A Study on MHD Nanofluid Flow and Heat Transfer with Convective Surface Temperature and Concentration

Consider the two dimensional steady incompressible MHD power-law nanofluid on a non-conducting stretching sheet y=0 embedded in porous medium. The flow of fluid is confined in the region y>0. The velocity, temperature and concentration of the stretching sheet are $u_{w}(x)=U_{0} x^{n}(n$ being the powerlaw index $), T_{f}$ and $C_{f}$ respectively. A magnetic field $B=$ $B_{0} x^{\frac{n-1}{2}}$ is applied normal to the sheet. Further, the fluid assumed to be slightly conducting, the magnetic Reynolds number is much less than unity and hence the induced magnetic field is negligible. The coordinate system describing the flow model is given in Figure 1.

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Figure 1. Flow geometry and coordinate system

The governing equations for the nanofluid can be written in Cartesian coordinates following Hayat et al. [17] with boundary conditions are given by

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$     (1)

$\rho\left(u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}\right)=\mu \frac{\partial^{2} u}{\partial y^{2}}-\sigma B_{0}^{2} u-\frac{\mu}{K p^{*}} u$     (2)

$u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}=\alpha \frac{\partial^{2} T}{\partial y^{2}}+\tau\left[D_{B} \frac{\partial C}{\partial y} \frac{\partial T}{\partial y}+\frac{D_{T}}{T_{\infty}}\left(\frac{\partial T}{\partial y}\right)^{2}\right]$

$-\frac{1}{(\rho C)_{f}} \frac{\partial q_{r}}{\partial y}+\frac{Q_{0}}{(\rho C)_{f}}\left(T-T_{\infty}\right)$     (3)

$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}={{D}_{B}}\frac{{{\partial }^{2}}C}{\partial {{y}^{2}}}+\frac{{{D}_{T}}}{{{T}_{\infty }}}\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}}-{{k}_{c}}\left( C-{{C}_{\infty }} \right)$     (4)

where, $u$ and $v$ are velocity components in the $x$ and $y$ directions, $T$ the temperature, $C$ the nanoparticle volume fraction, $k$ the thermal conductivity, $\sigma$ the electrical conductivity of the fluid, $\rho$ is the density, $\tau=\frac{(\rho C)_{p}}{(\rho C)_{f}}$ is the ratio of effective heat capacity of the nanoparticle material and heat capacity of the fluid, $k_{c}$ the rate of chemical reaction, $D_{B}$ and $D_{T}$ are the Brownian and thermophoresis diffusion coefficients respectively.

$\left. \begin{align}  & u={{u}_{w}}\left( x \right)={{U}_{0}}{{x}^{n}},v=0,\text{ }-k\frac{\partial T}{\partial y}={{h}_{f}}\left( {{T}_{f}}-T \right)\text{, }-{{D}_{m}}\frac{\partial C}{\partial y}={{k}_{m}}\left( {{C}_{f}}-C \right)\text{ }at\text{  }y=0 \\ & u=0,v=0,\text{ }T={{T}_{\infty }},\text{  }C={{C}_{\infty }}\text{  }as\text{  }y\to \infty  \\\end{align} \right\}$     (5)

We introduce the following similarity transformations

$\left. \begin{align}  & u={{U}_{0}}{{x}^{n}}{f}'\left( \eta  \right),v=-\sqrt{\frac{{{\upsilon }_{f}}{{U}_{0}}\left( n+1 \right)}{2}}{{x}^{\frac{n-1}{2}}}\left\{ f\left( \eta  \right)+\frac{n-1}{n+1}\eta {f}'\left( \eta  \right) \right\} \\ & \eta =y\sqrt{\frac{{{U}_{0}}\left( n+1 \right)}{2{{\upsilon }_{f}}}}{{x}^{\frac{n-1}{2}}},\theta \left( \eta  \right)=\frac{T-{{T}_{\infty }}}{{{T}_{f}}-{{T}_{\infty }}},\phi \left( \eta  \right)=\frac{C-{{C}_{f}}}{{{C}_{f}}-{{C}_{\infty }}} \\\end{align} \right\}$     (6)

Using Eq. (6), the equations (2)-(4) reduce to

$f^{\prime \prime \prime}+f f^{\prime \prime}-\frac{2 n}{n+1}\left(f^{\prime}\right)^{2}-\left(M+\frac{1}{K p}\right) f^{\prime}=0$     (7)

$(1+R) \theta^{\prime \prime}+\operatorname{Pr}\left[f \theta^{\prime}+N b \theta^{\prime} \phi^{\prime}+N t\left(\theta^{\prime}\right)^{2}+Q \theta\right]=0$     (8)

$\phi^{\prime \prime}+S c f \phi^{\prime}+\frac{N t}{N b} \theta^{\prime \prime}-\gamma S c \phi=0$     (9)

The non-dimensional boundary conditions (6) are

$\left. \begin{align}  & f\left( 0 \right)=0,{f}'\left( 0 \right)=1,{\theta }'\left( 0 \right)=-B{{i}_{t}}\left[ 1-\theta \left( 0 \right) \right],\text{ }{\phi }'\left( 0 \right)=-B{{i}_{c}}\left[ 1-\phi \left( 0 \right) \right] \\ & {f}'\left( \infty  \right)=0,\text{  }\theta \left( \infty  \right)=0,\text{  }\phi \left( \infty  \right)=0 \\\end{align} \right\}$     (10)

where, non-dimensional variables and parameters are

$\begin{align}  & M=\frac{2\sigma B_{0}^{2}}{{{\rho }_{f}}{{U}_{0}}\left( n+1 \right)},Kp=\frac{K{{p}^{*}}a}{\upsilon },\Pr =\frac{{{\upsilon }_{f}}}{\alpha },Sc=\frac{\upsilon }{{{D}_{B}}},B{{i}_{c}}=\frac{{{k}_{m}}}{{{D}_{m}}{{x}^{\frac{n-1}{2}}}}\sqrt{\frac{2{{\upsilon }_{f}}}{{{U}_{0}}\left( n+1 \right)}} \\ & Nb=\frac{\tau {{D}_{B}}\left( {{C}_{\infty }}-{{C}_{f}} \right)}{\upsilon },Nt=\frac{\tau {{D}_{T}}\left( {{T}_{\infty }}-{{T}_{f}} \right)}{\upsilon {{T}_{\infty }}},\gamma =\frac{2}{n+1}\frac{{{k}_{c}}}{{{U}_{0}}{{x}^{n-1}}},B{{i}_{t}}=\frac{{{h}_{f}}}{{{k}_{f}}{{x}^{\frac{n-1}{2}}}}\sqrt{\frac{2{{\upsilon }_{f}}}{{{U}_{0}}\left( n+1 \right)}} \\\end{align}$

The shearing stress, surface heat flux and surface mass flux are given by

$\tau_{w}=\mu\left(\frac{\partial u}{\partial y}\right)_{y=0}, q_{w}=-k\left(\frac{\partial T}{\partial y}\right)_{y=0}+\left(q_{r}\right)_{y=0}, q_{m}=-D_{B}\left(\frac{\partial C}{\partial y}\right)_{y=0}$

The non dimensional shear stress coefficient $C_{f}=\frac{\tau_{w}}{\rho U_{0}^{2} x^{2 n}},$ local Nusselt number $N u_{x}=\frac{-x q_{w}}{k\left(T_{f}-T_{\infty}\right)}$ local Sherwood number $S h_{x}=\frac{-x q_{m}}{D_{B}\left(C_{f}-C_{\infty}\right)}$ are given by

$\left. \begin{align}  & {{C}_{f}}{{\left( {{\operatorname{Re}}_{x}} \right)}^{^{{}^{1}/{}_{2}}}}=\left( -\sqrt{\frac{n+1}{2}} \right){f}''\left( 0 \right) \\ & N{{u}_{x}}{{\left( {{\operatorname{Re}}_{x}} \right)}^{^{{}^{-1}/{}_{2}}}}=-\left( \sqrt{\frac{n+1}{2}} \right)\left( 1+R \right){\theta }'\left( 0 \right)\text{ } \\ & S{{h}_{x}}{{\left( {{\operatorname{Re}}_{x}} \right)}^{^{{}^{-1}/{}_{2}}}}=-\left( \sqrt{\frac{n+1}{2}} \right){\phi }'\left( 0 \right) \\\end{align} \right\}$     (11)

where, $R e_{x}=\frac{U_{0} x^{\frac{n+1}{2}}}{v_{f}}$ is the local Reynolds number.

The numerical results achieved for velocity, temperature and concentration profiles for different values of pertinent parameters are illustrated through Figs. 3-8. Computations have been carried out for $M=0.5,Kp=2,R=n=1,Nb=0.4,Nt=0.3,B{{i}_{t}}=B{{i}_{c}}=0.3,Q=\gamma =0.1,$ $\Pr =1.3,$ and $Sc=1.1$. Fig. 3 displays the effect of magnetic parameter (M) and porosity parameter$\left( Kp \right)$on velocity distribution. The occurrence of magnetic field decreases the momentum boundary layer thickness. This is due to magnetic field produces a resistive force called Lorentz force which slows down the motion of the fluid. Therefore, an increase in M leads to decrease the velocity of the fluid. Further, porous matrix enhances the fluid flow. Fig. 4 reveals the influence of power-law index (n) on velocity, temperature and concentration profiles. It is observed that velocity decreases with an increase in power-law index whereas opposite trend is seen in case of temperature profile. On the other hand, there is no significant effect of power-law index on concentration of nanofluid.

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Figure 3. Effects of M and Kp on the velocity ${f}'\left( \eta  \right)$ profile

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Figure 4. Effect of n on the velocity ${f}'\left( \eta  \right)$, temperature $\theta \left( \eta  \right)$ and concentration $\phi \left( \eta  \right)$ profiles

The effects of thermal radiation parameter (R) and heat source/sink (Q) are shown in Fig. 5. Thermal radiation augments the thermal diffusivity of the nanofluid so that for higher values of R, heat will be added to the regime and temperature will be increased. It is noticed that, heat source parameter Q $\left( Q>0 \right)$ enhances the temperature of the nanofluid leading to an increase the thickness of the thermal boundary layer. But heat sink Q $\left( Q<0 \right)$ decreases the temperature profile. Fig. 6 portrays the Brownian motion (Nb) and thermophoresis parameter (Nt) on temperature and concentration profiles. It is noticed that temperature increases uniformly at all the layers as Nb and Nt increases. It is interesting to note that higher values of thermophoresis parameter increase the concentration but Brownian motion parameter decreases the concentration.

Fig. 7 is drawn to present the effects of thermal Biot number (Bi_t) and solutal Biot number (Bi_c) on temperature and concentration profiles. It is seen that both the parameters enhance the temperature and concentration profiles. Fig. 8 depicts the influence of Schmidt number (Sc) and chemical reaction parameter $\left( \gamma  \right)$ on concentration profile. Higher values of Schmidt number i.e. heavier species reduce the concentration due to lower diffusion of mass. It is also noticed that destructive chemical reaction parameter $\gamma $$\left( \gamma >0 \right)$ decreases the concentration and constructive chemical reaction parameter$\gamma $$\left( \gamma <0 \right)$increases concentration.

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Figure 5. Effects of R and Q on the temperature $\theta \left( \eta  \right)$ profile

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Figure 6. Effects of Nb and Nt on the temperature $\theta \left( \eta  \right)$and concentration $\phi \left( \eta  \right)$ profiles

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Figure 7. Effects of $B{{i}_{t}}$and $B{{i}_{c}}$ on the temperature $\theta \left( \eta  \right)$ and concentration $\phi \left( \eta  \right)$ profiles

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Figure 8. Effects of $Sc$ and $\gamma $on the concentration $\phi \left( \eta  \right)$ profile

Table 2 exhibits the values of skin-friction coefficient${{f}^{''}}\left( \text{0} \right)$, Nusselt number$-{{\theta }^{'}}\left( \text{0} \right)$and Sherwood number$-{{\phi }^{'}}\left( \text{0} \right)$. It is seen that skin friction at the wall increases as magnetic parameter and power-law index increase. But reverse tendency is observed due to porous matrix. It is interesting to note that Prandtl number, thermal radiation parameter and Schmidt number have no effects on skin friction. Further, it is noticed that increasing values of power-law index and radiation parameter increase the rate of heat transfer as well as mass transfer at the wall. Moreover, magnetic parameter, porous matrix, Prandtl number and Schmidt number have opposite effects on Nusselt number and Sherwood number.

Table 3 displays the values of Nusselt number$-{{\theta }^{'}}\left( \text{0} \right)$and Sherwood number$-{{\phi }^{'}}\left( \text{0} \right)$. The heat and mass fluxes at the wall decrease with an increase in thermophoresis parameter (Nt).  Brownian motion and solutal Biot number decrease the heat flux at the surface but adverse effect is observed in case of mass flux.  Further, it is noticed that Nusselt number increases with increase in thermal Biot number while Sherwood number decreases. One interesting result is that the rate of heat transfer at the wall decreases significantly with an increase in heat source parameter Q$\left( Q>0 \right)$whereas the adverse effect is observed in presence of heat sink Q$\left( Q<0 \right)$. Further, heat source increases the Sherwood number but heat sink reduces it.  On careful observation, it is seen that for destructive chemical reaction parameter$\gamma $$\left( \gamma >0 \right)$, the Sherwood number increases with an increase in$\gamma $ but decreases with an increase in constructive chemical reaction parameter$\gamma $$\left( \gamma <0 \right)$. Moreover, in an exothermic reaction, the rate of heat transfer at the wall increases while in endothermic reaction it decreases.

Table 2 Computation of skin friction coefficients, Nusselt number and Sherwood number when $Q=\gamma =0.1,Nb=Nt=0.5,\text{ }B{{i}_{t}}=B{{i}_{c}}=0.3$

Table 3 Computation of Nusselt number and Sherwood number when $M=0.5,Kp=2,Sc=0.6$ $n=\Pr =R=1$