3. Statement of the problem

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.

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]

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

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

The equation of continuity and momentum of MHD flow passing through porous space is

where velocity is represented by V, density by ρ, Cauchy stress tensor by T, magnetic body force by J � B, current density by J, magnetic field by B, and Darcy's resistance in the porous

For an incompressible and unsteady Jeffrey fluid the Cauchy stress tensor is defined as [9]

<sup>1</sup> <sup>þ</sup> <sup>λ</sup> <sup>A</sup> <sup>þ</sup> <sup>θ</sup> <sup>∂</sup><sup>A</sup>

<sup>∂</sup><sup>t</sup> <sup>þ</sup> ð Þ <sup>V</sup>:<sup>∇</sup> <sup>A</sup> 

¼ divT þ J � B þ R, (1)

, (2)

dV dt 

obtained results are analyzed graphically through various pertinent parameter.

discussed analytic solution of oscillating flow in a duct of Navier-Stokes equations.

has been done on MHD flow, see [15–19] and reference therein.

∇ � V ¼ 0, ρ

<sup>T</sup> ¼ �p<sup>I</sup> <sup>þ</sup> <sup>S</sup>, <sup>S</sup> <sup>¼</sup> <sup>μ</sup>

2. Governing equations

126 Porosity - Process, Technologies and Applications

given by [7]

medium by R.

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

$$w(\mathbf{x}, y, \mathbf{0}) = \partial\_{\mathbf{t}} w(\mathbf{x}, y, \mathbf{0}) = \mathbf{0},\tag{9}$$

<sup>ζ</sup><sup>m</sup> <sup>¼</sup> <sup>m</sup><sup>π</sup>

The Fourier transform Fmn(t) have to satisfy the initial conditions

We apply LT to Eq. (17) and using initial conditions (18) to get

Fmnð Þ¼ s

Fmnð Þ¼ s

ν<sup>p</sup>þ<sup>1</sup>λ<sup>s</sup> θq K<sup>p</sup>�<sup>r</sup> Hl

We apply the discrete inverse LT to Eq. (20), to obtain

�

Fmnð Þt

�

F xð Þ¼ ; y; t

4 dh X∞ m¼1

�X<sup>∞</sup> p¼0

�

ν<sup>p</sup>þ<sup>1</sup>λ<sup>s</sup> θq K<sup>p</sup>�<sup>r</sup> Hl

Eq. (19) in series form as

<sup>d</sup> , <sup>λ</sup><sup>n</sup> <sup>¼</sup> <sup>n</sup><sup>π</sup>

<sup>h</sup> and <sup>λ</sup>mn <sup>¼</sup> <sup>ζ</sup><sup>2</sup>

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

νλmnU <sup>1</sup> � �ð Þ<sup>1</sup> <sup>m</sup> ½ �ð Þ <sup>1</sup> <sup>þ</sup> iw<sup>θ</sup> <sup>1</sup> � �ð Þ<sup>1</sup> <sup>n</sup> ½ �

ð Þ �<sup>1</sup> �ð Þ <sup>p</sup>þqþrþsþ<sup>l</sup> <sup>q</sup>!r!s!l!Γð Þ<sup>p</sup> <sup>Γ</sup>ð Þ<sup>p</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>p</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>p</sup> sl�qþpþ<sup>1</sup> :

ð Þ �<sup>1</sup> �ð Þ <sup>p</sup>þqþrþsþ<sup>l</sup> <sup>q</sup>!r!s!l!Γð Þ<sup>p</sup> <sup>Γ</sup>ð Þ<sup>p</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>p</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>p</sup> <sup>Γ</sup><sup>l</sup> � <sup>q</sup> <sup>þ</sup> <sup>p</sup> <sup>þ</sup> <sup>1</sup> :

Taking the inverse Fourier sine transform we get the analytic solution of the velocity field

sin ð Þ ζmx sin ð Þ λny Fmnð Þ x; y; t

X∞ n¼1

ν<sup>p</sup>þ<sup>1</sup>λ<sup>s</sup> θq K<sup>p</sup>�<sup>r</sup> Hl

ð Þ �<sup>1</sup> �ð Þ <sup>p</sup>þqþrþsþ<sup>l</sup>

X∞ m¼1

X∞ l¼0

Γð Þ q � p Γð Þ r � p Γð Þ s þ p þ 1 Γð Þ l þ p þ 1 <sup>Γ</sup>ð Þ<sup>p</sup> <sup>Γ</sup>ð Þ<sup>p</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>p</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>p</sup> <sup>Γ</sup><sup>l</sup> � <sup>q</sup> <sup>þ</sup> <sup>p</sup> <sup>þ</sup> <sup>1</sup> :

We will apply the discrete inverse LT technique [20] to obtain analytic solution for the velocity fields and to avoid difficult calculations of residues and contour integrals, but first we express

> νλmnU <sup>1</sup> � �ð Þ<sup>1</sup> <sup>m</sup> ½ �ð Þ <sup>1</sup> <sup>þ</sup> iw<sup>θ</sup> <sup>1</sup> � �ð Þ<sup>1</sup> <sup>n</sup> ½ � ζmλnð Þ s � iw

> > ð Þ <sup>λ</sup>mn <sup>r</sup>þ<sup>1</sup>

<sup>¼</sup> <sup>e</sup>iwtU <sup>1</sup> � �ð Þ<sup>1</sup> <sup>m</sup> ½ � <sup>1</sup> � �ð Þ<sup>1</sup> <sup>n</sup> ½ �νλmnð Þ <sup>1</sup> <sup>þ</sup> iw<sup>θ</sup> ζmλ<sup>n</sup>

ð Þ <sup>λ</sup>mn <sup>r</sup>þ<sup>1</sup>

X∞ n¼1

<sup>¼</sup> <sup>4</sup>eiwtUð Þ <sup>1</sup> <sup>þ</sup> iw<sup>θ</sup> dh

> X∞ q¼0

X∞ r¼0

X∞ s¼0

<sup>m</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> n:

<sup>ζ</sup>mλnð Þ <sup>s</sup> � iw ½ � ð Þ <sup>1</sup> <sup>þ</sup> <sup>λ</sup> ð Þþ <sup>s</sup> <sup>þ</sup> <sup>H</sup> <sup>ν</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>θ</sup><sup>s</sup> ð Þ <sup>λ</sup>mn <sup>þ</sup> <sup>K</sup> : (19)

X∞ p¼0

Γð Þ q � p Γð Þ r � p Γð Þ s þ p þ 1 Γð Þ l þ p þ 1

X∞ p¼0

Γð Þ q � p Γð Þ r � p Γð Þ s þ p þ 1 Γð Þ l þ p þ 1 t

X∞ q¼0

X∞ r¼0

X∞ s¼0

<sup>1</sup> � �ð Þ<sup>1</sup> <sup>m</sup> ½ � <sup>1</sup> � �ð Þ<sup>1</sup> <sup>n</sup> ½ �sin ð Þ <sup>ζ</sup><sup>x</sup> sin ð Þ <sup>λ</sup>ny ζmλ<sup>n</sup>

> ð Þ <sup>λ</sup>mn <sup>r</sup>þ<sup>1</sup> t l�qþp

> > q!r!s!l!

X∞ l¼0

X∞ q¼0

X∞ r¼0

X∞ s¼0

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

X∞ l¼0

l�qþp

(20)

129

(21)

(22)

Fmnð Þ¼ 0 ∂tFmnð Þ¼ 0 0: (18)

$$w(0, y, t) = w(\mathbf{x}, 0, t) = w(d, y, t) = w(\mathbf{x}, h, t) = \mathcal{U} \cos(wt),\tag{10}$$

or

$$w(0, y, t) = w(\mathbf{x}, 0, t) = w(d, y, t) = w(\mathbf{x}, h, t) = \mathsf{U} \mathsf{si}(wt), \tag{11}$$
 
$$t > 0, 0 < \mathbf{x} < d \text{ and } 0 < y < h.$$

The solutions of problems (8)–(10) and (8), (9), (11) are denoted by u(x, y, t) and v(x, y,t) respectively. We define the complex velocity field

$$F(\mathbf{x}, y, t) = \mu(\mathbf{x}, y, t) + i\nu(\mathbf{x}, y, t),\tag{12}$$

which is the solution of the problem

$$\mathbf{F}(\mathbf{1} + \lambda)\mathbf{\dot{b}}F(\mathbf{x}, y, t) = \nu \left(\mathbf{1} + \theta \frac{\partial}{\partial t}\right) \left(\mathbf{\dot{\phi}}\_x^2 + \mathbf{\dot{\phi}}\_y^2\right) F(\mathbf{x}, y, t) - \nu \mathbf{K} \left(\mathbf{1} + \theta \frac{\partial}{\partial t}\right) F(\mathbf{x}, y, t) - H(\mathbf{1} + \lambda)F(\mathbf{x}, y, t), \tag{13}$$

$$F(\mathbf{x}, y, \mathbf{0}) = \partial\_t F(\mathbf{x}, y, \mathbf{0}) = \mathbf{0},\tag{14}$$

$$F(0, y, t) = F(d, y, t) = F(\mathbf{x}, 0, t) = F(\mathbf{x}, h, t) = \mathbf{U}e^{i\mathbf{w}t},\tag{15}$$

$$t>0, 0<\text{x}$$

The solution of the problem (13)–(15) will be obtained by means of the DFFST and LT.

The DFFST of function F(x, y, t) is denoted by

$$F\_{nm}(t) = \int\_0^d \int\_0^h \sin\left(\frac{m\pi x}{d}\right) \sin\left(\frac{n\pi y}{h}\right) F(x, y, t) dx dy, \text{ } m, n = 1, 2, 3, \dots \tag{16}$$
