Unsteady Magnetohydrodynamic Couette-Poiseuille Flow within Porous Plates Filled with Porous Medium in the Presence of a Moving Magnetic Field with Hall and Ion-slip Effects

Magnetohydrodynamic (MHD) flows through purely fluid medium and porous medium have drawn attention of many researchers due to its diverse applications in geophysical, astrophysical, biological, agricultural and engineering problems. An electrically conducting fluid permeated by magnetic fields induces many electromagnetic phenomena viz. Hall effect, ion-slip effect, Ohmic heating etc. (Cramer and Pai [1]). Such phenomena may have considerable effect on heat and mass transfer flows and flow dynamics. Hall effect becomes significant when an ionized fluid with low density (plasma) is permeated by a strong magnetic field (Sato [2]). Hall effect is likely to be important in underground energy storage systems, magnetometers, Hall effect sensors and spacecraft propulsion. In addition, when the magnetic field is strong, the diffusion velocity of ions may not be negligible in an ionized fluid and phenomenon of ion-slip arises (Cramer and Pai [1]). In such cases, Hall and ion-slip currents are included in generalized Ohm’s law. Mittal et al. [3] presented a study on influence of Hall and ion-slip currents on two-dimensional MHD flow in the entrance region of a rectangular channel. Soundalgekar et al. [4] presented an analysis of the MHD Couette flow with heat transfer by taking Hall and ion-slip effects into account. Ram et al. [5] considered the combined effects of Hall and ion-slip currents on convective flow in a rotating fluid with oscillating wall temperature. Takhar and Jha [6] investigated the effects of Hall and ion-slip currents on MHD flow past an impulsively started plate in a rotating system. Attia [7] discussed the unsteady Couette flow of an electrically conducting fluid bounded by two parallel non-conducting porous plates taking Hall and ion-slip currents into consideration. Naroua et al. [8] analysed transient hydromagnetic partially-ionized heat generating gas dynamics flow with Hall/ion-slip effects in a rotating system using finite element method. Beg et al. [9] presented a theoretical study of unsteady MHD viscous Hartmann-Couette laminar flow and heat transfer in a Darcian porous medium with Hall current, ion-slip, viscous and Joule heating effects. Jha and Apere [10] analysed the influence of Hall and ion-slip currents on unsteady MHD Couette flow in a rotating system in the presence of a transverse magnetic field which is either fixed relative to the fluid or to the moving plate. Subsequently Jha and Apere [11] investigated time-dependent MHD Couette flow in a rotating system taking Hall and ion-slip currents into consideration. Recently Singh et al. [12] studied effects of Hall current and ion-slip on unsteady hydromagnetic generalized Couette flow in a rotating Darcian channel in the presence of a transverse magnetic field which is fixed relative to the fluid. The equation governing the fluid motion through a porous medium is based on Darcy’s law, which determine the drag force exerted by porous medium (Ingham and Pop [13]). MHD flows through porous medium may find applications in many biological and engineering problems viz. the movement of water, oil and natural gases in reservoirs, in filtration and water purification process, to study underground water sources and seepage of water in river beds etc. Chamkha [14] investigated steady laminar flow of two viscous, incompressible, electrically conducting and heat generating/absorbing immiscible fluids in porous and non-porous channels. Subsequently Chamkha [15] discussed unsteady laminar fully-developed hydromagnetic flow and heat transfer of an electrically conducting fluid and heat generating/absorbing fluid in a porous channel filled with porous medium. Makinde and Mhone [16] presented combined effect of transverse magnetic field and radiative heat transfer to unsteady flow of a conducting optically thin fluid through a channel filled with porous medium. Beg et al. [17] performed a numerical study of magnetohydrodynamic viscous plasma flow in a rotating porous medium with Hall current in the presence of an inclined magnetic field. Israel-Cookey et al. [18] analysed MHD free convection and oscillatory flow of an optically thin fluid bounded by two porous walls in a porous medium. Chauhan and Agrawal [19] presented a study on effects of Hall current on MHD Couette flow in a channel partially filled with a porous medium in the presence of an inclined magnetic filed in a rotating system. Singh and Mathew [20] presented the effects of permeability and suction/injection on an oscillatory free convective flow through a highly porous medium bounded between two infinite vertical porous plates. Singh and Pathak [21] considered the influence of Hall current and rotation on mixed convection MHD flow through a porous medium filled in a vertical channel in the presence of thermal radiation. Subsequently, Singh et al. [22] discussed the effect of Hall current on visco-elastic MHD oscillatory convective flow through a porous medium in a vertical channel with heat radiation. Recently Ahmed and Chamkha [23] investigated Hartmann-Newtonian radiating MHD flow for a rotating vertical porous channel merged in a Darcian porous regime. Seth et al. [24] discussed steady hydromagnetic class-II Couette flow through a porous medium with Hall current and rotation. Singh et al. [12] considered medium filled within horizontal parallel plates as porous medium in their problem. 

The motive of the present investigation is to study the influence of Hall current, ion-slip and permeability on unsteady magnetohydrodynamic Couette-Poiseuille flow within porous plates filled with porous medium in the presence of a moving uniform transverse magnetic field which is fixed relative to the moving porous plate. The flow within porous plates filled with porous medium is generated due to movement of the lower plate along x-axis and by an applied constant pressure gradient acting along the x-axis. Two particular cases for the medium filled within the porous plates are considered i.e. (i) the case of small permeable regime and (ii) the case of pure fluid regime.

Now it is required to find the solution of Eqs (10) and (6) subject to the initial and boundary conditions (7) to (9), for this purpose writing Eqs (10) and (6) in compact form, we obtain

$\begin{aligned} \frac{\partial q}{\partial t}+v_{0} \frac{\partial q}{\partial y} &=-\frac{1}{\rho} \frac{\partial p^{\prime}}{\partial x}+v \frac{\partial^{2} q}{\partial y^{2}} \\ &-\frac{\sigma B_{o}^{2}}{\rho}\left[\frac{\left\{\left(1+B_{i} B_{e}\right)-i B_{e}\right\} q-U_{0} t^{n}}{\left(1+B_{t} B_{e}\right)^{2}+B_{e}^{2}}\right]-\frac{v q}{k} \end{aligned}$(11)

where $q=u+i w$

The initial and boundary conditions in compact form are represented by

$q=0 \qquad$ for $0 \leq y \leq h$ and $t \leq 0$(12)

$q=U_{o} t^{n} \qquad$ at $y=0$ for $t>0$(13)

$q=0 \qquad$ at $y=h$ for $t>0$(14)

Eq (11) describe the general flow characteristic for Couette-Poiseuille flow between two infinite porous plates filled with the porous medium in the presence of a uniform transverse magnetic field. In order to describe the specific flow characteristics we consider the case when $n=1$ which corresponds to the accelerated motion.

Defining the following non-dimensional variables

$x^{*}=\frac{x}{h}, y^{*}=\frac{y}{h}, q^{*}=\frac{q}{U_{0}}, p^{*}=\frac{p^{\prime}}{\rho U_{0}^{2}}$ and $t^{*}=\frac{t U_{0}}{h}$(15)

After putting $n=1$ and using non-dimensional variables defined in Eq (15), Eq (11) and initial and boundary conditions (12) to (14), in non-dimensional form, become     

$\frac{\partial q^{*}}{\partial t^{*}}+N_{t} \frac{\partial q^{*}}{\partial y^{*}}=R+\frac{1}{R_{e}} \frac{\partial^{2} q^{*}}{\partial y^{2}}$$\quad-\frac{H_{a}^{2}}{R_{e}}\left[\frac{\left\{\left(1+B_{i} B_{e}\right)-i B_{e}\right\} q^{*}-R_{1} t^{*}}{\left(1+B_{i} B_{e}\right)^{2}+B_{e}^{2}}\right]-\frac{q^{*}}{D_{a} R_{e}}$(16)

$q^{*}=0 \qquad$ for $0 \leq y^{*} \leq 1$ and $t^{*} \leq 0$(17)

$q^{*}=R_{1} t^{*}$ at $y^{*}=0$ for $t^{*}>0$(18)

$q^{*}=0 \qquad$ at $\quad y^{*}=1$ for $t^{*}>0$(19)

where $N_{t}=v_{0} / U_{0}$($N_{t}>0$ for suction and $N_{t}<0$ for injection) represents the mass of the fluid entering into the Darcian channel via the moving porous plate and coming out via the stationary porous plate, $R=-d p^{*} / d x^{*}$, $R_{e}=U_{0} h / v$ represents the relative strength of inertial force to the viscous force, $H_{a}^{2}=\sigma B_{0}^{2} h^{2} / \mu$ represents the relative strength of magnetic force to the viscous force,$D_{a}=k / h^{2}$ represents the relative permeability of  Darcian channel and $R_{1}=h / U_{0}$.

The exact solution of Eq (16) subject to the initial and boundary conditions (17) to (19) are derived using Laplace transform method. Taking Laplace transform of Eq (16), boundary conditions (18), (19) and using initial condition (17), we transform Eq (16) and boundary conditions (18), (19) to the following equations

$\frac{d^{2} \overline{q^{*}}}{d y^{-2}}-N_{t} R_{e} \frac{d \overline{q}^{*}}{d y^{*}}-R_{e}\left(s+m_{4}\right) \overline{q^{*}}=-\frac{R_{1} R_{e} m_{1}}{s^{2}}-\frac{R R_{e}}{s}$ (20)

$\overline{q^{*}}=\frac{R_{1}}{s^{2}} \qquad$ at $y^{*}=0$(21)

$\overline{q^{*}}=0 \qquad$ at $y^{*}=1$(22)

where $\overline{q^{*}}(y, s)=\int_{0}^{\infty} e^{-s t} q^{*}(y, t) d t$, $s(s>0)$ being the Laplace parameter and

$m_{1}=\frac{H_{a}^{2}}{R_{e}\left[\left(1+B_{i} B_{e}\right)^{2}+B_{e}^{2}\right]}, m_{2}=B_{e} m_{1}$

Solving Eq (20) subject to the boundary conditions (21) and (22), the solution of  Eq (20) is given by

$\overline{q}_{a}^{*}=\sum_{n=0}^{\infty}\left[\left(\frac{e^{-a \lambda_{1}}}{s^{2}}-\frac{e^{-b \lambda_{1}}}{s^{2}}\right)+(-1)^{n+1}\left\{\frac{m_{1}}{m_{4}}\left(\frac{e^{-c \lambda_{1}}}{s^{2}}+\frac{e^{-d \lambda_{1}}}{s^{2}}\right)\right.\right.$$-\left(\frac{m_{1}}{m_{4}^{2}}-\frac{R^{*}}{m_{4}}\right)\left(\frac{e^{-c \lambda_{1}}}{s}+\frac{e^{-d \lambda_{1}}}{s}-\frac{e^{-c \lambda_{1}}}{\left(s+m_{4}\right)}-\frac{e^{-d \lambda_{1}}}{\left(s+m_{4}\right)}\right) \} ]$$+\left\{\frac{m_{1}}{m_{4} s^{2}}-\left(\frac{m_{1}}{m_{4}^{2}}-\frac{R^{*}}{m_{4}}\right)\left(\frac{1}{s}-\frac{1}{\left(s+m_{4}\right)}\right)\right\}$ (23)

where

$a=2 n+y^{*}, b=2+2 n-y^{*}, c=n+y^{*}, d=n+1-y^{*}$,

$q_{a}^{*}=\frac{q^{*}}{R_{1}}=u_{a}^{*}+i w_{a}^{*}, X_{2}=\frac{N_{t}^{2} R_{e}}{4}+m_{3}$,

$\alpha, \beta=\frac{1}{\sqrt{2}}\left[\left\{X_{2}^{2}+m_{2}^{2}\right\}^{1 / 2} \pm\left(X_{2}\right)\right]^{1 / 2}, \quad R^{*}=\frac{R}{R_{1}}$,

$p=\alpha-i \beta, \lambda_{1}=\left(N_{t} R_{e} / 2\right)+\sqrt{R_{e}} \sqrt{s+p^{2}}$.

To invert Eq (23), the following Fourier-Mellin inversion integral formula is used: 

$\begin{aligned} L^{-1}\left[\frac{e^{-y^{*}(s+b)^{1 / 2}}}{\left(s-a^{*}\right)^{N+1}}\right] &=F\left(y^{*}, t^{*}, N, a^{*}, b^{*}\right) \\ &=\frac{1}{2 \pi i} \int_{B_{2}} \frac{e^{s t^{*}} e^{-y^{*}\left(s+b^{*}\right)^{1 / 2}}}{\left(s-a^{*}\right)^{N+1}} d s \end{aligned}$(24)

$\begin{aligned} F\left(y^{*}, t^{*}, N, a^{*}, b^{*}\right)=& \frac{d}{d b}\left[F\left(y^{*}, t^{*}, N-1, a^{*}, b^{*}\right)\right] \\ &+t^{*} F\left(y^{*}, t^{*}, N-1, a^{*}, b^{*}\right) \end{aligned}$ (25)

where $B r_{2}$ is the Bromwich path defined by McLachlan [26].

When $N=0$, from Eq (24), we obtain

  $F\left(y^{*}, t^{*}, 0, a^{*}, b^{*}\right)=\frac{e^{a^{*} t}}{2}\left(e^{-y \sqrt{\left(a^{*}+b^{\prime}\right)}} \operatorname{erfc}\left(\frac{y^{*}}{2 \sqrt{t^{*}}}-\sqrt{\left(a^{*}+b^{*}\right) t^{*}}\right)\right.$$+e^{y^{*} \sqrt{\left(a^{2}+b^{2}\right)}} \operatorname{erfc}\left(\frac{y^{*}}{2 \sqrt{t^{*}}}+\sqrt{\left(a^{*}+b^{*}\right) t^{*}}\right) )$ (26)

When $N=1$ from Eqs (24), (25) and (26), we get

$\begin{align}  & F\left( {{y}^{*}},\,\,{{t}^{*}},\,\,1,\,\,{{a}^{*}},\,\,{{b}^{*}} \right) \\  & =\frac{{{e}^{{{a}^{*}}{{t}^{*}}}}}{2}\left[ \left( {{t}^{*}}+\frac{{{y}^{*}}}{2\sqrt{\left( {{a}^{*}}+{{b}^{*}} \right)}} \right){{e}^{{{y}^{*}}\sqrt{\left( {{a}^{*}}+{{b}^{*}} \right)}}} \right.erfc\left( \frac{{{y}^{*}}}{2\sqrt{{{t}^{*}}}}+\sqrt{\left( {{a}^{*}}+{{b}^{*}} \right){{t}^{*}}} \right) \\ \end{align}$

$\left. +\left( {{t}^{*}}-\frac{{{y}^{*}}}{2\sqrt{\left( {{a}^{*}}+{{b}^{*}} \right)}} \right){{e}^{-y\sqrt{\left( {{a}^{*}}+{{b}^{*}} \right)}}}erfc\left( \frac{{{y}^{*}}}{2\sqrt{{{t}^{*}}}}-\sqrt{\left( {{a}^{*}}+{{b}^{*}} \right){{t}^{*}}} \right) \right].$ (27)

$q_{a}^{*}=\frac{1}{m_{4}}\left[m_{1} t^{*}-\left(\frac{m_{1}}{m_{4}}-R^{*}\right)\left(1-e^{-m_{t} t^{*}}\right)\right]$$+\sum_{n=0}^{\infty}\left[\left(e^{-\frac{a N_{t} R_{0}}{2}} F\left(a_{1}, t^{*}, 1,0, p^{2}\right)-e^{-\frac{b N_{t} R_{e}}{2}} F\left(b_{1}, t^{*}, 1,0, p^{2}\right)\right)\right.$

$+\frac{(-1)^{n+1}}{m_{4}}\left\{m_{1}\left(e^{-\frac{c N_{t} R_{e}}{2}} F\left(c_{1}, t^{*}, 1,0, p^{2}\right)\right.\right.$$+e^{-\frac{d V_{t} R_{e}}{2}} F\left(d_{1}, t^{*}, 1,0, p^{2}\right) )-\left(\frac{m_{1}}{m_{4}}-R^{*}\right)$

$\times\left(e^{-\frac{c N_{t} R_{e}}{2}} F\left(c_{1}, t^{*}, 0,0, p^{2}\right)+e^{-\frac{d N_{t} R_{e}}{2}} F\left(d_{1}, t^{*}, 0,0, p^{2}\right)\right.$$-e^{-\frac{C N_{t} R_{e}}{2}} F\left(c_{1}, t^{*}, 0,-m_{4}, p^{2}\right)$$-e^{-\frac{d N_{t} R_{e}}{2}} F\left(d_{1}, t^{*}, 0,-m_{4}, p^{2}\right) \} ]$ (28)

The solution for fluid velocity is obtained by inverting Eq (23) and displayed in the following form

where

$a_{1}=a \sqrt{R_{e}}, b_{1}=b \sqrt{R_{e}}, c_{1}=c \sqrt{R_{e}}, d_{1}=d \sqrt{R_{e}}$

To examine the effects of different flow parameters to the fluid flow problem the graphs for fluid velocity and the shear stress at the moving porous plate are computed and generated for various values of different flow parameters taking $H_{a}^{2}=81, \quad t^{*}=0.5, R^{*}=5$ and $R_{e}=4$.The numerical values of flow parameters corresponding to each Figure are given in the caption of the Figure and inside the Figure. Figures 2 and 3 show the influence of Hall current parameter $B_{e}$ on the fluid velocity in the primary and secondary flow directions respectively. It can be seen from Figure 2 that the fluid velocity in the primary flow direction $u_{a}^{*}$ decreases on increasing Hall current parameter $B_{e}$ in case of both the small permeable regime $\left(D_{a}=0.01\right)$ and pure fluid regime$\left(D_{a} \rightarrow \infty\right)$.Figure 3 exhibits that fluid velocity in the secondary flow direction $w_{a}^{*}$ increases on increasing Hall current parameter $B_{e}$ in case of both the small permeable regime and pure fluid regime. This implies that in case of both the small permeable regime and pure fluid regime, Hall current tends to reduce fluid velocity in the primary flow direction whereas it has reverse effect on the fluid velocity in the secondary flow direction. The usual nature of Hall current is to induce secondary flow in the flow-field, our result comply with it. Figures 4 and 5 illustrate the effect of ion-slip parameter $B_{i}$ on the fluid velocity in the primary and secondary flow directions respectively. Both the fluid velocity in the primary flow direction $u_{a}^{*}$ and fluid velocity in the secondary flow direction $w_{a}^{*}$ decrease on increasing ion-slip parameter $B_{i}$ in case of both the small permeable regime and pure fluid regime. This means that in case of both the small permeable regime and pure fluid regime, ion-slip tends to reduce fluid velocity in both the primary and secondary flow directions. Figure 6 illustrates the effect of permeability parameter $D_{a}$ on the fluid velocity in both the primary and secondary flow directions. Both the fluid velocity in the primary flow direction $u_{a}^{*}$ and fluid velocity in the secondary flow direction $w_{a}^{*}$ increase on increasing the permeability parameter$D_{a}$. This concludes that permeability tends to enhance fluid velocity in both the primary and secondary flow directions. Figures 7 and 8 exhibit the influence of suction/injection parameter $N_{t}$ on the fluid velocity in the primary and secondary flow directions respectively. Both the fluid velocity in the primary flow direction $u_{a}^{*}$ and fluid velocity in the secondary flow direction $w_{a}^{*}$ increase on increasing suction parameter $N_{t}$ and these decrease on increasing injection parameter $N_{t}$ in case of pure fluid regime. The fluid velocity in the primary flow direction $u_{a}^{*}$ decreases on increasing suction parameter $N_{t}$ and it increases on increasing injection parameter $N_{t}$ in the lower half of the channel whereas these effects are reversed in the upper half of the channel in case of small permeable regime. The fluid velocity in the secondary flow direction $w_{a}^{*}$ increases on increasing suction parameter $N_{t}$ and it decreases on increasing injection parameter $N_{t}$ in case of small permeable regime. This implies that in case of pure fluid regime, suction tends to enhance fluid velocity in both the primary and secondary flow directions whereas injection has reverse effect on these. In case of small permeable regime, suction tends to enhance fluid velocity in the primary flow direction in the upper half of the channel and fluid velocity in the secondary flow direction throughout the channel whereas injection has reverse effect on these. In case of small permeable regime the nature of fluid velocity in primary flow direction in the lower half of the channel is reversed due to permeability of the medium.

figure_2.png

Figure 2. Fluid velocity distribution in the primary flow direction when $\mathrm{B}_{i}=0.5$ and $N_{t}=1.0$

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Figure 3. Fluid velocity distribution in the secondary flow direction when $\mathrm{B}_{i}=0.5$ and $N_{t}=1.0$

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Figure 4. Fluid velocity distribution in the primary flow direction when $\mathrm{B}_{e}=0.25$ and $N_{t}=1.0$

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Figure 5. Fluid velocity distribution in the secondary flow direction when $\mathrm{B}_{e}=0.25$ and $N_{t}=1.0$

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Figure 6. Fluid velocity distribution in the primary and secondary flow directions when $\mathrm{B}_{e}=0.25, \mathrm{B}_{i}=0.5$ andand $N_{t}=1.0$

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Figure 7. Fluid velocity distribution in the primary flow direction when $\mathrm{B}_{e}=0.25$ and $\mathrm{B}_{i}=0.5$

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Figure 8. Fluid velocity distribution in the secondary flow direction when $\mathrm{B}_{e}=0.25$ and $\mathrm{B}_{i}=0.5$

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Figure 9. Shear stress distribution in the primary flow direction when $N_{t}=1.0$

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Figure 10. Shear stress distribution in the secondary flow direction when $N_{t}=1.0$

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Figure 11. Shear stress distribution in the primary and secondary flow directions when $B_{e}=0.25$ and $B_{i}=0.5$

Figures 9 and 10 display the effects of Hall current parameter $B_{e}$ and ion-slip parameter $B_{i}$ on shear stress at the moving porous plates in the primary flow direction $\tau_{x a}^{*}$ and shear stress at the moving porous plate in the secondary flow direction $\tau_{z a}^{*}$ respectively. Shear stress at the moving porous plate in the primary flow direction $\tau_{x a}^{*}$ decreases on increasing Hall current parameter $B_{e}$ in case of both the small permeable regime and pure fluid regime whereas it decreases on increasing ion-slip parameter $B_{i}$ in case of the small permeable regime and pure fluid regime when $B_{i}=0.5$ and 1.Shear stress at the moving porous plate in the secondary flow direction $\tau_{z a}^{*}$ increases on increasing Hall current parameter $B_{e}$ and it decreases on increasing ion-slip parameter $B_{i}$ in case of both the small permeable regime and pure fluid regime. This implies that in case of both the small permeable regime and pure fluid regime, Hall current tends to reduce shear stress at the moving porous plate in the primary flow direction whereas it has reverse effect on shear stress at the moving porous plate in the secondary flow direction. In case of pure fluid regime, ion-slip tends to reduce shear stress at the moving porous plate in the primary flow direction when $B_{i}=0.5$ and 1 and shear stress at the moving porous plate in secondary flow direction. In case of small permeable regime, ion-slip tends to reduce shear stress at the moving porous plate in both the primary and secondary flow directions. Figure 11 depicts the effects of permeability parameter $D_{a}$ and suction/injection parameter $N_{t}$ on the shear stress at the moving porous plate in the primary flow direction $\tau_{x a}^{*}$ and the shear stress at the moving porous plate in the secondary flow direction$\tau_{z a}^{*}$. Both the shear stress at the moving porous plate in the primary flow direction $\tau_{x a}^{*}$ and the shear stress at the moving porous plate in the secondary flow direction $\tau_{z a}^{*}$ increase on increasing permeability parameter $D_{a}$ and suction parameter $N_{t}$ while it decrease on increasing injection parameter $N_{t}$.This shows that permeability and suction tend to enhance shear stress at the moving porous plate in both the primary and secondary flow directions whereas injection has reverse effect on these.

It is noticed from Figures 2 to 11 that both the fluid velocity and the shear stress at the moving porous plate in case of pure fluid regime is more than the fluid velocity and shear stress at the moving porous plate in case of small permeable regime.

Authors are thankful to the reviewers for their valuable comments and suggestions. S. Ghousia Begum is thankful to University Grant Commission (ugc), New Delhi for providing financial assistance to carry-out this research work.