J. Non-Newtonian Fluid Mech. 132 (2005) 28–35 Analytical solutions for fully developed laminar flow of some viscoelastic liquids with a Newtonian solvent contribution D.O.A. Cruz a , F.T. Pinho b , P.J. Oliveira c,∗ a Depart. Eng. Mecˆ anica, Universidade Federal do Par´ a-UFPa, Campus Universit´ ario do Guam ´ a, 66075-900 Par´ a, Brasil b Centro de Estudos de Fen´ omenos de Transporte, DEM, Universidade do Minho, Campus de Azur´ em, 4800-058 Guimar ˜ aes, Portugal c Depart. Eng. Electromecˆ anica, Unidade Materiais Tˆ exteis e Papeleiros, Universidade da Beira Interior, 6201-001 Covilh˜ a, Portugal Received 6 June 2005; received in revised form 30 July 2005; accepted 18 August 2005 Abstract We present analytical solutions for fully developed pipe and channel flows of two viscoelastic fluids possessing a Newtonian solvent, where the polymer contribution is either described by the Phan-Thien–Tanner (PTT) or FENE-P models. We derive in detail the pipe flow solution for the PTT fluid, and present the final solutions for the remaining three cases. This constitutes an important addition to existing results where the presence of a solvent with Newtonian characteristics has been consistently overlooked, as it posed considerable difficulties to the task of obtaining a closed form solution. In addition, interesting aspects of the solutions are discussed. © 2005 Elsevier B.V. All rights reserved. Keywords: Analytical solution; Pipe flow; Channel flow; Fully developed; PTT; Solvent viscosity; FENE-P; Viscoelastic 1. Introduction The quest for analytical solutions of the most frequently used viscoelastic rheological models in relatively simple flows is, in our view, a matter of great importance but one which has been largely overlooked. As a justification for this statement we would like to highlight out the major relevance of the Poiseuille flow solution in the fluid mechanics of Newtonian fluids: it has a ped- agogical motivation, appearing in all books of fluid mechanics as the typical example of a closed-form analytical solution to the Navier-Stokes equations; it is employed in practical experimen- tal apparatus as a means to obtaining the viscosity of Newtonian liquids, for example in capillary-tube viscometers; and finally, it is often used to check numerical solutions, as a simple limiting test case, or to impose boundary conditions in the inlet or outlet of pipes and channels in complex geometries. A final point is that analytical solutions (when available) provide the simplest and most efficient way to perform parametric investigations of the effects of independent variables on output variables. ∗ Corresponding author. Fax: +351 275329952. E-mail addresses: doac@ufpa.br (D.O.A. Cruz), fpinho@dem.uminho.pt (F.T. Pinho), pjpo@ubi.pt (P.J. Oliveira). Similar motivations are valid when the fluid, instead of fol- lowing the simple linear stress/strain relationship of the Newto- nian model, follows more complex differential constitutive mod- els typical of non-Newtonian media possessing viscoelasticity. In the early days of Rheology, when some of the constitutive equations still in use today were devised, like the type A and B fluids developed by Oldroyd [1], such a need was realised, but a relatively simple graphical procedure was proposed by Oldroyd [2] and later improved upon by Walters [3,4] to solve fully devel- oped flows in pipes or channels for any fluid model, provided the steady viscosity function η(˙ γ ) was known in terms of the shear rate ˙ γ . The indirect procedure devised by Walters [3] consisted in eliminating the shear stress from the constitutive and motion equations in order to obtain an explicit equation giving the radial position r as a function of the shear rate ˙ γ . In this way the main independent variable is not r but the shear rate itself. Then by integrating the definition ˙ γ = du/dr he was able to obtain the desired velocity profile u(r) as u =  ˙ γ ˙ γ w ˙ γ (dr/d˙ γ )d˙ γ , where the wall shear rate ˙ γ w was taken as the main independent parameter. This approach can obviously be applied to both Generalised- Newtonian Fluids (GNF) and also to viscoelastic fluid models. In the latter case the viscosity function η(˙ γ ) is not known a pri- ori, but can be derived out for many models, leading to complex viscosity variations. No attempt was made to introduce such 0377-0257/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2005.08.013