Abstract
In this article, exact solutions of unsteady oscillatory generalized Burgers’ fluid are proposed for three different cases using Fourier transform approach. The fluid is electrically conducting under the influence of uniform transverse magnetic field and passing through the porous medium. MHD flows are induced by imposed periodic pressure gradients with smaller oscillations. Closed form solutions are obtained using Fourier sine transform, and several existing results are recovered as limiting cases. Furthermore, effects of different fluid parameters on the velocity profile are studied graphically. Analysis reveals that magnetic field and porosity parameter increases the velocity profile in case of Oldroyd-B and generalized Burgers’ fluid. It is also observed that magnetic field has more prominent effect on Burgers’ fluid as compared to Oldroyd-B fluid, while porosity parameter showed noticeable effect on Oldroyd-B fluid as compared to Burgers’ fluid.
1. Introduction
The mechanics of non-Newtonian fluids is much complicated and more nonlinear in comparison with that of the Newtonian fluids. Many problems dealing with the flow of non-Newtonian fluids have been studied by engineers and mathematicians in various geometrical configurations. Examples of these fluids are paste, gel, shampoo, soap, bead dough etc. The analysis of such flows finds important applications in engineering practice, particularly in chemical industries. A huge variety of consumer goods are now made from injection-molded plastics which often contain high concentrations of glass or carbon fibers. Many modern paints and lubricants contain polymer additives which are added to enhance their flow properties or the quality of the finished products. Also, many food stuffs (e.g. tomato sauce) and biological fluids (e.g. blood) are non-Newtonian.
Due to variety of fluids, there are several constitutive equations of non-Newtonian fluids in the literature. Among these the viscoelastic fluids have acquired the special status. In the literature, numerous works have been done to investigate the behavior of viscoelastic fluids. Some interesting flows in this direction are discussed by Rajagopal [1], Shamsuddin et al. [2, 3], Anwar et al. [4], Tan and Masuoka [5], Venkatesan and Ganesan [6], Fetecau and Vien [7], Fetecau and Fetecau [8], Lee [9], Fetecau and Agop [10], Ali et al. [11], Hayat et al. [12–14], and Nanganthran et al. [15].
All the above investigations have dealt with second grade and Oldroyd-B fluid models. Very little work have been reported to flow of a Burgers’ fluid. In light of the above literature review, it is noted that generalized Burgers’ fluid model with oscillatory motion in porous medium has not been solved using Fourier transform method. Novelty of this study is further elaborated through Table 1. Moreover, recent studies on Burgers’ fluid flow are also being considered in literature. Gangadhar et al. [16] have investigated Burgers’ fluid with convective heating. Generalized Burgers’ nanofluid with Cattaneo–Christov Relations is studied by Shahzad et al. [17] using Galerkin finite element method. Hayat et al. [18, 19] reported some analytical results for flows of a Burgers’ fluid under varying conditions. Khan et al. [20] investigated Burgers’ fluid under magnetization with stagnation point flow. The Burgers’ fluid is a viscoelastic fluid which has been used to characterize cheese, soil, asphalt etc [21–25]. This model has also been used in calculating the transient creep properties of the earth’s mantle and in the modeling of high temperature viscoelasticity of fine-grained poly-crystalline olivine [26, 27]. The Burgers’ fluid model has also been generalized and Waqas et al. [28] discussed some heat flux models to flow in this direction.
In order to solve fluid models various methodologies are used in literature. Fourier transform method is employed in this study to obtain exact solutions of the modeled problems. The Fourier transform can express an arbitrary a-periodic function as an infinite integral over a continuous range of frequencies. Firstly it was used in the treatment of single pulse phenomena by electrical engineers. The Fourier transform and the related operations of convolution and correlation has applications in optics, acoustics, scattering and diffraction of x-rays, neutrons and electrons, and a-periodic effects in electrical circuits. In various recent studies, Fourier transform method is used to solve mathematical problems effectively. Abro et al. [29] studied thermal effects on micro polar fluid with MHD and porosity by utilizing Fourier sine transform scheme. Numerical solution of Burgers’ equation was presented by Egidi et al. [30] by using Fourier transforms. Liu et al. [31] employed different transforms to study MHD flow and heat transfer of generalized Burgers’ fluid.
The main objective of this paper is to analyze the unsteady flows of generalized Burgers’ fluid induced by the cosine and sine oscillations of an infinite plate in a porous medium. The flow caused by an oscillating pressure gradient is also considered. The fluid is electrically conducting and occupies the porous structure. Graphical analysis for fluid velocity is given against varying fluid parameters. Generalization of fluid model into different fluids such as Newtonian, second grade, Maxwell, Oldroyd-B, and generalized fluid with varying material constants is depicted in tabular form. Such flows in porous media are important in enhanced oil recovery, paper and textile coating, and composite manufacturing processes. The whole analysis is given in the presence of a constant magnetic field which is very important. Finally, the results are discussed with the help of several graphs.
2. Governing Equations
The Cauchy stress in a generalized Burgers’ fluid is [28]. where indicates the indeterminate spherical stress, is first Rivlin-Ericksen tensor, is the dynamic viscosity, is the velocity gradient, and are the relaxation and retardation times, is the extra stress tensor, are material constants and is the upper convected time derivative defined by where is the material derivative and
The flows under consideration have the following properties
Unsteady velocity in -direction and fluid is at rest at . (i)Uniform magnetic field is assumed along with Darcian Porous medium
The fluid velocity vector in this case is where and are the unit vector and velocity parallel to the -axis, respectively. The velocity field (5) automatically satisfies the incompressibility condition. Since is a function of and , the stress field will also depend upon and . Now, Eq.(2) together with the initial condition (the fluid being at rest up to the moment =0) yields and
The balance of linear momentum for MHD fluid in a porous medium is where is the fluid density, is the current density, is the total magnetic field, is applied magnetic field, is the induced magnetic fields and is the Darcy resistance for a generalized Burgers’ fluid in the porous medium. Neglecting the displacement current, the Maxwell equations and Ohms’ law are where is the electric field, is the magnetic permeability and is the electrical conductivity. For small magnetic Reynolds number, the induced magnetic field is neglected. It is also assumed that .
The Darcy’s law holds for flows of viscous fluid with low speed, in an unbounded porous medium. This law gives relation among velocity and pressure drop induced by frictional drag while ignoring boundary effects on the flow. A direct proportionality between induced pressure and Darcian velocity is observed by this law. Brinkman proposed an equation describing the locally averaged flow for the porous medium with boundaries. In literature, there are various modified Darcy’s law applied for viscous flows. Very little attention has been given to macroscopic models for viscoelastic flows in a porous medium. The following law for both relaxation and retardation phenomenon in an unbounded porous medium holds [5]: in which is permeability, is the Darcian velocity and is the porosity of porous medium. Note that for , Eq. (10) reduces to well known Darcy’s law of viscous fluid.
By the analogy with constitutive Eq (2), the following law for unidirectional flow of a generalized Burgers’ fluid has been suggested:
The pressure gradient in above equation can also be interpreted as a measure of resistance to flow in the bulk of porous medium and is a measure of flow resistance offered by the solid matrix. Thus can be inferred from Eq.(11) to satisfy the following equation:
Upon use of the stated assumptions, Eq.(8) yields where the pressure gradient in direction has been ignored and is the kinematic viscosity.
3. Stokes’ Second Problem
This section deals with the MHD flow of a generalized Burgers’ fluid in a porous space The fluid is bounded by a rigid boundary at Initially, both fluid and boundary are at rest. For , the boundary starts to oscillate in its own plane. In absence of pressure gradient, the equation which governs the flow is (13). The appropriate boundary and initial conditions are in which is the imposed frequency.
In order to find the solution we define the Fourier sine transform pair as
3.1. When and
Taking Fourier sine transform of Eqs. (13)–(15) and then solving them in the -plane, we get the expression for the starting solution as follows where and indicate the transient and steady state solutions, respectively, and are given by where
Fourier inversion of Eqs. (19) and (20) yields
Note that for large times and becomes [see appendix]
Introducing the following dimensionless quantities
Eq.(24) takes the following form where
3.2. For and
For this case we have the following expression where
Adopting the same methodology of solution as for Eqs. (23) and (31), gives where
It is worth mentioning to note that for the Eq. (37) reduces to the solutions for an Oldroyd-B fluid. Moreover, Eq. (37) recovers the results of second grade fluid [8] when and ( is the material parameter of second grade fluid).
3.3. For and
Employing the similar procedure as for the case of the transient and steady state solutions are
The Eq. (40) in dimensionless variables now gives
3.4. For and
Here we have
After finding the above integral, the solution in dimensionless variables is obtained as follows
The above equation also reduces to the result of second grade fluid [8] for and
4. Modified Stokes’ Second Problem
Here, we consider the MHD flow between two infinite plates distance apart. The lower plate at oscillates in its own plane for while the upper plate at is stationary. The problem which governs the flow consists of Eqs. (13) and (14)
Following the same method of solution as in the previous section we have.
4.1. For and
where
4.2. For and
we have where
Note that the results of second grade fluid can be obtained by choosing and in Eq. (49).
4.3. For and
we obtain
4.4. For and
we get
The above equation gives the solution of second grade fluid [8] for and
5. Time-Periodic Plane Poiseuille Flow
In this section the flow between the two stationary plates is induced by an oscillating pressure gradient in the -direction. Initially the fluid and plates are at rest. The pressure gradient is of the following form
The flow is governed by Eq. (45) and
The solutions here are given by.
5.1. When and
then where
5.2. When and
then