Exact solutions for the rotational flow of a generalized Maxwell fluid between two circular cylinders W. Akhtar a , I. Siddique b,⇑ , A. Sohail c a Institute of Space Technology, Department of Aeronautics and Aitronautics, Islamabad, Pakistan b Department of Mathematical Sciences, COMSATS Institute of Information Technology, Lahore, Pakistan c Abdus Salam School of Mathematical Sciences GC University, Lahore, Pakistan article info Article history: Received 19 August 2010 Received in revised form 15 October 2010 Accepted 16 October 2010 Available online 26 October 2010 Keywords: Generalized Maxwell fluids Velocity field Shear stress Circular cylinders abstract Unsteady flow of an incompressible generalized Maxwell fluid between two coaxial circu- lar cylinders is studied by means of the Laplace and finite Hankel transforms. The motion of the fluid is produced by the rotation of cylinders around their common axis. The solutions that have been obtained, written in integral and series form in terms of the generalized G a,b,c (, t)-functions, are presented as a sum of the Newtonian solutions and the correspond- ing non-Newtonian contributions. They satisfy all imposed initial and boundary conditions and for k ? 0 reduce to the solutions corresponding to the Newtonian fluids performing the same solution. Furthermore, the corresponding solutions for ordinary Maxwell fluids are also obtained for b = 1. Finally, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity field x(r, t) have been depicted against r and t for different values of the material and fractional parameters. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction The motion of fluid in the neighborhood of a rotating body is of interest in many engineering applications, such as the flow of drilling fluid in well-bore and lubrication studies. The flow between rotating cylinders, start form rest, has applica- tions in the food industry and is one of the most important problem of motion near rotating bodies. Unsteady pressure- driven flow of a classical Maxwell fluid in a pipe is studied by Rahaman and Ramkissoon [1]. Hayat et al. [2] obtained the velocity fields for some simple flows of Oldroyd-B fluids by using Fourier transform, also they studied the unsteady flows due to non-coaxial rotations of a disc fluid at infinity in [3–6]. Recently, Fetecau et al. [7] obtained the velocity fields and the associated tangential stresses corresponding to some helical flows of Oldroyd-B fluids between two infinite coaxial circular cylinders and within an infinite circular cylinder. Our purpose here is to study the motion of a Maxwell fluid with fractional derivatives between two infinite concentric circular cylinders, both are rotating around their common axis. In recent years, due to their applications in different areas of physics and engineering including the complex dynamics, considerable interest in such fluids has been stimulated. The governing equations with fractional derivatives are proved to be a valuable tool to handle viscoelastic properties [8–12]. Especially, they have an important role in describing the properties of polymeric solutions and melts. In particular, it has been shown that the predictions of a fractional derivative Maxwell model are in excellent agreement with the linear visco- elastic date in glass transition and a-relaxation zeros [13,14], but the list of the applications of fractional calculus is long. In the following, the velocity and the associated shear stress corresponding to the motion of an incompressible Maxwell fluid with fractional derives between two concentric circular cylinders are determined using Laplace and finite Hankel 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.10.026 ⇑ Corresponding author. E-mail addresses: imransiddique@ciitlahore.edu.pk, imransmsrazi@gmail.com (I. Siddique). Commun Nonlinear Sci Numer Simulat 16 (2011) 2788–2795 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns