Exact solutions of accelerated flows for a Burgers’ fluid. I. The case c < k 2 =4 M. Khan a, * , S. Hyder Ali a , C. Fetecau b a Department of Mathematics, Quaid-i-Azam University, 45320 Islamabad, Pakistan b Department of Mathematics, Technical University of Iasi, R-6600 Iasi, Romania article info Keywords: Burgers’ fluid Exact solutions Accelerating flows Relaxation times abstract In this paper, we derive the exact solutions of accelerated flows for a Burgers’ fluid. The expressions for the velocity field and the associated tangential stress to the flows of a Bur- gers’ fluid induced by the accelerating flat plate are established when the relaxation times satisfy the condition c < k 2 =4. Employing the Laplace transform, the exact solutions are developed for the following two problems: (i) constantly accelerating flow and (ii) variable accelerating flow. The corresponding solutions for a Newtonian, second grade, Maxwell and Oldroyd-B fluids appear as the limiting cases of the presented analysis. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction The study of the non-Newtonian fluids has a host of well-established applications in a number of processes that occur in engineering and industry particularly in extrusion of crude oil from petroleum products. For these fluids the classical Navier– Stokes’ theory is inadequate. Because of complexity, there are several models of non-Newtonian fluids in the literature. In the category of non-Newtonian fluids, the fluids of differential type have acquired special status as well as much controversy [1]. These fluids cannot describe the influence of relaxation and retardation times. Amongst the non-Newtonian fluids models that are capable of describing the effects of relaxation and retardation phenomena is the rate type model. Moreover, the rate type viscoelastic fluid models can also describe stress relaxation, creep and the normal stress differences that develop during sim- ple shear flows. The first rate type model, which is still used widely, is due to Maxwell [2]. Recently, Rajagopal and Srinivasa [3], based on the seminal work of Maxwell, developed a systematic thermodynamic framework within which models for a variety of rate type viscoelastic fluids can be obtained. Among them, the Oldroyd-B fluid model seems to be amenable to anal- ysis and more importantly experimental. Actually, Oldroyd [4] was the first who developed systematically three dimensional rate type models, that satisfies the requirements of frame indifference, but he did not concern it with thermodynamical issues. There is a growing body of literature concerning Maxwell and Oldroyd fluids (subclasses of rate type model). Some interesting recent studies regarding these fluids are presented in Refs. [5–18]. Very recently, another subclass of the rate type viscoelastic fluids proposed by Burgers’ [19] has become very popular amongst the researchers. This is primarily because of its success such as asphalt in geomechanics and food products such as cheese, etc. There are numerous examples of the use of Burgers’ model to study asphalt and asphalt mixes [20]. Burgers’ model has also been used extensively in calculating the transient creep properties of the earth’s mantle. High temperature of fine-grained polycrystalline olivine has also been modeled using Burgers’ model. Actually, in his paper, Burgers’ developed an one-dimensional linear model, namely, r þ k 1 _ r þ k 2 € r ¼ g 1 _ e þ g 2 € e; 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.05.121 * Corresponding author. E-mail addresses: mkhan@qau.edu.pk, mkhan_21@yahoo.com (M. Khan). Applied Mathematics and Computation 203 (2008) 881–894 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc