J. Non-Newtonian Fluid Mech. 141 (2007) 85–98 Fully-developed pipe and planar flows of multimode viscoelastic fluids D.O.A. Cruz a , F.T. Pinho b,c,∗ a Departamento de Engenharia Mecˆ anica, Universidade Federal do Par´ a- UFPa, Campus Universit´ ario do Guam ´ a, 66075-900 Bel´ em, Par´ a, Brazil b Centro de Estudos de Fen´ omenos de Transporte, Faculdade de Engenharia da, Universidade do Porto, 4200-465 Porto, Portugal c Universidade do Minho, Largo do Pa¸ co, 4704-553 Braga, Portugal Received 2 May 2006; received in revised form 19 July 2006; accepted 1 September 2006 Abstract Two solutions are presented for fully-developed pipe and planar flows of multimode viscoelastic models. The fluids have a Newtonian solvent contribution and the polymer modes are described by the Phan-Thien—Tanner (PTT), the FENE-P or the Giesekus equation. The first solution is exact and can handle any number of modes, but is only semi-analytical. The second solution, which is presented only for the PTT model with a linear stress coefficient and the FENE-P model, can also handle any number of modes. It is based on a truncated series expansion and is completely analytical, but provides only an approximated solution. The complexity of the multimode solutions is investigated first with the exact semi-analytical method and it is shown that at high Deborah number flows the high-order stresses can become as important as the stress of the first mode. It is also under these conditions that the approximated analytical solution deviates from the exact semi-analytical solution. A criterion for the accurate use of the approximated solution is presented. Fortran codes are provided to obtain these solutions at the internet address at the end. © 2006 Elsevier B.V. All rights reserved. Keywords: Multimode viscoelastic models; PTT; FENE-P; Giesekus; Pipe flow; Planar flow 1. Introduction Nonlinear differential constitutive equations are increasingly used to describe the rheology of viscoelastic fluids and in solv- ing fluid mechanics problems of relevance to polymer melts and solutions. Analytical solutions of such problems provide strong insight and are also useful for validation and verification purposes. However, analytical solutions can only be obtained for simple constitutive models and/or under simplifying flow conditions, such as flow symmetry and fully-developed flow conditions, which lead to integrable expressions. As a conse- quence, most of the studies in the literature concern single-mode models. A few examples for viscoelastic constitutive equations are: Beris et al. [1] for concentric and eccentric annular flow of Maxwell, White–Metzner and Criminale–Eriksen–Filbey (CEF) fluid models, Cruz and Pinho [2] for skewed Poiseuille-Couette flows of Phan-Thien—Tanner (PTT) fluids, Oliveira [3] for pipe and planar flows of a FENE-P fluid (Finitely-Extensible- ∗ Corresponding author at: Centro de Estudos de Fen´ omenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal. Tel.: +351 967 170 674; fax: +351 225082153/1445. E-mail addresses: doac@ufpa.br (D.O.A. Cruz), fpinho@fe.up.pt (F.T. Pinho). Nonlinear-Elastic dumbbell model with Peterlin approxima- tion), for Giesekus fluids, the works of Schleiniger and Weinacht [4] for pipe and planar flows and of Yoo and Choi [5] for Couette and planar flows. Van Schaftingen and Crochet [6] have developed an analytical solution for the Poiseuille flow of Johnson–Segalman fluid model. The various works of Oliveira, Pinho and co-workers [7–9] and of Hashemabadi et al. [10,11] are for isothermal and non-isothermal pipe and planar flows of PTT and FENE-P fluids. Whereas these works were aimed at investigating the charac- teristics of steady flow conditions, other analytical contributions investigated stability issues. It is the case of Hulsen [12] and Siline and Leonov [13] for the Giesekus and Leonov models, but also of Georgiou [14] for the Oldroyd-B model and of Georgiou and Vlassopoulos [15] for the model of Johnson and Segalman [16] in steady flow and of Fyrillas et al. [17] for unsteady flows. Many other analytical contributions can be found in the spe- cialized journals, such as the Journal of Non-Newtonian Fluid Mechanics and Rheologica Acta, or earlier in the Quaterly Jour- nal of Mechanics and Applied Mathematics, among others. However, the complex rheology of viscoelastic polymer melts and polymer solutions usually requires the use of multimode models for an adequate description of the fluid behaviour. The coupling between the various modes and the flow kinematics 0377-0257/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2006.09.001