Analytical solutions for Newtonian and inelastic non-Newtonian flows with wall slip L.L. Ferrás a , J.M. Nóbrega a,⇑ , F.T. Pinho b a IPC – Institute for Polymers and Composites, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal b Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal article info Article history: Received 14 November 2011 Received in revised form 8 February 2012 Accepted 10 March 2012 Available online 21 March 2012 Keywords: Analytical solutions Couette and Poiseulle flows Slip boundary condition Generalized Newtonian fluid abstract This work presents analytical solutions for both Newtonian and inelastic non-Newtonian fluids with slip boundary conditions in Couette and Poiseuille flows using the Navier linear and non-linear slip laws and the empirical asymptotic and Hatzikiriakos slip laws. The non-Newtonian constitutive equation used is the generalized Newtonian fluid model with the viscosity described by the power law, Bingham, Herschel–Bulkley, Sisko and Robertson–Stiff models. While for the linear slip model it was always possi- ble to obtain closed form analytical solutions, for the remaining non-linear models it is always necessary to obtain the numerical solution of a transcendent equation. Solutions are included with different slip laws or different slip coefficients at different walls. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction Wall slip occurs in many industrial applications, such as in poly- mer extrusion processes, thus affecting the throughput and the quality of the final product [1]. Therefore, analytical solutions of slip in shear flows are important to solve relevant industrial prob- lems and better understand them, but also for the assessment of computational codes used in fluid flow simulations. There are many exact solutions for fluid flow in the literature [2,3] some of which are very simple, and others that use complex rheological models [3]. Even though the simple exact solutions seem trivial, they are the building blocks to the understanding of more complex solutions. They usually rely on the Dirichlet type (no-slip) bound- ary condition (u = 0, where u stands for the velocity at the wall). However, there is experimental evidence suggesting that some flu- ids do not obey this condition at the wall [4], and show instead slip along the wall. For a review on wall slip with non-Newtonian flu- ids, including slip laws and techniques to measure this property, the works of Denn [1], Lauga et al. [4] and Hatzikiriakos [5] are strongly advised. Meijer and Verbraak [6] and Potente et al. [7,8] present analyti- cal solutions for Poiseuille flow in extrusion using wall slip for Newtonian and power law fluids. Chatzimina et al. [9] solves for non-linear slip in annular flows and analyses its stability. Ellahi et al. [10] presents an analytical solution for viscoelastic fluids de- scribed by the 8-constant Oldroyd constitutive equation with non-linear wall slip. Wu et al. [11] investigated analytically the pressure driven transient flow of Newtonian fluids in microtubes with Navier slip, whereas Mathews and Hill [12] presented analyt- ical solutions for pipe, annular and channel flows with the slip boundary conditions given by Thompson and Troian [13]. Yang and Zhu [14], and the references cited therein, report analytical solutions and theoretical studies of squeeze flow with the Navier slip boundary condition. It is also worth mentioning the works on the well-posedness of the Stokes equations with leak, slip and threshold boundary conditions [15,16], which also included their numerical implementation. In spite of the wealth of solutions in the literature, there is a wide range of slip conditions, which have not been addressed ana- lytically. With the exception of the simple linear Navier slip, for most other slip laws in the literature the analytical solutions for the so-called indirect problem are missing. Here, the results are dependent on the imposed flow rate. For the direct problem the lit- erature is rich on the solutions [6–11] but lack the corresponding reverse case, and this is not just a matter of inverting the final expressions because of the non-linearity of the slip models and of the constitutive equations. In fact, the inverse problem is invari- ably more difficult to obtain than the solution of the direct prob- lem. The main purpose of this paper is precisely to address these issues and report some new analytical solutions in particular for the inverse problem. The remainder of this paper is organized as follows: Section 2 presents the governing equations and the employed slip models. The study of Newtonian fluid flows with slip is presented first in Section 3, starting with the simple Couette flow for the sake of understanding and this is followed by the Poiseuille flow using lin- ear and non-linear slip boundary conditions and different slip coef- ficients at the upper and bottom walls (the existing relevant 0377-0257/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnnfm.2012.03.004 ⇑ Corresponding author. Tel.: +351 253510320; fax: +351 253510339. E-mail addresses: luis.ferras@dep.uminho.pt (L.L. Ferrás), mnobrega@dep. uminho.pt (J.M. Nóbrega), fpinho@fe.up.pt (F.T. Pinho). Journal of Non-Newtonian Fluid Mechanics 175–176 (2012) 76–88 Contents lists available at SciVerse ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: http://www.elsevier.com/locate/jnnfm