New Analytical Study of the Effects Thermo-Diffusion, Diffusion-Thermo and Chemical Reaction of Viscous Fluid on Magneto Hydrodynamics Flow in Divergent and Convergent Channels
Abstract: In this paper, the magneto hydrodynamic (MHD) flow of viscous fluid in a channel with non-parallel plates is studied. The governing partial differential equation was transformed into a system of dimensionless non-similar coupled ordinary differential equation. The transformed conservations equations were solved by using new algorithm. Basically, this new algorithm depends mainly on the Taylor expansion application with the coefficients of power series resulting from integrating the order differential equation. Results obtained from new algorithm are compared with the results of numerical Range-Kutta fourth-order algorithm with help of the shooting algorithm. The comparison revealed that the resulting solutions were excellent agreement. Thermo-diffusion and diffusion-thermo effects were investigated to analyze the behavior of temperature and concentration profile. Also the influences of the first order chemical reaction and the rate of mass and heat transfer were studied. The computed analytical solution result for the velocity, temperature and concentration distribution with the effect of various important dimensionless parameters was analyzed and discussed graphically.

1. Introduction

2. Mathematical Formulation

Consider the flow of an incompressible fluid due to source or sink that is located at the intersection of two rigid plane walls angled $2\alpha$ apart. Radial and symmetric nature of the flow is taken into consideration. Induced magnetic field is ignored and an applied magnetic field is considered that is applied across the flow direction. Under the aforesaid assumptions velocity field takes the form $V=\left[{u}_{r},0,0\right]$, where ${u}_{r}$ is a function of both r and $\theta$. Soret and Dufour effects are also considered that are incorporated in energy and concentration equations respectively. The fluid is also assumed to be chemically reacting. Also, the temperature and concentration are also the function of both r and $\theta$. While the results of angle opening $\alpha$ on concentration profile show that the increase in angle gives a decrease in concentration profile. The governing equations for mass, motion, energy, and mass transfer in polar coordinates under imposed assumptions become   (Figure 1).

$\frac{1}{r}\frac{\partial }{\partial r}\left(r{u}_{r}\right)=0,$ (1)

${u}_{r}\frac{\partial {u}_{r}}{\partial r}=-\frac{1}{\rho }\frac{\partial p}{\partial r}+\upsilon \left[\frac{{\partial }^{2}{u}_{r}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {u}_{r}}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}{u}_{r}}{\partial {\theta }^{2}}-\frac{{u}_{r}}{{r}^{2}}\right]-\frac{\sigma {B}_{0}^{2}}{\rho }{u}_{r},$ (2)

$-\frac{1}{\rho r}\frac{\partial p}{\partial \theta }+\frac{2\upsilon }{{r}^{2}}\frac{\partial {u}_{r}}{\partial \theta }=0,$ (3)

$\begin{array}{c}\rho {c}_{p}{u}_{r}\frac{\partial T}{\partial r}=k\left[\frac{{\partial }^{2}T}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}T}{\partial {\theta }^{2}}\right]+\mu \left[4{\left(\frac{\partial {u}_{r}}{\partial r}\right)}^{2}+{\left(\frac{\partial {u}_{r}}{\partial \theta }\right)}^{2}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{D{K}_{T}}{{C}_{s}}\frac{{\partial }^{2}C}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial C}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}C}{\partial {\theta }^{2}}\end{array}$ (4)

${u}_{r}\frac{\partial C}{\partial r}=D\left[\frac{{\partial }^{2}C}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial C}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}C}{\partial {\theta }^{2}}\right]+\frac{D{K}_{T}}{{T}_{m}}\left[\frac{{\partial }^{2}T}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}T}{\partial {\theta }^{2}}\right]$ (5)

the boundary conditions are,

${u}_{r}=U,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial {u}_{r}}{\partial \theta }=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial T}{\partial \theta }=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial C}{\partial \theta }=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}\text{ }\theta =\alpha ,$

Figure 1. Schematic diagram of the flow problem.

${u}_{r}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}T={T}_{m},\text{\hspace{0.17em}}\text{\hspace{0.17em}}C={C}_{w},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}\text{ }\theta =\alpha ,$ (6)

where p is the fluid pressure, $\upsilon =\frac{\mu }{\rho }$ is kinematic viscosity, D in order are the

specific heat and coefficient of mass diffusivity. $K,{K}_{T},D$ correspondingly are the thermal conductivity, thermal-diffusion ratio and coefficient of mass diffusivity. Further ${C}_{s},{T}_{w},{T}_{m},{C}_{w},{K}_{1}$ represent the concentration susceptibility, temperature at wall, mean fluid temperature, concentration at the wall and the chemical reaction constant respectively. From the continuity Equation (1), we can write

$F\left(\theta \right)=r{u}_{r}\left(r,\theta \right),$ (7)

with the use of dimensionless parameters award 

$f\left(\eta \right)=\frac{F\left(\theta \right)}{rU},\text{ }\eta =\frac{\theta }{\alpha },\text{ }\beta \left(\eta \right)=\frac{T}{{T}_{w}},\text{ }\varphi \left(\eta \right)=\frac{C}{{C}_{w}}.$ (8)

Eliminating p from Equations (1) and (2) using Equations (7) and (8), we get a system of nonlinear ordinary differential equation for the normalized velocity profile $f\left(\eta \right)$, temperature profile $\beta \left(\eta \right)$ and concentration profile $\varphi \left(\eta \right)$

${f}^{‴}\left(\eta \right)+2\alpha Ref\left(\eta \right){f}^{\prime }\left(\eta \right)+\left(4-Ha\right){\alpha }^{2}{f}^{\prime }\left(\eta \right)=0,$

${\beta }^{″}\left(\eta \right)+{E}_{c}{P}_{r}\left[4{\alpha }^{2}{f}^{2}\left(\eta \right)+{\left({f}^{\prime }\left(\eta \right)\right)}^{2}\right]+{D}_{f}{P}_{r}{\varphi }^{″}\left(\eta \right)=0,$

${\varphi }^{″}\left(\eta \right)+{S}_{c}{S}_{r}{\beta }^{″}-{S}_{c}\gamma {\alpha }^{2}\varphi \left(\eta \right)=0,$ (9)

$f\left(0\right)=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{f}^{\prime }\left(0\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(1\right)=0,$

${\beta }^{\prime }\left(0\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta \left(1\right)=1,$

$\varphi \left(0\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi \left(1\right)=1,$ (10)

where,

$Re=\frac{Ur\alpha }{\mu }\left\{\begin{array}{l}\text{divergent}\text{\hspace{0.17em}}\text{channel}:\alpha >0,\text{\hspace{0.17em}}U>0\\ \text{convergent}\text{\hspace{0.17em}}\text{channel}:\alpha <0,\text{\hspace{0.17em}}U<0\end{array}$, $Ha=\sqrt{\frac{\sigma {B}_{0}^{2}}{\mu }}$, ${P}_{r}=\frac{\mu {c}_{p}}{K}$.

${E}_{c}=\frac{{U}^{2}}{{c}_{p}{T}_{w}}$, ${D}_{f}=\frac{D{K}_{T}{C}_{w}}{\upsilon {c}_{p}{C}_{s}{T}_{w}}$, ${S}_{c}=\frac{\upsilon }{D}$, ${S}_{r}=\frac{D{K}_{T}{T}_{w}}{\upsilon {T}_{w}{C}_{w}}$, represent Reynolds, Hartmman, Prandtl, Eckert, Dufour, Schmidt and Soret number respectively while $\gamma =\frac{{K}_{1}}{\upsilon }$ is the first order chemical reaction parameter. The local Nusselt and Sherwood numbers are define by

$Nu=\frac{{{q}_{w}|}_{\eta =1}}{K{T}_{w}}=-\frac{1}{\alpha }{\beta }^{\prime }\left(1\right),$ (11)

$Sh=\frac{{{M}_{w}|}_{\eta =1}}{D{C}_{w}}=-\frac{1}{\alpha }{\varphi }^{\prime }\left(1\right).$ (12)

3. Description of the New Algorithm

This section describes how to obtain a new scheme to calculate the coefficients of the power series solution resulting from solving nonlinear ordinary differential equations to find analytical-approximate solution. These coefficients are important basis to construct the solution formula, therefore they can be computed recursively by differentiation ways. To illustrative the computation and operations for these coefficients and derivation the new scheme, we summarized the detail a new outlook in the following steps.

Step (1): Consider the non-linear differential equation as follows:

$H\left(f\left(\eta \right),{f}^{\prime }\left(\eta \right),{f}^{″}\left(\eta \right),{f}^{‴}\left(\eta \right),\cdots ,{f}^{\left(n-1\right)}\left(\eta \right),{f}^{\left(n\right)}\left(\eta \right)\right)=0,$ (13)

integrating Equation (13) with respect to $\eta$ on $\left[0,\eta \right]$ yield

$f\left(\eta \right)=f\left(0\right)+{f}^{\prime }\left(0\right)\eta +{f}^{″}\left(0\right)\frac{{\eta }^{2}}{2!}+\cdots +{f}^{\left(n-1\right)}\left(0\right)\frac{{\eta }^{n-1}}{\left(n-1\right)!}+{L}^{-1}G\left[f\left(\eta \right)\right],$ (14)

where,

$G\left[f\left(\eta \right)\right]=H\left(f\left(\eta \right),{f}^{\prime }\left(\eta \right),{f}^{″}\left(\eta \right),\cdots ,{f}^{\left(n-1\right)}\left(\eta \right)\right),{L}^{-1}={\int }_{0}^{\eta }{\int }_{0}^{\eta }\cdots {\int }_{0}^{\eta }{\left(\text{d}\eta \right)}^{n}.$ (15)

Step (2): We take Taylor series expansion of the function $G\left[f\left(\eta \right)\right]$ about $\eta ={\eta }_{0}$ as follows

$G\left[f\left(\eta \right)\right]=\underset{n=0}{\overset{\infty }{\sum }}\frac{{\left(\Delta \eta \right)}^{n}}{n!}\frac{{\text{d}}^{n}G\left({f}_{0}\left(\eta \right)\right)}{\text{d}{\eta }^{n}},$ (16)

rewriting the Equation (16)

$\begin{array}{c}G\left[f\left(\eta \right)\right]=G\left[{f}_{0}\left(\eta \right)\right]+\frac{\Delta \eta }{1!}{G}^{\prime }\left[{f}_{0}\left(\eta \right)\right]+\frac{{\left(\Delta \eta \right)}^{2}}{2!}{G}^{″}\left[{f}_{0}\left(\eta \right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{\left(\Delta \eta \right)}^{3}}{3!}{G}^{‴}\left[{f}_{0}\left(\eta \right)\right]+\cdots \end{array}$ (17)

Now, we assume that $\Delta \eta =\mathrm{max}\left\{\eta ,{\eta }_{0}\right\}$ and substituting Equation (17) in Equation (14), we obtain

$f\left(\eta \right)={f}_{0}+{f}_{1}+{f}_{2}+{f}_{3}+{f}_{4}+\cdots ,$ (18)

where,

${f}_{0}=f\left(0\right)+{f}^{\prime }\left(0\right)\eta +{f}^{″}\left(0\right)\frac{{\eta }^{2}}{2!}+\cdots +{f}^{\left(n-1\right)}\left(0\right)\frac{{\eta }^{\left(n-1\right)}}{\left(n-1\right)!},\text{ }{f}_{1}={L}^{-1}G\left[{f}_{0}\left(\eta \right)\right],$

${f}_{2}={L}^{-1}\frac{\mathrm{max}\left\{\eta ,{\eta }_{0}\right\}}{1!}{G}^{\prime }\left[{f}_{0}\left(\eta \right)\right],\text{ }{f}_{3}={L}^{-1}\frac{{\left(\mathrm{max}\left\{\eta ,{\eta }_{0}\right\}\right)}^{2}}{2!}{G}^{″}\left[{f}_{0}\left(\eta \right)\right],$

${f}_{4}={L}^{-1}\frac{{\left(\mathrm{max}\left\{\eta ,{\eta }_{0}\right\}\right)}^{3}}{3!}{G}^{‴}\left[{f}_{0}\left(\eta \right)\right]\cdots$ (19)

Step (3): We focus on computing the derivatives of G with respect to $\eta$ which is the crucial part of the proposed method. Let start calculating

$G\left[f\left(\eta \right)\right],{G}^{\prime }\left[f\left(\eta \right)\right],{G}^{″}\left[f\left(\eta \right)\right],{G}^{‴}\left[f\left(\eta \right)\right],\cdots$.

$G\left[f\left(\eta \right)\right]=H\left(f\left(\eta \right),{f}^{\prime }\left(\eta \right),{f}^{″}\left(\eta \right),{f}^{‴}\left(\eta \right),{f}^{‴}\text{'}\left(\eta \right),\cdots ,{f}^{\left(n-1\right)}\left(\eta \right)\right),$ (20)

${G}^{\prime }\left[f\left(\eta \right)\right]=\frac{\text{d}G\left[f\left(\eta \right)\right]}{\text{d}\eta }={G}_{1f}\cdot {f}_{\eta }+{G}_{1{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }+\cdots +{G}_{{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)},$ (21)

$\begin{array}{l}{G}^{″}\left[f\left(\eta \right)\right]=\frac{{\text{d}}^{2}G\left[f\left(\eta \right)\right]}{\text{d}{z}^{2}}\\ ={G}_{ff}\cdot {\left({f}_{\eta }\right)}^{2}+{G}_{f{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }{f}_{\eta }+{G}_{f{f}^{″}}\cdot {f}_{\eta }{\left({f}_{\eta }\right)}^{\prime \text{​}\prime }+\cdots \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{f{f}^{\left(n-1\right)}}\cdot \left({f}_{\eta }\right){\left({f}_{\eta }\right)}^{\left(n-1\right)}+{G}_{f}\cdot {f}_{\eta \eta }+{G}_{{f}^{\prime }f}\cdot {\left({f}_{\eta }\right)}^{\prime }\cdot {f}_{\eta }+{G}_{{f}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\cdots +{G}_{{f}^{\prime }{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{\prime }{\left({f}_{\eta }\right)}^{\left(n-1\right)}+{G}_{{f}^{\prime }}\cdot {\left({f}_{\eta \eta }\right)}^{\prime }+{G}_{{f}^{″}f}\cdot {\left({f}_{\eta }\right)}^{\prime \text{​}\prime }\cdot {f}_{\eta }\end{array}$

$\begin{array}{l}\text{ }+{G}_{{{f}^{\prime }}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }{\left({f}_{\eta }\right)}^{\prime \text{​}\prime }+{G}_{{f}^{″}{f}^{″}}\cdot {\left({f}_{\eta }\right)}^{\prime \text{​}\prime }{}^{2}+{G}_{{f}^{″}{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{\prime \text{​}\prime }{\left({f}_{\eta }\right)}^{\left(n-1\right)}\\ \text{ }+G{f}^{″}\cdot {\left({f}_{\eta \eta }\right)}^{\prime \text{​}\prime }+\cdots +{G}_{{f}^{\left(n-1\right)}f}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}\cdot {f}_{\eta }+{G}_{{f}^{\left(n-1\right)}{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}\cdot {\left({f}_{\eta }\right)}^{\prime }\\ \text{ }+\cdots +{G}_{{f}^{\left(n-1\right)}{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)2}+{G}_{{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta \eta }\right)}^{\left(n-1\right)}\end{array}$ (22)

$\begin{array}{l}{G}^{‴}\left[f\left(\eta \right)\right]=\frac{{\text{d}}^{3}G\left[f\left(\eta \right)\right]}{\text{d}{\eta }^{3}}\\ ={G}_{fff}\cdot {\left({f}_{\eta }\right)}^{3}+{G}_{ff{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{2}{\left({f}_{\eta }\right)}^{\prime }+\cdots +{G}_{ff{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{2}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{ff}\cdot 2\left({f}_{\eta }\right)\cdot {f}_{\eta \eta }+{G}_{f{f}^{\prime }f}\cdot {\left({f}_{\eta }\right)}^{\prime }{\left({f}_{\eta }\right)}^{2}+{G}_{f{f}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{2}\left({f}_{\eta }\right)+\cdots \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{f{f}^{\prime }{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{\prime }\left({f}_{\eta }\right)\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}+{G}_{f{f}^{\prime }}\cdot \left[{\left({f}_{\eta \eta }\right)}^{\prime }\cdot {f}_{\eta }+{\left({f}_{\eta }\right)}^{\prime }\cdot {f}_{\eta \eta }\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{f{f}^{″}f}\cdot {\left({f}_{\eta }\right)}^{\prime \text{​}\prime }{\left({f}_{\eta }\right)}^{2}+{G}_{f{f}^{″}{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime \text{​}\prime }\left({f}_{\eta }\right)\cdot {\left({f}_{\eta }\right)}^{\prime }+\cdots \end{array}$

$\begin{array}{l}\text{ }+{G}_{f{f}^{″}{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{\prime \text{​}\prime }\left({f}_{\eta }\right)\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}+{G}_{f{f}^{″}}\cdot \left[{f}_{\eta \eta }\cdot {\left({f}_{\eta }\right)}^{\prime \text{​}\prime }+{f}_{\eta }\cdot {\left({f}_{\eta \eta }\right)}^{\prime \text{​}\prime }\right]+\cdots \\ \text{ }+{G}_{f{f}^{\left(n-1\right)}f}\cdot {\left({f}_{\eta }\right)}^{2}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}+{G}_{f{f}^{\left(n-1\right)}{f}^{\prime }}\cdot \left({f}_{\eta }\right)\cdot {\left({f}_{\eta }\right)}^{\prime }\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}+\cdots \\ \text{ }+{G}_{f{f}^{\left(n-1\right)}{f}^{\left(n-1\right)}}\cdot \left({f}_{\eta }\right)\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)2}+{G}_{f{f}^{\left(n-1\right)}}\cdot \left[\left({f}_{\eta \eta }\right)\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}+\left({f}_{\eta }\right){\left({f}_{\eta \eta }\right)}^{\left(n-1\right)}\right]\end{array}$

$\begin{array}{l}\text{ }+{G}_{ff}\cdot {f}_{\eta \eta }\cdot \left({f}_{\eta }\right)+{G}_{f{f}^{\prime }}\cdot {f}_{\eta \eta }\cdot {\left({f}_{\eta }\right)}^{\prime }+\cdots +{G}_{f{f}^{\left(n-1\right)}}\cdot {f}_{\eta \eta }\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}+{G}_{\eta }\cdot {f}_{\eta \eta \eta }\\ \text{ }+{G}_{{f}^{\prime }ff}\cdot {\left({f}_{\eta }\right)}^{\prime }{\left({f}_{\eta }\right)}^{2}+{G}_{{f}^{\prime }{f}^{\prime }f}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{2}\left({f}_{\eta }\right)+\cdots +{G}_{{f}^{\prime }f{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{\prime }\left({f}_{\eta }\right)\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}\\ \text{ }+{G}_{{f}^{\prime }f}\cdot \left[{\left({f}_{\eta \eta }\right)}^{\prime }\cdot {f}_{\eta }+{\left({f}_{\eta }\right)}^{\prime }\cdot {f}_{\eta \eta }\right]+{G}_{{f}^{\prime }{f}^{\prime }f}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{2}\cdot {f}_{z}+{G}_{{f}^{\prime }{f}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{3}+\cdots \\ \text{ }+{G}_{{f}^{\prime }{f}^{\prime }{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{2}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}+{G}_{{f}^{\prime }{f}^{\prime }}\cdot 2{\left({f}_{\eta }\right)}^{\prime }\cdot {\left({f}_{\eta \eta }\right)}^{\prime }+\cdots \\ \text{ }+{G}_{{f}^{\left(n-1\right)}{f}^{\left(n-1\right)}f}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)2}\cdot {f}_{\eta }+{G}_{{f}^{\left(n-1\right)}{f}^{\left(n-1\right)}{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)2}\cdot {\left({f}_{\eta }\right)}^{\prime }+\cdots \end{array}$

$\begin{array}{l}\text{ }+{G}_{{f}^{\left(n-1\right)}{f}^{\left(n-1\right)}{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)3}+{G}_{{f}^{\left(n-1\right)}{f}^{\left(n-1\right)}}\cdot 2\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}\cdot {\left({f}_{\eta \eta }\right)}^{\left(n-1\right)}\\ \text{ }+{G}_{{f}^{\left(n-1\right)}f}\cdot {\left({f}_{\eta \eta }\right)}^{\left(n-1\right)}\cdot {f}_{\eta }+{G}_{{f}^{\left(n-1\right)}{f}^{\prime }}\cdot {\left({f}_{\eta \eta }\right)}^{\left(n-1\right)}\cdot {\left({f}_{\eta }\right)}^{\prime }+\cdots \\ \text{ }+{G}_{{f}^{\left(n-1\right)}{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta \eta }\right)}^{\left(n-1\right)}\cdot {\left({f}_{\eta }\right)}^{\left(n-1\right)}+{G}_{{f}^{\left(n-1\right)}}\cdot {\left({f}_{\eta \eta \eta }\right)}^{\left(n-1\right)}\end{array}$ (23)

$⋮$

The calculations are more complicated in the second and third derivatives because of the product rules. Consequently, the systematic structure on calculation is extremely important. Fortunately, due to the assumption that the operator G and the solution f are analytic functions, then the mixed derivatives are equivalence.

We note that the derivatives function to f unknown, so we suggest the following hypothesis

${f}_{\eta }={f}_{1}={L}^{-1}G\left[{f}_{0}\left(\eta \right)\right],\text{ }{f}_{\eta \eta }={f}_{2}={L}^{-1}\frac{\mathrm{max}\left\{\eta ,{\eta }_{0}\right\}}{1!}{G}^{\prime }\left[{f}_{0}\left(\eta \right)\right],$

${f}_{\eta \eta \eta }={f}_{3}={L}^{-1}\frac{{\left(\mathrm{max}\left\{\eta ,{\eta }_{0}\right\}\right)}^{2}}{2!}{G}^{″}\left[{f}_{0}\left(\eta \right)\right],$

${f}_{\eta \eta \eta \eta }={f}_{4}={L}^{-1}\frac{{\left(\mathrm{max}\left\{\eta ,{\eta }_{0}\right\}\right)}^{3}}{3!}{G}^{‴}\left[{f}_{0}\left(\eta \right)\right],$

${f}_{\eta \eta \eta \eta \eta }={f}_{5}={L}^{-1}\frac{{\left(\mathrm{max}\left\{\eta ,{\eta }_{0}\right\}\right)}^{4}}{4!}{G}^{‴}\text{'}\left[{f}_{0}\left(\eta \right)\right],\cdots$ (24)

Therefore Equations (20)-(23) are evaluated by

$G\left[{f}_{0}\left(\eta \right)\right]=H\left({f}_{0}\left(\eta \right),{{f}^{\prime }}_{0}\left(\eta \right),{{f}^{″}}_{0}\left(\eta \right),\cdots ,{f}_{0}^{\left(n-1\right)}\left(\eta \right)\right),$ (25)

${G}^{\prime }\left[{f}_{0}\left(\eta \right)\right]={G}_{{f}_{0}}\cdot {f}_{1}+{G}_{{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }+{G}_{{{f}^{″}}_{0}}\cdot {\left({f}_{1}\right)}^{\prime \text{​}\prime }+\cdots +{G}_{{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)},$ (26)

$\begin{array}{l}{G}^{″}\left[{f}_{0}\left(\eta \right)\right]\\ ={G}_{{f}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{2}+{G}_{{f}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{f}_{1}+{G}_{{f}_{0}{{f}^{″}}_{0}}\cdot {f}_{1}{\left({f}_{1}\right)}^{\prime \text{​}\prime }+\cdots +{G}_{{f}_{0}{f}_{0}^{\left(n-1\right)}}\cdot \left({f}_{1}\right){\left({f}_{1}\right)}^{\left(n-1\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{{f}_{0}}\cdot {f}_{2}+{G}_{{{f}^{\prime }}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }\cdot {f}_{1}+{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{2}+\cdots +{G}_{{{f}^{\prime }}_{0}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\prime }{\left({f}_{1}\right)}^{\left(n-1\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{{{f}^{\prime }}_{0}}\cdot {\left({f}_{2}\right)}^{\prime }+{G}_{{{f}^{″}}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{\prime \text{​}\prime }\cdot {f}_{1}+{G}_{{{f}^{″}}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{\left({f}_{1}\right)}^{\prime \text{​}\prime }+{G}_{{{f}^{″}}_{0}{{f}^{″}}_{0}}\cdot {\left({f}_{1}\right)}^{\prime \text{​}\prime }{}^{2}\end{array}$

$\begin{array}{l}\text{ }+\cdots +{G}_{{{f}^{″}}_{0}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\prime \text{​}\prime }{\left({f}_{1}\right)}^{\left(n-1\right)}+{G}_{{{f}^{″}}_{0}}\cdot {\left({f}_{2}\right)}^{\prime \text{​}\prime }+{G}_{{f}^{\left(n-1\right)}f}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}\cdot {f}_{1}\\ \text{ }+{G}_{{f}_{0}^{\left(n-1\right)}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}\cdot {\left({f}_{1}\right)}^{\prime }+\cdots +{G}_{{f}_{0}^{\left(n-1\right)}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)2}+\cdots +{G}_{{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{2}\right)}^{\left(n-1\right)},\end{array}$ (27)

$\begin{array}{l}{G}^{‴}\left[{f}_{0}\left(\eta \right)\right]\\ ={G}_{{f}_{0}{f}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{3}+{G}_{{f}_{0}{f}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{2}{\left({f}_{1}\right)}^{\prime }+\cdots +{G}_{{f}_{0}{f}_{0}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{2}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{{f}_{0}{f}_{0}}\cdot 2\left({f}_{1}\right)\cdot {f}_{2}+{G}_{{f}_{0}{{f}^{\prime }}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{\left({f}_{1}\right)}^{2}+{G}_{{f}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{2}\left({f}_{1}\right)+\cdots \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{{f}_{0}{{f}^{\prime }}_{0}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\prime }\left({f}_{1}\right)\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}+{G}_{{f}_{0}{{f}^{\prime }}_{0}}\cdot \left[{\left({f}_{2}\right)}^{\prime }\cdot {f}_{z}+{\left({f}_{1}\right)}^{\prime }\cdot {f}_{2}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{{f}_{0}{{f}^{″}}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{\prime \text{​}\prime }{\left({f}_{1}\right)}^{2}+{G}_{{f}_{0}{{f}^{″}}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime \text{​}\prime }\left({f}_{1}\right)\cdot {\left({f}_{1}\right)}^{\prime }+\cdots \end{array}$

$\begin{array}{l}\text{ }+{G}_{{f}_{0}{{f}^{″}}_{0}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\prime \text{​}\prime }\left({f}_{1}\right)\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}+{G}_{{f}_{0}{{f}^{″}}_{0}}\cdot \left[{f}_{2}\cdot {\left({f}_{1}\right)}^{\prime \text{​}\prime }+{f}_{1}\cdot {\left({f}_{2}\right)}^{\prime \text{​}\prime }\right]+\cdots \\ \text{ }+{G}_{{f}_{0}{f}_{0}^{\left(n-1\right)}{f}_{0}}\cdot {\left({f}_{1}\right)}^{2}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}+{G}_{{f}_{0}{f}_{0}^{\left(n-1\right)}{{f}^{\prime }}_{0}}\cdot \left({f}_{1}\right)\cdot {\left({f}_{1}\right)}^{\prime }\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}+\cdots \\ \text{ }+{G}_{{f}_{0}{f}_{0}^{\left(n-1\right)}{f}_{0}^{\left(n-1\right)}}\cdot \left({f}_{1}\right)+\cdots +{G}_{{f}_{0}{f}_{0}^{\left(n-1\right)}}\cdot {f}_{2}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}+{G}_{{f}_{0}}\cdot {f}_{3}+{G}_{{{f}^{\prime }}_{0}{f}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{\left({f}_{1}\right)}^{2}\\ \text{ }+{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{2}\left({f}_{1}\right)+\cdots +{G}_{{{f}^{\prime }}_{0}{f}_{0}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\prime }\left({f}_{1}\right)\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}\\ \text{ }+{G}_{{{f}^{\prime }}_{0}{f}_{0}}\cdot \left[{\left({f}_{2}\right)}^{\prime }\cdot {f}_{1}+{\left({f}_{1}\right)}^{\prime }\cdot {f}_{1}\right]+{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{2}\cdot {f}_{1}+{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{3}+\cdots \end{array}$

$\begin{array}{l}\text{ }+{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{2}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}+{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot 2{\left({f}_{1}\right)}^{\prime }{\left({f}_{2}\right)}^{\prime }+\cdots +{G}_{{f}_{0}^{\left(n-1\right)}{f}_{0}^{\left(n-1\right)}{f}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{3}\\ \text{ }+\cdots +{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{2}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}+{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot 2{\left({f}_{1}\right)}^{\prime }\cdot {\left({f}_{2}\right)}^{\prime }+\cdots \\ \text{ }+{G}_{{f}_{0}^{\left(n-1\right)}{f}_{0}^{\left(n-1\right)}{f}_{0}}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)2}\cdot {f}_{1}+{G}_{{f}_{0}^{\left(n-1\right)}{f}_{0}^{\left(n-1\right)}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)2}\cdot {\left({f}_{1}\right)}^{\prime }+\cdots \end{array}$

$\begin{array}{l}\text{ }+{G}_{{f}_{0}^{\left(n-1\right)}{f}_{0}^{\left(n-1\right)}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)3}+{G}_{{f}_{0}^{\left(n-1\right)}{f}_{0}^{\left(n-1\right)}}\cdot 2\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}\cdot {\left({f}_{2}\right)}^{\left(n-1\right)}\\ \text{ }+{G}_{{f}_{0}^{\left(n-1\right)}{f}_{0}}\cdot {\left({f}_{2}\right)}^{\left(n-1\right)}\cdot {f}_{1}+{G}_{{f}_{0}^{\left(n-1\right)}{{f}^{\prime }}_{0}}\cdot {\left({f}_{2}\right)}^{\left(n-1\right)}\cdot {\left({f}_{1}\right)}^{\prime }+\cdots \\ \text{ }+{G}_{{f}_{0}^{\left(n-1\right)}{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{2}\right)}^{\left(n-1\right)}\cdot {\left({f}_{1}\right)}^{\left(n-1\right)}+{G}_{{f}_{0}^{\left(n-1\right)}}\cdot {\left({f}_{3}\right)}^{\left(n-1\right)}\end{array}$ (28)

$\begin{array}{l}⋮\end{array}$

Step (4): Substituting Equations (25)-(28) in Equation (18) we get the required analytical-approximate solution for the Equation (13).

4. Application of the New Algorithm to the Magneto Hydrodynamic (MHD) Flow of Viscous Fluid in a Channel with Non-Parallel Plates

The new algorithm described in the previous section can be used as a powerful solver to the nonlinear differential Equations (9)-(10) and to find new an analytical-approximate solution. From step (1) we have

$f\left(\eta \right)=f\left(0\right)+{f}^{\prime }\left(0\right)\eta +{f}^{″}\left(0\right)\frac{{\eta }^{2}}{2!}+{L}_{1}^{-1}\left[-2\alpha Ref\left(\eta \right){f}^{\prime }\left(\eta \right)-\left(4-Ha\right){\alpha }^{2}{f}^{\prime }\left(\eta \right)\right],$

$\beta \left(\eta \right)=\beta \left(0\right)+{\beta }^{\prime }\left(0\right)\eta +{L}_{2}^{-1}\left[-{E}_{c}{P}_{r}\left[4{\alpha }^{2}{f}^{2}\left(\eta \right)-{\left({f}^{\prime }\left(\eta \right)\right)}^{2}\right]-{D}_{f}{P}_{r}{\varphi }^{″}\left(\eta \right)\right],$

$\varphi \left(\eta \right)=\varphi \left(0\right)+{\varphi }^{\prime }\left(0\right)\eta +{L}_{3}^{-1}\left[-{S}_{c}{S}_{r}{\beta }^{″}+{S}_{c}\gamma {\alpha }^{2}\varphi \left(\eta \right)\right],$ (29)

rewrite the Equation (29) as follows

$f\left(\eta \right)={A}_{1}+{A}_{2}\eta +{A}_{3}\frac{{\eta }^{2}}{2!}+{L}^{-1}{G}_{1}\left[f\left(\eta \right)\right],$

$\beta \left(\eta \right)={B}_{1}+{B}_{2}\eta +{L}^{-1}{G}_{2}\left[\beta \left(\eta \right)\right],$

$\varphi \left(\eta \right)={C}_{1}+{C}_{2}\eta +{L}^{-1}{G}_{3}\left[\varphi \left(\eta \right)\right],$ (30)

where,

${A}_{1}=f\left(0\right),\text{ }{A}_{2}={f}^{\prime }\left(0\right),\text{ }{A}_{3}={f}^{″}\left(0\right),$

${B}_{1}=\beta \left(0\right),\text{ }{B}_{2}={\beta }^{\prime }\left(0\right),\text{ }{C}_{1}=\varphi \left(0\right),\text{ }{C}_{2}={\varphi }^{\prime }\left(0\right),$

${G}_{1}\left[f\right]=-2\alpha Ref\left(\eta \right){f}^{\prime }\left(\eta \right)-\left(4-Ha\right){\alpha }^{2}{f}^{\prime }\left(\eta \right),$

${G}_{2}\left[\beta \right]=-{E}_{c}{P}_{r}\left[4{\alpha }^{2}{f}^{2}\left(\eta \right)-{\left({f}^{\prime }\left(\eta \right)\right)}^{2}\right]-{D}_{f}{P}_{r}{\varphi }^{″}\left(\eta \right),$

${G}_{3}\left[\varphi \right]=-{S}_{c}{S}_{r}{\beta }^{″}+{S}_{c}\gamma {\alpha }^{2}\varphi \left(\eta \right),$

and ${L}_{1}^{-1}\left(.\right)={\int }_{0}^{\eta }{\int }_{0}^{\eta }{\int }_{0}^{\eta }{\left(\text{d}\eta \right)}^{3},{L}_{2}^{-1}\left(.\right)={L}_{3}^{-1}\left(.\right)={\int }_{0}^{\eta }{\int }_{0}^{\eta }{\left(\text{d}\eta \right)}^{2}.$ (31)

From the boundary conditions the Equation (30) becomes

$f\left(\eta \right)=1+{A}_{3}\frac{{\eta }^{2}}{2!}+{L}^{-1}{G}_{1}\left[f\left(\eta \right)\right],$

$\beta \left(\eta \right)={B}_{1}+{L}^{-1}{G}_{2}\left[\beta \left(\eta \right)\right],$

$\varphi \left(\eta \right)={C}_{1}+{L}^{-1}{G}_{3}\left[\varphi \left(\eta \right)\right].$ (32)

From step (2) suppose that $\Delta \eta =\mathrm{max}\left\{1,0\right\}=1$, yield

${f}_{0}=1+{A}_{3}\frac{{\eta }^{2}}{2!}\text{ }{f}_{1}={L}_{1}^{-1}{G}_{1}\left[{f}_{0}\left(\eta \right)\right],\text{ }{f}_{2}={L}_{2}^{-1}{{G}^{\prime }}_{1}\left[{f}_{0}\left(\eta \right)\right],\cdots ,$

${\beta }_{0}={B}_{1},\text{ }{\beta }_{1}={L}_{2}^{-1}{G}_{2}\left[{\beta }_{0}\left(\eta \right)\right],\text{ }{\beta }_{2}={L}_{2}^{-1}{{G}^{\prime }}_{2}\left[{\beta }_{0}\left(\eta \right)\right],\cdots ,$

${f}_{0}={C}_{1},\text{ }{\varphi }_{1}={L}_{3}^{-1}{G}_{3}\left[{\varphi }_{0}\left(\eta \right)\right],\text{ }{\varphi }_{2}={L}_{3}^{-1}{{G}^{\prime }}_{3}\left[{\varphi }_{0}\left(\eta \right)\right],\cdots ,$ (33)

and the analytical-approximate solution are

$f\left(\eta \right)={f}_{0}+{f}_{1}+{f}_{2}+{f}_{3}+\cdots ,$

$\beta \left(\eta \right)={\beta }_{0}+{\beta }_{1}+{\beta }_{2}+{\beta }_{3}+\cdots ,$

$\varphi \left(\eta \right)={\varphi }_{0}+{\varphi }_{1}+{\varphi }_{2}+{\varphi }_{3}+,\cdots$ (34)

From step (3) yields

${G}_{1}\left[f\left(\eta \right)\right]=-2\alpha Ref\left(\eta \right){f}^{\prime }\left(\eta \right)-\left(4-Ha\right){\alpha }^{2}{f}^{\prime }\left(\eta \right),$

${G}_{2}\left[\beta \left(\eta \right)\right]=-{E}_{c}{P}_{r}\left[4{\alpha }^{2}{f}^{2}\left(\eta \right)-{\left({f}^{\prime }\left(\eta \right)\right)}^{2}\right]-{D}_{f}{P}_{r}{\varphi }^{″}\left(\eta \right),$

${G}_{3}\left[\varphi \left(\eta \right)\right]=-{S}_{c}{S}_{r}{\beta }^{″}+{S}_{c}\gamma {\alpha }^{2}\varphi \left(\eta \right),$ (35)

${{G}^{\prime }}_{1}\left[f\left(\eta \right)\right]=\frac{\text{d}{G}_{1}\left[f\left(\eta \right)\right]}{\text{d}\eta }={G}_{1f}\cdot {f}_{\eta }+{G}_{1{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime },$

${{G}^{\prime }}_{2}\left[\beta \left(\eta \right)\right]=\frac{\text{d}{G}_{2}\left[\beta \left(\eta \right)\right]}{\text{d}\eta }={G}_{2f}\cdot {f}_{\eta }+{G}_{2{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }+{G}_{2}{\varphi }^{″}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime },$

${{G}^{\prime }}_{3}\left[\varphi \left(\eta \right)\right]=\frac{\text{d}{G}_{3}\left[\varphi \left(\eta \right)\right]}{\text{d}\eta }={G}_{3{\beta }^{″}}\cdot {\left({\beta }_{\eta }\right)}^{\prime \text{​}\prime }+{G}_{3\varphi }\cdot \left({\varphi }_{\eta }\right),$ (36)

$\begin{array}{l}{{G}^{″}}_{1}\left[f\left(\eta \right)\right]=\frac{{\text{d}}^{2}{G}_{1}\left[f\left(\eta \right)\right]}{\text{d}{\eta }^{2}}\\ ={G}_{ff}\cdot {\left({f}_{\eta }\right)}^{2}+2\cdot {G}_{f{f}^{\prime }}\cdot {f}_{\eta }{\left({f}_{\eta }\right)}^{\prime }+{G}_{{f}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{2}+{G}_{f}\cdot {f}_{\eta \eta }+{G}_{{f}^{\prime }}\cdot {\left({f}_{\eta \eta }\right)}^{\prime }\end{array}$

$\begin{array}{l}{{G}^{″}}_{2}\left[\beta \left(\eta \right)\right]=\frac{{\text{d}}^{2}{G}_{2}\left[\beta \left(\eta \right)\right]}{\text{d}{\eta }^{2}}\\ ={G}_{ff}\cdot {\left({f}_{\eta }\right)}^{2}+2\cdot {G}_{f{f}^{\prime }}\cdot {f}_{\eta }{\left({f}_{\eta }\right)}^{\prime }+{G}_{{f}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{2}+{G}_{f}\cdot {f}_{\eta \eta }+{G}_{{f}^{\prime }}\cdot {\left({f}_{\eta \eta }\right)}^{\prime }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{G}_{2{\varphi }^{″}{\varphi }^{″}}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }{}^{2}+2\cdot {G}_{2{\varphi }^{″}f}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {f}_{\eta }+2\cdot {G}_{2{\varphi }^{″}{f}^{\prime }}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {{f}^{\prime }}_{\eta }+{G}_{2{\varphi }^{″}}\cdot {\left({\varphi }_{\eta \eta }\right)}^{\prime \text{​}\prime },\end{array}$

$\begin{array}{l}{{G}^{″}}_{3}\left[\varphi \left(\eta \right)\right]=\frac{{\text{d}}^{2}{G}_{3}\left[\varphi \left(\eta \right)\right]}{\text{d}{\eta }^{2}}\\ ={G}_{3{\beta }^{″}{\beta }^{″}}\cdot {\left({\beta }_{\eta }\right)}^{\prime \text{​}\prime }{}^{2}+{G}_{3\varphi \varphi }\cdot {\left({\varphi }_{\eta }\right)}^{2}+2{G}_{3{\beta }^{″}\varphi }\cdot {\left({\beta }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {\varphi }_{\eta }+{G}_{3\varphi }\cdot \left({\varphi }_{\eta \eta }\right)+{G}_{3{\beta }^{″}}\cdot {\left({\beta }_{\eta \eta }\right)}^{\prime \text{​}\prime },\end{array}$ (37)

$\begin{array}{l}{{G}^{‴}}_{1}\left[f\left(\eta \right)\right]=\frac{{\text{d}}^{3}{G}_{1}\left[f\left(\eta \right)\right]}{\text{d}{\eta }^{3}}\\ ={G}_{1fff}{\left({f}_{\eta }\right)}^{3}+3\cdot {G}_{1ff{f}^{\prime }}{\left({f}_{\eta }\right)}^{2}\cdot {\left({f}_{\eta }\right)}^{\prime }+3\cdot {G}_{1f{f}^{\prime }{f}^{\prime }}\cdot \left({f}_{\eta }\right){\left({f}_{\eta }\right)}^{\prime }{}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{1{f}^{\prime }{f}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{3}+3\cdot {G}_{1ff}\cdot {f}_{\eta \eta }\cdot {f}_{\eta }+3\cdot {G}_{1f{f}^{\prime }}\cdot {f}_{\eta \eta }\cdot {\left({f}_{\eta }\right)}^{\prime }+{G}_{1f}\cdot {f}_{\eta \eta \eta }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3\cdot {G}_{1{f}^{\prime }f}\cdot {\left({f}_{\eta \eta }\right)}^{\prime }\left({f}_{\eta }\right)+3\cdot {G}_{1{f}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta \eta }\right)}^{\prime }{\left({f}_{\eta }\right)}^{\prime }+{G}_{1{f}^{\prime }}\cdot {\left({f}_{\eta \eta \eta }\right)}^{\prime },\end{array}$

$\begin{array}{l}{{G}^{‴}}_{2}\left[\beta \left(\eta \right)\right]=\frac{{\text{d}}^{3}{G}_{1}\left[\beta \left(\eta \right)\right]}{\text{d}{\eta }^{3}}\\ ={G}_{2fff}{\left({f}_{\eta }\right)}^{3}+3\cdot {G}_{2ff{f}^{\prime }}{\left({f}_{\eta }\right)}^{2}\cdot {\left({f}_{\eta }\right)}^{\prime }+3\cdot {G}_{2f{f}^{\prime }{f}^{\prime }}\cdot \left({f}_{\eta }\right){\left({f}_{\eta }\right)}^{\prime }{}^{2}+{G}_{2{f}^{\prime }{f}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3\cdot {G}_{2ff}\cdot {f}_{\eta \eta }\cdot {f}_{\eta }+3\cdot {G}_{2f{f}^{\prime }}\cdot {f}_{\eta \eta }\cdot {\left({f}_{\eta }\right)}^{\prime }+{G}_{2f}\cdot {f}_{\eta \eta \eta }+3\cdot {G}_{2{f}^{\prime }f}\cdot {\left({f}_{\eta \eta }\right)}^{\prime }\left({f}_{\eta }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3\cdot {G}_{2{f}^{\prime }{f}^{\prime }}\cdot {\left({f}_{\eta \eta }\right)}^{\prime }{\left({f}_{\eta }\right)}^{\prime }+{G}_{2{f}^{\prime }}\cdot {\left({f}_{\eta \eta \eta }\right)}^{\prime }+{G}_{2{\varphi }^{″}{\varphi }^{″}{\varphi }^{″}}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }{}^{2}\end{array}$

$\begin{array}{l}\text{ }+3\cdot {G}_{2{\varphi }^{″}{\varphi }^{″}f}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }{}^{2}\cdot \left({f}_{\eta }\right)+3\cdot {G}_{2{\varphi }^{″}{\varphi }^{″}{f}^{\prime }}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }{}^{2}\cdot {\left({f}_{\eta }\right)}^{\prime }+3\cdot {G}_{2{\varphi }^{″}{\varphi }^{″}}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {\left({\varphi }_{\eta \eta }\right)}^{\prime \text{​}\prime }\\ \text{ }+3\cdot {G}_{{\varphi }^{″}f}\cdot {\left({\varphi }_{\eta \eta }\right)}^{\prime \text{​}\prime }\cdot {f}_{\eta }+3\cdot {G}_{{\varphi }^{″}{f}^{\prime }}\cdot {\left({\varphi }_{\eta \eta }\right)}^{\prime \text{​}\prime }\cdot {\left({f}_{\eta }\right)}^{\prime }+4\cdot {G}_{2{\varphi }^{″}f{f}^{\prime }}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {f}_{\eta }\cdot {\left({f}_{\eta }\right)}^{\prime }\\ \text{ }+2\cdot {G}_{{\varphi }^{″}ff}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {\left({f}_{\eta }\right)}^{2}+2\cdot {G}_{{\varphi }^{″}{f}^{\prime }{f}^{\prime }}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {\left({f}_{\eta }\right)}^{\prime }{}^{2}+2\cdot {G}_{2{\varphi }^{″}f}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {f}_{\eta \eta }\\ \text{ }+2\cdot {G}_{2{\varphi }^{″}{f}^{\prime }}\cdot {\left({\varphi }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {\left({f}_{\eta \eta }\right)}^{\prime }+{G}_{2{\varphi }^{″}}\cdot {\left({\varphi }_{\eta \eta \eta }\right)}^{\prime \text{​}\prime },\end{array}$

$\begin{array}{l}{{G}^{‴}}_{3}\left[\varphi \left(\eta \right)\right]=\frac{{\text{d}}^{3}{G}_{3}\left[\varphi \left(\eta \right)\right]}{\text{d}{\eta }^{3}}\\ ={G}_{3{\beta }^{″}{\beta }^{″}{\beta }^{″}}\cdot {\left({\beta }_{\eta }\right)}^{\prime \text{​}\prime }{}^{3}+3\cdot {G}_{3{\beta }^{″}{\beta }^{″}\varphi }\cdot {\left({\beta }_{\eta }\right)}^{\prime \text{​}\prime }{}^{2}{\varphi }_{\eta }+3\cdot {G}_{3{\beta }^{″}{\beta }^{″}}\cdot {\left({\beta }_{\eta }\right)}^{\prime \text{​}\prime }\cdot {\left({\beta }_{\eta \eta }\right)}^{\prime \text{​}\prime }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{3\varphi \varphi \varphi }\cdot {\left({\varphi }_{\eta }\right)}^{3}+3\cdot {G}_{3\varphi \varphi {\beta }^{″}}\cdot {\left({\varphi }_{\eta }\right)}^{2}\cdot {{\beta }^{″}}_{\eta }+3\cdot {G}_{3\varphi \varphi }\cdot \left({\varphi }_{\eta }\right){\varphi }_{\eta \eta }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3\cdot {G}_{3\varphi \beta }\cdot \left({\varphi }_{\eta \eta }\right)\cdot {\beta }_{\eta }+{G}_{3\varphi }\cdot \left({\varphi }_{\eta \eta \eta }\right)+3\cdot {G}_{3{\beta }^{″}\varphi }\cdot {\left({\beta }_{\eta \eta }\right)}^{\prime \text{​}\prime }\cdot {\varphi }_{\eta }+{G}_{3{\beta }^{″}}\cdot {\left({\beta }_{\eta \eta \eta }\right)}^{\prime \text{​}\prime }.\end{array}$ (38)

$\begin{array}{l}⋮\end{array}$

We note that the derivatives of f with respect $\eta$ that are given in (19), can be computing by Equations (35)-(38) as

${G}_{1}\left[{f}_{0}\left(\eta \right)\right]=-2\alpha Re{f}_{0}\left(\eta \right){{f}^{\prime }}_{0}\left(\eta \right)-\left(4-Ha\right){\alpha }^{2}{{f}^{\prime }}_{0}\left(\eta \right),$

${G}_{2}\left[{\beta }_{0}\left(\eta \right)\right]=-{E}_{c}{P}_{r}\left[4{\alpha }^{2}{f}_{0}^{2}\left(\eta \right)-{\left({{f}^{\prime }}_{0}\left(\eta \right)\right)}^{2}\right]-{D}_{f}{P}_{r}{{\varphi }^{″}}_{0}\left(\eta \right),$

${G}_{3}\left[{\varphi }_{0}\left(\eta \right)\right]=-{S}_{c}{S}_{r}{{\beta }^{″}}_{0}+{S}_{c}\gamma {\alpha }^{2}{\varphi }_{0}\left(\eta \right),$ (39)

${{G}^{\prime }}_{1}\left[{f}_{0}\left(\eta \right)\right]={G}_{1{f}_{0}}\cdot {f}_{1}+{G}_{1{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime },$

${{G}^{\prime }}_{2}\left[{\beta }_{0}\left(\eta \right)\right]={G}_{2{f}_{0}}\cdot {f}_{1}+{G}_{2{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }+{G}_{2{{\varphi }^{″}}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime },$

${{G}^{\prime }}_{3}\left[{\varphi }_{0}\left(\eta \right)\right]={G}_{3{{\beta }^{″}}_{0}}\cdot {\left({\beta }_{1}\right)}^{\prime \text{​}\prime }+{G}_{3{\varphi }_{0}}\cdot \left({\varphi }_{1}\right),$ (40)

$\begin{array}{c}{{G}^{″}}_{1}\left[{f}_{0}\left(\eta \right)\right]={G}_{{f}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{2}+2\cdot {G}_{{f}_{0}{{f}^{\prime }}_{0}}\cdot {f}_{1}{\left({f}_{1}\right)}^{\prime }+{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{{f}_{0}}\cdot {f}_{2}+{G}_{{{f}^{\prime }}_{0}}.{\left({f}_{2}\right)}^{\prime }\end{array}$

$\begin{array}{c}{{G}^{″}}_{2}\left[{\beta }_{0}\left(\eta \right)\right]={G}_{{f}_{0}{f}_{0}}\cdot {\left({f}_{1}\right)}^{2}+2\cdot {G}_{{f}_{0}{{f}^{\prime }}_{0}}\cdot {f}_{1}{\left({f}_{1}\right)}^{\prime }+{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{1}\right)}^{\prime }{}^{2}+{G}_{{f}_{0}}\cdot {f}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{{{f}^{\prime }}_{0}}\cdot {\left({f}_{2}\right)}^{\prime }+{G}_{2{{\varphi }^{″}}_{0}{{\varphi }^{″}}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }{}^{2}+2\cdot {G}_{2{{\varphi }^{″}}_{0}{f}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }\cdot {f}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\cdot {G}_{2{{\varphi }^{″}}_{0}{{f}^{\prime }}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }\cdot {\left({f}_{1}\right)}^{\prime }+{G}_{2{{\varphi }^{″}}_{0}}\cdot {\left({\varphi }_{2}\right)}^{\prime \text{​}\prime }\end{array}$

$\begin{array}{c}{{G}^{″}}_{3}\left[{\varphi }_{0}\left(\eta \right)\right]={G}_{3{{\beta }^{″}}_{0}{{\beta }^{″}}_{0}}\cdot {\left({\beta }_{1}\right)}^{\prime \text{​}\prime }{}^{2}+{G}_{3{\varphi }_{0}{\varphi }_{0}}\cdot {\left({\varphi }_{1}\right)}^{2}+2{G}_{3{{\beta }^{″}}_{0}{\varphi }_{0}}\cdot {\left({\beta }_{1}\right)}^{\prime \text{​}\prime }\cdot {\varphi }_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{3{\varphi }_{0}}\cdot \left({\varphi }_{2}\right)+{G}_{3{{\beta }^{″}}_{0}}\cdot {\left({\beta }_{2}\right)}^{\prime \text{​}\prime }\end{array}$ (41)

$\begin{array}{c}{{G}^{‴}}_{1}\left[{f}_{0}\left(\eta \right)\right]={G}_{1{f}_{0}{f}_{0}{f}_{0}}{\left({f}_{1}\right)}^{3}+3\cdot {G}_{1{f}_{0}{f}_{0}{{f}^{\prime }}_{0}}{\left({f}_{1}\right)}^{2}.{\left({f}_{1}\right)}^{\prime }+3\cdot {G}_{1{f}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot \left({f}_{1}\right){\left({f}_{1}\right)}^{\prime }{}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{1{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{3}\right)}^{\prime }{}^{3}+3\cdot {G}_{1{f}_{0}{f}_{0}}\cdot {f}_{2}\cdot {f}_{1}+3\cdot {G}_{1{f}_{0}{{f}^{\prime }}_{0}}\cdot {f}_{2}\cdot {\left({f}_{1}\right)}^{\prime }+{G}_{{f}_{0}}\cdot {f}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3\cdot {G}_{1{{f}^{\prime }}_{0}{f}_{0}}\cdot {\left({f}_{2}\right)}^{\prime }\left({f}_{1}\right)+3\cdot {G}_{1{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{2}\right)}^{\prime }{\left({f}_{1}\right)}^{\prime }+{G}_{1{{f}^{\prime }}_{0}}\cdot {\left({f}_{3}\right)}^{\prime },\end{array}$

$\begin{array}{c}{{G}^{‴}}_{2}\left[{\beta }_{0}\left(\eta \right)\right]={G}_{2{f}_{0}{f}_{0}{f}_{0}}{\left({f}_{1}\right)}^{3}+3\cdot {G}_{2{f}_{0}{f}_{0}{{f}^{\prime }}_{0}}{\left({f}_{1}\right)}^{2}\cdot {\left({f}_{1}\right)}^{\prime }+3\cdot {G}_{2{f}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot \left({f}_{1}\right){\left({f}_{1}\right)}^{\prime }{}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{2{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}.{\left({f}_{1}\right)}^{\prime }{}^{3}+3\cdot {G}_{2{f}_{0}{f}_{0}}\cdot {f}_{2}\cdot {f}_{1}+3\cdot {G}_{2{f}_{0}{{f}^{\prime }}_{0}}\cdot {f}_{2}\cdot {\left({f}_{1}\right)}^{\prime }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{2f}\cdot {f}_{3}+3\cdot {G}_{2{{f}^{\prime }}_{0}{f}_{0}}\cdot {\left({f}_{2}\right)}^{\prime }\left({f}_{1}\right)+3\cdot {G}_{2{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot {\left({f}_{2}\right)}^{\prime }{\left({f}_{1}\right)}^{\prime }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{2{{f}^{\prime }}_{0}}\cdot {\left({f}_{3}\right)}^{\prime }+{G}_{2{{\varphi }^{″}}_{0}{{\varphi }^{″}}_{0}{{\varphi }^{″}}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }{}^{2}+3\cdot {G}_{2{{\varphi }^{″}}_{0}{{\varphi }^{″}}_{0}{f}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }{}^{2}\cdot \left({f}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3\cdot {G}_{2{{\varphi }^{″}}_{0}{{\varphi }^{″}}_{0}{{f}^{\prime }}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }{}^{2}\cdot {\left({f}_{1}\right)}^{\prime }+3\cdot {G}_{2{{\varphi }^{″}}_{0}{{\varphi }^{″}}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }\cdot {\left({\varphi }_{2}\right)}^{\prime \text{​}\prime }\end{array}$

$\begin{array}{c}\text{ }+3\cdot {G}_{{{\varphi }^{″}}_{0}{f}_{0}}\cdot {\left({\varphi }_{2}\right)}^{\prime \text{​}\prime }\cdot {f}_{1}+3\cdot {G}_{{{\varphi }^{″}}_{0}{{f}^{\prime }}_{0}}\cdot {\left({\varphi }_{2}\right)}^{\prime \text{​}\prime }\cdot {\left({f}_{1}\right)}^{\prime }\\ \text{ }+4\cdot {G}_{2{{\varphi }^{″}}_{0}{f}_{0}{{f}^{\prime }}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }\cdot {f}_{1}\cdot {\left({f}_{1}\right)}^{\prime }+2\cdot {G}_{{{\varphi }^{″}}_{0}{f}_{0}{f}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }\cdot {\left({f}_{1}\right)}^{2}\\ \text{ }+2\cdot {G}_{{{\varphi }^{″}}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }\cdot {\left({f}_{1}\right)}^{\prime }{}^{2}+2\cdot {G}_{2{{\varphi }^{″}}_{0}{f}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }\cdot {f}_{2}\\ \text{ }+2\cdot {G}_{2{{\varphi }^{″}}_{0}{{f}^{\prime }}_{0}}\cdot {\left({\varphi }_{1}\right)}^{\prime \text{​}\prime }\cdot {\left({f}_{2}\right)}^{\prime }+{G}_{2{{\varphi }^{″}}_{0}}\cdot {\left({\varphi }_{3}\right)}^{\prime \text{​}\prime },\end{array}$

$\begin{array}{c}{{G}^{‴}}_{3}\left[{\varphi }_{0}\left(\eta \right)\right]={G}_{3{{\beta }^{″}}_{0}{{\beta }^{″}}_{0}{{\beta }^{″}}_{0}}\cdot {\left({\beta }_{1}\right)}^{\prime \text{​}\prime }{}^{3}+3\cdot {G}_{3{{\beta }^{″}}_{0}{{\beta }^{″}}_{0}{\varphi }_{0}}\cdot {\left({\beta }_{1}\right)}^{\prime \text{​}\prime }{}^{2}{\varphi }_{1}+3\cdot {G}_{3{{\beta }^{″}}_{0}{{\beta }^{″}}_{0}}\cdot {\left({\beta }_{1}\right)}^{\prime \text{​}\prime }\cdot {\left({\beta }_{2}\right)}^{\prime \text{​}\prime }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{G}_{3{\varphi }_{0}{\varphi }_{0}{\varphi }_{0}}\cdot {\left({\varphi }_{1}\right)}^{3}+3\cdot {G}_{3{\varphi }_{0}{\varphi }_{0}{{\beta }^{″}}_{0}}\cdot {\left({\varphi }_{1}\right)}^{2}\cdot {{\beta }^{″}}_{1}+3\cdot {G}_{3{\varphi }_{0}{\varphi }_{0}}\cdot \left({\varphi }_{1}\right){\varphi }_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3\cdot {G}_{3{\varphi }_{0}{\beta }_{0}}\cdot \left({\varphi }_{2}\right)\cdot {\beta }_{1}+{G}_{3{\varphi }_{0}}\cdot {\varphi }_{3}+3\cdot {G}_{3{{\beta }^{″}}_{0}{\varphi }_{0}}\cdot {\left({\beta }_{2}\right)}^{\prime \text{​}\prime }\cdot {\varphi }_{1}+{G}_{3{{\beta }^{″}}_{0}}\cdot {\left({\beta }_{3}\right)}^{\prime \text{​}\prime }.\end{array}$ (42)

$⋮$

The extraction of the first derivatives of G can be represented as:

${G}_{1{f}_{0}}=-2\alpha Re{{f}^{\prime }}_{0}\left(\eta \right),\text{ }{G}_{{f}_{0}{f}_{0}}=0,\text{ }{G}_{{f}_{0}{{f}^{\prime }}_{0}}=-2\alpha Re,$

${G}_{1{f}_{0}{f}_{0}{f}_{0}}={G}_{1{f}_{0}{{f}^{\prime }}_{0}{f}_{0}}={G}_{1{f}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}={G}_{1{f}_{0}{f}_{0}{{f}^{\prime }}_{0}}=0,$

${G}_{1{{f}^{\prime }}_{0}}=-2\alpha Re{f}_{0}\left(\eta \right)-\left(4-Ha\right){\alpha }^{2},\text{ }{G}_{1{{f}^{\prime }}_{0}{f}_{0}}=-2\alpha Re,\text{ }{G}_{1{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}=0,$

${G}_{1{{f}^{\prime }}_{0}{f}_{0}{f}_{0}}={G}_{1{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{f}_{0}}={G}_{1{{f}^{\prime }}_{0}{f}_{0}{{f}^{\prime }}_{0}}={G}_{1{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}=0,$ (43)

${G}_{2{f}_{0}}=-8{E}_{c}{P}_{r}{\alpha }^{2}{f}_{0}\left(\eta \right),\text{ }{G}_{2{f}_{0}{f}_{0}}=-8{E}_{c}{P}_{r}{\alpha }^{2},\text{ }{G}_{2{f}_{0}{{f}^{\prime }}_{0}}=0,$

${G}_{2{f}_{0}{f}_{0}{f}_{0}}={G}_{2{f}_{0}{{f}^{\prime }}_{0}{f}_{0}}={G}_{2{f}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}={G}_{2{f}_{0}{f}_{0}{{f}^{\prime }}_{0}}=0,{G}_{2{{\varphi }^{″}}_{0}}=-{D}_{f}{P}_{r},$

${G}_{2{{f}^{\prime }}_{0}}=-8{E}_{c}{P}_{r}{\alpha }^{2}{{f}^{\prime }}_{0}\left(\eta \right),\text{ }{G}_{{{f}^{\prime }}_{0}{f}_{0}}=0,\text{ }{G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}=-8{E}_{c}{P}_{r}{\alpha }^{2},$

${G}_{{{f}^{\prime }}_{0}{f}_{0}}={G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{f}_{0}}={G}_{{{f}^{\prime }}_{0}{f}_{0}{{f}^{\prime }}_{0}}={G}_{{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}{{f}^{\prime }}_{0}}=0,{G}_{2{{\varphi }^{″}}_{0}{f}_{0}}=0,{G}_{2{{\varphi }^{″}}_{0}{{\varphi }^{″}}_{0}}=0$ (44)

${G}_{{{\beta }^{″}}_{0}}=-{S}_{c}{S}_{r},\text{ }{G}_{{{\beta }^{″}}_{0}{{\beta }^{″}}_{0}}=0,\text{ }{G}_{{{\beta }^{″}}_{0}{\varphi }_{0}}=0,{G}_{{\varphi }_{0}}={S}_{c}\gamma {\alpha }^{2},$

${G}_{{\varphi }_{0}{\varphi }_{0}}=0,{G}_{{\varphi }_{0}{\varphi }_{0}{\varphi }_{0}}={G}_{{{\beta }^{″}}_{0}{\varphi }_{0}{\varphi }_{0}}={G}_{{\varphi }_{0}{{\beta }^{″}}_{0}{\varphi }_{0}}={G}_{{\varphi }_{0}{\varphi }_{0}{{\beta }^{″}}_{0}}=0,$ (45)

from Equation (33) by using Equations (39)-(42), gives the following,

${f}_{0}=\frac{1}{2}{A}_{3}{\eta }^{2}+1,$

${\beta }_{0}={B}_{1},$

${\varphi }_{0}={C}_{1},$ (46)

${f}_{1}=-\frac{1}{120}\alpha Re{A}_{3}^{2}{\eta }^{6}-\left(\frac{1}{12}\alpha Re{A}_{3}+\frac{1}{6}{\alpha }^{2}{A}_{3}-\frac{1}{24}{\alpha }^{2}Ha{A}_{3}\right){\eta }^{4},$

${\beta }_{1}=-\frac{1}{30}{\alpha }^{2}{P}_{r}{E}_{c}{A}_{3}^{2}{\eta }^{6}-\frac{1}{12}\left(4{A}_{3}{\alpha }^{2}+{A}_{3}^{2}\right){P}_{r}{E}_{c}{\eta }^{4}-2{\alpha }^{2}{P}_{r}{E}_{c}{\eta }^{2},$

${\varphi }_{1}=\frac{1}{2}\gamma {\alpha }^{2}{S}_{c}{C}_{1},$ (47)

$\begin{array}{c}{f}_{2}=\frac{1}{10800}{\alpha }^{2}R{e}^{2}{A}_{3}^{3}{\eta }^{10}+\left(\frac{1}{280}{\alpha }^{3}Re{A}_{3}^{2}-\frac{1}{1120}{\alpha }^{3}ReHa{A}_{3}^{2}+\frac{1}{560}{\alpha }^{2}R{e}^{2}{A}_{3}^{2}\right){\eta }^{8}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(\frac{1}{180}{\alpha }^{2}R{e}^{2}{A}_{3}+\frac{1}{45}{\alpha }^{3}Re{A}_{3}-\frac{1}{180}{\alpha }^{3}ReHa{A}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{45}{\alpha }^{4}{A}_{3}+\frac{1}{90}{\alpha }^{4}ReHa{A}_{3}+\frac{1}{720}{\alpha }^{4}ReH{a}^{2}{A}_{3}\right){\eta }^{6},\end{array}$

$\begin{array}{c}{\beta }_{2}=\frac{1}{2700}{\alpha }^{2}R{e}^{2}{A}_{3}^{3}{P}_{r}{E}_{c}{\eta }^{10}+\frac{1}{1680}\alpha {A}_{3}^{2}{P}_{r}{E}_{c}\left(-5Ha{\alpha }^{3}+12{\alpha }^{2}Re\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+20{\alpha }^{3}+3{A}_{3}Re\right){\eta }^{4}+2{S}_{r}{S}_{c}P-r{E}_{c}{\alpha }^{2}{\eta }^{2},\end{array}$

${\varphi }_{2}=\frac{1}{30}{\alpha }^{2}{P}_{r}{E}_{c}{S}_{r}{S}_{c}{A}_{3}^{2}{\eta }^{6}+\left(\frac{1}{12}{P}_{r}{E}_{c}{S}_{r}{S}_{c}\left(4{A}_{3}{\alpha }^{2}+{A}_{3}^{2}\right)+\frac{1}{24}{S}_{c}^{2}{\gamma }^{2}{\alpha }^{4}{C}_{1}\right){\eta }^{4},$ (48)

$\begin{array}{c}{f}_{3}=-\frac{1}{1572480}{\alpha }^{3}R{e}^{3}{A}_{3}^{4}{\eta }^{14}-\left(\frac{359}{9979200}{\alpha }^{4}R{e}^{2}{A}_{3}^{3}-\frac{359}{39916800}{\alpha }^{4}R{e}^{2}Ha{A}_{3}^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{359}{19958400}{\alpha }^{3}R{e}^{3}{A}_{3}^{3}\right){\eta }^{12}+\left(\frac{29}{226800}{\alpha }^{4}R{e}^{2}Ha{A}_{3}^{2}+\frac{29}{113400}{\alpha }^{5}ReHa{A}_{3}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{29}{907200}{\alpha }^{5}ReH{a}^{2}{A}_{3}^{2}-\frac{29}{56700}{\alpha }^{4}R{e}^{2}{A}_{3}^{2}-\frac{29}{226800}{\alpha }^{3}R{e}^{3}{A}_{3}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{29}{56700}{\alpha }^{5}Re{A}_{3}^{2}\right){\eta }^{10}-\left(\frac{1}{10080}{\alpha }^{3}R{e}^{3}{A}_{3}+\frac{1}{1680}{\alpha }^{4}R{e}^{2}{A}_{3}\end{array}$

$\begin{array}{c}\text{ }-\frac{1}{6720}{\alpha }^{4}R{e}^{2}Ha{A}_{3}+\frac{1}{840}{\alpha }^{5}Re{A}_{3}-\frac{1}{1680}{\alpha }^{5}ReHa{A}_{3}\\ \text{ }+\frac{1}{13440}{\alpha }^{5}ReH{a}^{2}{A}_{3}+\frac{1}{1260}{\alpha }^{6}{A}_{3}-\frac{1}{1680}{\alpha }^{6}Ha{A}_{3}\\ \text{ }+\frac{1}{6720}{\alpha }^{6}H{a}^{2}{A}_{3}-\frac{1}{80640}{\alpha }^{6}H{a}^{3}{A}_{3}\right){\eta }^{8},\end{array}$

$\begin{array}{c}{\beta }_{3}=-\frac{1}{393120}{P}_{r}{E}_{c}{\alpha }^{4}R{e}^{2}{A}_{3}^{4}{\eta }^{14}-\frac{1}{9979200}{P}_{r}{E}_{c}{A}_{3}^{3}{\alpha }^{2}Re\left(-345Ha{\alpha }^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+718Re{\alpha }^{2}+1380{\alpha }^{3}+259{A}_{3}Re\right){\eta }^{12}+\frac{1}{453600}{P}_{r}{E}_{c}{A}_{3}^{2}{\alpha }^{2}\left(-49H{a}^{2}{\alpha }^{4}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+214HaRe{\alpha }^{3}+392Ha{\alpha }^{4}+120{A}_{3}HaRe\alpha -232R{e}^{2}{\alpha }^{2}-856Re{\alpha }^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-784{\alpha }^{4}-240{A}_{3}R{e}^{2}-480{A}_{3}Re\alpha \right){\eta }^{10}-\frac{1}{20160}{P}_{r}{E}_{c}{A}_{3}{\alpha }^{3}\left(2H{a}^{2}{\alpha }^{4}\end{array}$

$\begin{array}{l}\text{ }+13{A}_{3}H{a}^{2}{\alpha }^{2}-8HaRe{\alpha }^{3}-16Ha{\alpha }^{4}-52{A}_{3}HaRe\alpha -104{A}_{3}Ha{\alpha }^{2}\\ \text{ }+8R{E}^{2}{\alpha }^{2}+32Re{\alpha }^{3}+32{\alpha }^{4}+52{A}_{3}R{E}^{2}+208{A}_{3}Re\alpha +208{A}_{3}{\alpha }^{2}\right){\eta }^{8}\\ \text{ }-\frac{1}{60}{P}_{r}^{2}{D}_{f}{S}_{r}{S}_{c}{E}_{c}{\alpha }^{2}{A}_{3}^{2}{\eta }^{6}-\frac{1}{24}{P}_{r}{D}_{f}\left({S}_{r}{S}_{c}{P}_{r}{E}_{c}{A}_{3}\left(4{\alpha }^{2}+{A}_{3}\right)\underset{\text{ }}{\overset{\text{ }}{\text{ }}}\\ \text{ }+\frac{1}{2}{S}_{2}{\gamma }^{2}{\alpha }^{4}{C}_{1}\right){\eta }^{4}-P{r}^{2}{D}_{f}{S}_{r}{S}_{c}{E}_{c}{\alpha }^{2}{\eta }^{2}\end{array}$

$\begin{array}{c}{\varphi }_{3}=-\frac{1}{5400}{S}_{r}{S}_{c}{P}_{r}{E}_{c}{\alpha }^{3}Re{A}_{3}^{3}{\eta }^{10}+\frac{1}{16}\left(-\frac{1}{210}{S}_{r}{S}_{c}{P}_{r}{E}_{c}\alpha {A}_{3}^{2}\left(-5Ha{\alpha }^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+12Re{\alpha }^{2}+20{\alpha }^{3}+3{A}_{3}Re\right)+a\frac{1}{210}{S}_{c}^{2}\gamma {\alpha }^{4}{S}_{r}{P}_{r}{E}_{c}{A}_{3}^{2}\right){\eta }^{8}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{12}\left(\frac{1}{15}{S}_{r}{S}_{c}{P}_{r}{E}_{c}\alpha {A}_{3}\left(Ha{\alpha }^{3}+{A}_{3}Ha\alpha -2Re{\alpha }^{2}-4{\alpha }^{3}-2{A}_{3}\alpha \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{5}{S}_{c}\gamma {\alpha }^{2}\left(\frac{1}{12}{S}_{r}{S}_{c}{P}_{r}{A}_{3}\left(4{\alpha }^{2}+{A}_{3}\right)+\frac{1}{24}{S}_{c}^{2}{\gamma }^{2}{\alpha }^{4}{C}_{1}\right)\right){\eta }^{6}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{12}{S}_{c}^{2}\gamma S-r{P}_{r}{E}_{c}{\alpha }^{4}{\eta }^{4}+\frac{1}{4}{S}_{r}{S}_{c}{P}_{r}{D}_{f}\gamma {\alpha }^{2}Re{C}_{1}{\eta }^{2}\end{array}$ (49)

$⋮$

From step (4) substitution Equations (46)-(42) in Equation (34), the analytical-approximate solution can be resulted as follows:

$\begin{array}{c}f\left(\eta \right)=1+\frac{1}{2}{A}_{3}{\eta }^{2}-\left(\frac{1}{12}\alpha Re{A}_{3}+\frac{1}{6}{\alpha }^{2}{A}_{3}-\frac{1}{24}{\alpha }^{2}Ha{A}_{3}\right){\eta }^{4}+\left(\frac{1}{180}{\alpha }^{2}R{e}^{2}{A}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{120}\alpha Re{A}_{3}^{2}+\frac{1}{45}{\alpha }^{3}Re{A}_{3}-\frac{1}{180}{\alpha }^{3}ReHa{A}_{3}+\frac{1}{45}{\alpha }^{4}{A}_{3}+\frac{1}{90}{\alpha }^{4}ReHa{A}_{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{720}{\alpha }^{4}+ReH{a}^{2}{A}_{3}\right){\eta }^{6}+\left(\frac{1}{280}{\alpha }^{3}Re{A}_{3}^{2}-\frac{1}{1120}{\alpha }^{3}ReHa{A}_{3}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{560}{\alpha }^{2}R{e}^{2}{A}_{3}^{2}-\left(\frac{1}{10080}{\alpha }^{3}R{e}^{3}{A}_{3}+\frac{1}{1680}{\alpha }^{4}R{e}^{2}{A}_{3}-\frac{1}{6720}{\alpha }^{4}R{e}^{2}Ha{A}_{3}\end{array}$

$\begin{array}{c}\text{ }+\frac{1}{840}{\alpha }^{5}Re{A}_{3}-\frac{1}{1680}{\alpha }^{5}ReHa{A}_{3}+\frac{1}{13440}{\alpha }^{5}ReH{a}^{2}{A}_{3}+\frac{1}{1260}{\alpha }^{6}{A}_{3}\\ \text{ }-\frac{1}{1680}{\alpha }^{6}Ha{A}_{3}+\frac{1}{6720}{\alpha }^{6}H{a}^{2}{A}_{3}-\frac{1}{80640}{\alpha }^{6}H{a}^{3}{A}_{3}\right){\eta }^{8}\\ \text{ }+\frac{1}{10800}{\alpha }^{2}R{e}^{2}{A}_{3}^{3}{\eta }^{10}+\cdots \end{array}$

$\begin{array}{c}\beta \left(\eta \right)={B}_{1}+\left(-2{\alpha }^{2}{P}_{r}{E}_{c}+2{S}_{r}{S}_{c}{P}_{r}{E}_{c}{\alpha }^{2}\right){\eta }^{2}+\left(-\frac{1}{12}\left(4{A}_{3}{\alpha }^{2}+{A}_{3}^{2}\right){P}_{r}{E}_{c}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{1680}\alpha {A}_{3}^{2}{P}_{r}{E}_{c}\left(-5Ha{\alpha }^{3}+12{\alpha }^{2}Re+20{\alpha }^{3}+3{A}_{3}Re\right)\right){\eta }^{4}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{30}{\alpha }^{2}{P}_{r}{E}_{c}{A}_{3}^{2}{\eta }^{6}+\frac{1}{2700}{\alpha }^{2}R{e}^{2}{A}_{3}^{3}{P}_{r}{E}_{c}{\eta }^{10}+\cdots \end{array}$

$\begin{array}{c}\varphi \left(\eta \right)={C}_{1}+\frac{1}{2}\gamma {\alpha }^{2}{S}_{c}{C}_{1}{\eta }^{2}+\left(\frac{1}{12}{P}_{r}{E}_{c}{S}_{r}{S}_{c}\left(4{A}_{3}{\alpha }^{2}+{A}_{3}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{24}l{S}_{c}^{2}{\gamma }^{2}{\alpha }^{4}{C}_{1}\right){\eta }^{4}+\frac{1}{30}{\alpha }^{2}{P}_{r}{E}_{c}{S}_{r}{S}_{c}{A}_{3}^{2}{\eta }^{6}+\cdots \end{array}$ (50)

5. The Analysis of Convergence

Here, the analysis of convergence for the analytical-approximate solution (50) that was resulted from the application of new power series algorithm for solving the problem has been extensively studied.

Definition (1): Suppose that H is Banach space, R is the real numbers and $G\left[F,H,P\right]=\left({G}_{1}\left[F\right],{G}_{2}\left[H\right],{G}_{2}\left[P\right]\right)$ is a nonlinear operators defined by $G\left[F,H,P\right]:{H}^{3}\to {R}^{3}$. Then the sequence of the solutions generated from a new algorithm can be written as

$\begin{array}{l}{F}_{n+1}={G}_{1}\left[{F}_{n}\right],\text{ }{F}_{n}=\underset{k=0}{\overset{n}{\sum }}\text{ }{f}_{k},\text{ }n=0,1,2,3,\cdots \\ {H}_{n+1}={G}_{2}\left[{H}_{n}\right],\text{ }{H}_{n}=\underset{k=0}{\overset{n}{\sum }}\text{ }{\beta }_{k},\text{ }n=0,1,2,3,\cdots \\ {P}_{n+1}={G}_{3}\left[{P}_{n}\right],\text{ }{P}_{n}=\underset{k=0}{\overset{n}{\sum }}\text{ }{\varphi }_{k},\text{ }n=0,1,2,3,\cdots \end{array}$ (51)

Definition (2): 1 Suppose that $G\left[F,H,P\right]$ satisfies Lipschitz condition such that for $0\le {\gamma }_{1},{\gamma }_{2},{\gamma }_{3}<1$, ${\gamma }_{1},{\gamma }_{2},{\gamma }_{3}\in R$, we have

$\begin{array}{l}‖{G}_{1}\left[{F}_{n}\right]-{G}_{1}\left[{F}_{n-1}\right]‖\le {\gamma }_{1}‖{F}_{n}-{F}_{n-1}‖,\\ ‖{G}_{2}\left[{H}_{n}\right]-{G}_{2}\left[{H}_{n-1}\right]‖\le {\gamma }_{2}‖{H}_{n}-{H}_{n-1}‖,\\ ‖{G}_{3}\left[{P}_{n}\right]-{G}_{3}\left[{P}_{n-1}\right]‖\le {\gamma }_{3}‖{P}_{n}-{P}_{n-1}‖,\end{array}$ (52)

Now, we assume that $G\left[{F}_{n},{H}_{n},{P}_{n}\right]=G\left(n\right)$ for simplify with $\gamma ={\gamma }_{1}+{\gamma }_{2}+{\gamma }_{3}$, $0\le \gamma <1$ yield,

$‖G\left(n\right)-G\left(n-1\right)‖\le \gamma ‖\left({F}_{n},{H}_{n},{P}_{n}\right)-\left({F}_{n-1},{H}_{n-1},{P}_{n-1}\right)‖.$ (53)

The sufficient condition for convergent of the series analytical-approximate solutions ${F}_{n}$, ${H}_{n}$, ${P}_{n}$ is given in the following theorems.

Theorem (1): 2 The series of the analytical-approximate solution ${\left\{{S}_{n}=\left({F}_{n},{H}_{n},{P}_{n}\right)\right\}}_{0}^{\infty }$ generated from new algorithm converge if the following condition is satisfied:

$‖{S}_{n}-{S}_{m}‖\to 0,\text{ }\text{as}\text{\hspace{0.17em}}\text{\hspace{0.17em}}m\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{ }0\le \gamma <1,$ (54)

Proof. From the above definition, the next equation can be written as

$\begin{array}{c}‖{S}_{n}-{S}_{m}‖=‖\left({F}_{n},{H}_{n},{P}_{n}\right)-\left({F}_{m},{H}_{m},{P}_{m}\right)‖\\ =‖\left(\underset{k=0}{\overset{n}{\sum }}\text{ }{f}_{k},\underset{k=0}{\overset{n}{\sum }}{\beta }_{k},\underset{k=0}{\overset{n}{\sum }}\text{ }{\varphi }_{k}\right)-\left(\underset{k=0}{\overset{m}{\sum }}\text{ }{f}_{k},\underset{k=0}{\overset{m}{\sum }}\text{ }{\beta }_{k},\underset{k=0}{\overset{m}{\sum }}\text{ }{\varphi }_{k}\right)‖\\ =‖\left({f}_{0}+{L}^{-1}\underset{k=0}{\overset{n}{\sum }}\frac{{\Delta }^{k}}{k!}\frac{{\text{d}}^{\left(k\right)}{G}_{1}\left[{f}_{0}\left(\eta \right)\right]}{\text{d}{z}^{\left(\eta \right)}},{\beta }_{0}+{L}^{-1}\underset{k=0}{\overset{n}{\sum }}\frac{{\Delta }^{k}}{k!}\frac{{\text{d}}^{\left(k\right)}{G}_{2}\left[{\beta }_{0}\left(\eta \right)\right]}{\text{d}{\eta }^{\left(k\right)}},\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}{\varphi }_{0}+{L}^{-1}\underset{k=0}{\overset{n}{\sum }}\frac{{\Delta }^{k}}{k!}\frac{{\text{d}}^{\left(k\right)}{G}_{3}\left[{\varphi }_{0}\left(\eta \right)\right]}{\text{d}{\eta }^{\left(k\right)}}-\left({f}_{0}+{L}^{-1}\underset{k=0}{\overset{m}{\sum }}\frac{{\Delta }^{k}}{k!}\frac{{\text{d}}^{\left(k\right)}{G}_{1}\left[{f}_{0}\left(\eta \right)\right]}{k!\text{d}{\eta }^{\left(k\right)}},\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{0}+{L}^{-1}\underset{k=0}{\overset{m}{\sum }}\frac{{\Delta }^{k}}{k!}\frac{{\text{d}}^{\left(k\right)}{G}_{2}\left[{\beta }_{0}\left(\eta \right)\right]}{k!\text{d}{\eta }^{\left(k\right)}},{\varphi }_{0}+{L}^{-1}\underset{k=0}{\overset{m}{\sum }}\frac{{\Delta }^{k}}{k!}\frac{{\text{d}}^{\left(k\right)}{G}_{3}\left[{\varphi }_{0}\left(\eta \right)\right]}{k!\text{d}{\eta }^{\left(k\right)}}\right)‖\end{array}$

$\begin{array}{l}=‖{L}^{-1}G{\left[\underset{k=0}{\overset{n-1}{\sum }}{f}_{k},\underset{k=0}{\overset{n-1}{\sum }}{\beta }_{k},\underset{k=0}{\overset{n-1}{\sum }}\text{ }\varphi \right]}_{k}-{L}^{-1}G\left[\underset{k=0}{\overset{m-1}{\sum }}\text{ }{f}_{k},\underset{k=0}{\overset{m-1}{\sum }}\text{ }{\beta }_{k},\underset{k=0}{\overset{m-1}{\sum }}\text{ }{\varphi }_{k}\right]‖\\ \le |{L}^{-1}|‖G\left[\underset{k=0}{\overset{n-1}{\sum }}{f}_{k},\underset{k=0}{\overset{n-1}{\sum }}{\beta }_{k},\underset{k=0}{\overset{n-1}{\sum }}\text{ }{\varphi }_{k}\right]-G\left[\underset{k=0}{\overset{m-1}{\sum }}{f}_{k},\underset{k=0}{\overset{m-1}{\sum }}\text{ }{\beta }_{k},\underset{k=0}{\overset{m-1}{\sum }}\text{ }{\varphi }_{k}\right]‖\\ \le |{L}^{-1}|‖G\left[{F}_{n-1},{H}_{n-1},{P}_{n-1}\right]-G\left[{F}_{m-1},{H}_{m-1},{P}_{m-1}\right]‖\\ \le \gamma ‖\left({F}_{n-1},{H}_{n-1},{P}_{n-1}\right)-\left({F}_{m-1},{H}_{m-1},{P}_{m-1}\right)‖\\ =\gamma ‖{S}_{n-1}-{S}_{m-1}‖,\end{array}$ (55)

since G[F,H,P] satisfies Lipschitz condition. Let $n=m+1$, then

$\begin{array}{l}‖{F}_{m+1}-{F}_{m}‖\le {\gamma }_{1}‖{F}_{m}-{F}_{m-1}‖,\\ ‖{H}_{m+1}-{H}_{m}‖\le {\gamma }_{2}‖{H}_{m}-{H}_{m-1}‖,\\ ‖{P}_{m+1}-{P}_{m}‖\le {\gamma }_{3}‖{P}_{m}-{P}_{m-1}‖,\end{array}$ (56)

hence,

$\begin{array}{l}‖{F}_{m}-{F}_{m-1}‖\le {\gamma }_{1}‖{F}_{m-1}-{F}_{m-2}‖\le \cdots \le {\gamma }_{1}^{m-1}‖{F}_{1}-{F}_{0}‖,\\ ‖{H}_{m}-{H}_{m-1}‖\le {\gamma }_{2}‖{H}_{m-1}-{H}_{m-2}‖\le \cdots \le {\gamma }_{2}^{m-1}‖{H}_{1}-{H}_{0}‖,\\ ‖{P}_{m}-{P}_{m-1}‖\le {\gamma }_{3}‖{P}_{m-1}-{P}_{m-2}‖\le \cdots \le {\gamma }_{3}^{m-1}\le ‖{P}_{1}-{P}_{0}‖,\end{array}$ (57)

from Equation (57) we get

$\begin{array}{l}‖{F}_{2}-{F}_{1}‖\le {\gamma }_{1}‖{F}_{1}-{F}_{0}‖,‖{H}_{2}-{H}_{1}‖\le {\gamma }_{2}‖{H}_{1}-{H}_{0}‖,‖{P}_{2}-{P}_{1}‖\le {\gamma }_{3}‖{P}_{1}-{P}_{0}‖,\\ ‖{F}_{3}-{F}_{2}‖\le {\gamma }_{1}^{2}‖{F}_{1}-{F}_{0}‖,‖{H}_{3}-{H}_{2}‖\le {\gamma }_{2}^{2}‖{H}_{1}-{H}_{0}‖,‖{F}_{3}-{F}_{2}‖\le {\gamma }_{3}^{2}‖{F}_{1}-{F}_{0}‖,\\ ‖{F}_{4}-{F}_{3}‖\le {\gamma }_{1}^{3}‖{F}_{1}-{F}_{0}‖,‖{H}_{4}-{H}_{3}‖\le {\gamma }_{2}^{3}‖{H}_{1}-{H}_{0}‖,‖{P}_{4}-{P}_{3}‖\le {\gamma }_{3}^{3}‖{P}_{1}-{P}_{0}‖\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ ‖{F}_{m}-{F}_{m-1}‖\le {\gamma }_{1}^{m-1}‖{F}_{1}-{F}_{0}‖,‖{H}_{m}-{H}_{m-1}‖\le {\gamma }_{2}^{m-1}\le ‖{H}_{1}-{H}_{0}‖,\\ ‖{P}_{m}-{P}_{m-1}‖\le {\gamma }_{3}^{m-1}‖{P}_{1}-{P}_{0}‖\end{array}$ (58)

By using triangle inequality, we find that as $m\to \infty$, we have $‖{S}_{n}-{S}_{m}‖\to 0$, then ${S}_{n}$ is a Cauchy sequence in Banach space H3.

Theorem (2): 3 Let $G=\left({G}_{1},{G}_{2},{G}_{3}\right)$ be a nonlinear operator satisfies Lipschitz condition from H3 to H3. If the series analytical-approximate solution $\left\{{S}_{n}\right\}$ converges, then it is converged to the solution of the problem (9)-(10).

Proof.

$\begin{array}{l}‖G\left[{S}_{2}\right]-G\left[{S}_{1}\right]‖\\ =‖\left({G}_{1}\left[{F}_{2}\right],{G}_{2}\left[{H}_{2}\right],{G}_{3}\left[{P}_{2}\right]\right)-\left({G}_{1}\left[{F}_{1}\right],{G}_{2}\left[{H}_{1}\right],{G}_{3}\left[{P}_{1}\right]\right)‖\\ =‖{G}_{1}\left[{F}_{2}\right]-{G}_{1}\left[{F}_{1}\right],{G}_{2}\left[{H}_{2}\right]-{G}_{2}\left[{H}_{1}\right],{G}_{3}\left[{P}_{2}\right]-{G}_{3}\left[{P}_{1}\right]‖\\ =‖{G}_{1}\left[{F}_{2}\right]-{G}_{1}\left[{F}_{1}\right]‖+‖{G}_{2}\left[{H}_{2}\right]-{G}_{2}\left[{H}_{1}\right]‖+‖{G}_{3}\left[{P}_{2}\right]-{G}_{3}\left[{P}_{1}\right]‖\end{array}$

$\begin{array}{l}\le {\gamma }_{1}‖{F}_{2}-{F}_{1}‖+{\gamma }_{2}‖{H}_{2}-{H}_{1}‖+{\gamma }_{3}‖{P}_{2}-{P}_{1}‖\\ \le \left({\gamma }_{1}+{\gamma }_{2}+{\gamma }_{3}\right)‖\left({F}_{2},{H}_{2},{P}_{2}\right)-\left({F}_{1},{H}_{1},{P}_{1}\right)‖\\ =\gamma ‖{S}_{2}-{S}_{1}‖\end{array}$

Therefore, from the Banach fixed-point theorem, there is a unique solution of the problem (9)-(10). We will prove that $\begin{array}{l}{\left\{{S}_{n}\right\}}_{0}^{\infty }\end{array}$ converges to S.

$G\left[S\right]=G\left[\underset{k=0}{\overset{\infty }{\sum }}{S}_{k}\right]={\mathrm{lim}}_{n\to \infty }G\left[\underset{k=0}{\overset{n}{\sum }}{S}_{k}\right]={\mathrm{lim}}_{n\to \infty }G\left[{S}_{n}\right]={\mathrm{lim}}_{n\to \infty }{S}_{n+1}=S.$

In practice, the theorems (1) and (2) suggest to compute the value of ${\gamma }_{1},{\gamma }_{2},{\gamma }_{3}$, as described in the following definition.

Definition (1): 4 for $k=1,2,3,\cdots$

${\gamma }_{1}^{k}=\left(\begin{array}{ll}\frac{‖{F}_{k+1}-{F}_{k}‖}{‖{F}_{1}-{F}_{0}‖}=\frac{‖{f}_{k+1}‖}{‖{f}_{1}‖},\hfill & ‖{f}_{1}‖\ne 0,\hfill \\ 0,\hfill & ‖{f}_{1}‖=0,\hfill \end{array}$

${\gamma }_{2}^{k}=\left(\begin{array}{ll}\frac{‖{H}_{k+1}-{H}_{k}‖}{‖{H}_{1}-{H}_{0}‖}=\frac{‖{\beta }_{k+1}‖}{‖{\beta }_{1}‖},\hfill & ‖{\beta }_{1}‖\ne 0,\hfill \\ 0,\hfill & ‖{\beta }_{1}‖=0,\hfill \end{array}$

${\gamma }_{3}^{k}=\left(\begin{array}{ll}\frac{‖{P}_{k+1}-{P}_{k}‖}{‖{P}_{1}-{P}_{0}‖}=\frac{‖{\varphi }_{k+1}‖}{‖{\varphi }_{1}‖},\hfill & ‖{\varphi }_{1}‖\ne 0,\hfill \\ 0,\hfill & ‖{\varphi }_{1}‖=0.\hfill \end{array}$ (59)

Now, the definition (1) can be applied on the magneto hydrodynamic (MHD) flow of viscous fluid in a channel with non-parallel plates to find convergence, then to obtain for examples as below.

If we choose $Re=30$, $\alpha ={3}^{\circ }$, $Ha=500$, ${S}_{r}={S}_{c}=0.1$, ${P}_{r}=0.1$, ${D}_{f}=0.01$, ${E}_{c}=0.01$, $\gamma =0.04$ then obtain:

$\begin{array}{l}{‖{F}_{2}-{F}_{1}‖}_{2}\le {\gamma }_{1}{‖{F}_{1}-{F}_{0}‖}_{2}⇒{\gamma }_{1}=0.08954317<1,\\ {‖{F}_{3}-{F}_{2}‖}_{2}\le {\gamma }_{1}^{2}{‖{F}_{1}-{F}_{0}‖}_{2}⇒{\gamma }_{1}^{2}=0.005099704<1,\\ {‖{F}_{4}-{F}_{3}‖}_{2}\le {\gamma }_{1}^{3}{‖{F}_{1}-{F}_{0}‖}_{2}⇒{\gamma }_{1}^{3}=0.0001956583630<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{H}_{2}-{H}_{1}‖}_{2}\le {\gamma }_{2}{‖{H}_{1}-{H}_{0}‖}_{2}⇒{\gamma }_{2}=0.2496673982<1,\\ {‖{H}_{3}-{H}_{2}‖}_{2}\le {\gamma }_{2}^{2}{‖{H}_{1}-{H}_{0}‖}_{2}⇒{\gamma }_{2}^{2}=0.03173895000<1,\\ {‖{H}_{4}-{H}_{3}‖}_{2}\le {\gamma }_{2}^{3}{‖{H}_{1}-{H}_{0}‖}_{2}⇒{\gamma }_{2}^{3}=0.002169482685<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{P}_{2}-{P}_{1}‖}_{2}\le {\gamma }_{3}{‖{P}_{1}-{P}_{0}‖}_{2}⇒{\gamma }_{3}=0.73232384<1,\\ {‖{P}_{3}-{P}_{2}‖}_{2}\le {\gamma }_{3}^{2}{‖{P}_{1}-{P}_{0}‖}_{2}⇒{\gamma }_{3}^{2}=0.09141845239<1,\\ {‖{P}_{4}-{P}_{3}‖}_{2}\le {\gamma }_{3}^{3}{‖{P}_{1}-{P}_{0}‖}_{2}⇒{\gamma }_{3}^{3}=0.003873822310<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{F}_{2}-{F}_{1}‖}_{+\infty }\le \gamma {‖{F}_{1}-{F}_{0}‖}_{+\infty }⇒\gamma =0.07399291<1,\\ {‖{F}_{3}-{F}_{2}‖}_{+\infty }\le {\gamma }^{2}{‖{F}_{1}-{F}_{0}‖}_{+\infty }⇒{\gamma }^{2}=0.004719598<1,\\ {‖{F}_{4}-{F}_{3}‖}_{+\infty }\le {\gamma }^{3}{‖{F}_{1}-{F}_{0}‖}_{+\infty }⇒{\gamma }^{3}=0.0001557525428<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{H}_{2}-{H}_{1}‖}_{+\infty }\le \gamma {‖{H}_{1}-{H}_{0}‖}_{+\infty }⇒\gamma =0.2384340096<1,\\ {‖{H}_{3}-{H}_{2}‖}_{+\infty }\le {\gamma }^{2}{‖{H}_{1}-{H}_{0}‖}_{+\infty }⇒{\gamma }^{2}=0.02467876<1,\\ {‖{H}_{4}-{H}_{3}‖}_{+\infty }\le {\gamma }^{3}{‖{H}_{1}-{H}_{0}‖}_{+\infty }⇒{\gamma }^{3}=0.001715583134<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{P}_{2}-{P}_{1}‖}_{+\infty }\le \gamma {‖{P}_{1}-{P}_{0}‖}_{+\infty }⇒\gamma =0.732255116<1,\\ {‖{P}_{3}-{P}_{2}‖}_{+\infty }\le {\gamma }^{2}{‖{P}_{1}-{P}_{0}‖}_{+\infty }⇒{\gamma }^{2}=0.087297019<1,\\ {‖{P}_{4}-{P}_{3}‖}_{+\infty }\le {\gamma }^{3}{‖{P}_{1}-{P}_{0}‖}_{+\infty }⇒{\gamma }^{3}=0.003011824761<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

Also, if we get $\alpha =-{2}^{\circ }$, $Re=10$, $Ha=110$, ${P}_{r}={D}_{f}=0.2$, $\gamma =0.4$, ${E}_{c}=0.01$.

$\begin{array}{l}{‖{F}_{2}-{F}_{1}‖}_{2}\le {\gamma }_{1}{‖{F}_{1}-{F}_{0}‖}_{2}⇒{\gamma }_{1}=0.03060256592<1,\\ {‖{F}_{3}-{F}_{2}‖}_{2}\le {\gamma }_{1}^{2}{‖{F}_{1}-{F}_{0}‖}_{2}⇒{\gamma }_{1}^{2}=0.1113608591<1,\\ {‖{F}_{4}-{F}_{3}‖}_{2}\le {\gamma }_{1}^{3}{‖{F}_{1}-{F}_{0}‖}_{2}⇒{\gamma }_{1}^{3}=0.4872382121<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{H}_{2}-{H}_{1}‖}_{2}\le {\gamma }_{2}{‖{H}_{1}-{H}_{0}‖}_{2}⇒{\gamma }_{2}=0.0004683459276<1,\\ {‖{H}_{3}-{H}_{2}‖}_{2}\le {\gamma }_{2}^{2}{‖{H}_{1}-{H}_{0}‖}_{2}⇒{\gamma }_{2}^{2}=0.005594660398<1,\\ {‖{H}_{4}-{H}_{3}‖}_{2}\le {\gamma }_{2}^{3}{‖{H}_{1}-{H}_{0}‖}_{2}⇒{\gamma }_{2}^{3}=0.02712979918<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{P}_{2}-{P}_{1}‖}_{2}\le {\gamma }_{3}{‖{P}_{1}-{P}_{0}‖}_{2}⇒{\gamma }_{3}=0.000005903459683<1,\\ {‖{P}_{3}-{P}_{2}‖}_{2}\le {\gamma }_{3}^{2}{‖{P}_{1}-{P}_{0}‖}_{2}⇒{\gamma }_{3}^{2}=0.0001292820508<1,\\ {‖{P}_{4}-{P}_{3}‖}_{2}\le {\gamma }_{3}^{3}{‖{P}_{1}-{P}_{0}‖}_{2}⇒{\gamma }_{3}^{3}=0.0004542885489<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{F}_{2}-{F}_{1}‖}_{+\infty }\le {\gamma }_{1}{‖{F}_{1}-{F}_{0}‖}_{+\infty }⇒{\gamma }_{1}=0.02756015475<1,\\ {‖{F}_{3}-{F}_{2}‖}_{+\infty }\le {\gamma }_{1}^{2}{‖{F}_{1}-{F}_{0}‖}_{+\infty }⇒{\gamma }_{1}^{2}=0.1104543115<1,\\ {‖{F}_{4}-{F}_{3}‖}_{+\infty }\le {\gamma }_{1}^{3}{‖{F}_{1}-{F}_{0}‖}_{+\infty }⇒{\gamma }_{1}^{3}=0.4872217326<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{H}_{2}-{H}_{1}‖}_{+\infty }\le {\gamma }_{2}{‖{H}_{1}-{H}_{0}‖}_{+\infty }⇒{\gamma }_{2}=0.00041798914<1,\\ {‖{H}_{3}-{H}_{2}‖}_{+\infty }\le {\gamma }_{2}^{2}{‖{H}_{1}-{H}_{0}‖}_{+\infty }⇒{\gamma }_{2}^{2}=0.005302953669<1,\\ {‖{H}_{4}-{H}_{3}‖}_{+\infty }\le {\gamma }_{2}^{3}{‖{H}_{1}-{H}_{0}‖}_{+\infty }⇒{\gamma }_{2}^{3}=0.02690804181<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

$\begin{array}{l}{‖{P}_{2}-{P}_{1}‖}_{+\infty }\le {\gamma }_{3}{‖{P}_{1}-{P}_{0}‖}_{+\infty }⇒{\gamma }_{3}=0.00000534538<1,\\ {‖{P}_{3}-{P}_{2}‖}_{+\infty }\le {\gamma }_{3}^{2}{‖{P}_{1}-{P}_{0}‖}_{+\infty }⇒{\gamma }_{3}^{2}=0.00010743626<1,\\ {‖{P}_{4}-{P}_{3}‖}_{+\infty }\le {\gamma }_{3}^{3}{‖{P}_{1}-{P}_{0}‖}_{+\infty }⇒{\gamma }_{3}^{3}=0.00043058757<1,\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\end{array}$

Then ${\sum }_{k=0}^{\infty }\text{ }{f}_{k}\left(\eta \right)$, ${\sum }_{k=0}^{\infty }\text{ }{\beta }_{k}\left(\eta \right)$ and ${\sum }_{k=0}^{\infty }\text{ }{\varphi }_{k}\left(\eta \right)$ converge to the solutions $f\left(\eta \right)$, $\beta \left(\eta \right)$ and $\varphi \left(\eta \right)$ respectively when $0\le {\gamma }_{1}^{k},{\gamma }_{2}^{k},{\gamma }_{3}^{k}<1$, $k=1,2,\cdots$.

6. Results and Discussions

This section is dedicated to study the influence of various non dimensional physical parameters on velocity field $f\left(\eta \right)$, temperature field $\beta \left(\eta \right)$ and concentration field $\varphi \left(\eta \right)$. Also the influence of different parameters on rate of heat transfer and rate of mass transfer are under observation for diverging and converging channels. In Table 1 and Table 2 proof convergence the values ${A}_{3}$, ${B}_{1}$ and ${C}_{1}$ of initial solutions. The stability of these values can be clearly distinguished from the fourth approximation. Table 3 and Table 4 present a comparison of the solutions obtained by new algorithm and numerical algorithm. These tables show that an excellent between the solutions. On the other hand Tables 5-13 are explained to analyze the behavior of Nusselt number and Sherwood number with variation parameters. As for can say that impotent to mention that Nusselt number gives a description of heat transfer rate at the wall, while Sherwood number represents the rate of mass transfer at the wall. In Table 5 and Table 6, Nusselt number with very Reynolds number Re and channel opening $\alpha$ are discussed. One can clearly observe that for diverging channel, Nusselt number a rise with increasing in Reynolds number, but for increasing Re lead to the rate of heat transfer a drop for converging channel. Sherwood number appears to be increasing as the Reynolds number grows for converging channel and decreases for diverging channel. The behavior of Nusselt number in channel opening is quite opposite for Reynolds number. Table 7 gives description of variation in heat transfer rate and mass transfer rate at the wall with the rising values of Hartmman number. Increase in heat transfer rate and mass transfer rate are observed for increasing Hartmman number Ha in the case of both converging and diverging channels. Table 8 and Table 13 portray that the rate of heat transfer increasing for both diverging and converging channels an increase in Schmidt, Soret, Prandtl, Eckert, Dufour numbers and chemical reaction parameter respectively. Furthermore these tables for both channels clear that the rate of mass transfer at the walls decreases with an increase in Schmidt, Soret, Prandtl, Eckert, Dufour numbers and chemical reaction parameter.

Table 1. $Re=30,Ha=500,{P}_{r}=0.1,{D}_{f}=0.02,{S}_{c}={S}_{r}=0.1,\gamma =0.04,\alpha ={3}^{\circ }$.

Table 2. $Re=10,Ha=110,{P}_{r}=0.2,{D}_{f}=0.2,{S}_{c}={S}_{r}=0.2,\gamma =0.4,\alpha =-{2}^{\circ }$.

Table 3. Comparison between new scheme and $\left(R-K4\right)$ scheme for the analytical solutions $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ when $Re=30$, $Ha=500$, $\alpha ={3}^{\circ }$, ${P}_{r}=0.1$, ${S}_{c}={S}_{r}=0.1$, ${E}_{c}=0.01$, ${D}_{f}=0.04$.

Table 4. Comparison between new scheme and $\left(R-K4\right)$ scheme for the analytical solutions $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ when $Re=10$, $Ha=110$, ${P}_{r}=0.2$, ${D}_{f}=0.2$, ${S}_{c}={S}_{r}=0.2$, $\gamma =0.4$, $\alpha =-{2}^{\circ }$.

Table 5. Variation in Nusselt number and Sherwood number with varying $Re$ when $Ha=100,{E}_{c}=1,{S}_{c}={S}_{r}=0.2,{D}_{f}={P}_{r}=0.5,\gamma =0.1$.

Table 6. Variation in Nusselt number and Sherwood number with varying $\alpha$ when $Re=25,Ha=100,{E}_{c}=1,{S}_{c}={S}_{r}=0.2,{D}_{f}={P}_{r}=0.5,\gamma =0.1$.

Table 7. Variation in Nusselt number and Sherwood number with varying $Ha$ when $Re=25,{E}_{c}=1,{S}_{c}={S}_{r}=0.2,{D}_{f}={P}_{r}=0.5,\gamma =0.1$.

Table 8. Variation in Nusselt number and Sherwood number with varying ${S}_{c}$ when $Re=25,{E}_{c}=1,{S}_{r}=0.2,{D}_{f}={P}_{r}=0.5,\gamma =0.1$.

Table 9. Variation in Nusselt number and Sherwood number with varying ${S}_{r}$ when $Re=25,{E}_{c}=1,{S}_{c}=0.2,{D}_{f}={P}_{r}=0.5,\gamma =0.1$.

Table 10. Variation in Nusselt number and Sherwood number with varying ${P}_{r}$ when $Re=25,{E}_{c}=1,{S}_{r}={S}_{c}=0.2,{D}_{f}=0.5,\gamma =0.1$.

Table 11. Variation in Nusselt number and Sherwood number with varying ${E}_{c}$ when $Re=25,{S}_{r}={S}_{c}=0.2,{D}_{f}={P}_{r}=0.5,\gamma =0.1$.

Table 12. Variation in Nusselt number and Sherwood number with varying ${D}_{f}$ when $Re=25,{E}_{c}=1,{S}_{r}={S}_{c}=0.2,{P}_{r}=0.5,\gamma =0.1$.

Table 13. Variation in Nusselt number and Sherwood number with varying $\gamma$ when $Re=25,{E}_{c}=1,{S}_{r}={S}_{c}=0.2,{D}_{f}={P}_{r}=0.5$.

In addition to this section highlights the major outcomes of the analytical study presented by new algorithm. The analysis of the variations in temperature and concentration profiles for different parameters is prepared. For that purpose, Figures are plotted for varying several parameters. Moreover we have been divided this section into two subsections follow as.

・ Channel divergent ( $\alpha >0$ ).

In Figures 2-9 are plotted to show the behavior curves of velocity, temperature and concentration profiles under the impact of different physical parameters. An increasing the opening angle $\alpha$ gives variations in velocity, temperature and concentration profiles as displayed in Figure 2. The influence of parameter $\alpha$ on the velocity field $f\left(\eta \right)$ for divergent channel causes more effect at the middle channel as well as it represents as a maximum position at the central line (when $\eta =0$ ). It also has least effect in part near the walls (when $\eta =-1,1$ ). High temperature in the central region of the channel is conspicuously clear and the maximum temperature lies there too. While the results of angle opening $\alpha$ on concentration profile show that the increase in $\alpha$ gives a decreased concentration profile. The central portion of the channel is more effect, while the portion of the near of walls the concentration is less affected. Effects of the increase Reynolds number Re in $f\left(\eta \right),\beta \left(\eta \right)$ and $\varphi \left(\eta \right)$ are observed in Figure 3. The velocity field $f\left(\eta \right)$ is decreasing with increasing Reynold number and clearly shows that the highest level reaches the central part (when $\eta =0$ ). A rise in temperature profile $\beta \left(\eta \right)$ leads to an increase in Re as for the concentration profile $\varphi \left(\eta \right)$, it is in a state of decrease when there is an increase in Re. The prominent appearance in the central part can be observed in the lowest effect. The behavior of temperature distribution under the influence of Hartmann number Ha can be seen from Figure 4. The velocity distribution remains unchanged with the increase Hartmann. This figure gives a clear picture of how a stronger magnetic field can lead to a change in the temperature of the fluid. But Hartmann number gives a simple rise to the concentration $\varphi \left(\eta \right)$ although this increase is simple, however can be a way to control the concentration of the fluid. In Figure 5 the change in temperature and concentration with an increase in Soret number Sr are plotted. A rise in temperature and a drop in concentration at the central portion of the channel are observed. In fact that stronger viscous forces are responsible for these phenomena. Impact of the Schmidt number Sc on temperature and concentration are demonstrated in Figure 6, with note that the effect Sr is similar to the effect of the Soret number $Sr$.Figure 7 is plotted the effect of growing values of Prandtl number on temperature profile. Arise in temperature of the fluid with increasing Pr can be seen. Figure 8 explains that change in temperature with rising Eckert number Ec. Again the increase in temperature was observed to increase Ec. The effect of Eckert number on $\varphi \left(\eta \right)$ for diverging channel, an increase in Ec decreases the concentration profile. This shows, the strong viscous forces are responsible for a rise temperature of the fluid and this rise is a lot of effect at the central portion. In Figure 9 the changes of chemical reaction $\gamma$ and Dufour number ${D}_{f}$ on $\beta$ and $\varphi \left(\eta \right)$ are presented. Dufour number arises due to the concentration gradient present in energy equation. The main variations in temperature are in central portion of the channel, where the variation in temperature near the walls is almost negligible, which can be said, the stronger concentration results in higher temperature values. Also this figure demonstrates that the increasing of $\gamma$ lead to decrease the concentration profile.

・ Channel convergent ( $\alpha <0$ ).

For the converging channel, the variations in velocity, temperature and concentration profile due to the varying parameters are depicted in Figures 10-17, the behavior of velocity and temperature for changing angle opening $\alpha$ and Reynolds number $Re$ is quite opposite to the behavior of $f\left(\eta \right)$, $\beta \left(\eta \right)$ and $\varphi \left(\eta \right)$ in diverging channel as seen in Figure 10 and Figure 11. On that other hand, Figures 11-17 tell the effect of Harmann, Eckert.

Figure 2. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, when the angle $\alpha$ is varied.

Figure 3. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Ha=100$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha ={3}^{\circ }$ when the Reynolds number $Re$ is varied.

Figure 4. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha ={3}^{\circ }$ when the Harmann number $Ha$ is varied.

Figure 5. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sc=0.2$, $\gamma =0.1$, $\alpha ={3}^{\circ }$ when the Soret number $Sr$ is varied.

Figure 6. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sr=0.2$, $\gamma =0.1$, $\alpha ={3}^{\circ }$ when the Schmidt number $Sc$ is varied.

Figure 7. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, ${D}_{f}=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha ={3}^{\circ }$ when the Prandtl number $Pr$ is varied.

Figure 8. $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr={D}_{f}=0.5$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha ={3}^{\circ }$ when the Eckert number $Ec$ is varied.

Figure 9. $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha ={3}^{\circ }$ when the Dufour number ${D}_{f}$ and chemical reaction parameter $\gamma$ are varied.

Figure 10. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Ha=100$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha =-{3}^{\circ }$ when the Reynolds number $Re$ is varied.

Figure 11. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, when the angle $\alpha$ is varied.

Figure 12. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=30$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha =-{3}^{\circ }$ when the Harmann number $Ha$ is varied.

Figure 13. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sc=0.2$, $\gamma =0.1$, $\alpha =-{3}^{\circ }$ when the Soret number $Sr$ is varied.

Figure 14. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr={D}_{f}=0.5$, $Ec=1$, $Sr=0.2$, $\gamma =0.1$, $\alpha =-{3}^{\circ }$ when the schmidt number $Sc$ is varied.

Figure 15. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, ${D}_{f}=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha =-{3}^{\circ }$ when the Prandtl number $Pr$ is varied.

Figure 16. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr={D}_{f}=0.5$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha =-{3}^{\circ }$ when the Eckert number $Ec$ is varied.

Figure 17. $f\left(\eta \right)$, $\beta \left(\eta \right)$, $\varphi \left(\eta \right)$ for the value $Re=50$, $Ha=100$, $Pr=0.5$, $Ec=1$, $Sc=Sr=0.2$, $\gamma =0.1$, $\alpha =-{3}^{\circ }$ when the Dufour number ${D}_{f}$ and chemical reaction parameter $\gamma$ are varied.

Prandtl and Dufour numbers on the temperature profile is similar for the effect in diverging channel. Also these Figures demonstrated the concentration profile possess same effect when there are changing in Hartmann number, Schimdt number, Soret number and chemical reaction parameter. Physical explanations can be provided the temperature profile show that the temperature at the central region increases with increasing angle opening. This can be attributed to that for fixed Reynold number, increasing angle opening leads to increase the cross-sectional flow area. This in turn leads to decrease the flow velocity and this mean the flow will be decelerated. Therefore, the heat dissipation will be reduced which leads to increase the temperature of the fluid. The inertia force of the fluid increases with increasing Reynold number which leads to enhance the parabolic behavior (increasing central temperature) for diverging channel with the opposite view in converging channel. Hartmman number increase in this case Lorentz force is also increasing for diverging and converging channels. This force imports extra drag to the flow. Therefore the temperature profile becomes more flat which means decreasing the temperature within the central region. The thin boundary layers that are near to the wall lead to that the temperature gradient at the highest level. Furthermore to the existence of the thick boundary layer in central region lead to that the temperature gradient at low level. Eckert number increases with increased temperature and thus produces an increase in Kinetic energy. The change of the temperature profile with Prandtl number, and the increase of temperature with Prandtl number result from increasing of the momentum diffusivity. The Dufour number shows to increase shows less effect on temperature, increasing Dufour leads to increase the thermal energy of the fluid thus the temperature increase. The rate of most chemical reactions increases with a decrease the concentration of reactants. As for temperature, it increases if a reaction is heat-emitting and decreases when the reaction absorbs heat.

7. Conclusions

In this paper, the unsteady and two-dimensional magneto hydrodynamic (MHD) flow of viscous fluid in a channel with non-parallel plates is studied analytically using a new algorithm. The solution obtained by new algorithm is an infinite power series for appropriate initial approximation. The construction of this algorithm possessed good convergent series and the convergence of the results is explicitly shown. Graphical results and tables are presented to investigate the influence of physical parameters on velocity, temperature and concentration. Analysis of the converge confirms that the new algorithm is an efficient technique as compared to Range-Kutta algorithm with help of Shooting algorithm. The new algorithm that is widely applied to solve ordinary differential equations lead to the solutions resulting from this algorithm is compatible with numerical solution. Effects of different parameters on temperature and concentration profiles are analyzed and presented graphically. The conclusions can be drawn from the analysis presented:

・ The behavior of temperature and concentration profiles are the same results $\alpha$, $Re$, ${E}_{c}$, ${P}_{r}$ and ${D}_{f}$ for diverging channel.

・ Hartmann number $Ha$ can be used to reduce the temperature of the flow fluid. Also, concentration of the fluid can also be controlled by employing a strong magnetic field.

・ For converging channel, the variations in temperature are opposite for diverging channel with an increase in channel opening $\alpha$ and $Re$.

・ For diverging channel, Nusselt number drops with a rise in angle opening and increases with a rise in Reynolds number and behaves oppositely for convergent channel.

・ Increase in heat transfer rate is observed for increasing ${P}_{r}$, ${E}_{c}$, ${S}_{r}$, ${S}_{c}$, ${D}_{f}$ and $\gamma$ in both channels.

・ Increase in Reynolds number and Angle opening gives a drop to mass transfer rate for diverging channel and a rise for the converging channel.

・ The rate of mass transfer decreased for both channels with an increase in Schmidt, Soret, Prandtl, Eckert, Dufour numbers and chemical reaction parameter.

・ Results obtained by new algorithm are in excellent agreement with numerical solution obtained.

Cite this paper: Al-Saif, A. and Jasim, A. (2019) New Analytical Study of the Effects Thermo-Diffusion, Diffusion-Thermo and Chemical Reaction of Viscous Fluid on Magneto Hydrodynamics Flow in Divergent and Convergent Channels. Applied Mathematics, 10, 268-300. doi: 10.4236/am.2019.104020.
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