Homotopy analysis for the influence of Navier slip flow in a vertical channel with cross diffusion effects

Abstract

This research is to examine the laminar, incompressible free convective Navier slip flow between vertical plates with cross diffusion effects. This investigation includes the first order chemical reaction also. The resulting equations with boundary conditions are reduced into dimensionless form using suitable transformations. Homotopy analysis method has been applied to solve the system. The influence of emerging parameters on fluid flow quantities have been presented graphically. In addition, the nature of physical quantities are shown in tabular form.

Introduction

Combined heat and mass transfer in free convection flow between vertical parallel plates has significant importance in many applications. Kairi and Murthy [1] presented the applications and past theoretical investigations on free convection flow in a channel. Fattahi et al. [2] applied the lattice Boltzmann technique to study the nature of free convection flow of nano fluids. Huelsz and Rechtman [3] also applied the lattice Boltzmann technique for heat transfer in an inclined square cavity due to natural convection. Most recently, Terekhov et al. [4] studied the laminar free convection heat transfer between vertical isothermal plates.

Generally, the no-slip condition has been accepted for the fluid over a solid surface boundary condition. At the solid boundary, the general boundary conditions for slip flow has been proposed by Navier [5]. Eegunjobi and Makinde [6] presented the early literature and application of the slip flow in various geometries. Recently, Rundora and Makinde [7] analyzed the influence of Navier slip and variable viscosity through a porous medium with asymmetric convective boundary conditions. Most recently, Ng [8] investigated the nature of starting flow in channels with boundary slip.

The Soret and Dufour effects are encountered in the areas of chemical engineering, geosciences, etc. The applications and early literature can be seen in [9]. Recently, Srinivasacharya et al. [10] studied the mixed convection flow along a vertical wavy surface with the effects of cross diffusion and variable properties in a porous medium. Most recently, Umar et al. [11] presented the effects of cross diffusions and chemical reaction flow in converging and diverging channels.

Most of the chemical reactions comprise both homogeneous and heterogeneous reactions. The homogeneous reaction takes place in bulk of the fluid, while heterogeneous reaction occurs on some catalytic surfaces. Generally, the interaction between the homogeneous and heterogeneous reactions are very complex and is involved in the production and consumption of reactant species at various rates, both on the catalytic surfaces and within the fluid. The applications and early literature has been reported by Kothandapani and Prakash [12]. Most recently, RamReddy and Pradeepa [13] used the spectral quasi-linearization method to study the effect of homogeneous–heterogeneous reactions on non-linear convection flow of micropolar fluid in a porous medium with convective boundary condition.

In this paper, the natural convection flow in a vertical channel saturated with Navier slip condition has been investigated in the presence of cross diffusions and the rate of chemical reaction. The survey clearly shows that the combined effects with slip flow condition through a vertical channel has not been presented elsewhere. In view of applications and importance, the authors are motivated to take this problem. Homotopy analysis method (HAM) [14,15,16,17,18,19,20,21,22,23] has been used for the solution of the present problem. Convergence of the resulting series solution is explained. Then the discussion with respect to the pertinent parameters of the present study on the solutions of velocity, temperature and concentration components.

Mathematical modelling

We consider two-dimensional free convection fluid flow with Navier slip boundary in a vertical channel. Flow configuration is presented in Fig. 1. All the fluid properties are considered to be constant with the exception that the density in the buoyancy term of the balance of momentum equation. Since the plates are infinitely long, all physical variables depend on y only. The governing two-dimensional steady flow equations can be put into the form:

$$\begin{aligned} v_ y=0 \,\, \Rightarrow \,\, v = v_0 = \text{ constant } \end{aligned}$$

(1)

$$\begin{aligned} v_0 \rho \frac{\partial u}{\partial y}=\rho g \left( \beta _T(T-T_1)+ \beta _C(C-C_1)\right) +\mu \frac{\partial ^2 u}{\partial y^2} \end{aligned}$$

(2)

$$\begin{aligned} \rho C_P v_0 \frac{\partial T}{\partial y}=K_f \frac{\partial ^2 T}{\partial y^2}+ \mu \left( \frac{\partial u}{\partial y}\right) ^2+ \frac{DK_T}{C_S C_P} \frac{\partial ^2 C}{\partial y^2} \end{aligned}$$

(3)

$$\begin{aligned} v_0 \frac{\partial C}{\partial y}=D \frac{\partial ^2 C}{\partial y^2}+\frac{D K_T}{T_m}\frac{\partial ^2 T}{\partial y^2}-K_1 (C-C_1) \end{aligned}$$

(4)

with

$$\begin{aligned} u=\gamma _1 \frac{\partial u}{\partial y}, \quad T=T_1, \quad C=C_1 \quad \text{ at }\quad y=-d \end{aligned}$$

(5a)

$$\begin{aligned} u=\gamma _2 \frac{\partial u}{\partial y}, \quad T=T_2, \quad C=C_2 \quad \text{ at }\quad y=d \end{aligned}$$

(5b)

where the velocity components in x- and y- are u, v, respectively, g is the acceleration due to gravity, $\rho$ is the density, $C_p$ is the specific heat, $\mu$ is the coefficient of viscosity, $\beta _T$ is the coefficient of thermal expansion, $\beta _C$ is the coefficient of solutal expansion, $\alpha$ is the thermal diffusivity, D is the mass diffusivity, $K_T$ is the coefficient of thermal conductivity, $C_S$ is the concentration susceptibility, $T_m$ is the mean fluid temperature, $K_f$ is the thermal diffusion ratio, $K_1$ is the rate of chemical reaction, $\gamma _1$ and $\gamma _2$ are the slip coefficients at both the plates.

Introducing the following dimensionless variables

$$\begin{aligned} \eta =\frac{y}{d}, u=\displaystyle {\frac{\nu Gr}{d} f}, \theta =\displaystyle {\frac{T-T_1}{T_2-T_1}}, \phi =\displaystyle {\frac{C-C_1}{C_2-C_1}} \end{aligned}$$

(6)

in Eqs. (2)–(4), we obtain the governing dimensionless equations as

$$\begin{aligned} f'' + \theta + N \phi -Re f'=0 \end{aligned}$$

(7)

$$\begin{aligned} \theta ''- RePr \theta '+ Br Gr^2(f')^2+Pr D_f \phi ''=0 \end{aligned}$$

(8)

$$\begin{aligned} \phi ''- ReSc \phi '+ Sc Sr\theta ''-K Sc \phi =0 \end{aligned}$$

(9)

with

$$\begin{aligned} \eta= & {} -1: f- \beta _1 f'= \theta = \phi =0 \nonumber \\ \eta= & {} 1: f- \beta _2 f'=0 \quad \text{ and } \quad \theta = \phi =1 \end{aligned}$$

(10)

where the primes represents differentiation with respect to $\eta$, $\displaystyle {N=\frac{\beta _C(C_2-C_1)}{\beta _T(T_2-T_1)}}$ is the buoyancy parameter, $\displaystyle {Re=\frac{\rho v_0 d}{\mu }}$ is the Reynolds number, $\displaystyle {Sc=\frac{\nu }{D}}$ is the Schmidth number, $Gr= \displaystyle {\frac{g_a \beta _T (T_2-T_1)d^3}{\nu ^2}}$ is the Grashof number, $\displaystyle {Pr=\frac{\mu C_p}{K_f}}$ is the Prandtl number, $\displaystyle {Br=\frac{\mu \nu ^2}{K_f d^2 (T_2-T_1)}}$ is the Brinkman number, $\displaystyle {K=\frac{K_1 d^2}{\nu }}$ is the chemical reaction parameter, $S_r= \displaystyle {\frac{DK_T(T_2-T_1)}{\nu T_m(C_2-C_1)}}$ is the thermo-diffusion parameter and $\displaystyle {D_f=\frac{DK_T(C_2-C_1)}{\nu C_S C_P (T_2-T_1)}}$ is the Dufour number. $\displaystyle {\beta _1=\frac{\gamma _1}{d}}$, $\displaystyle {\beta _2=\frac{\gamma _2}{d}}$ are the slip parameters.

The skin friction coefficients$(C_f)$, heat (Nu) and mass (Sh) fluxes at the vertical walls can be given by

$$\begin{aligned} \frac{Re^2}{Gr} {C_f}_1&= 2 f'(-1), \, \frac{Re^2}{Gr} {C_f}_2=2 f'(1)\nonumber \\ Nu_{1,2}&= -\left. \theta '(\eta ) \right| _{\eta =-1,1}, \, Sh_{1,2}=-\left. \phi '(\eta ) \right| _{\eta =-1,1} \end{aligned}$$

(11)

where

$$\begin{aligned} \displaystyle {C_f=\frac{\tau _w}{\rho v_0^2}}, \, Nu= \displaystyle {\frac{q_w d}{K_f(T_2-T_1)}}, \, \displaystyle {Sh=\frac{q_m d}{D (C_2-C_1)}}\nonumber \\ \end{aligned}$$

(12)

and

$$\begin{aligned} \tau _w=\mu \left. \frac{\partial u}{\partial y}\right| _{y=\pm d}; q_w= & {} \left. -K_f \frac{\partial T}{\partial y}\right] _{y=\pm d}\nonumber \\ q_m= & {} \left. -D \frac{\partial C}{\partial y}\right] _{y=\pm d} \end{aligned}$$

(13)

Effect of the various parameters involved in the investigation on physical coefficients are discussed in the following section.

Fig. 1

Physical model and coordinate system.

Homotopy solution

For HAM solutions, we choose the initial approximations of $f(\eta )$, $\theta (\eta )$ and $\phi (\eta )$ as follows:

$$\begin{aligned} f_0(\eta )=0, \quad \theta _0(\eta )=\frac{1+\eta }{2}, \quad \phi _0(\eta )=\frac{1+\eta }{2} \end{aligned}$$

(14)

with the auxiliary linear operator

$$\begin{aligned} L =\frac{\partial ^2}{\partial \eta ^2}\quad \text{ such } \text{ that } \, \, L_1(c_1\eta +c_2)=0 \end{aligned}$$

(15)

in which $c_1$ and $c_2$ are the arbitrary constants. $h_1$, $h_2$ and $h_3$ (the convergence control parameters) are introduced in zeroth-order deformations as

$$\begin{aligned} (1-p)L[f(\eta ;p)-f_0(\eta )]=p h_1 N_1[f(\eta ;p)] \end{aligned}$$

(16)

$$\begin{aligned} (1-p)L[\theta (\eta ;p)-\theta _0(\eta )]=p h_2 N_2[\theta (\eta ;p)] \end{aligned}$$

(17)

$$\begin{aligned} (1-p)L[\phi (\eta ;p)-\phi _0(\eta )]=p h_3 N_3[\phi (\eta ;p)] \end{aligned}$$

(18)

subject to the boundary conditions

$$\begin{aligned} f(-1;p)-\beta _1 f'(-1;p)&= 0, f(1;p)-\beta _2 f'(1;p)=0\nonumber \\ \theta (-1;p)=0, \theta (1;p)&= 1, \phi (-1;p)=0, \phi (1;p)=1\nonumber \\ \end{aligned}$$

(19)

where $p \in [0, 1]$ is the embedding parameter and the non-linear operators $N_1$, $N_2$ and $N_3$ are defined as:

$$\begin{aligned} N_1[f(\eta ,p),\theta (\eta ,p),\phi (\eta ,p)]= & {} f''(\eta ,p) + \theta (\eta ,p)\nonumber \\&+\, N \phi (\eta ,p) -Re f'(\eta ,p) \end{aligned}$$

(20)

$$\begin{aligned} N_2[f(\eta ,p),\theta (\eta ,p),\phi (\eta ,p)]&= \theta ''(\eta ,p)- RePr \theta '(\eta ,p)\nonumber \\&\quad +\, Br Gr^2(f'(\eta ,p))^2+Pr D_f \phi ''(\eta ,p)\nonumber \\ \end{aligned}$$

(21)

$$\begin{aligned} N_3[f(\eta ,p),\theta (\eta ,p),\phi (\eta ,p)]&= \phi ''(\eta ,p)- ReSc \phi '(\eta ,p)\nonumber \\&\quad+\, Sc Sr\theta ''(\eta ,p)-K Sc \phi (\eta ,p)\nonumber \\ \end{aligned}$$

(22)

For $p=0$, we have the initial guess approximations

$$\begin{aligned} f(\eta ;0)=f_0(\eta ), \, \theta (\eta ;0)=\theta _0(\eta ), \, \phi (\eta ;0)=\phi _0(\eta ) \end{aligned}$$

(23)

When $p=1$, Eqs. (16)–(18) are same as (7)–(9), respectively, therefore, at $p = 1$, we get the final solutions

$$\begin{aligned} f(\eta ;1)=f(\eta ), \, \theta (\eta ;1)=\theta (\eta ), \, \phi (\eta ;1)=\phi (\eta ) \end{aligned}$$

(24)

Hence, the process of giving an increment to p from 0 to 1 is the process of $f(\eta ;p)$ varying continuously from the initial guess $f_0(\eta )$ to the final solution $f(\eta )$ (similar for $\theta (\eta ,p)$ and $\phi (\eta ,p)$). This kind of continuous variation is called deformation in topology so that we call system of Eqs. (16)–(19), the zeroth-order deformation equation. Next, the mth-order deformation equations follow as

$$\begin{aligned} L_1[f_m(\eta )-\chi _m f_{m-1}(\eta )]=h_1 R_m^f(\eta ) \end{aligned}$$

(25)

$$\begin{aligned} L[\theta _m(\eta )-\chi _m \theta _{m-1}(\eta )]=h_2 R_m^\theta (\eta ) \end{aligned}$$

(26)

$$\begin{aligned} L[\phi _m(\eta )-\chi _m \phi _{m-1}(\eta )]=h_3 R_m^\phi (\eta ) \end{aligned}$$

(27)

with the boundary conditions

$$\begin{aligned} f_m(-1)=0,\quad f_m(1)=0, \quad \theta _m(-1)=0\nonumber \\ \theta _m(1)=0, \quad \phi _m(-1)=0, \quad \phi _m(1)=0, \end{aligned}$$

(28)

where

$$\begin{aligned} R_m^f(\eta )=f'' + \theta + N \phi -Re f' \end{aligned}$$

(29)

$$\begin{aligned} R_m^\theta (\eta )=&\theta ''- RePr \theta '+ Br Gr^2 \sum _{n=0}^{m-1}f'_{m-1-n}f'_n + Pr D_f \phi ''\nonumber \\ \end{aligned}$$

(30)

$$\begin{aligned} R_m^\theta (\eta )=\phi ''- ReSc \phi '+ Sc Sr\theta ''-K Sc \phi \end{aligned}$$

(31)

for m being integer

$$\begin{aligned} \chi _m&= 0 \quad \text{ for } \quad m \le 1 \nonumber \\ & = 1 \quad \text{ for } \quad m > 1 \end{aligned}$$

(32)

The initial guess approximations $f_0(\eta )$, $\theta _0(\eta )$ and $\phi _0(\eta )$, the linear operators L and the auxiliary parameters $h_1, h_2$ and $h_3$ are assumed to be selected such that Eqs. (16)–(19) have solution at each point $p \in [0, 1]$ and also with the help of Taylor’s series and due to Eq. (23); $f(\eta ;p)$, $\theta (\eta ;p)$ and $\phi (\eta ;p)$ can be expressed as

$$\begin{aligned} f(\eta ;p)=f_0(\eta )+\sum _{m=1}^\infty f_m(\eta )p^m \end{aligned}$$

(33)

$$\begin{aligned} \theta (\eta ;p)=\theta _0(\eta )+\sum _{m=1}^\infty \theta _m(\eta )p^m \end{aligned}$$

(34)

$$\begin{aligned} \phi (\eta ;p)=\phi _0(\eta )+\sum _{m=1}^\infty \phi _m(\eta )p^m \end{aligned}$$

(35)

in which $h_1$, $h_2$ and $h_3$ are chosen in such a way that the series (33)–(35) are convergent [15] at $p=1$. Therefore, we have from (24) that

$$\begin{aligned} f(\eta )=f_0(\eta )+\sum _{m=1}^\infty f_m(\eta ) \end{aligned}$$

(36)

$$\begin{aligned} \theta (\eta )=\theta _0(\eta )+\sum _{m=1}^\infty \theta _m(\eta ) \end{aligned}$$

(37)

$$\begin{aligned} \phi (\eta )=\phi _0(\eta )+\sum _{m=1}^\infty \phi _m(\eta ) \end{aligned}$$

(38)

for which we presume that the initial guesses to f, $\theta$ and $\phi$, the auxiliary linear operators L and the non-zero auxiliary parameters $h_1$, $h_2$ and $h_3$ are so properly selected that the deformation $f(\eta ,p)$, $\theta (\eta ,p)$ and $\phi (\eta ,p)$ are smooth enough and their mth-order derivatives with respect to p in Eqs. (36)–(38) exist and are given, respectively, by       $\left. f_m(\eta )=\frac{1}{m!}\frac{\partial ^m f(\eta ;p)}{\partial p^m}\right| _{p=0}$,    $\left. \theta _m(\eta )=\frac{1}{m!}\frac{\partial ^m \theta (\eta ;p)}{\partial p^m}\right| _{p=0}$ and    $\left. \phi _m(\eta )=\frac{1}{m!}\frac{\partial ^m \phi (\eta ;p)}{\partial p^m}\right| _{p=0}$. It is clear that the convergence of Taylor series at $p= 1$ is a prior assumption, whose justification is provided via a theorem [25], so that the system in (36)–(38) holds true. The formulae in (36)–(38) provide us with a direct relationship between the initial guesses and the exact solutions. All the effects of interaction of the magnetic field as well as of the heat transfer, Hall and Ion effects and couple stress flow field can be studied from the exact formulas (36)–(38). Moreover, a special emphasize should be placed here that the mth-order deformation system (25)–(28) is a linear differential equation system with the auxiliary linear operators L whose fundamental solution is known.

Table 1 Optimal value of $h_1$ at different order of approximations

Table 2 Optimal value of $h_2$ at different order of approximations

Table 3 Optimal value of $h_3$ at different order of approximations

Convergence of the HAM solution

The expressions for f, $\theta$ and $\phi$ contain the auxiliary parameters $h_1, h_2$ and $h_3$. As pointed out by Liao [14], the convergence and the rate of approximation for the HAM solution strongly depend on the values of auxiliary parameter h. For this purpose, h-curves are plotted to choose $h_1, h_2$ and $h_3$ in such a manner that the solutions (33)–(35) ensure convergence [14]. Here to see the admissible values of $h_1, h_2$ and $h_3$, the h-curves are plotted for 15th-order of approximation in Figs. 2, 3 and 4 by taking the values of the parameters $N=2, Re=2, Pr=0.71, Sc=0.22, Br=0.5, Gr=0.5, Sr=0.5, Df=0.5, K=1.0$. It is clearly noted from Fig. 2 that the range for the admissible values of $h_1$ is $-1.25< h_1 < -0.6$. From Fig. 3, it can be seen that the h-curve has a parallel line segment that corresponds to a region $-1.15< h_2 < -0.65$. Figure 4 depicts that the admissible value of $h_3$ are $-1.2< h_3 < -0.6$. A wide valid zone is evident in these figures ensuring convergence of the series. To choose optimal value of auxiliary parameters, the average residual errors (see Ref. [15] for more details) are defined as

$$\begin{aligned} E_{f,m}=\frac{1}{2K}\sum _{i=-K}^K\left( N_1\left[ \sum _{j=0}^m f_j(i\triangle t)\right] \right) ^2 \end{aligned}$$

(39)

$$\begin{aligned} E_{\theta ,m}=\frac{1}{2K}\sum _{i=-K}^K\left( N_2\left[ \sum _{j=0}^m \theta _j(i\triangle t)\right] \right) ^2 \end{aligned}$$

(40)

$$\begin{aligned} E_{\phi ,m}=\frac{1}{2K}\sum _{i=-K}^K\left( N_3\left[ \sum _{j=0}^m \phi _j(i\triangle t)\right] \right) ^2 \end{aligned}$$

(41)

where $\triangle t = 1/K$ and K = 5. At different order of approximations (m), minimum of average residual errors are shown in Tables 1, 2 and 3. It is clear from Table 1 that the average residual error for f is minimum at $h_1=-0.93$. It can be seen from Table 2 that the minimum of average residual error for $\theta$ attains at $h_2=-0.93$. Table 3 depicts that at $h_3=-0.93$, $E_\phi$ attains minimum. Therefore, the optimum values of convergence control parameters are taken as $h_1=-0.93,\, h_2=-0.93,\, h_3=-0.93$.

To see the accuracy of the solutions, the residual errors are defined for the system as

$$\begin{aligned} \hbox {RE}_f=f_n''(\eta ) + \theta _n(\eta )+ N \phi _n(\eta ) -Re f_n'(\eta ) \end{aligned}$$

(42)

$$\begin{aligned} \hbox {RE}_\theta= & {} \theta _n''(\eta )- RePr \theta _n'(\eta )\nonumber \\&+ Br Gr^2(f_n'(\eta ))^2+Pr D_f \phi _n''(\eta ) \end{aligned}$$

(43)

$$\begin{aligned} \hbox {RE}_\theta= & {} \phi _n''(\eta )- ReSc \phi _n'(\eta )\nonumber \\&+ Sc Sr\theta _n''(\eta )-K Sc \phi _n(\eta ) \end{aligned}$$

(44)

where $f_n(\eta )$, $\theta _n(\eta )$ and $\phi _n(\eta )$ are the HAM solutions for $f(\eta )$, $\theta (\eta )$ and $\phi (\eta )$, respectively. For optimality of the convergence control parameters, residual error [24] for different values of h in the convergence region displayed in Figs. 5, 6 and 7. We examine that $h_1=-0.93,\, h_2=-0.93,\, h_3=-0.93$ gives a better solution. Table 4 establishes the convergence of the obtained series solution. It is found from the above observations that the series given by (33)–(35) converge in the whole region of $\eta$ when $h_1=-0.93,\, h_2=-0.93,\, h_3=-0.93$

To pursue the convergence of the HAM solutions to the exact ones, the graphs for the ratio (following the recent work of [25])

$$\begin{aligned} \beta _f=\left| {\frac{f_m(h)}{f_{m-1}(h)}}\right| , \beta _\theta =\left| {\frac{\theta _m(h)}{\theta _{m-1}(h)}}\right| , \beta _\phi =\left| {\frac{\phi _m(h)}{\phi _{m-1}(h)}}\right| \nonumber \\ \end{aligned}$$

(45)

against the number of terms m in the homotopy series is presented in Figs. 8, 9 and 10. Figures strongly indicate that a finite limit of $\beta$ will be attained in the limit of $m\rightarrow \infty$, which will remain less than or equals to unity. The velocity and temperature solutions seem to converge in an oscillatory manner requiring more terms in the homotopy series. Thus, the convergence to the exact solution is assured by the HAM method.

Fig. 2

The h curve of $f(\eta )$

Fig. 3

The h curve of $\theta (\eta )$

Fig. 4

The h curve of $\phi (\eta )$

Fig. 5

Residual error of $f(\eta )$ when $\beta _h= 2, \beta _i=2, \alpha = 0.5, Ha=5$

Fig. 6

Residual error of $\theta (\eta )$ when $\beta _h= 2, \beta _i=2, \alpha = 0.5, Ha=5$

Fig. 7

Residual error of $\phi (\eta )$ when $\beta _h= 2, \beta _i=2, \alpha = 0.5, Ha=5$

Fig. 8

The ratio $\beta _f$ from the theorem to reveal the convergence of the HAM solutions

Fig. 9

The ratio $\beta _\theta$ from the theorem to reveal the convergence of the HAM solutions

Fig. 10

The ratio $\beta _\phi$ from the theorem to reveal the convergence of the HAM solutions

Fig. 11

Variation $f(\eta )$ with $\beta _1$ when $\beta _2= 0.1, Sr=0.5, D_f=0.5, K=1.0$

Fig. 12

Variation $f(\eta )$ with $\beta _2$ when $\beta _1= 0.1, Sr=0.5, D_f=0.5, K=1.0$

Discussion of results

In the absence of Brinkman $(Br=0)$ number, the system of Eqs. (7)–(9) reduces to the linear system of equations for which exact solution exist. Comparison between analytical solution and HAM solution is shown in Table 5. The comparisons are found to be in a very good agreement. Therefore, the HAM code can be used with great confidence to study the problem considered in this paper.

The profiles of velocity ($f(\eta )$), temperature ($\theta (\eta )$) and concentration ($\phi (\eta )$) are computed and presented through plots in Figs. 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 and 21 with different values of $\beta _1, \beta _2$, $Sr, D_f, K$. Computations were carried out by fixing the parameters $Pr=0.71, N=2, Br=0.5, Da=0.5, Re=2, Sc=0.22, Gr=0.5$ to analyze the effects of the emerging parameters $\beta _1$, $\beta _2$, Sr, $D_f$ and K.

Figure 11 presents the influence of the slip flow parameter $\beta _1$ on $f(\eta )$. It is clear that increase in the parameter $\beta _1$ leads to increase the flow velocity $f(\eta )$ at the injection wall and there is slight reverse trend can be found at the terminal wall. Figure 12 shows that the effect of $\beta _2$ on the flow velocity $f(\eta )$. It can be seen from this figure that as $\beta _2$ increases the flow velocity decreases negotiably at the initial wall but appreciable decrease in the velocity is noticed at the suction wall.

The effect of Soret parameter on velocity, temperature and concentration profiles are presented in Figs. 13, 14 and 15. It is clear from Fig. 13 that the higher values of Soret number Sr increases the velocity of the flow. Since the values of $S_r$ increases due to either an decrease in the temperature difference or increase in the concentration difference, with this lowest peak of the flow velocity compatible with the lowest Soret number. Figure 14 shows the influence of Sr on $\theta$. It is observed that the temperature decreases as Sr increases. The nature of $\phi$ with the influence of Sr is shown in Fig. 15. It is noted that the dimensionless concentration increases as an increase in the Soret number. The results clearly exhibits that the flow field is significantly influenced by the Soret number.

The effect of Dufour number ($D_f$) on dimensionless velocity can be seen in Fig. 16. It is clear that the velocity of the fluid increases with an increase in diffusion thermo parameter. Since the values of $D_f$ increases due to either an increase in the temperature difference or decrease in the concentration difference, i.e., the lowest peak of the flow velocity compatible with the lowest Dufour number. The effect of diffusion thermo parameter on dimensionless temperature is presented in Fig. 17. It is noted that the temperature profile increases with an increase in Dufour number. Figure 18 demonstrates that the concentration of the fluid decreases as Dufour effect increases. This is because of temperature gradients contribution to diffusion of the species.

The chemical reaction parameter K effect on velocity, temperature and concentration profiles are shown in Figs. 19, 20 and 21. It is observed from these figures that the fluid flow velocity decreases as K increases. The temperature increases and concentration of the fluid decreases when the chemical reaction parameter increases. Since the chemical molecular diffusivity drop down when chemical reaction is high, i.e., lesser diffusion. Hence, they are procured by the transfer of species. The higher K will decrease the concentration species. Therefore, with the increase in K suppresses the distribution of the concentration at all the points of the fluid flow. With this, it can be claimed that on the distribution of the concentration, massive effect is higher with heavier diffusing species.

Variation of slip parameters ($\beta _1$ and $\beta _2$), thermal diffusion parameter (Sr), diffusion thermo parameter ($D_f$), together with the chemical reaction parameter K are shown in Table 6 by fixing the remaining parameters. It is seen that the friction factor decreases at both the walls with an increase in the slip parameters $\beta _1$ and $\beta _2$. It is clear that increase in $\beta _1$ leads to increase in heat transfer coefficient at the plates where as the mass transfer rate increases at the initial wall and decreases at the terminal wall. It can be observed that at the initial plate the heat transfer rate decreases and at the second plate it is found to be increasing while the reverse trend is perceived on mass transfer rate with an increase in $\beta _2$. It is observed from this table that the skin friction coefficient, heat transfer rate increases at the injection wall and decreases at the terminal wall with the increase of Soret parameter (Sr), where as mass transfer rate have reverse trend at both the plates. It is noticed that higher values of Dufour parameter ($D_f$) increases the friction factor and mass transfer rate at the initial plate and decreases at the suction wall, while heat transfer rate shows the reverse trend at two walls. The effect of chemical reaction parameter K on skin friction, heat and mass transfer rates are presented in Table 6. It is seen from the table that all the physical parameters are decreasing at the injection wall and those are increasing at the suction wall with the enhance in K. It is also seen from this table that the higher values of K increases the mass transfer rate at $\eta =-1$ and decreases at $\eta =1$. The influence of the emerging parameters are patently obvious from the Table 6, and hence are not discussed for conciseness.

Conclusions

Two-dimensional natural convection flow in a vertical channel in the presence of chemical reaction and the cross diffusion effects with Navier slip. Homotopy analysis method has been employed to solve the final system of equations. The markable findings are summarized as:

• Flow velocity and the rate of mass transfer increases at the injection wall, decreases slightly at the suction wall and the rate of heat transfer increases at both the walls with an enhance in slip parameter $\beta _1$.

• As $\beta _2$ increases, the fluid flow velocity and friction factors decrease (slightly at the injection wall and marginally at the suction wall). Where as heat transfer rate disintegrates at the injection wall and increases at the end wall, while the opposite nature is noticed on the rate of mass transfer with an enhance in $\beta _2$.

• It is noticed that the velocity and concentration of the fluid increases and temperature profile decreases with an increase in Soret parameter. In addition, it is identified that the friction factor, Nusselt number increases at the injection plate and decreases at the end wall as Sr increases, where as mass transfer rate shows the reverse trend at both the walls.

• Fluid flow velocity, $\theta$ increases and the dimensionless concentration profile decreases as $D_f$ increases. It is seen that as Dufour parameter increases, the skin friction and mass transfer rate at $\eta =-1$ and decreases at $\eta =1$ wall, while the reverse trend is noticed in case of heat transfer rate at both the walls.

• As reaction parameter increases, the velocity and concentration of the fluid decreases and the temperature profile decreases. It is also observed that the values of $f'(0)$, Nusselt and Sherwood numbers decrease at the initial wall and those increase at the terminal plate with the increase in K.

Fig. 13

Soret effect on $f(\eta )$ when $\beta _1=0.1, \beta _2=0.1, D_f=0.5, K=1.0$

Fig. 14

Soret effect on $\theta (\eta )$ when $\beta _1=0.1, \beta _2=0.1, D_f=0.5, K=1.0$

Fig. 15

Soret effect on $\phi (\eta )$ when $\beta _1=0.1, \beta _2=0.1, D_f=0.5, K=1.0$

Fig. 16

Dufour effect on $f(\eta )$ when $\beta _1=0.1, \beta _2=0.1, Sr=0.5, K=1.0$

Fig. 17

Dufour effect on $\theta (\eta )$ when $\beta _1=0.1, \beta _2=0.1, Sr=0.5, K=1.0$

Fig. 18

Dufour effect on $\phi (\eta )$ when $\beta _1=0.1, \beta _2=0.1, Sr=0.5, K=1.0$

Fig. 19

Chemical reaction effect on $f(\eta )$ when $\beta _1=0.1, \beta _2=0.1, Sr=0.5, D_f=0.5$

Fig. 20

Chemical reaction effect on $\theta (\eta )$ when $\beta _1=0.1, \beta _2=0.1, Sr=0.5, D_f=0.5$

Fig. 21

Chemical reaction effect on $\phi (\eta )$ when $\beta _1=0.1, \beta _2=0.1, Sr=0.5, D_f=0.5$

Table 4 Convergence of HAM solutions for different order of approximations

Table 5 Comparison of exact solution ($Br=0$) with HAM solution

Table 6 Effects of skin friction coefficient, heat and mass transfer rates for various values of Sr, $D_f$, $\beta _1$, $\beta _2$ and K