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

Abstract

This chapter presents some new exact solutions corresponding to unsteady magnetohydrodynamic (MHD) flow of Jeffrey fluid in a long porous rectangular duct oscillating parallel to its length. The exact solutions are established by means of the double finite Fourier sine transform (DFFST) and discrete Laplace transform (LT). The series solution of velocity field, associated shear stress and volume flow rate in terms of Fox H-functions, satisfying all imposed initial and boundary conditions, have been obtained. Also, the obtained results are analyzed graphically through various pertinent parameter.

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] 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] 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, 16, 17, 18, 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.

2. Governing equations

The equation of continuity and momentum of MHD flow passing through porous space is given by [7]

$$\nabla ·V=0,\rho \left(\frac{d\mathbf{V}}{\mathit{dt}}\right)=\mathit{div}\mathbf{T}+\mathbf{J}\times \mathbf{B}+\mathbf{R},$$E1

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 medium by R.

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

$$\mathbf{T}=-p\mathbf{I}+\mathbf{S},\mathbf{S}=\frac{\mu }{1+\mathrm{\lambda }}\left[\mathbf{A}+\mathrm{\theta }\left(\frac{\partial \mathbf{A}}{\partial \mathbf{t}}+\left(\mathbf{V}.\nabla \right)\mathbf{A}\right)\right],$$E2

where S and pI represents the extra stress tensor and the indeterminate spherical stress, the dynamic viscosity is denoted by μ, A = L + L^T 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

$$\mathbf{J}\times \mathit{B}=-\sigma {\beta }_o^2\mathbf{V},$$E3

where σ represents electrical conductivity and β_o the strength of magnetic field. For the Jeffrey fluid the Darcy’s resistance satisfies the following equation

$$\mathbf{R}=-\frac{\mathit{\mu \varphi }}{\kappa \left(1+\lambda \right)}\left(1+\theta \frac{\partial }{\partial t}\right)\mathbf{V},$$E4

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

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

$$\mathbf{V}=\left(0,0,w\left(x,y,t\right)\right),\mathbf{S}=\mathbf{S}\left(x,y,t\right)$$E5

where w is the velocity in the z-direction. The continuity equation for such flows is automatically satisfied. Also, at t = 0, the fluid being at rest is given by

$$\mathbf{S}\left(x,y,0\right)=\mathbf{0},$$E6

therefore from Eqs. (2), (5) and (6), it results that S_xx = S_yy = S_yz = S_zz = 0 and the relevant equations

$${\tau }_1=\frac{\mu }{\left(1+\lambda \right)}\left(1+\theta \frac{\partial }{\partial t}\right){\partial }_{x}w\left(x,y,t\right),{\tau }_2=\frac{\mu }{\left(1+\lambda \right)}\left(1+\theta \frac{\partial }{\partial t}\right){\partial }_{y}w\left(x,y,t\right),$$E7

where τ₁ = S_xy and τ₂ = S_xz are the tangential stresses. In the absence of pressure gradient in the flow direction, the governing equation leads to

$$\left(1+\lambda \right){\partial }_{t}w\left(x,y,t\right)=\nu \left(1+\theta \frac{\partial }{\partial t}\right)\left({\partial }_x^2+{\partial }_y^2\right)w\left(x,y,t\right)-\mathit{\nu K}\left(1+\theta \frac{\partial }{\partial t}\right)w\left(x,y,t\right)-H\left(1+\lambda \right)w\left(x,y,t\right),$$E8

where $H=\frac{\sigma B_0^2}{\rho }$ is the magnetic parameter, $K=\frac{\varphi }{\kappa }$ is the porosity parameter and ν = μ/ρ is the kinematic viscosity.

3. Statement of the problem

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⁺ 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

$$w\left(x,y,0\right)={\partial }_{t}w\left(x,y,0\right)=0,$$E9

$$w\left(0,y,t\right)=w\left(x,0,t\right)=w\left(d,y,t\right)=w\left(x,h,t\right)=\mathit{\text{Ucos}}\left(\mathit{wt}\right),$$E10

or

$$w\left(0,y,t\right)=w\left(x,0,t\right)=w\left(d,y,t\right)=w\left(x,h,t\right)=\mathit{\text{Usin}}\left(\mathit{wt}\right),$$E11

$$t>0,0<x<d\mathit{and}0<y<h.$$

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

$$F\left(x,y,t\right)=u\left(x,y,t\right)+\mathit{iv}\left(x,y,t\right),$$E12

which is the solution of the problem

$$\left(1+\lambda \right){\partial }_{t}F\left(x,y,t\right)=\nu \left(1+\theta \frac{\partial }{\partial t}\right)\left({\partial }_x^2+{\partial }_y^2\right)F\left(x,y,t\right)-\mathit{\nu K}\left(1+\theta \frac{\partial }{\partial t}\right)F\left(x,y,t\right)-H\left(1+\lambda \right)F\left(x,y,t\right),$$E13

$$F\left(x,y,0\right)={\partial }_{t}F\left(x,y,0\right)=0,$$E14

$$F\left(0,y,t\right)=F\left(d,y,t\right)=F\left(x,0,t\right)=F\left(x,h,t\right)=U{e}^{\mathit{iwt}},$$E15

$$t>0,0<x<d\mathit{and}0<y<h.$$

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

$$F_{\mathit{mn}}\left(t\right)={\int }_0^d‍{\int }_0^h‍\mathit{sin}\left(\frac{\mathit{m\pi x}}{d}\right)\mathit{sin}\left(\frac{\mathit{n\pi y}}{h}\right)F\left(x,y,t\right)\mathit{\text{dxdy}},m,n=1,2,3,..$$E16

4. Calculation of the velocity field

Multiplying both sides of Eq. (13) by $\mathit{sin}\left(\frac{\mathit{m\pi x}}{d}\right)$ and $\mathit{sin}\left(\frac{\mathit{n\pi y}}{h}\right)$, integrating w.r.t x and y over [0, d] × [0, h] and using Eq. (16), we get

$$\begin{array}{c}\left(1+\lambda \right)\frac{\partial F_{mn}\left(t\right)}{\partial t}+\nu {\lambda }_{mn}\left(1+\theta \frac{\partial }{\partial t}\right)F_{mn}\left(t\right)+H\left(1+\lambda \right)F_{mn}\left(t\right)+\nu K\left(1+\theta \frac{\partial }{\partial t}\right)F_{mn}\left(t\right)\\ =\nu {\lambda }_{mn}U\frac{\left[1-{\left(-1\right)}^m\right]\left[1-{\left(-1\right)}^n\right]}{{\zeta }_m{\lambda }_n}\left(1+iw\theta \right)e^{iwt},\end{array}$$E17

where

$${\zeta }_m=\frac{\mathit{m\pi }}{d},{\lambda }_n=\frac{\mathit{n\pi }}{h}\mathit{and}{\lambda }_{\mathit{mn}}={\zeta }_m^2+{\lambda }_n^2.$$

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

$$F_{\mathit{mn}}\left(0\right)={\partial }_t{F}_{\mathit{mn}}\left(0\right)=0.$$E18

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

$${\overline{F}}_{\mathit{mn}}\left(s\right)=\frac{\nu {\lambda }_{\mathit{mn}}U\left[1-{\left(-1\right)}^m\right]\left(1+\mathit{iw\theta }\right)\left[1-{\left(-1\right)}^n\right]}{{\zeta }_m{\lambda }_n\left(s-\mathit{iw}\right)\left[\left(1+\lambda \right)\left(s+H\right)+\nu \left(1+\mathit{\theta s}\right)\left({\lambda }_{\mathit{mn}}+K\right)\right]}.$$E19

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 Eq. (19) in series form as

$$\begin{array}{c}{\overline{F}}_{\mathit{mn}}\left(s\right)=\frac{\nu {\lambda }_{\mathit{mn}}U\left[1-{\left(-1\right)}^m\right]\left(1+\mathit{iw\theta }\right)\left[1-{\left(-1\right)}^n\right]}{{\zeta }_m{\lambda }_n\left(s-\mathit{iw}\right)}\sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\sum _{l=0}^{\infty }\hfill \\ \times \frac{{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{H}^l{\left({\lambda }_{\mathit{mn}}\right)}^{r+1}\mathrm{\Gamma }\left(q-p\right)\mathrm{\Gamma }\left(r-p\right)\mathrm{\Gamma }\left(s+p+1\right)\mathrm{\Gamma }\left(l+p+1\right)}{{\left(-1\right)}^{-\left(p+q+r+s+l\right)}q!r!s!l!\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }\left(1+p\right)\mathrm{\Gamma }\left(1+p\right)s^{l-q+p+1}}.\hfill \end{array}$$E20

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

$$\begin{array}{c}{F}_{\mathit{mn}}\left(t\right)\hfill \\ =\frac{e^{\mathit{iwt}}U\left[1-{\left(-1\right)}^m\right]\left[1-{\left(-1\right)}^n\right]\nu {\lambda }_{\mathit{mn}}\left(1+\mathit{iw\theta }\right)}{{\zeta }_m{\lambda }_n}\sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\sum _{l=0}^{\infty }‍\hfill \\ \times \frac{{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{H}^l{\left({\lambda }_{\mathit{mn}}\right)}^{r+1}\mathrm{\Gamma }\left(q-p\right)\mathrm{\Gamma }\left(r-p\right)\mathrm{\Gamma }\left(s+p+1\right)\mathrm{\Gamma }\left(l+p+1\right)t^{l-q+p}}{{\left(-1\right)}^{-\left(p+q+r+s+l\right)}q!r!s!l!\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }\left(1+p\right)\mathrm{\Gamma }\left(1+p\right)\mathrm{\Gamma }l-q+p+1}.\hfill \end{array}$$E21

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

$$\begin{array}{cc}F\left(x,y,t\right)\hfill & =\frac{4}{\mathit{dh}}\sum _{m=1}^{\infty }‍\sum _{n=1}^{\infty }‍\mathit{sin}\left({\zeta }_{m}x\right)\mathit{sin}\left({\lambda }_{n}y\right)F_{\mathit{mn}}\left(x,y,t\right)\hfill \\ \hfill & \hspace{1em}=\frac{4e^{\mathit{iwt}}U\left(1+\mathit{iw\theta }\right)}{\mathit{dh}}\sum _{m=1}^{\infty }‍\sum _{n=1}^{\infty }‍\frac{\left[1-{\left(-1\right)}^m\right]\left[1-{\left(-1\right)}^n\right]\mathit{sin}\left(\mathit{\zeta x}\right)\mathit{sin}\left({\lambda }_{n}y\right)}{{\zeta }_m{\lambda }_n}\hfill \\ \hfill & \hspace{1em}\times \sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\sum _{l=0}^{\infty }‍\frac{{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{H}^l{\left({\lambda }_{\mathit{mn}}\right)}^{r+1}{t}^{l-q+p}}{{\left(-1\right)}^{-\left(p+q+r+s+l\right)}q!r!s!l!}\hfill \\ \hfill & \hspace{1em}\times \frac{\mathrm{\Gamma }\left(q-p\right)\mathrm{\Gamma }\left(r-p\right)\mathrm{\Gamma }\left(s+p+1\right)\mathrm{\Gamma }\left(l+p+1\right)}{\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }\left(1+p\right)\mathrm{\Gamma }\left(1+p\right)\mathrm{\Gamma }l-q+p+1}.\hfill \end{array}$$E22

To obtain a more compact form of velocity field we write Eq. (22) in terms of Fox H-function,

$$\begin{array}{cc}F\left(x,y,t\right)\hfill & =\frac{4e^{\mathit{iwt}}U\left(1+\mathit{iw\theta }\right)}{\mathit{dh}}\sum _{m=1}^{\infty }‍\sum _{n=1}^{\infty }‍\frac{\mathit{sin}\left({\zeta }_{m}x\right)\mathit{sin}\left({\lambda }_{n}y\right)\left[1-{\left(-1\right)}^m\right]\left[1-{\left(-1\right)}^n\right]}{{\zeta }_m{\lambda }_n}\hfill \\ \hfill & \hspace{1em}\times \sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\frac{{\left(-1\right)}^{p+q+r+s}{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{\left({\lambda }_{\mathit{mn}}\right)}^{r+1}{t}^{-q+p}}{q!r!s!}\hfill \\ \hfill & \hspace{1em}\times H_{4,6}^{1,4}[\mathit{Ht}\left|\begin{array}{l}\left(1-q+p,0\right),\left(1-r+p,0\right),\left(-s-p,0\right),\left(-p,1\right)\\ \left(0,1\right),\left(1-p,0\right),\left(1-p,0\right),\left(-p,0\right),\left(-p,0\right),\left(q-p,1\right)\end{array}].\right.\hfill \end{array}$$E23

or

$$\begin{array}{cc}F\left(x,y,t\right)\hfill & =\frac{16e^{\mathit{iwt}}U\left(1+\mathit{iw\theta }\right)}{\mathit{dh}}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{\mathit{sin}\left({\zeta }_{c}x\right)\mathit{sin}\left({\lambda }_{e}y\right)}{{\zeta }_c{\lambda }_e}\hfill \\ \hfill & \hspace{1em}\times \sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\frac{{\left(-1\right)}^{p+q+r+s}{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{\left({\lambda }_{\mathit{ce}}\right)}^{r+1}{t}^{-q+p}}{q!r!s!}\hfill \\ \hfill & \times H_{4,6}^{1,4}[\mathit{Ht}\left|\begin{array}{l}\left(1-q+p,0\right),\left(1-r+p,0\right),\left(1-s-p,0\right),\left(-p,1\right)\\ \left(0,1\right),\left(1-p,0\right),\left(1-p,0\right),\left(-p,0\right),\left(-p,0\right),\left(q-p,1\right)\end{array}],\right.\hfill \end{array}$$E24

where

$${\zeta }_c=\left(2m+1\right)\frac{\pi }{d},{\lambda }_e=\left(2n+1\right)\frac{\pi }{h},c=2m+1,e=2n+1.$$

From Eq. (24), we obtain the velocity field due to cosine oscillations of the duct

$$\begin{array}{cc}u\left(x,y,t\right)\hfill & =\frac{16U\left(\mathit{cos}\left(\mathit{wt}\right)-\mathit{w\theta }\mathit{sin}\left(\mathit{wt}\right)\right)}{\mathit{dh}}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{\mathit{sin}\left({\zeta }_{c}x\right)\mathit{sin}\left({\lambda }_{e}y\right)}{{\zeta }_c{\lambda }_e}\hfill \\ \hfill & \hspace{1em}\times \sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\frac{{\left(-1\right)}^{p+q+r+s}{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{\left({\lambda }_{\mathit{ce}}\right)}^{r+1}{t}^{-q+p}}{q!r!s!}\hfill \\ \hfill & \hspace{1em}\times H_{4,6}^{1,4}[\mathit{Ht}\left|\begin{array}{l}\left(1-q+p,0\right),\left(1-r+p,0\right),\left(1-s-p,0\right),\left(-p,1\right)\\ \left(0,1\right),\left(1-p,0\right),\left(1-p,0\right),\left(-p,0\right),\left(-p,0\right),\left(q-p,1\right)\end{array}],\right.\hfill \end{array}$$E25

and the velocity field due to sine oscillations of the duct

$$\begin{array}{cc}v\left(x,y,t\right)\hfill & =\frac{16U\left(\mathit{sin}\left(\mathit{wt}\right)-\mathit{w\theta }\mathit{cos}\left(\mathit{wt}\right)\right)}{\mathit{dh}}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{\mathit{sin}\left({\zeta }_{c}x\right)\mathit{sin}\left({\lambda }_{e}y\right)}{{\zeta }_c{\lambda }_e}\hfill \\ \hfill & \hspace{1em}\times \sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\frac{{\left(-1\right)}^{p+q+r+s}{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{\left({\lambda }_{\mathit{ce}}\right)}^{r+1}{t}^{-q+p}}{q!r!s!}\hfill \\ \hfill & \hspace{1em}\times H_{4,6}^{1,4}[\mathit{Ht}\left|\begin{array}{l}\left(1-q+p,0\right),\left(1-r+p,0\right),\left(1-s-p,0\right),\left(-p,1\right)\\ \left(0,1\right),\left(1-p,0\right),\left(1-p,0\right),\left(-p,0\right),\left(-p,0\right),\left(q-p,1\right)\end{array}].\right.\hfill \end{array}$$E26

We use the following property of the Fox H-function [21] in the above equation

$$\begin{array}{c}\hfill H_{p,q+1}^{1,p}[-\chi \left|\begin{array}{l}\left(1-a_1,A_1\right),\left(1-a_2,A_2\right),\dots ,\left(1-a_p,A_p\right)\\ \left(1,0\right),\left(1-b_1,B_1\right),\dots ,\left(1-b_q,B_q\right)\end{array}]\right.\\ =\sum _{k=0}^{\infty }‍\frac{\mathrm{\Gamma }\left(a_1+A_{1}k\right)\dots \mathrm{\Gamma }\left(a_p+A_{p}k\right)}{k!\mathrm{\Gamma }\left(b_1+B_{1}k\right)\dots \mathrm{\Gamma }\left(b_q+B_{q}k\right)}{\chi }^k.\hfill \end{array}$$

5. Calculation of the shear stress

We denote the tangential tensions for the cosine oscillations of the duct by τ_1c(x, y, t), τ_2c(x, y, t) and for sine oscillations by τ_1s(x, y, t), τ_2s(x, y, t).

If we introduce

$${\tau }_1\left(x,y,t\right)={\tau }_{1c}\left(x,y,t\right)+i{\tau }_{1s}\left(x,y,t\right),$$E27

$${\tau }_2\left(x,y,t\right)={\tau }_{2c}\left(x,y,t\right)+i{\tau }_{2s}\left(x,y,t\right),$$E28

in Eq. (7), we get

$${\tau }_1\left(x,y,t\right)=\frac{\mu }{\left(1+\lambda \right)}\left(1+\theta \frac{\partial }{\partial t}\right){\partial }_{x}F\left(x,y,t\right),$$E29

$${\tau }_2\left(x,y,t\right)=\frac{\mu }{\left(1+\lambda \right)}\left(1+\theta \frac{\partial }{\partial t}\right){\partial }_{y}F\left(x,y,t\right).$$E30

We apply LT to Eqs. (29) and (30), to obtain

$${\overline{\tau }}_1\left(x,y,s\right)=\frac{\mu \left(1+\mathit{\theta s}\right)}{1+\lambda }{\partial }_x\overline{F}\left(x,y,s\right),$$E31

$${\overline{\tau }}_2\left(x,y,s\right)=\frac{\mu \left(1+\mathit{\theta s}\right)}{1+\lambda }{\partial }_y\overline{F}\left(x,y,s\right).$$E32

Taking the inverse Fourier transform of Eq. (19) to get $\overline{F}\left(x,y,s\right)$ and then by putting it into Eq. (31), we get

$$\begin{array}{cc}{\overline{\tau }}_1\left(x,y,s\right)\hfill & =\frac{4\mu \left(1+\mathit{\theta s}\right)}{\mathit{dh}\left(1+\lambda \right)}\sum _{m=1}^{\infty }‍\sum _{n=1}^{\infty }‍\frac{\mathit{cos}\left({\zeta }_{m}x\right)\mathit{sin}\left({\lambda }_{n}y\right)\left[1-{\left(-1\right)}^m\right]\left[1-{\left(-1\right)}^n\right]}{\left[\left(1+\lambda \right)\left(s+H\right)+\nu \left(1+\mathit{\theta s}\right)\left({\lambda }_{\mathit{mn}}+K\right)\right]}\hfill \\ \hfill & \times \frac{\mathit{U\nu }{\lambda }_{\mathit{mn}}\left(1+\mathit{iw\theta }\right)}{{\lambda }_n\left(s-\mathit{iw}\right)},\hfill \end{array}$$E33

or

$${\overline{\tau }}_1\left(x,y,s\right)=\frac{16\mu \left(1+\mathit{\theta s}\right)}{\mathit{dh}\left(1+\lambda \right)}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{\mathit{cos}\left({\zeta }_{c}x\right)\mathit{sin}\left({\lambda }_{e}y\right)\mathit{U\nu }{\lambda }_{\mathit{ce}}\left(1+\mathit{iw\theta }\right)}{{\lambda }_e\left(s-\mathit{iw}\right)\left[\left(1+\lambda \right)\left(s+H\right)+\nu \left(1+\mathit{\theta s}\right)\left({\lambda }_{\mathit{ce}}+K\right)\right]},$$E34

where

$${\zeta }_c=\left(2m+1\right)\frac{\pi }{d},{\lambda }_e=\left(2n+1\right)\frac{\pi }{h},c=2m+1,e=2n+1.$$

We express Eq. (34) in series form in order to obtain a more suitable form of τ₁,

$$\begin{array}{cc}{\overline{\tau }}_1\left(x,y,s\right)\hfill & =\frac{16\mathit{\mu U}\left(1+\mathit{iw\theta }\right)}{\mathit{dh}\left(s-\mathit{iw}\right)}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{\mathit{cos}\left({\zeta }_{c}x\right)\mathit{sin}\left({\lambda }_{e}y\right)}{{\lambda }_e}\sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\sum _{l=0}^{\infty }\hfill \\ \hfill & \hspace{1em}\times \frac{{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{H}^l{\left({\lambda }_{\mathit{ce}}\right)}^{r+1}\mathrm{\Gamma }\left(q-p-1\right)\mathrm{\Gamma }\left(r-p\right)\mathrm{\Gamma }\left(s+p+2\right)\mathrm{\Gamma }\left(l+p+1\right)}{{\left(-1\right)}^{-\left(p+q+r+s+l\right)}q!r!s!l!\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }\left(p+1\right)\mathrm{\Gamma }\left(2+p\right)\mathrm{\Gamma }\left(1+p\right)s^{l-q+p+1}}.\hfill \end{array}$$E35

Using the inverse LT of the last equation, we obtain

$$\begin{array}{cc}{\tau }_1\left(x,y,t\right)\hfill & =\frac{16\mu e^{\mathit{iwt}}U\left(1+\mathit{iw\theta }\right)}{\mathit{dh}}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{\mathit{cos}\left({\zeta }_{c}x\right)\mathit{sin}\left({\lambda }_{e}y\right)}{{\lambda }_e}\sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }\hfill \\ \hfill & \hspace{1em}\times \sum _{l=0}^{\infty }‍\frac{{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{H}^l{\left({\lambda }_{\mathit{ce}}\right)}^{r+1}{t}^{l-q+p}\mathrm{\Gamma }\left(q-p-1\right)\mathrm{\Gamma }\left(s+p+2\right)}{{\left(-1\right)}^{-\left(p+q+r+s+l\right)}q!r!s!l!\mathrm{\Gamma }\left(p\right)\mathrm{\Gamma }\left(p+1\right)\mathrm{\Gamma }\left(2+p\right)\mathrm{\Gamma }\left(1+p\right)}\hfill \\ \hfill & \hspace{1em}\times \frac{\mathrm{\Gamma }\left(l+p+1\right)}{\mathrm{\Gamma }\left(l-q+p+1\right)}.\hfill \end{array}$$E36

Lastly, we write the stress field in a more compact form by using Fox H-function

$$\begin{array}{cc}{\tau }_1\left(x,y,t\right)\hfill & =\frac{16\mu e^{\mathit{iwt}}U\left(1+\mathit{iw\theta }\right)}{\mathit{dh}}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{\mathit{cos}\left({\zeta }_{c}x\right)\mathit{sin}\left({\lambda }_{e}y\right)}{{\lambda }_e}\sum _{p=0}^{\infty }\hfill \\ \hfill & \hspace{1em}\times \sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\frac{{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{\left({\lambda }_{\mathit{mn}}\right)}^{r+1}{t}^{-q+p}}{{\left(-1\right)}^{-\left(p+q+r+s\right)}q!r!s!}\hfill \\ \hfill & \hspace{1em}\times H_{4,6}^{1,4}[\mathit{Ht}\left|\begin{array}{l}\left(2-q+p,0\right),\left(1-r+p,0\right),\left(-1-s-p,0\right),\left(-p,1\right)\\ \left(0,1\right),\left(-p,0\right),\left(-1-p,0\right),\left(-p,0\right),\left(-p,0\right),\left(q-p,1\right)\end{array}].\right.\hfill \end{array}$$E37

From Eq. (37), we obtain the tangential tension due to cosine oscillations of the duct

$$\begin{array}{cc}{\tau }_{1c}\left(x,y,t\right)\hfill & =\frac{16\mathit{U\mu }\left(\mathit{cos}\left(\mathit{wt}\right)-\mathit{w\theta }\mathit{sin}\left(\mathit{wt}\right)\right)}{\mathit{dh}}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{\mathit{cos}\left({\zeta }_{c}x\right)\mathit{sin}\left({\lambda }_{e}y\right)}{{\lambda }_e}\hfill \\ \hfill & \hspace{1em}\times \sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\frac{{\left(-1\right)}^{p+q+r+s}{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{\left({\lambda }_{\mathit{mn}}\right)}^{r+1}{t}^{-q+p}}{q!r!s!}\hfill \\ \hfill & \times H_{4,6}^{1,4}[\mathit{Ht}\left|\begin{array}{l}\left(2-q+p,0\right),\left(1-r+p,0\right),\left(-1-s-p,0\right),\left(-p,1\right)\\ \left(0,1\right),\left(-p,0\right),\left(-1-p,0\right),\left(-p,0\right),\left(-p,0\right),\left(q-p,1\right)\end{array}],\right.\hfill \end{array}$$E38

and the tangential tension corresponding to sine oscillations of the duct

$$\begin{array}{cc}{\tau }_{1s}\left(x,y,t\right)\hfill & =\frac{16\mathit{U\mu }\left(\mathit{sin}\left(\mathit{wt}\right)-\mathit{w\theta }\mathit{cos}\left(\mathit{wt}\right)\right)}{\mathit{dh}}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{\mathit{cos}\left({\zeta }_{c}x\right)\mathit{sin}\left({\lambda }_{e}y\right)}{{\lambda }_e}\sum _{p=0}^{\infty }\hfill \\ \hfill & \hspace{1em}\times \sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\frac{{\left(-1\right)}^{p+q+r+s}{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{\left({\lambda }_{\mathit{mn}}\right)}^{r+1}{t}^{-q+p}}{q!r!s!}\hfill \\ \hfill & \hspace{1em}\times H_{4,6}^{1,4}[\mathit{Ht}\left|\begin{array}{l}\left(2-q+p,0\right),\left(1-r+p,0\right),\left(-1-s-p,0\right),\left(-p,1\right)\\ \left(0,1\right),\left(-p,0\right),\left(-1-p,0\right),\left(-p,0\right),\left(-p,0\right),\left(q-p,1\right)\end{array}].\right.\hfill \end{array}$$E39

In the similar fashion we can find τ_2c(x, y, t) and τ_2s(x, y, t) from Eqs. (19) and (32).

6. Volume flux

The volume flux due to cosine oscillations is given by

$$Q_c\left(x,y,t\right)={\int }_0^d‍{\int }_0^h‍u\left(x,y,t\right)\mathit{\text{dxdy}},$$E40

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

$$\begin{array}{cc}u\left(x,y,t\right)\hfill & =\frac{64U\left(\mathit{cos}\left(\mathit{wt}\right)-\mathit{w\theta }\mathit{sin}\left(\mathit{wt}\right)\right)}{\mathit{dh}}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{1}{{\left({\zeta }_c{\lambda }_e\right)}^2}\hfill \\ \hfill & \hspace{1em}\times \sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\frac{{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{\left({\lambda }_{\mathit{ce}}\right)}^{r+1}{t}^{-q+p}}{{\left(-1\right)}^{-\left(p+q+r+s\right)}q!r!s!}\hfill \\ \hfill & \hspace{1em}\times H_{4,6}^{1,4}[\mathit{Ht}\left|\begin{array}{l}\left(1-q+p,0\right),\left(1-r+p,0\right),\left(1-s-p,0\right),\left(-p,1\right)\\ \left(0,1\right),\left(1-p,0\right),\left(1-p,0\right),\left(-p,0\right),\left(-p,0\right),\left(q-p,1\right)\end{array}].\right.\hfill \end{array}$$E41

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

$$\begin{array}{cc}v\left(x,y,t\right)\hfill & =\frac{64U\left(\mathit{sin}\left(\mathit{wt}\right)-\mathit{w\theta }\mathit{cos}\left(\mathit{wt}\right)\right)}{\mathit{dh}}\sum _{c=0}^{\infty }‍\sum _{e=0}^{\infty }‍\frac{1}{{\left({\zeta }_c{\lambda }_e\right)}^2}\hfill \\ \hfill & \hspace{1em}\times \sum _{p=0}^{\infty }‍\sum _{q=0}^{\infty }‍\sum _{r=0}^{\infty }‍\sum _{s=0}^{\infty }‍\frac{{\nu }^{p+1}{\lambda }^s{\theta }^q{K}^{p-r}{\left({\lambda }_{\mathit{ce}}\right)}^{r+1}{t}^{-q+p}}{{\left(-1\right)}^{-\left(p+q+r+s\right)}q!r!s!}\hfill \\ \hfill & \hspace{1em}\times H_{4,6}^{1,4}[\mathit{Ht}\left|\begin{array}{l}\left(1-q+p,0\right),\left(1-r+p,0\right),\left(1-s-p,0\right),\left(-p,1\right)\\ \left(0,1\right),\left(1-p,0\right),\left(1-p,0\right),\left(-p,0\right),\left(-p,0\right),\left(q-p,1\right)\end{array}].\right.\hfill \end{array}$$E42

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, 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.

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.

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.

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.