The Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative model

https://doi.org/10.1016/j.nonrwa.2005.09.007Get rights and content

Abstract

The Rayleigh–Stokes problem for a generalized second grade fluid subject to a flow on a heated flat plate and within a heated edge was investigated. For description of such a viscoelastic fluid, fractional calculus approach in the constitutive relationship model was used. Exact solutions of the velocity and temperature fields were obtained using the Fourier sine transform and the fractional Laplace transform. The well-known solutions of the Stokes’ first problem for a viscous Newtonian fluid, as well as those corresponding to a second grade fluid, appear in limiting cases of our results.

Introduction

There are very few cases in which the exact solutions of Navier–Stokes equations can be obtained. These are even rare if the constitutive relations for viscoelastic fluids are considered. However, the interest in viscoelastic flows has grown considerably, due largely to the demands of such diverse areas as biorheology, geophysics, chemical and petroleum industries [26]. Because of the difficulty to suggest a single model, which exhibits all properties of viscoelastic fluids, they cannot be described as simply as Newtonian fluids. For this reason many models of constitutive equations have been proposed. Recently fractional calculus has encountered much success in the description of viscoelasticity. The starting point of the fractional derivative model of viscoelastic fluid is usually a classical differential equation which is modified by replacing the time derivative of an integer order by the so-called Riemann–Liouville fractional calculus operator. This generalization allows one to define precisely non-integer order integrals or derivatives. Bagley [1], Friedrich [5], Huang Junqi [7], He Guangyu [6], Xu [24], [25] and Tan [19], [20], [21], [23] have sequentially introduced the fractional calculus approach into various rheology problems. Fractional derivatives have been found to be quite flexible in describing viscoelastic behaviors.

The first problem of Stokes for the flat plate as well as the Rayleigh–Stokes problem for an edge has received much attention because of its practical importance [17], [18], [27]. This unsteady flow problem examines the diffusion of vorticity in a half-space filled with a viscous incompressible fluid that is set to motion when an infinite flat plate suddenly assumes a constant velocity parallel to itself from rest. By means of the similarity by transformation of variables, the exact solution corresponding to a Newtonian fluid was obtained in an elegant form by Stokes. But for a second grade fluid, a strict similarity solution does not exist [16]. Further, the equation of motion for such a fluid is a higher order than the Navier–Stokes equation and thus, in general one needs conditions in addition to the usual adherence boundary condition. Rajagopal firstly investigated this problems and gave a few exact solutions [2], [10], [11], [12], [13], [14].

However, the determination of the temperature distribution within a fluid when the internal friction is not negligible is of utmost importance. The thermal convection of a second grade fluid subject to some unidirectional flows was studied by Bandelli [2]. Recently, Fetecau extended the Rayleigh–Stokes problem to that for a heated second grade fluids [4]. In this paper, the Rayleigh–Stokes problem for a heated generalized second grade fluid was investigated. The temperature distribution in a generalized second grade fluid subject to a flow on a heated flat plate and within a heated edge was determined using the Fourier sine transform and fractional Laplace transformuential fractional derivatives. Some classical and previous results can be regarded as particular cases of our results, such as the classical solution of the first problem of Stokes for Newtonian viscous fluid and that for a heated second grade fluid.

Section snippets

Constitutive equations

For a second grade fluid, the extra stress tensor T is given by the constitutive equation [15]:T=-pI+μA1+α1A2+α2A12,where T is the Cauchy stress tensor, p is the hydrostatic pressure, I the identity tensor. α1 and α2 are normal stress moduli. A1 and A2 are the kinematical tensors defined throughA1=gradV+(gradV)T,A2=dA1dt+A1(gradV)+(gradV)TA1,where d/dt denotes the material time derivative, V is the velocity and grad the gradient operator.

The second grade fluid given by Eq. (1) is compatible

Stokes’ first problem for a heated flat plate

Suppose that a generalized second grade fluid, at rest, occupies the space above an infinitely extended plate in the (y,z)—plane. At time t=0+ the plane suddenly moves in its plane with a constant velocity U. Let T0 denotes temperature of the plate for t0, and suppose the temperature of the fluid at the moment t=0 is zero. By the influence of shear and of heat conduction the fluid, above the plate, is gradually moved and heated. The velocity field will be of the formV=u(x,t)j,where u is the

The Rayleigh–Stokes problem for a heated edge

Let us now consider a generalized second grade fluid at rest occupies the space of the first dial of a rectangular edge (x0,-<y<,z0). At the moment t=0+ the extended edge is impulsively brought to the constant velocity U. The two walls of the edge will be again maintained to the temperature T0. The velocity field will be of the formV=u(x,z,t)jand the temperature field θ=θ(x,z,t).

Similar to Section 3, the balance of linear momentum and the energy equation reduce tou(x,z,t)t=(ν+αDtβ)2x2+2

Conclusions

In this work, we have presented some results about the generalized second fluid on a heated flat plate and within a heated edge. Exact solutions of the velocity and temperature fields are obtained using the Fourier sine transform and fractional Laplace transform. The temperature distribution in a generalized second grade fluid subject to a linear flow on a heated flat plate and within a heated edge was determined. Some previous and classical results can be considered as particular cases of our

Acknowledgements

We would like to thank Prof. Rajagopal and the referee for their useful comments and suggestions regarding an earlier version of this paper. This work was supported by National Natural Science Foundation of China (10372007, 10572006) and Japan Society for the Promotion of Science (P02325).

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