Flow of a generalized Oldroyd-B fluid due to a constantly accelerating plate
Introduction
The flow characteristics of non-Newtonian fluids are quite different of those from the Newtonian fluids. Therefore, several constitutive equations have been proposed for these fluids. Among them, the models of rate type have received much attention. As early as Oldroyd [1], developed a systematic procedure for developing rate type viscoelastic fluid models. He was careful to build into his framework the invariance requirements that the model ought to meet, but no thermodynamical issue has been taken into consideration. Recently, Rajagopal and Srinivasa [2] have built a systematic thermodynamic framework within which models for a variety of rate type viscoelastic fluids can be obtained. Amid them, the Oldroyd-B model has had some success in describing the response of some polymeric liquids. This model is amenable to analysis and more importantly experimental. For this reason many papers, regarding these fluids, have already appeared in the literature.
Recently, the fractional calculus has encountered much success in the description of viscoelasticity. Especially, the rheological constitutive equations with fractional derivatives play an important role in the description of the behavior of the polymer solutions and melts. The constitutive equations corresponding to the generalized non-Newtonian fluids are obtained from those for non-Newtonian fluids by replacing the time derivatives of an integer order by the so called Rieman–Liouville fractional operators. More exactly, the ordinary derivatives of first, second or higher order are replaced by fractional derivatives of noninteger order [3]. During the last years, many researchers have studied different problems related to such fluids. Here, we shall refer only to those regarding rate type fluids [4], [5], [6], [7], [8], [9], [10], [11]. Some of the obtained results have been also extended to magnetohydrodynamic flows in porous media [12], [13], [14], [15], [16].
The aim of this paper is to establish exact solutions for the unsteady flow of an incompressible generalized Oldroyd-B fluid (GOF) due to an infinite constantly accelerating plate. These solutions, obtained by means of the Fourier sine and Laplace transforms, are presented under integral and series forms in terms of the generalized G and R functions [17]. The similar solutions for Maxwell fluids with fractional derivatives as well as those for the ordinary models are established as limiting cases of our solutions. We would like to emphasize that the exact solutions for different problems with technical relevance are very important. In addition to serving as approximations to some specific initial-boundary value problems, they can be used as tests to verify numerical schemes that are developed to study more complex unsteady flow problems.
Section snippets
Governing equations
The constitutive equations for an incompressible GOF, are given by [9]where is the Cauchy stress tensor, denotes the indeterminate spherical stress, the extra-stress tensor, the first Rivlin–Ericksen tensor, where is the velocity gradient, is the dynamic viscosity of the fluid, and are the relaxation and retardation times andIn the above relations is the velocity, is the gradient operator and
Exact solution for the velocity field
Multiplying both sides of Eq. (8) by , integrating then with respect to y from 0 to and taking into account the initial and boundary conditions (9), (10), (11), we find that:where the Fourier sine transform of , has to satisfy the conditions:Denoting bythe image function of , applying the Laplace transform to Eq. (12), using the
Calculation of the shear stress
Applying the Laplace transform to Eq. (6) and using the initial condition (5), we find that:The image function of can be easy obtained from Eqs. (18), (19), (20) and (A.4). Introducing the result into (21), it results that:
Limiting cases
1. Making into Eqs. (18), (32), with , and given by (24), (28), (30), we attain to the velocity field:and the associated shear stress:corresponding to a generalized Maxwell fluid are recovered. In view
Conclusions
In this paper, closed-form expressions for the velocity field and the shear stress corresponding to the unsteady flow induced by an infinite constantly accelerating plate in an incompressible generalized Oldroyd-B fluid, have been determined using Fourier sine and Laplace transforms. In order to avoid the lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method is applied. The solutions that have been obtained satisfy all imposed
Acknowledgements
The authors acknowledge the support from the Ministry of Education and Research, CNCSIS, through PN II-Ideas, Grant No. 26/28-09-2007, CNCSIS code ID-593. They would also like to express their gratitude to the referees for their careful assessment and for letting know about the similar results in the field.
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