MHD Flow of Micropolar Fluid via Porous Medium Within the Rotating Frame of Reference

In comparison to Newtonian fluids, non-Newtonian fluids have several interesting applications in various areas due to the physics of the problems. From these fluids microploar fluid is became popular in the mind of young researchers for their varied applications in industries as well as in engineering. Though the movement of the fluid particles is randomly oriented and there is a translation as well as rotation occurs in each volume element of the fluid, it revealed several interesting phenomena. Eringen [1, 2] originated the theoretical concept of micropolar fluids and thermo micropolar fluids for the characteristics in particular animal blood, exotic lubricants, polymeric fluids, liquid crystals, etc. due to microrotation, these fluids exhibit microscopic effects arising from local structure of the fluid particles. Ariman et al. [3, 4] provides excellent reviews about the mechanics of the micropolar fluid.

The exploration of heat transport phenomena is vital in the case of moving fluid with incorporating the study of both exothermic and endothermic chemical reaction. Flow over a stretching surface for the interaction of chemical reaction that takes due to the combine effects of working fluid and the foreign mass in the various chemical engineering processes. Depending upon the several physical properties its order varies. In general, reaction rate is related to the species concentration is known as first-order reaction, one of the simplest and widely used chemical reactions. Time-dependent flow of polar fluid in a rotating frame of reference with the chemical reaction and heat source was organized by Bakr [5]. He has proposed his investigation by considering oscillatory plate velocity in conjunction with constant suction. Further, flow past an oscillatory vertical plate for a micropolar fluid with the Newtonian heating boundary conditions is worked out by Hussanan et al. [6]. Sheri and Shamshuddin [7] have analyzed the influence of dissipative heat transport phenomena of MHD polar fluid in conjunction with reactive agents. Rout et al. [8] have studied the behavior of magnetic parameter applied in the normal direction for the flow of a conducting micropolar fluid. The proposed boundary value problem is conducted over the plate placed vertically where the wall temperature and concentration varies with time. In viewing to the importance of aforesaid discussion, Animasaun [9] has analyzed the flow of micropolar fluid with the inclusion of temperature dependent viscosity and thermal conductivity. The crux of their investigation is the study of melting boundary condition.

Mahmoud [10] has proposed the flow of a non-Newtonian fluid past an extended surface embedding with porous matrix. In addition, an enhanced flow characteristic is analyzed by incorporating the surface slip, and variable viscosity for the behavior of reactive agents. In a recent study, Khan [11] has used a non-Darcy Jeffry model using modified Darcy’s law. In his investigation the proposed model is based on fractional calculus for the non-Newtonian fluid. Dessie and Kishan [12] have studied the influenced heat characteristics incorporating viscous dissipation in a MHD boundary layer flow past an extended surface. In their work, they have proposed the effects of variable viscosity in conjunction with external heat source/sink. Mishra et al. [13] have carried out the flow of a conducting viscoelastic fluid for the behavior of chemical reaction. Further, Mishra and his co-workers [14-16] have worked out several studies on the flow properties of various non-Newtonian fluid for several physical parameters with different geometries.

From various applications, the crude oil extraction is one in which the role of porosity plays a vital role. In addition, ground water hydrology, geothermal systems, packed bed, etc. are the example of different processes for those the medium is filled with porous materials. It is the measure of the empty spaces in the material and in a total volume; it is the fraction of empty volume. In a flow channel it varies from location to location depending upon the pattern. It can be used in various fields i.e. metallurgy, materials, manufacturing, mechanics and engineering etc. To the best of author’s knowledge, the flow of MHD micropolar fluid with heat and mass transfer in a rotating frame of reference in presence of porous medium has not been addressed so far. Therefore, we have extended the work of Bakr [5] by considering the flow in a porous medium. So, our study is now confined to heat transport mechanism of a conducting polar fluid within a rotating frame embedding with porous medium.

We have now considered

${{w}^{*}}=-{{w}_{0}}$   (10)

Which satisfy Eq. (1) with unvarying w₀ indicates the standard velocity at the plate and denotes suction/injection depending upon the sign. Following non-dimensional quantities are assumed for the transformation of the governing equations.

$\begin{align}  & z=\frac{{{z}^{*}}U{}_{r}}{\upsilon }\,,\,u=\frac{{{u}^{*}}}{{{U}_{r}}},\,v=\frac{{{v}^{*}}}{{{U}_{r}}},\,t=\frac{{{t}^{*}}U_{r}^{2}}{\nu },n=\frac{{{n}^{*}}\upsilon }{U_{r}^{2}},\delta =\frac{\alpha }{\rho \upsilon }, \\ & {{n}_{1}}=\frac{\upsilon }{U_{r}^{2}}n_{1}^{*},{{n}_{2}}=\frac{\upsilon }{U_{r}^{2}}n_{2}^{*},\,\,\theta =\frac{T-{{T}_{\infty }}}{{{T}_{w}}-{{T}_{\infty }}},\,\,{{K}_{p}}=\frac{K{{U}_{r}}^{2}}{{{\upsilon }^{2}}},R=\frac{2\Omega \upsilon }{U_{r}^{2}}, \\ & \,\,\phi =\frac{C-{{C}_{\infty }}}{{{C}_{w}}-{{C}_{\infty }}},\,\,Sc=\frac{\upsilon }{D},Pr=\frac{\mu {{C}_{p}}}{k},Gr=\frac{g\beta \upsilon ({{T}_{w}}-{{T}_{\infty }})}{U_{r}^{3}},\, \\ & s=\frac{{{w}_{o}}}{{{U}_{r}}},Gm=\frac{g\hat{\beta }\upsilon ({{C}_{w}}-{{C}_{\infty }})}{U_{r}^{3}},S=\frac{{{Q}_{H}}{{\upsilon }^{2}}}{U_{r}^{2}k},Kc=\frac{{{K}_{l}}\upsilon }{{{U}_{r}}},\,\, \\ & M=\frac{\sigma B_{0}^{2}\upsilon }{\rho U_{r}^{2}},\,\lambda =\frac{Kc}{u{{\lambda }^{*}}}=(1+\frac{1}{2}\delta ), \\\end{align}$

Imposing above quantities and Eq. (10) the non-dimensional form of the governing equations is,

$\begin{align}  & \frac{\partial u}{\partial t}-s\frac{\partial u}{\partial z}-Rv=(1+\delta )\frac{{{\partial }^{2}}u}{\partial {{z}^{2}}}+ \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Gr\theta +Gm\phi -(M+\frac{1}{Kp})u-\delta \frac{\partial {{n}_{2}}}{\partial z} \\\end{align}$  (11)

$\frac{\partial v}{\partial t}-s\frac{\partial v}{\partial z}+Ru=(1+\delta )\frac{{{\partial }^{2}}v}{\partial {{z}^{2}}}-(M+\frac{1}{Kp})v-\delta \frac{\partial {{n}_{1}}}{\partial z}$   (12)

$\frac{\partial {{n}_{1}}}{\partial t}-s\frac{\partial {{n}_{1}}}{\partial z}=\lambda \frac{{{\partial }^{2}}{{n}_{1}}}{\partial {{z}^{2}}}$  (13)

$\frac{\partial {{n}_{2}}}{\partial t}-s\frac{\partial {{n}_{2}}}{\partial z}=\lambda \frac{{{\partial }^{2}}{{n}_{2}}}{\partial {{z}^{2}}}$  (14)

$\frac{\partial \theta }{\partial t}-s\frac{\partial \theta }{\partial z}=\frac{1}{Pr}\frac{{{\partial }^{2}}\theta }{\partial {{z}^{2}}}-\frac{S}{Pr}\theta $   (15)

$\frac{\partial \phi }{\partial t}-s\frac{\partial \phi }{\partial z}=\frac{1}{Sc}\frac{{{\partial }^{2}}\phi }{\partial {{z}^{2}}}-Kc\phi $   (16)

The corresponding initial and boundary conditions in non-dimensional form are

$u=0,v=0,{{n}_{1}}=0,{{n}_{2}}=0,\theta =0,\phi =0\,\,for\,\,t\le 0$  (17)

$\left. \begin{align}  & u=1+\frac{\varepsilon }{2}({{e}^{int}}+{{e}^{-int}}),v=0,{{n}_{1}}=-\frac{1}{2}\frac{\partial v}{\partial z}, \\ & {{n}_{2}}=\frac{1}{2}\frac{\partial u}{\partial z},\theta =1,\phi =1\,\,at\,\,z=0 \\ & u=v={{n}_{1}}={{n}_{2}}=\theta =\phi =0\,\,as\,\,z\to \infty  \\\end{align} \right\}\,\,\,for\,t>0$   (18)

To solve the equations (11)-(16), we have considered

$U=u+iv\,\,,\,\,N={{n}_{1}}+i{{n}_{2}}$  (19)

Now with the help of Eq. (19), Eqns. (11)-(16) are reduced to

$\begin{align}  & \frac{\partial U}{\partial t}-s\frac{\partial U}{\partial z}+iRU=(1+\delta )\frac{{{\partial }^{2}}U}{\partial {{z}^{2}}}+ \\ & \,\,\,\,\,\,\,\,\,\,\,Gr\theta +Gm\phi -\left( M+\frac{1}{Kp} \right)U+i\delta \frac{\partial N}{\partial z} \\\end{align}$,  (20)

$\frac{\partial N}{\partial t}-s\frac{\partial N}{\partial z}=\lambda \frac{{{\partial }^{2}}N}{\partial {{z}^{2}}}$,  (21)

$\frac{\partial \theta }{\partial t}-s\frac{\partial \theta }{\partial z}=\frac{1}{Pr}\frac{{{\partial }^{2}}\theta }{\partial {{z}^{2}}}-\frac{S}{Pr}\theta $, (22)

$\frac{\partial \phi }{\partial t}-s\frac{\partial \phi }{\partial z}=\frac{1}{Sc}\frac{{{\partial }^{2}}\phi }{\partial {{z}^{2}}}-Kc\phi $,  (23)

The associated boundary conditions are

$U=0,N=0,\theta =0,\phi =0\,\,\,for\,\,t\le 0$, (24)

$\left. \begin{align}  & U=1+\frac{\varepsilon }{2}({{e}^{int}}+{{e}^{-int}}),N=\frac{i}{2}\frac{\partial U}{\partial z},\theta =1,\phi =1\,\,at\,\,z=0 \\ & U=N=\theta =\phi =0\,\,\,as\,\,z\to \infty  \\\end{align} \right\}\,\,for\,\,t>0$  (25)

In order to solve Eqns. (20)-(23) satisfying Eq. (25), we have defined the followings according to Ganapathy [17].

$U(z,t)={{U}_{0}}(z)+\frac{\varepsilon }{2}\left( {{e}^{int}}{{U}_{1}}(z)+{{e}^{-int}}{{U}_{2}}(z) \right)$  (26)

$N(z,t)={{N}_{0}}(z)+\frac{\varepsilon }{2}\left( {{e}^{int}}{{N}_{1}}(z)+{{e}^{-int}}{{N}_{2}}(z) \right)$ (27)

$\theta (z,t)={{\theta }_{0}}(z)+\frac{\varepsilon }{2}\left( {{e}^{int}}{{\theta }_{1}}(z)+{{e}^{-int}}{{\theta }_{2}}(z) \right)$  (28)

$\phi (z,t)={{\phi }_{\,0}}(z)+\frac{\varepsilon }{2}\left( {{e}^{int}}{{\phi }_{\,1}}(z)+{{e}^{-int}}{{\phi }_{\,2}}(z) \right)$  (29)

Substituting Eqns. (26)-(29) in Eqns. (20)-(23) and equating the coefficients of the harmonic and non-harmonic terms and neglecting the terms of $\varepsilon^{2}$, we have the following set of equations.

$\begin{align}  & (1+\delta ){{U}_{0}}^{\prime \prime }+s{{U}_{0}}^{\prime }-\left( M+\frac{1}{Kp}+iR \right){{U}_{0}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+Gr{{\theta }_{0}}+Gm{{\phi }_{0}}+i\delta {{N}_{0}}^{\prime }=0 \\\end{align}$  (30)

$\lambda {{N}_{0}}^{\prime \prime }+s{{N}_{0}}^{\prime }=0$   (31)

${{\theta }_{0}}^{\prime \prime }+sPr{{\theta }_{0}}^{\prime }-S{{\theta }_{0}}=0$  (32)

${{\phi }_{\,0}}^{\prime \prime }+sSc{{\phi }_{\,0}}^{\prime }-\lambda Sc{{\phi }_{\,0}}=0$  (33)

$\begin{align}  & (1+\delta ){{U}_{1}}^{\prime \prime }+s{{U}_{1}}^{\prime }-\left( M+\frac{1}{Kp}+i(R+n) \right){{U}_{1}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+Gr{{\theta }_{1}}+Gm{{\phi }_{1}}+i\delta {{N}_{1}}^{\prime }=0 \\\end{align}$   (34)

$\lambda {{N}_{1}}^{\prime \prime }+s{{N}_{1}}^{\prime }-in{{N}_{1}}=0$  (35)

${{\theta }_{1}}^{\prime \prime }+sPr{{\theta }_{1}}^{\prime }-(S+inPr){{\theta }_{1}}=0$  (36)

${{\phi }_{1}}^{\prime \prime }+sSc{{\phi }_{1}}^{\prime }-(in+\lambda )Sc{{\phi }_{1}}=0$ (37)

$\begin{align}  & (1+\delta ){{U}_{2}}^{\prime \prime }+s{{U}_{2}}^{\prime }-\left( M+\frac{1}{Kp}+i(R-n) \right){{U}_{2}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+Gr{{\theta }_{2}}+Gm{{\phi }_{\,2}}+i\delta {{N}_{2}}^{\prime }=0 \\\end{align}$   (38)

$\lambda {{N}_{2}}^{\prime \prime }+s{{N}_{2}}^{\prime }+in{{N}_{2}}=0$  (39)

${{\theta }_{2}}^{\prime \prime }+sPr{{\theta }_{2}}^{\prime }-(S-inPr){{\theta }_{2}}=0$ (40)

${{\phi }_{\,2}}^{\prime \prime }+sSc{{\phi }_{\,2}}^{\prime }+(in-\lambda )Sc{{\phi }_{2}}=0$ (41)

where the primes denote differentiation with respect to z.

The associated boundary conditions are

$\left. \begin{align}  & {{U}_{0}}=1,{{N}_{0}}=\frac{i}{2}{{U}_{0}}^{\prime },{{\theta }_{0}}=1,{{\phi }_{\,0}}=1\,\,at\,\,z=0 \\ & {{U}_{0}}=0,{{N}_{0}}=0,{{\theta }_{0}}=0,{{\phi }_{\,0}}=0\,\,at\,\,z\to \infty  \\\end{align} \right\}$   (42)

$\left. \begin{align}  & {{U}_{1}}=1,{{N}_{1}}=\frac{i}{2}{{U}_{1}}^{\prime },{{\theta }_{1}}=0,{{\phi }_{\,1}}=0\,\,at\,\,z=0 \\ & {{U}_{1}}=0,{{N}_{1}}=0,{{\theta }_{1}}=0,{{\phi }_{\,1}}=0\,\,as\,\,z\to \infty  \\\end{align} \right\}$  (43)

$\left. \begin{align}  & {{U}_{2}}=1,{{N}_{2}}=\frac{i}{2}{{U}_{2}}^{\prime },{{\theta }_{2}}=0,{{\phi }_{\,2}}=0\,\,at\,\,z=0 \\ & {{U}_{2}}=0,{{N}_{2}}=0,{{\theta }_{2}}=0,{{\phi }_{\,2}}=0\,\,as\,\,z\to \infty  \\\end{align} \right\}$  (44)

The solutions of the Eqns. (30)-(41) satisfying the conditions given in Eqns. (42)-(44) are given by

$\begin{align}  & U={{A}_{1}}{{e}^{-{{m}_{1}}z}}+{{A}_{2}}{{e}^{-{{m}_{2}}z}}+{{A}_{3}}{{e}^{-{{m}_{3}}z}}+{{h}_{3}}{{e}^{-(s/\lambda )z}} \\ & \,\,\,\,\,\,\,\,\,+\frac{\varepsilon }{2}\left( ({{A}_{4}}{{e}^{-{{m}_{5}}z}}+{{h}_{6}}{{e}^{-{{m}_{4}}z}}){{e}^{int}} \right. \\ & \left. \,\,\,\,\,\,\,\,\,+({{A}_{5}}{{e}^{-{{m}_{7}}z}}+{{h}_{9}}{{e}^{-{{m}_{6}}z}}){{e}^{-int}} \right), \\\end{align}$  (45)

$N={{h}_{2}}{{e}^{-(s/\lambda )z}}+\frac{\varepsilon }{2}\left( {{h}_{5}}{{e}^{int-{{m}_{4}}z}}+{{h}_{8}}{{e}^{-(int+{{m}_{6}}z)}} \right)$  (46)

$\theta ={{e}^{-{{m}_{2}}z}}$  (47)

$\phi ={{e}^{-{{m}_{1}}z}}$ (48)

Furthermore, the shear stress at the plate may be written as

$\tau _{wx}^{*}={{\left[ (\mu +\alpha )\frac{\partial {{u}^{*}}}{\partial {{z}^{*}}}+\alpha n_{1}^{*} \right]}_{{{z}^{*}}=0}}=\rho U_{r}^{2}{{\left[ \left( 1+\delta  \right)\frac{\partial u}{\partial z}-\frac{\delta }{2}\frac{\partial v}{\partial z} \right]}_{z=0}}$ ,

$\tau _{wy}^{*}={{\left[ (\mu +\alpha )\frac{\partial {{v}^{*}}}{\partial {{z}^{*}}}+\alpha n_{2}^{*} \right]}_{{{z}^{*}}=0}}=\rho U_{r}^{2}{{\left[ \left( 1+\delta  \right)\frac{\partial v}{\partial z}+\frac{\delta }{2}\frac{\partial u}{\partial z} \right]}_{z=0}}$ ,

$\begin{align}  & \tau _{w}^{*}=\tau _{wx}^{*}+i\tau _{wy}^{*}=\rho U_{r}^{2}{{\left[ (1+\delta )\frac{\partial U}{\partial z}+\frac{i\delta }{2}\frac{\partial U}{\partial z} \right]}_{z=0}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\rho U_{r}^{2}{{\left[ \left[ 1+\delta (1+\frac{i}{2}) \right]{U}' \right]}_{z=0}} \\\end{align}$.

Therefore, the local coefficients for the quantities of interest are given by

${{C}_{f}}=\frac{\tau _{w}^{*}}{\rho U_{r}^{2}}=\left[ 1+\delta (1+\frac{i}{2}) \right]{U}'(0)$  (49)

${{\left. {{M}_{wx}}=\gamma {{\left( \frac{\partial n_{1}^{*}}{\partial {{z}^{*}}} \right)}_{{{z}^{*}}=0}}=\frac{\gamma U_{r}^{3}}{{{\upsilon }^{2}}}\frac{\partial {{n}_{1}}}{\partial z},\,\,\,\,\,{{C}_{wx}}=\frac{{{M}_{wx}}{{\upsilon }^{2}}}{\gamma U_{r}^{3}}=\frac{\partial {{n}_{1}}}{\partial z} \right|}_{z=0}},$

${{\left. {{M}_{wy}}=\gamma {{\left( \frac{\partial n_{2}^{*}}{\partial {{z}^{*}}} \right)}_{{{z}^{*}}=0}}=\frac{\gamma U_{r}^{3}}{{{\upsilon }^{2}}}\frac{\partial {{n}_{2}}}{\partial z},\,\,\,\,\,{{C}_{wy}}=\frac{{{M}_{wy}}{{\upsilon }^{2}}}{\gamma U_{r}^{3}}=\frac{\partial {{n}_{2}}}{\partial z} \right|}_{z=0}},$

${{M}_{w}}={{M}_{wx}}+i{{M}_{wy}}=\frac{\gamma U_{r}^{3}}{{{\upsilon }^{2}}}\left( {{\left. \frac{\partial {{n}_{1}}}{\partial z} \right|}_{z=0}}+{{\left. i\frac{\partial {{n}_{2}}}{\partial z} \right|}_{z=0}} \right)$,

${{C}_{w}}={{C}_{wx}}+i{{C}_{wy}}={{\left. \frac{\partial {{n}_{1}}}{\partial z} \right|}_{z=0}}+{{\left. i\frac{\partial {{n}_{2}}}{\partial z} \right|}_{z=0}}={N}'(0)$.  (50)

The rate of heat transfer at the surface is

$Nu={{\left. \frac{\chi }{{{T}_{\infty }}-{{T}_{w}}}\frac{\partial T}{\partial z} \right|}_{z=0}},\,\,Nu=-R{{e}_{x}}{\theta }'(0),$ (51)

where, $R{{e}_{x}}={{U}_{r}}\chi /\upsilon $  is the Reynolds number.

The rate of mass transfer at the surface is

$Sh={{\left. \frac{\chi }{{{C}_{w}}^{*}-{{C}_{\infty }}^{*}}\frac{\partial {{C}^{*}}}{\partial {{z}^{*}}} \right|}_{z=0}},\,\,Sh=-R{{e}_{x}}{\phi }'(0)$   (52)

In the present study we have theoretically discussed the heat transfer phenomena on the time-dependent MHD flow of micropolar fluid in a porous medium within a rotating frame, considering oscillatory plate velocity with constant suction and first-order chemical reaction. The concluding remarks are:

• The applied transverse magnetic field sets up the Lorentz force which reduces the fluid velocity.
• The angular velocity attenuates with the growth in the porosity parameter but impact opposes for the action of rotation parameter.
• The angular velocity decreases with the increase in microrotation viscosity.
• The retardation in the thickness of the thermal boundary layer is marked with the rise in Prandtl number.
• The heavier diffusing species favors in to retard the fluid concentration.
• An increase in reactive agents decelerates the shear stress and couple stress but increases the Sherwood number.