Abstract

Based on a modified Darcy law, the decay of potential vortex and diffusion of temperature in a generalized Oldroyd-B fluid with fractional derivatives through a porous medium is studied. Exact solutions of the velocity and temperature fields are obtained in terms of the generalized Mittag-Leffler function by using the Hankel transform and discrete Laplace transform of the sequential fractional derivatives. One of the solutions is the sum of the Newtonian solutions and the non-Newtonian contributions. As limiting cases of the present solutions, the corresponding solutions of the fractional Maxwell fluid and classical Maxwell fluids are given. The influences of the fractional parameters, material parameters, and the porous space on the decay of the vortex are interpreted by graphical results.

1. Introduction

In recent years, non-Newtonian fluids that cannot be described by the classical Navier-Stokes equations increasingly play an important role in many industries and technological applications. For the well-known fact that all non-Newtonian fluids are so complicated that they cannot be described by a single constitutive equation, various models of non-Newtonian fluid have been suggested in the literature. Amongst those models, the rate type fluid models, by the pioneering work of Frohlick and Sack [1] and Oldroyd [2], have received much more attention. As one of the rate type models, the Oldroyd-B model [3–6] has succeeded in describing important responses of some polymeric liquids.

Additionally, fractional calculus has been successful in the description of the viscoelastic properties of non-Newtonian fluids. The starting point of the fractional derivative model of a non-Newtonian fluid is usually a classical differential equation, which is modified by replacing the time derivative of an integer order by the fractional calculus operators. These fractional models have been proved to be valuable tools in characterizing the properties of polymeric solutions and melts [7–10].

The studies on vortex flow of viscoelastic fluid and heat flow for viscoelastic fluids are motivated by many practical applications, such as petroleum, chemical, and food industries and bioengineering. The corresponding problem for Newtonian fluids has been investigated by Zierep [11], by means of the similarity transformation, and for second grade fluids it was studied by Fetecau et al. [12]. Later, the results were extended to the generalized second grade fluids [13], fractional Maxwell model [14], Oldroyd-B fluids [15], and generalized Oldroyd-B fluids [16].

Moreover, special attention is given to the flow of viscoelastic fluid through porous media, due to the demands of such diverse areas as biorheology, geophysics, and chemical and petroleum industries [17–19]. In these references, the classical Darcy law was used to describe the effects of the pores on the velocity of a viscoelastic fluid.

In this paper, we consider vortex velocity and temperature fields for a generalized Oldroyd-B fluid with fractional derivatives through porous medium. Based on a modified Darcy law for a viscoelastic fluid, exact solutions are obtained in terms of generalized Mittag-Leffler function by using the Hankel transform and discrete Laplace transform of the sequential fractional derivatives.

2. Basic Equations

We begin by recalling the definition of the constitutive relationship of an incompressible fluid of Oldroyd-B type (also called the Jeffreys model). The extra stress tensor is given by the constitutive equation [15]: where denotes the indeterminate spherical stress due to the constraint of incompressibility, is the extra stress tensor, is the viscosity of the fluid, and and are constant relaxation and retardation times, respectively. It is assumed that . is the first Rivlin-Ericksen tensor, is the velocity gradient, is the velocity vector, and is defined as follows: Due to the symmetry of the considered circular motion for an Oldroyd-B fluid, the velocity field, in a system of cylindrical coordinates , can be assumed as ,  , and  . This flow field automatically satisfies the constraint of incompressibility. We suppose the initial distribution of the velocity is that of a potential of circulation and the flow is symmetrical with respect to the axis through a porous medium. Having in mind the initial condition , we can obtain the constitutive relationship: for the Oldroyd-B fluid performing this flow.

A direct approach to create fractional rheological constitutive equations is to replace the regular time derivatives of ordinary equation by fractional time derivatives of noninteger order. Therefore, according to the constitutive relationship of the classical Oldroyd-B fluid, the constitutive relationship of the generalized Oldroyd-B is given by where and are fractional derivatives of order and with respect to , respectively, and are defined as [20]

The new material constants and in (4) have the dimensions of and , respectively. Some of the papers use and instead of and . However, for the sake of simplicity, we keep the same notations as in the ordinary case. This model includes the ordinary Oldroyd-B fluid as a special case for   and includes the fractional Maxwell and classical Maxwell fluid for ,   and for ,  , respectively.

It is well known that in the flow of viscous Newtonian fluid at a low speed through a porous medium the pressure drop caused by the frictional drag is directly proportional to velocity, which is the Darcy law. Recently many modified Darcy’s laws have been presented for viscous Newtonian fluid flows in porous media. However, for viscoelastic fluid in porous media, only a few mathematical macroscopic filtration models were proposed. By analogy with Oldroyd-B fluid’s constitutive relationship, the following phenomenological model, which relates pressure drop and velocity for a viscoelastic fluid in an unbounded porous medium, has been introduced [17]: where is the permeability and is the porosity of the porous medium. Equation (6) can be simplified to Darcy’s law when . In the same way, by analogy with generalized Oldroyd-B fluid’s constitutive relationship equation (4), we introduce a modified phenomenological Darcy law with fractional derivatives as follows:

In consideration of the balance of forces acting on a volume element of fluid, the local volume average balance of linear momentum can be given by where is the material time derivative, denotes the density, and is Darcy’s resistance for a generalized Oldroyd-B fluid in the porous medium and can be inferred from (7) to satisfy the following equation [17]:

Then we obtain the momentum equation through a porous medium in the form: where the pressure gradient in the direction is ignored.

Substituting (4) into (10), we obtain in which is the kinematic viscosity. The initial and natural conditions are

Further, we consider that there exists temperature field during the vortex flow in a generalized Oldroyd-B fluid. Its initial distribution and natural conditions are similarly assumed to be in the same form of previous studies [13, 18] with

The energy equation, when the Fourier law of heat conduction is applied, may be written in the form [13, 18]: where is the radiant heating which is neglected in this paper, is the specific heat, and is the conductivity (be assumed to be constant).

3. Velocity Field

Let us introduce dimensionless variables ,  ,  and and dimensionless parameters and  . Then (11) and (12) can be changed into dimensionless equations as follows (for brevity the dimensionless mark “” is omitted here): where   ( is the Reynolds number in porous media).

In order to get the exact solution of above equations, we introduce the Hankel transform of order and its inverse transform as follows [21]: where is the first kind of the Bessel function of order . Applying the Hankel transform of order one () to (16) and (17) and having in mind condition (18), we can obtain

The Laplace transform is defined as follows:

Then, applying the Laplace transform to (17) and considering the formula [19]: we get

In order to avoid the burdensome calculations of residues and contour integrals, we apply the discrete inverse Laplace transform method to give . Firstly, we rewrite (23) as a series form

Then, applying the inverse Laplace transform term by term to (24) and considering the following property of the generalized Mittag-Leffler function [20]: we have

Finally, applying the inverse Hankel transform to (26), we obtain the exact solution to the velocity field

In addition, we can also rewrite (23) as another form of series as follows:

Applying the discrete inverse Laplace transform to the above equation and considering (25), we get where

Then, applying the inverse Hankel transform to (29), we obtain the equivalent solution of vortex velocity as follows: where (cf. [19, equation ]) is the velocity field corresponding to a Newtonian fluid performing the same motion through a porous medium.

4. Temperature Field

Equations (13)~(15) can be nondimensionalized by introducing dimensionless variables ,  ,  ,  and  . The dimensionless equation with the initial and natural boundary conditions can be given as follows (for brevity, the dimensionless mark “” is omitted here): where , , and

Making the Hankel transform of zero order with respect to in (33) and (34), and having in mind the condition (35), we obtain in which is given by

To obtain the equation above, we substituted (27) into (36) and used the following property of the Bessel function:

Solving (37) with (38), we have

Then, applying the inverse Hankel transform of zero order to (41), we get the exact solution to the temperature field in which is given by (39).

5. Special Cases and Numerical Results

When , (27) and (31) can be simplified to the dimensionless solutions of vortex velocity field for generalized Oldroyd-B fluids in nonporous cases. Equivalent expression for velocity in terms of function is obtained by Fetecau et al. [16]. And when ,  , the above results can be simplified to the solution for ordinary Oldroyd-B fluid through porous medium.

Making ,   to (27) and (42), we have vortex velocity of the fractional Maxwell fluid through porous medium as follows: and the temperature field of the fluid as in which is given by

Also when making ,   to (31), we get the solutions for the fractional Maxwell fluid through porous medium. When , (46) is the same result as that obtained by Fetecau et al. [16].

Further making to (43), (44), and (45), we, respectively, have the velocity field and the temperature field where for the classical Maxwell fluid through porous medium.

When , (23) reduces to

Separately applying the inverse Laplace and inverse Hankel transform to and in the above equation, we have the solution of vortex velocity field which is the same as the result obtained by Shen et al. [13], when , and temperature field is where for the generalized second grade fluid through a porous medium.