;

X∞ l¼0

(35)

(36)

(37)

(38)

X∞ p¼0

cosð Þ ζcx sin ð Þ λey λe

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>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>p</sup> <sup>Γ</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>p</sup>

cosð Þ ζcx sin ð Þ λey λe

ð Þ 2 � q þ p; 0 ,ð Þ 1 � r þ p; 0 ,ð Þ �1 � s � p; 0 ,ð Þ �p; 1

ð Þ 0; 1 ,ð Þ �p; 0 ,ð Þ �1 � p; 0 ,ð Þ �p; 0 ,ð Þ �p; 0 , qð Þ � p; 1

X∞ e¼0

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

ð Þ 0; 1 ,ð Þ �p; 0 ,ð Þ �1 � p; 0 ,ð Þ �p; 0 ,ð Þ �p; 0 , qð Þ � p; 1

q!r!s!

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

> X∞ c¼0

<sup>4</sup>, <sup>6</sup> Ht ð Þ <sup>2</sup> � <sup>q</sup> <sup>þ</sup> <sup>p</sup>; <sup>0</sup> ,ð Þ <sup>1</sup> � <sup>r</sup> <sup>þ</sup> <sup>p</sup>; <sup>0</sup> ,ð Þ �<sup>1</sup> � <sup>s</sup> � <sup>p</sup>; <sup>0</sup> ,ð Þ �p; <sup>1</sup>

ð Þ �<sup>1</sup> �ð Þ <sup>p</sup>þqþrþ<sup>s</sup> <sup>q</sup>!r!s!

ð Þ �<sup>1</sup> <sup>p</sup>þqþrþ<sup>s</sup>

The volume flux due to cosine oscillations is given by

$$Q\_c(\mathbf{x}, y, t) = \int\_0^d \int\_0^h \mu(\mathbf{x}, y, t) d\mathbf{x} dy,\tag{40}$$

putting u(x, y,t) from Eq. (25) into the above equation, we obtain the volume flux of the rectangular duct due to cosine oscillations

$$\begin{split} u(x,y,t) &= \frac{64U(\cos\left(wt\right)-w\theta\sin\left(wt\right))}{dh} \sum\_{c=0}^{\infty} \sum\_{\epsilon=0}^{\infty} \frac{1}{\left(\zeta\_{\epsilon}\lambda\_{\epsilon}\right)^{2}} \\ &\times \sum\_{p=0}^{\infty} \sum\_{q=0}^{\infty} \sum\_{r=0}^{\infty} \sum\_{s=0}^{\infty} \frac{\nu^{p+1}\lambda^{s}\theta^{q}K^{p-r}(\lambda\_{\epsilon\varepsilon})^{r+1}t^{-q+p}}{(-1)^{-(p+q+r+s)}q!r!s!} \\ &\times H\_{4,6}^{1,4}\left[H\right] \begin{matrix} (1-q+p,0),(1-r+p,0),(1-s-p,0),(-p,1) \\ (0,1),(1-p,0),(1-p,0),(-p,0),(-p,0),(q-p,1) \end{matrix} \right]. \end{split} \tag{41}$$

Similarly, we obtain the volume flux of the rectangular duct due to the sine oscillations

$$\begin{split} v(\mathbf{x}, y, t) &= \frac{64il(\sin(wt) - w\theta\cos(wt))}{d\hbar} \sum\_{c=0}^{\infty} \sum\_{\epsilon=0}^{\infty} \frac{1}{(\underline{\zeta}\_c \lambda\_\epsilon)^2} \\ &\times \sum\_{p=0}^{\infty} \sum\_{q=0}^{\infty} \sum\_{r=0}^{\infty} \sum\_{s=0}^{\infty} \frac{\nu^{p+1} \lambda^s \theta^q H^{p-r} (\lambda\_{cc})^{r+1} t^{-q+p}}{(-1)^{-(p+q+r+s)} q! r! s!} \\ &\times H\_{4,6}^{1,4} \left[ H \middle| \begin{matrix} (1-q+p, 0), (1-r+p, 0), (1-s-p, 0), (-p, 1) \\ (0, 1), (1-p, 0), (1-p, 0), (-p, 0), (-p, 0), (q-p, 1) \end{matrix} \right]. \end{split} \tag{42}$$

### 7. Numerical results and discussion

We have presented flow problem of MHD Jeffrey fluid passing through a porous rectangular duct. Exact analytical solutions are established for such flow problem using DFFST and LT technique. The obtained solutions are expressed in series form using Fox H-functions. Several graphs are presented here for the analysis of some important physical aspects of the obtained solutions. The numerical results show the profiles of velocity and the adequate shear stress for the flow. We analyze these results by variating different parameters of interest.

The effects of relaxation time λ of the model are important for us to be discuss. In Figure 1 we depict the profiles of velocity and shear stress for three different values of λ. It is observed from these figures that the flow velocity as well as the shear stress decreases with increasing λ, which corresponds to the shear thickening phenomenon. Figure 2 are sketched to show the velocity and the shear stress profiles at different values of retardation time θ. It is noticeable that velocity as well as the shear stress decreases by increasing θ. In order to study the effect of frequency of oscillation ω, we have plotted Figure 3, where it appears that the velocity is also a strong function of ω of the Jeffrey fluid. The effect of frequency of oscillation on the velocity profile for cosine oscillation is same as that of the retardation time θ. The effect of magnetic parameter H of the model is important for us to be discussed. In Figure 4,

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>0</sup>

t

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>0</sup>

t

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>0</sup>

t

=0.5 =0.7

=0.9

Figure 3. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of ω.

Figure 4. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of H.

Figure 5. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of K.

Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, H = 0.5, λ = 1.4, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, λ = 1.4, K = 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

K=2 K=4

K=6

Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, H = 0.5, K = 0.6, d = 1, h = 2, θ = 0.6, λ = 1.4 and ν = 0.1.

H=0.1 H=0.5

H=0.9

0 1 2 3

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

t

0 1 2 3

t

0 1 2 3

t

=0.5 =0.7 135

=0.9

H=0.1 H=0.5

H=0.9

K=2 K=4

K=6

<sup>4</sup> 5 10

<sup>4</sup> 5 10

<sup>4</sup> 1 10

1 10 <sup>4</sup>

2 10 <sup>4</sup>

3 10 <sup>4</sup>

0

5 10 <sup>4</sup>

1 10 <sup>3</sup>

1.5 10 <sup>3</sup>

0

5 10 <sup>4</sup>

1 10 <sup>3</sup>

1.5 10 <sup>3</sup>

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

0

1 10 <sup>4</sup>

5 10 <sup>5</sup>

5 10 <sup>5</sup>

1 10 <sup>4</sup>

u

1.5 10 <sup>4</sup>

2 10 <sup>4</sup>

1 10 <sup>4</sup>

u

1.5 10 <sup>4</sup>

2 10 <sup>4</sup>

2 10 <sup>4</sup>

u

3 10 <sup>4</sup>

Figure 1. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of λ. Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, H = 0.5, K = 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

Figure 2. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of θ. Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, H = 0.5, K = 0.6, d = 1, h = 2, λ = 1.4, ω = 0.5 and ν = 0.1.

Unsteady Magnetohydrodynamic Flow of Jeffrey Fluid through a Porous Oscillating Rectangular Duct http://dx.doi.org/10.5772/intechopen.70891 135

technique. The obtained solutions are expressed in series form using Fox H-functions. Several graphs are presented here for the analysis of some important physical aspects of the obtained solutions. The numerical results show the profiles of velocity and the adequate shear stress for

The effects of relaxation time λ of the model are important for us to be discuss. In Figure 1 we depict the profiles of velocity and shear stress for three different values of λ. It is observed from these figures that the flow velocity as well as the shear stress decreases with increasing λ, which corresponds to the shear thickening phenomenon. Figure 2 are sketched to show the velocity and the shear stress profiles at different values of retardation time θ. It is noticeable that velocity as well as the shear stress decreases by increasing θ. In order to study the effect of frequency of oscillation ω, we have plotted Figure 3, where it appears that the velocity is also a strong function of ω of the Jeffrey fluid. The effect of frequency of oscillation on the velocity profile for cosine oscillation is same as that of the retardation time θ. The effect of magnetic parameter H of the model is important for us to be discussed. In Figure 4,

0 1 2 3

t

0 1 2 3

t

=1.0 =1.4

=1.8

=0.4 =0.6

=0.8

<sup>3</sup> 5 10

<sup>4</sup> 5 10

5 10 <sup>4</sup>

1 10 <sup>3</sup>

1.5 10 <sup>3</sup>

0

Figure 1. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of λ.

Figure 2. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of θ.

Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, H = 0.5, K = 0.6, d = 1, h = 2, λ = 1.4, ω = 0.5 and ν = 0.1.

Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, H = 0.5, K = 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

=0.4 =0.6

=0.8

5 10 <sup>3</sup>

0.01

0.015

0

the flow. We analyze these results by variating different parameters of interest.

=1.0 =1.4

=1.8

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>0</sup>

t

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>0</sup>

t

5 10 <sup>4</sup>

5 10 <sup>4</sup>

1 10 <sup>3</sup>

1.5 10 <sup>3</sup>

u

u

1 10 <sup>3</sup>

1.5 10 <sup>3</sup>

134 Porosity - Process, Technologies and Applications

Figure 3. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of ω. Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, H = 0.5, K = 0.6, d = 1, h = 2, θ = 0.6, λ = 1.4 and ν = 0.1.

Figure 4. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of H. Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, λ = 1.4, K = 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

Figure 5. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of K. Other parameters are taken as x = 0.5, y = 0.3, U = 0.2, H = 0.5, λ = 1.4, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

we depict the profiles of velocity and shear stress for three different values of H. It is observed from these figures that the flow velocity as well as the shear stress decreases with increasing H, which corresponds to the shear thickening phenomenon. Figure 5 is sketched to show the velocity and the shear stress profiles at different values of K. It is noticeable that velocity as well as the shear stress increases by increasing K. In order to study the effects of t, we have plotted Figure 6, where it appears that the velocity is also a strong function of t of the Jeffrey fluid. It can be observed that the increase of t acts as an increase of the magnitude of velocity components near the plate, and this corresponds to the shear-thinning behavior of the examined non-Newtonian fluid. Figure 7 presents the velocity field and the shear stress profiles at different values of y. It is noticeable that velocity and shear stress both decreases by increasing y.

Author details

Pakhtunkhwa, Pakistan

Pakistan

References

Amir Khan1,2\*, Gul Zaman1 and Obaid Algahtani<sup>3</sup>

Acta Mechanica. 1995;113:233-239

Physics of Fluids. 2005;17:023101

Mass Transfer. 2004;40:203-209

1995;124:365-375

Zeitschrift für Naturforschung. 2010;65a:540-548

medium. Journal of Porous Media. 2007;10:473-487

\*Address all correspondence to: amir.maths@gmail.com

1 Department of Mathematics, University of Malakand, Chakdara, Dir (Lower), Khyber

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

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

137

2 Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhwa,

[1] Rajagopal KR, Srinivasa A. Exact solutions for some simple flows of an Oldroyd-B fluid.

[2] Tan WC, Masuoka T. Stoke's first problem for second grade fluid in a porous half space.

[3] Tan WC, Masuoka T. Stoke's first problem for an Oldroyd-B fluid in a porous half space.

[4] Khadrawi AF, Al-Nimr MA, Othman A. Basic viscoelastic fluid problems using the

[5] Chen CI, Chen CK, Yang YT. Unsteady unidirectional flow of an Oldroyd-B fluid in a circular duct with different given volume flow rate. International Journal of Heat and

[6] Nadeem S, Hussain A, Khan M. Stagnation flow of a Jeffrey fluid over a shrinking sheet.

[7] Khan M. Partial slip effects on the oscillatory flows of a fractional Jeffrey fluid in a porous

[8] Hayat T, Khan M, Fakhar K, Amin N. Oscillatory rotating flows of a fractional Jeffrey

[9] Khan M, Iftikhar F, Anjum A. Some unsteady flows of a Jeffrey fluid between two side

[10] Gardner LRT, Gardner GA. A two-dimensional bi-cubic B-spline finite element used in a study of MHD duct flow. Computer Methods in Applied Mechanics and Engineering.

fluid filling a porous medium. Journal of Porous Media. 2010;13(1):29-38

walls over a plane wall. Zeitschrift für Naturforschung. 2011;66(a):745-752

3 Department of Mathematics, Science College, King Saud University, Saudi Arabia

International Journal of Non-Linear Mechanics. 2005;40:515-522

Jeffreys model. Chemical Engineering Science. 2005;60:7131-7136

Figure 6. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of t. Other parameters are taken as x = 0.5, λ = 1.4, U = 0.2, H = 0.5, K = 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

Figure 7. Velocity and shear stress profiles corresponding to the cosine oscillations of the duct for different values of y. Other parameters are taken as x = 0.5, λ = 1.4, U = 0.2, H = 0.5, K = 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.
