /C 1 ¼0.5, the interaction of the two normal stress differences results in a stable flow field. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction According to the classical fluid mechanics theories, the fully developed Newtonian flow inside the straight ducts is always rectilinear. In 1956, Ericksen [1] studied the possibility of recti- linear flow for non-Newtonian fluids. He used the Rivlin consti- tutive equation and found that the viscoelastic flow in straight ducts with non-circular shape of cross section is not rectilinear. For instance, in fully developed viscoelastic flows inside straight ducts with the square cross section, four pairs of counter rotating secondary flows appear in the flow field. He attributed the existence of these secondary flows to the second normal stress difference of viscoelastic materials, which causes an anisotropic behavior in the flow field. A similar structure for the secondary flows is reported by Green and Rivlin [2] for the viscoelastic flow in straight ducts with elliptic cross section. These vortices are also visualized in the experimental observations [3–6]. Fosdick and Serrin [7] improved the results of Ericksen [1] via deriving a precise theorem about the general condition for non-rectilinear flows in non-circular ducts. Truesdell and Noll [8] have captured the corner vortices using the perturbation method. McLeod [9] studied the over determined systems and presented two theorem about the rectilinear steady flow of simple fluids, which cover the results of Fosdick and Serrin [7]. Oldroyd [10] deduced the general conditions for steady viscoelastic flow in straight ducts and explained that the non-zero second normal stress difference and sharp corner presence in the cross section are necessary but not sufficient conditions for the secondary flows existence. He affirmed that secondary flows are not generated in the viscoelastic flows in which the second normal stress difference is proportional to the viscosity. The necessary conditions for the existence of secondary flows in any straight ducts with arbitrary shape of cross section are also affirmed by Huang and Rajagopal [11] to be the non-zero second normal stress difference and shear stress. The general criteria about the direction of corner vortices rotation are recently presented by Yue et al. [12]. There is an interesting analogy between the non-Newtonian laminar flow and Newtonian turbu- lent flow in straight ducts. Similar conditions for the generation of secondary flows in turbulent flow have been presented by Huang and Rajagopal [13]. Rajagopal and Huilgol [14] studied the rectilinear shear flow of second order fluid between two parallel plates and obtained the upper and lower error bounds for pressure. Rajagopal and Zhang [15] obtained the pattern of secondary deformation in a cylinder under the axial and cross-sectional deformations. They used the perturbation technique by considering the cross sectional devia- tion from circularity as the perturbation parameter. Mollica and Rajagopal [16] have found the secondary deformations of viscoe- lastic materials between eccentric cylinders under the pressure gradient. Baldoni et al. [17] investigated the third order fluid flow Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics 0020-7462/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2011.08.006 n Corresponding author. Tel.: þ98 21 64543412; fax: þ98 21 66419736. E-mail address: mrnobari@cic.aut.ac.ir (M.R.H. Nobari). International Journal of Non-Linear Mechanics 47 (2012) 14–25