Chemical Reaction Effects on MHD Slip Flow of Convective Nanofluid over a Vertical Plate Embedded in Porous Medium with Heat Source/Sink

Consider the steady laminar two dimensional flow of a viscous incompressible nanofluid over a semi-infinite vertical plate embedded in a porous medium. Cartesian co-ordinate system, as shown in Figure 1 is chosen such that x,y axes, respectively, are in the vertical upward and perpendicular directions on the plane of the vertical plate-surface. The ambient temperature, concentration and volume fraction are mentioned in the Figure 1 as T_∞, C_∞, and f_∞, respectively. The plate is heated/cooled depending upon T_f<T_∞/T_f>T_∞, where T_f is the fluid temperature.

1.jpg

Figure 1. Flow configuration

By employing Oberbeck-Boussinesq approximation, making use of standard boundary layer approximations and eliminating pressure, the equation of continuity, motion, heat energy, volume fraction and concentration for the nanofluid following RamReddy [11], Kandasamy et al. [17], and Das et al. [18] are given by

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

$\rho_{f_{\infty}}\left(u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}\right)=\mu \frac{\partial^{2} u}{\partial y^{2}}+g \rho_{f_{\infty}}\left(1-\phi_{\infty}\right)$

$\left[\beta_{T}\left(T-T_{\infty}\right)+\beta_{C}\left(C-C_{\infty}\right)\right]-\left(\rho_{p}-\rho_{f_{\infty}}\right) g\left(\phi-\phi_{\infty}\right)-\sigma B_{0}^{2} u-\frac{\mu}{K_{p}^{*}} u$  (2)

\(\begin{align}  & u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}={{\alpha }_{m}}\frac{{{\partial }^{\text{2}}}T}{\partial {{y}^{\text{2}}}}+\tau \left[ {{D}_{B}}\frac{\partial \phi }{\partial y}\frac{\partial T}{\partial y}+\frac{{{D}_{T}}}{{{T}_{\infty }}}{{\left( \frac{\partial T}{\partial y} \right)}^{\text{2}}} \right]-\frac{1}{{{(\rho C)}_{f}}}\frac{\partial {{q}_{r}}}{\partial y} \\ & \text{                       +}\frac{\sigma B_{0}^{2}{{u}^{2}}}{\rho {{C}_{p}}}+\frac{\mu }{\rho {{C}_{p}}}{{\left( \frac{\partial u}{\partial y} \right)}^{2}}+{{Q}_{0}}\left( T-{{T}_{\infty }} \right) \\\end{align}\)  (3)

$u \frac{\partial \phi}{\partial x}+v \frac{\partial \phi}{\partial y}=D_{B} \frac{\partial^{2} \phi}{\partial y^{2}}+\frac{D_{T}}{T_{\infty}} \frac{\partial^{2} T}{\partial y^{2}}$  (4)

$u \frac{\partial C}{\partial x}+v \frac{\partial C}{\partial y}=D_{s} \frac{\partial^{2} C}{\partial y^{2}}+D_{C T} \frac{\partial^{2} T}{\partial y^{2}}-K c^{*}\left(C-C_{\infty}\right)$  (5)

From the momentum transport Eq. (2), it is evident that the base fluid is a Newtonian viscous fluid, the term $\mu \frac{\partial^{2} u}{\partial y^{2}}$, presents the contribution due to viscosity. Hence, thin liquid, liquid of low viscosity may be preferred to act as a base fluid for example water, kerosene, ethyl benzene and any metallic/non-metallic oxides etc. which exhibit the Newtonian characteristics.

The boundary conditions following Aziz [19] as

$\left.\begin{array}{l}y=0: \quad u=0, \quad v=0,-k \frac{\partial T}{\partial y}=h_{f}\left(T_{f}-T\right), \phi=\phi_{w}, C=C_{w} \\ y \rightarrow \infty: u \rightarrow u_{\infty}, T \rightarrow T_{\infty}, \quad \phi \rightarrow \phi_{\infty}, C \rightarrow C_{\infty}\end{array}\right\}$  (6)

By using the following similarity transformations

$\eta=\frac{y}{\sqrt{2} x} \operatorname{Re}_{x}^{1 / 2}, f(\eta)=\frac{1}{\sqrt{2} v} \operatorname{Re}_{x}^{-1 / 2} \psi(\eta)$

$\theta(\eta)=\frac{T-T_{\infty}}{T_{w}-T_{\infty}}, \gamma(\eta)=\frac{\phi-\phi_{\infty}}{\phi_{w}-\phi_{\infty}}, S(\eta)=\frac{C-C_{\infty}}{C_{w}-C_{\infty}}$

with stream function $\psi(x, y)$ such that $u=\frac{\partial \psi}{\partial y}$ and $v=-\frac{\partial \psi}{\partial x}$, the Eqns. (2) - (5) reduce to

$f^{\prime \prime \prime}+f f^{\prime \prime}-\left(M+K_{P}\right) f^{\prime}+2 \lambda[\theta+N c S-N r \gamma]=0$  (7)

\(\left( 1+R \right){\theta }''+\Pr \left[ \begin{align}  & f{\theta }'+Nb{\theta }'{\gamma }'+Nt{{{{\theta }'}}^{2}} \\ & +Ec\left\{ {{\left( {{f}''} \right)}^{2}}+M{{\left( {{f}'} \right)}^{2}} \right\}+Q\theta  \\\end{align} \right]=0\)   (8)

$\gamma^{\prime \prime}+L e f \gamma^{\prime}+\frac{N t}{N b} \theta^{\prime \prime}=0$  (9)

$\frac{1}{S c} S^{\prime \prime}+S_{T} \theta^{\prime \prime}+f S^{\prime}-2 K c S=0$  (10)

Physical significance of Eq. (9): Buongiorno [3] concluded that for laminar flow thermophoresis and Brownian diffusion are important mechanism in nanofluid analysis. Based on this the continuity Eq. (4), non-dimensional form of Eq. (9), is derived for nanoparticles. The Eq. (10) represents the solutal concentration. This gives rise to solutal concentration boundary layer and needs separate analysis.

The corresponding boundary conditions (6) reduce to

$\left.\begin{array}{l}\eta=0: f(0)=0, f^{\prime}(0)=0, \theta^{\prime}(0)=-B i[1-\theta(0)], \gamma(0)=1, S(0)=1 \\ \eta \rightarrow \infty: f^{\prime}(\infty) \rightarrow 1, \theta(\infty) \rightarrow 0, \gamma(\infty) \rightarrow 0, S(\infty) \rightarrow 0\end{array}\right\}$  (11)

where, non-dimensional variables and parameters are

$M=\frac{2 \sigma B_{0}^{2} x}{\rho_{f_{\infty}} u_{\infty}}, K_{p}=\frac{K_{p}^{*} u_{\infty}}{2 v x}, Q=\frac{2 Q_{0} x}{u_{\infty}}, \operatorname{Pr}=\frac{v}{\alpha_{m}}$

$L e=\frac{\alpha_{m}}{D_{B}}, S c=\frac{v}{D_{s}}, \lambda=\frac{G r_{x}}{\operatorname{Re}_{x}^{2}}, K c=\frac{x K c^{*}}{u_{\infty}}, \operatorname{Re}_{x}=\frac{u_{\infty} x}{v}$

$R=\frac{16 \sigma^{*} T_{\infty}^{3}}{3 k k^{*}}, N b=\frac{(\rho c)_{p} D_{B}\left(\phi_{w}-\phi_{\infty}\right)}{(\rho c)_{f} v}$

$N c=\frac{\beta_{C}\left(C_{w}-C_{\infty}\right)}{\beta_{T}\left(T_{f}-T_{\infty}\right)}, N r=\frac{\left(\rho_{p}-\rho_{f \infty}\right)\left(\phi_{w}-\phi_{\infty}\right)}{\rho_{f \infty} \beta_{T}\left(T_{f}-T_{\infty}\right)\left(1-\phi_{\infty}\right)}$

$S_{T}=\frac{D_{c T}\left(T_{f}-T_{\infty}\right)}{v\left(C_{w}-C_{\infty}\right)}, N t=\frac{(\rho c)_{p} D_{T}\left(T_{f}-T_{\infty}\right)}{(\rho c)_{f} v T_{\infty}}$

$G r_{x}=\frac{g \beta_{T}\left(1-\phi_{\infty}\right)\left(T_{f}-T_{\infty}\right) x^{3}}{v^{2}}, E c=\frac{u_{\infty}^{2}}{C_{p}\left(T_{f}-T_{\infty}\right)}$

And $B i=\frac{c}{k} \sqrt{\frac{2 v}{u_{\infty}}}$ is the Biot number. This boundary condition will be free from the local variable x by choosing $h_{f}=c x^{\frac{-1}{2}}$.

The most important physical quantities for the problem are shearing stress, local heat, local nanoparticle mass and local regular mass fluxes which can be obtained from

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

$q_{n}=-D_{B}\left[\frac{\partial \phi}{\partial y}\right]_{y=0} \operatorname{and} q_{m}=-D_{s}\left[\frac{\partial C}{\partial y}\right]_{y=0}$

The non-dimensional shear stress coefficient $C_{f}=\frac{\tau_{w}}{\rho_{f_{\infty}} u_{\infty}^{2}}$, the Nusselt number $N u_{x}=\frac{q_{w} x}{k\left(T_{f}-T_{\infty}\right)}$, the nanoparticle Sherwood number $N S h_{x}=\frac{q_{n} x}{D_{B}\left(\phi_{w}-\phi_{\infty}\right)}$ and the regular Sherwood number $S h_{x}=\frac{q_{m} x}{D_{s}\left(C_{w}-C_{\infty}\right)}$ are given by

$C_{f}=\frac{f^{\prime \prime}(0)}{\left(2 \operatorname{Re}_{r}\right)^{1 / 2}}, N u_{x}=-\left(\frac{\operatorname{Re}_{x}}{2}\right)^{1 / 2}(1+R) \theta^{\prime}(0)$

$N S h_{x}=-\left(\frac{\operatorname{Re}_{x}}{2}\right)^{1 / 2} \gamma^{\prime}(0), S h_{x}=-\left(\frac{\operatorname{Re}_{x}}{2}\right)^{1 / 2} S^{\prime}(0)$

To check the accuracy of the numerical method, computations for -θ'(0) is carried out for various values of Bi, λ, Pr and compared with the results of Kandasamy et al. [17] in Table 1 and they are found to be good in agreement. From Figure 2 it is seen that the numerical solution obtained by Runge-Kutta method agrees well with RamReddy et al. [11]. This proves the consistency of the numerical method used for the solution of the governing equations. 

Table 1. Comparison of -θ’(0) for various values of Bi, λ and Pr

2.png

Figure 2. Comparison of temperature profiles for Bi and Nr with RamReddy et al. [11]

Figure 3 exhibits the effect of magnetic parameter (M), porosity parameter (K_p) and chemical reaction parameter (Kc) on velocity distribution. The interaction of magnetic field with an electrically conducting flowing fluid induces a voltage across the magnetic field. The current generated by the induced voltage interacts with magnetic field and producing a resistive force to slow down the motion of the fluid. Therefore, an increase in M leads to decrease the velocity profiles in the flow domain. Further, porous medium resists the fluid flow also. It is also seen that the velocity increases with an increasing values of constructive chemical reaction parameter Kc (Kc<0) whereas reverse effect is observed is case of destructive chemical reaction parameter Kc (Kc>0).

3.png

Figure 3. Velocity profiles for various values of M, K_p and Kc with λ=Nr=Nc=Bi=Pr=S_T=1.0, Nt=Nb=0.45, Sc=0.6, Le=5.0, Q=R=Ec=0.1

Figure 4 reveals the influence of nanoparticle buoyancy ratio (Nr) and regular buoyancy ratio (Nc) on velocity distribution. It is observed that thermal boundary layer decreases with an increasing value of Nr and increases with increasing values of Nc.

4.png

Figure 4. Velocity profiles for various values of Nc and Nr with M=K_p=0.5, λ=Bi=Pr=S_T=1.0, Nt=Nb=0.45, Sc=0.6, Le=5.0, Q=R=Ec=Kc=0.1

5.png

Figure 5. Velocity profiles for various values of Bi and λ with M=K_p=0.5, Nc=Nr=Pr=S_T=1.0, Nt=Nb=0.45, Sc=0.6, Le=5.0, Q=R=Ec=Kc=0.1

6.png

Figure 6. Temperature profiles for various values of Nb and Nt with M=K_p=Q=R=Ec=Kc=0.1, λ=Bi=Pr=S_T=Nr=Nc=1.0, Sc=0.6, Le=10.0

Figure 5 exhibits the impact of Biot number (Bi) and mixed convection parameter (λ) on velocity profiles. It is observed that both the parameters enhance the velocity. The Biot number is the ratio of momentum diffusivity through v and thermal diffusivity through k. Here, momentum diffusivity is accelerated overriding the thermal conductivity of nanofluid, resulting accelerated fluid motion.

From Figure 6 it is clear that Brownian motion (Nb) and thermophoresis parameter Nt increase the temperature indicating more dispersion of thermal energy into flow domain. It is also revealed that thermal energy is dissipated into flow due to concentration and temperature differences between boundary layers and corresponding ambient conditions as well as Brownian diffusion coefficient and thermophoresis rather than kinematic viscosity (υ) of the base fluid responsible for momentum diffusivity.

7.png

Figure 7. Temperature profiles for various values of M, Q and Kc with Nb=Nt= 0.45, λ=Bi=Pr=S_T=Nr=N_C=1.0, Sc=0.6,Le=10.0, R=Ec=K_P=0.1

8.png

Figure 8. Temperature profiles for various values of R and Ec with M=K_p=Q=K_C=0.1,λ=Bi=Pr S_T=Nr=Nc=1.0, Nb=Nt= 0.45, Sc=0.6, Le=10.0

Figure 7 depicts the influence of magnetic parameter (M), heat source/sink parameter (Q) and chemical reaction parameter (Kc) on temperature distributions. It is noticed that an increase in the value of magnetic parameter enhances the temperature of the fluid. The temperature profiles increase with heat source parameter Q(Q>0) on the contrary, the temperature decreases with heat sink parameter Q(Q<0). It is also seen that the temperature increases with an increase in the destructive chemical reaction parameter Kc(Kc>0) and decreases in case of constructive chemical reaction parameter Kc(Kc<0). It is natural in the process of exothermic reaction; the heat energy is evolved that enhances the fluid temperature. The reverse thermo-physical conditions is observed for endothermic reaction.

Figure 8 presents the effect of thermal radiation parameter (R) and Eckert number (Ec) on the temperature profiles. The effect of radiation parameter near the bounding surface (0<η<0.65) is to decrease the temperature and beyond, the reverse effect is observed.  The point of intersection of the profiles drifted away from the boundary with higher value of Ec.

9.png

Figure 9. Volume fraction profiles for various values of Nb and Nt with M=K_p=Q=K_C=R=Ec=0.1, λ=Bi=Pr S_T=Nr=Nc=1.0, Sc=0.6, Le=10.0

Figure 9 is plotted to explain the effects of Brownian motion (Nb) and thermophoresis parameter (Nt) on volume fraction profiles. A sharp fall in volume fraction is seen with the variation of Nb and Nt except some anomalies in case of slightly increased value of Nt equals to 0.5, 0.9 where, for which the fall slows down when 0.6<η<1.9. This indicates that increase in Brownian motion resulting higher energy exchange is prevented.

10.png

Figure 10. Volume fraction profiles for various values of Le and Kc with M=K_p=Q= R=Ec=0.1, Nb=Nt= 0.45, Bi=Pr S_T=Nr=Nc=1.0, Sc=0.6

11.png

Figure 11. Concentration profiles for various values of Kc and S_T with M=K_p=Q=R=Ec=0.1, Nb=Nt= 0.45, λ=Bi=Pr Nr=Nc=1.0, Sc=0.6, Le=10.0

Figure 10 presents the effects of Lewis number (Le) and chemical reaction parameter (Kc) on volume fraction profiles. A sharp fall in volume fraction occurs due to higher value of Lewis number. One remarkable feature is that the volume fraction increases under the influence of exothermic reaction (Kc>0) and decreases for endothermic reaction (Kc<0).

Figure 11 depicts the effects of chemical reaction parameter (Kc) and Soret number (S_T) on concentration profiles. It is observed that destructive chemical reaction parameter Kc (Kc>0) decreases the concentration profiles while concentration increases with increasing values of generative chemical reaction parameter Kc (Kc<0). It is also seen that an increase in the Soret number results in a significant increase in the concentration distributions within the solutal boundary layer. 

Figure 12 displays the results of Schmidt number (Sc) and mixed convection parameter (λ) on concentration profiles.  The higher Sc representing heavier species lowers down the concentration level at all the layers when λ<1, whereas reverse effect is observed in case of λ>1.

12.png

Figure 12. Concentration profiles for various values of Sc and λ with M=K_p=Q=R=Ec=Kc=0.1, Nb=Nt= 0.45, Bi=Pr= Nr=Nc=1.0, Le=10

Table 2 exhibits the values of skin-friction coefficient f’’(0), heat transfer coefficient -θ’(0), regular Sherwood number -S’(0) and nanoparticle Sherwood number -γ’(0). It is found that all these surface criteria increase with increasing values of mixed convection parameter and Biot number whereas reverse effect is observed in case of magnetic parameter. Further, it is noticed that Nusselt number increases with increase in radiation parameter while skin-friction coefficient, regular Sherwood number and Sherwood number decrease with increase in thermal radiation parameter. On careful observation, it is seen that a significant reduction in heat transfer rate at the plate increases with Eckert number. On the other hand the skin friction coefficient, regular Sherwood number and Sherwood number increase with the increasing values of Eckert number. It is also noted that for destructive chemical reaction parameter Kc (Kc>0), the regular Sherwood number increases with an increase in Kc but skin friction coefficient, Nusselt number and nanoparticle Sherwood number increase with increase in constructive chemical reaction parameter Kc (Kc<0). One interesting result is that the rate of heat transfer at the wall decreases significantly with an increase in the heat source parameter Q (Q>0) whereas the reverse effect is observed in presence of heat sink parameter (Q>0). The temperature at the plate is θ’(0)=-Bi[1-θ(0)]. For Ec=0.1, θ(0)=0.8. Hence θ’(0)<0 when Bi>0. This indicates that due to higher value of Bi (Bi=5), the negative temperature at the plate could not be compensated due to the presence of heat source. Therefore, it is suggested that the value of Biot number to be reduced so that rate of heat transfer at the plate will increase.

Table 2. Variation of skin friction coefficients, heat transfer, nanoparticle mass transfer and regular mass transfer rates when Nr=Nc=Pr=S_T=1.0, K_p=Nb=Nt=0.5, Sc=0.6, Le=10