1. Introduction

Considerable progress has been made in studying flows of non-Newtonian fluids throughout the last few decades. Due to their viscoelastic nature non-Newtonian fluids, such as oils, paints, ketchup, liquid polymers and asphalt exhibit some remarkable phenomena. Amplifying interest of many researchers has shown that these flows are imperative in industry, manufacturing of food and paper, polymer processing and technology. Dissimilar to the Newtonian fluid, the flows of non-Newtonian fluids cannot be explained by a single constitutive model. In general the rheological properties of fluids are specified by their so-called constitutive equations. Exact recent solutions for constitutive equations of viscoelastic fluids are given by Rajagopal and Bhatnagar [1], Tan and Masuoka [2, 3], Khadrawi et al. [4] and Chen et al. [5] etc. Among non-Newtonian fluids the Jeffrey model is considered to be one of the simplest type of model which best explain the rheological effects of viscoelastic fluids. The Jeffrey model is a relatively simple linear model using the time derivatives instead of convected derivatives. Nadeem et al. [6] obtained analytic solutions for stagnation flow of Jeffrey fluid over a shrinking sheet. Khan [7] investigated partial slip effects on the oscillatory flows of fractional Jeffrey fluid in a porous medium. Hayat et al. [8]

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

examined oscillatory rotating flows of a fractional Jeffrey fluid filling a porous medium. Khan et al. [9] discussed unsteady flows of Jeffrey fluid between two side walls over a plane wall.

where S and pI represents the extra stress tensor and the indeterminate spherical stress, the dynamic viscosity is denoted by μ, A= L + L<sup>T</sup> is the first Rivlin-Ericksen tensor, L is the velocity gradient, λ is relaxation time and θ is retardation time. The Lorentz force due to magnetic field is

Unsteady Magnetohydrodynamic Flow of Jeffrey Fluid through a Porous Oscillating Rectangular Duct

<sup>J</sup> � <sup>B</sup> ¼ �σβ<sup>2</sup>

<sup>R</sup> ¼ � μϕ

In the following problem we consider a velocity field and extra stress of the form

<sup>∂</sup>xw xð Þ ; <sup>y</sup>; <sup>t</sup> , <sup>τ</sup><sup>2</sup> <sup>¼</sup> <sup>μ</sup>

fluid the Darcy's resistance satisfies the following equation

cally satisfied. Also, at t = 0, the fluid being at rest is given by

∂t

<sup>x</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> y

is given by Eq. (8). The associated initial and boundary conditions are

∂t <sup>∂</sup><sup>2</sup>

<sup>ρ</sup> is the magnetic parameter, <sup>K</sup> <sup>¼</sup> <sup>ϕ</sup>

<sup>τ</sup><sup>1</sup> <sup>¼</sup> <sup>μ</sup>

ð Þ <sup>1</sup> <sup>þ</sup> <sup>λ</sup> <sup>∂</sup>tw xð Þ¼ ; <sup>y</sup>; <sup>t</sup> <sup>ν</sup> <sup>1</sup> <sup>þ</sup> <sup>θ</sup> <sup>∂</sup>

0

3. Statement of the problem

where <sup>H</sup> <sup>¼</sup> <sup>σ</sup>B<sup>2</sup>

kinematic viscosity.

ð Þ <sup>1</sup> <sup>þ</sup> <sup>λ</sup> <sup>1</sup> <sup>þ</sup> <sup>θ</sup> <sup>∂</sup>

flow direction, the governing equation leads to

where σ represents electrical conductivity and β<sup>o</sup> the strength of magnetic field. For the Jeffrey

<sup>κ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>λ</sup> <sup>1</sup> <sup>þ</sup> <sup>θ</sup> <sup>∂</sup>

where w is the velocity in the z-direction. The continuity equation for such flows is automati-

therefore from Eqs. (2), (5) and (6), it results that Sxx = Syy = Syz = Szz = 0 and the relevant equations

where τ<sup>1</sup> = Sxy and τ<sup>2</sup> = Sxz are the tangential stresses. In the absence of pressure gradient in the

w xð Þ� ; <sup>y</sup>; <sup>t</sup> <sup>ν</sup><sup>K</sup> <sup>1</sup> <sup>þ</sup> <sup>θ</sup> <sup>∂</sup>

We take an incompressible flow of Jeffrey fluid in a porous rectangular duct under an imposed transverse magnetic field whose sides are at x = 0, x = d, y = 0, and y = h. At time t = 0<sup>+</sup> the duct begins to oscillate along z-axis. Its velocity is of the form of Eq. (5) and the governing equation

where κ(>0) and ϕ(0 < ϕ < 1) are the permeability and the porosity of the porous medium.

∂t

V ¼ ð Þ 0; 0; w xð Þ ; y; t , S ¼ Sð Þ x; y; t (5)

ð Þ <sup>1</sup> <sup>þ</sup> <sup>λ</sup> <sup>1</sup> <sup>þ</sup> <sup>θ</sup> <sup>∂</sup>

∂t

Sð Þ¼ x; y; 0 0, (6)

∂t

w xð Þ� ; <sup>y</sup>; <sup>t</sup> <sup>H</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>λ</sup> w xð Þ ; <sup>y</sup>; <sup>t</sup> , (8)

<sup>κ</sup> is the porosity parameter and ν = μ/ρ is the

<sup>∂</sup>yw xð Þ ; <sup>y</sup>; <sup>t</sup> , (7)

<sup>o</sup>V, (3)

http://dx.doi.org/10.5772/intechopen.70891

127

<sup>V</sup>, (4)

Much attention has been given to the flows of rectangular duct because of its wide range applications in industries. Gardner and Gardner [10] discussed magnetohydrodynamic (MHD) duct flow of two-dimensional bi-cubic B-spline finite element. Fetecau and Fetecau [11] investigated the flows of Oldroyd-B fluid in a channel of rectangular cross-section. Nazar et al. [12] examined oscillating flow passing through rectangular duct for Maxwell fluid using integral transforms. Unsteady magnetohydrodynamic flow of Maxwell fluid passing through porous rectangular duct was studied by Sultan et al. [13]. Tsangaris and Vlachakis [14] discussed analytic solution of oscillating flow in a duct of Navier-Stokes equations.

In the last few decades the study of fluid motions through porous medium have received much attention due to its importance not only to the field of academic but also to the industry. Such motions have many applications in many industrial and biological processes such as food industry, irrigation problems, oil exploitation, motion of blood in the cardiovascular system, chemistry and bio-engineering, soap and cellulose solutions and in biophysical sciences where the human lungs are considered as a porous layer. Unsteady MHD flows of viscoelastic fluids passing through porous space are of considerable interest. In the last few years a lot of work has been done on MHD flow, see [15–19] and reference therein.

According to the authors information up to yet no study has been done on the MHD flow of Jeffrey fluid passing through a long porous rectangular duct oscillating parallel to its length. Hence, our main objective in this note is to make a contribution in this regard. The obtained solutions, expressed under series form in terms of Fox H-functions, are established by means of double finite Fourier sine transform (DFFST) and Laplace transform (LT). Finally, the obtained results are analyzed graphically through various pertinent parameter.
