ORIGINAL CONTRIBUTION An approximate solution for the Couette–Poiseuille flow of the Giesekus model between parallel plates Ahmadreza Raisi & Mahmoud Mirzazadeh & Arefeh Sadat Dehnavi & Fariborz Rashidi Received: 11 January 2007 / Accepted: 10 May 2007 / Published online: 18 June 2007 # Springer-Verlag 2007 Abstract An approximate analytical solution is derived for the Couette–Poiseuille flow of a nonlinear viscoelastic fluid obeying the Giesekus constitutive equation between parallel plates for the case where the upper plate moves at constant velocity, and the lower one is at rest. Validity of this approximation is examined by comparison to the exact solution during a parametric study. The influence of Deborah number (De) and Giesekus model parameter (α) on the velocity profile, normal stress, and friction factor are investigated. Results show strong effects of viscoelastic parameters on velocity profile and normal stress. In addition, five velocity profile types were obtained for different values of α, De, and the dimensionless pressure gradient (G). Keywords Giesekus constitutive equation . Couette–Poiseuille flow . Viscoelastic fluid . Parallel plates Introduction Through the years, many authors have tried to analyze the flow of different classes of materials in ducts and channels using various constitutive equations such as inelastic and linear/nonlinear viscoelastic models. An extensive literature review on this subject can be found in Pinho and Oliveira (2000), Alves et al. (2001), Escudier et al. (2002), and Oliveira (2002). The flow of non-Newtonian fluids between parallel plates is also a problem of considerable practical interest. Because of their simplicity and originality, parallel plates are often used to simulate the actual flow domain conditions in single screw extruders. Because a narrow gap exists between the barrel and the screw of the extruder, the assumption of fluid flowing between parallel plates leads to meaningful and accurate results. Tadmor and Gogos (1979) published an extensive compilation of the available information and mathematical explanation for single screw extrusion. Giesekus (1982) has developed a three-parameter model using molecular ideas that is nonlinear in the stresses. This model has gained prominence because it describes the power-law regions for viscosity and normal-stress coeffi- cients; it also gives a reasonable description of the elonga- tional viscosity and the complex viscosity. This model incorporates shear-thinning shear viscosity, nonvanishing normal-stress differences, extensional viscosity with finite asymptotic value, and nonexponential stress relaxation and start-up curves. Thus, it reproduces many characteristics of the rheology of polymer solutions as well as other liquids. The Giesekus model is employed increasingly to predict the flow and heat transfer of viscoelastic fluids. Yoo and Choi (1989) studied the Giesekus model (without retarda- tion time) in plane Couette–Poiseuille flows and obtained analytic solutions. Later, Schleiniger and Weinacht (1991) studied the steady Poiseuille flow in channels and pipes for the same model (with and without solvent contribution). They showed the possibility of the infinite number of weak solutions for the boundary value problem, in addition to the classical ones of Yoo and Choi (1989). They also showed that only the classical solutions are physically relevant solutions. Mostafaiyan et al. (2004) used the Giesekus model and obtained solution for the axial annular flow of a Rheol Acta (2008) 47:75–80 DOI 10.1007/s00397-007-0212-9 A. Raisi : M. Mirzazadeh : A. S. Dehnavi : F. Rashidi (*) Department of Chemical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Hafez Ave., P.O. Box 15875-4413, Tehran, Iran e-mail: Rashidi@aut.ac.ir