In this paper, magnetohydrodynamics natural convection of nanofluid flow over a vertical circular cone immersed in a porous medium under the influence of thermal radiation is investigated using multi-step differential transformation method. The accuracies of the analytical solutions are established through the verifications of the results of the present study with the results of the numerical solutions and the past studies. The approximate analytical solutions are used to examine the impacts of cone angle, flow medium porosity, magnetic field, nanoparticles volume-fraction and shape on the flow and heat transfer behaviours of the Copper (II) Oxide-water nanofluid. It is hoped that this study will enhance better understanding of flow process for the design of flow and heat transfer equipment.

Keywords: free convection, cone angle, porous medium, nanoparticle, magnetic field, multi-step differential transformation method

c_p, specific heat capacity; k, thermal conductivity; m,shape factor; Pr, Prandtl number; µ, velocity component in x-direction; v, velocity component in z-direction; y, axis perpendicular to plates; x, axis along the horizontal direction; y, axis along the vertical direction

Symbols: β, volumetric extension coefficients; ρ, density of the fluid; µ,dynamic viscosity; η,similarity variable; λ, sphericity; Φ , volume fraction or concentration of the nanofluid; θ,dimensionless temperature

Subscript: f, fluid; s, solid; nf, nanofluid

There are various applications of free convection flow and heat transfer in aeronautics, reactor fluidization, glass-fiber production processes, aerodynamic, cooling of gas turbine rotor blades, drawing of a polymer sheet, food stuff processing, melt spinning, mechanical forming processes, cooling of metallic plates, wire and fiber coating, extrusion of plastic sheets, continuous casting, rolling, annealing, and tinning of copper wires. Since the quality of products in the processes depends considerably on the flow and heat transfer characteristics in the various applications, the analysis of fluid flow and heat transfer is very essential Earlier studies on free convections flow established its importance in science and engineering applications.^1–11 However, the analyses of these studies are based on the approximations. Therefore, in some recent studies, improved approximated analyses of the flow and heat transfer models have been presented. Na & Habib¹² adopted parameter differentiation method while Merkin¹³ presented the similarity solutions for free convection on a vertical plate. Merkin& Pop¹⁴ and Ali et al.,¹⁵ applied numerical methods for the flow process. In a very recent paper, Motsa et al.,^16,17 utilized homotopy analysis method and spectral local linearization approach to analyze the free convection boundary layer flow with heat and mass transfer. Also, Ghotbi et al.,¹⁸ presented the analytical solutions to the free convection flow using the homotopy analysis method.

The simple procedures in the theory and applications of differential transformation method (DTM) have shown the effectiveness of the method in solving both linear and nonlinear equations. The method was introduced by Ζhou.¹⁹ In the free flow analysis, over a vertical surface, Yu & Chen²⁰ and Kuo^21,22 applied the method. However, it should be noted that the DTM solutions diverge for some differential equations that extremely have nonlinear behaviors or have boundary-conditions at infinity. This is because the series solution method is limited to small domain. Therefore, Rashidi et al.,²³ applied DTM coupled with Padé-approximant technique to the flow problem. However, the Padé-approximant technique comes with large volume of calculations and computations. Therefore, another technique for an improved rate of convergence and the radius of convergence of power series solution is required. Among the newly developed techniques, is the multi-step differential transform method (MDTM). The method can be applied directly to nonlinear differential equations of infinite boundary conditions without the use of after-treatment techniques and domain transformation techniques.^24–44

The previous studies on the problem under investigation are based on the flow of viscous fluid over a vertical surface without considering the effects of magnetic field and porosity of the flow medium. To the best of the author’s knowledge, a study on free convection flow of nanofluid over a vertical circular cone using multi-step differential transform method. Therefore, in this work, magnetohydrodynamics natural convection of nanofluid flow over a vertical circular cone immersed in a porous medium under the influence of thermal radiation is investigated using multi-step differential transformation method. The approximate analytical solutions are used to examine the impacts of cone angle, flow medium porosity, magnetic field, nanoparticles volume-fraction, shape and type on the flow and heat transfer behaviours of various nanofluids. Also, the accuracies of the analytical solutions are established through the verifications of the results with the results of the numerical solutions and the results in the past studies.

Problem formulation and mathematical analysis

Consider a free-convection flow of an incompressible electrically conducting nanofluid past a cone embedded in a porous medium which is inclined from the vertical with an acute angle γ measured in the clockwise direction and situated in an otherwise quiescent ambient fluid at temperature T_w as shown in Figure 1. A transverse magnetic field of strength B_o is applied normal to the inclined plate. The plate coincides with the plane y=0 and the flow being confined to y >0. Assuming that the flow in the laminar boundary layer is two-dimensional and steady, the equations for continuity and motion are given as 

$\frac{\partial \left(ru\right)}{\partial x}+\frac{\partial \left(rv\right)}{\partial y}=0$   (1)

${\rho }_{nf}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)={\mu }_{nf}\frac{{\partial }^{2}u}{\partial y^2}+g{\left(\rho \beta \right)}_{nf}\left(T-T_{\infty }\right)cos\gamma -\sigma B_o^{2}u-\frac{\mu u}{K}$   (2)

${\left(\rho c_p\right)}_{nf}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=k_{nf}\frac{{\partial }^{2}T}{\partial y^2}-\frac{\partial q_r}{\partial y}$   (3)

Assuming no slip conditions, the appropriate boundary conditions are given as

$u=0,v=0,T=T_{s}aty=0$   (4a)

$u=0,T=T_w,aty\to \infty$   (4b)

where the various physical and thermal properties in the Eq. (1-3) are given as

${\rho }_{nf}={\rho }_f\left(1-\varphi \right)+{\rho }_s\varphi$   (5a)

${\left(\rho c_p\right)}_{nf}={\left(\rho c_p\right)}_f\left(1-\varphi \right)+{\left(\rho c_p\right)}_s\varphi$   (5b)

${\left(\rho \beta \right)}_{nf}={\left(\rho \beta \right)}_f\left(1-\varphi \right)+{\left(\rho \beta \right)}_s\varphi$   (5c)

${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}}$   (5d)

$k_{nf}=k_f\left[\frac{k_s+\left(m-1\right)k_f-\left(m-1\right)\varphi \left(k_f-k_s\right)}{k_s+\left(m-1\right)k_f+\varphi \left(k_f-k_s\right)}\right]$   (6)

${\sigma }_{nf}={\sigma }_f\left[1+\frac{3\left\{\frac{{\sigma }_s}{{\sigma }_f}-1\right\}\varphi }{\left\{\frac{{\sigma }_s}{{\sigma }_f}+2\right\}\varphi -\left\{\frac{{\sigma }_s}{{\sigma }_f}-1\right\}\varphi }\right],$   (7)

The above model in Eq. (6) is Hamilton Crosser’s model. The “m” in the model represents the shape factor which its numerical values for different shapes are given in Table 1. It should be noted that the shape factor, $m=\frac{3}{\lambda }$ , where λ is the sphericity (the ratio of the surface area of the sphere and the surface area of the real particles with equal volumes). For sphericity of sphere, platelet, cylinder, laminar and brick are 1.000, 0.526, 0.625, 0.185 and 0.811, respectively. It should be noted that the Hamilton Crosser’s model becomes a Maxwell-Garnett’s model, when the shape factor of the nanoparticle, m=3. Tables 2&3 present the physical and thermal properties of the base fluid and the nanoparticles, respectively. SWCNTs mean single-walled carbon nanotubes.

However, the present study focusses on Copper (II) Oxide-water nanofluid.

$r=xsin\lambda ,$   (8)

Moreover γ is the half angle of the frustum cone. Because the boundary layer-thickness is small, the local radius to a point in the boundary layer r can be represented by the local radius of the cone.

Going back to Eq. (1), (2) and (3) and if one introduces a stream function, $\psi \left(x,y\right)$  such that

$ru=\frac{\partial \psi }{\partial y},rv=-\frac{\partial \psi }{\partial x},$   (9)

and use the following similarity and dimensionless variables

$\eta ={\left[\frac{{\rho }_f^2\left(g{\beta }_f\left(T_w-T_{\infty }\right)\right)}{4{\mu }_f^{2}x}\right]}^{\frac{1}{4}}y,\psi =\frac{4{\mu }_f}{{\rho }_f}{\left[\frac{{\rho }_f^2\left(g{\beta }_f\left(T_w-T_{\infty }\right)\right)x^3}{4{\mu }_f^2}\right]}^{1/4}f\left(\eta \right),$
$\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }},Pr=\frac{{\mu }_f{c}_p}{k_f},$   (10)

one arrives at fully coupled third and second orders ordinary differential equations

$f^{‴}\left(\eta \right)+{\left(1-\varphi \right)}^{2.5}\left\{\begin{array}{l}\left[\left(1-\varphi \right)+\varphi \left(\frac{{\rho }_s}{{\rho }_f}\right)\right]\left(3f\left(\eta \right)f^{″}\left(\eta \right)-2{\left(f^{\prime }\left(\eta \right)\right)}^2\right)\\ +\left[\left(1-\varphi \right)+\varphi \left[{\left(\rho \beta \right)}_s/{\left(\rho \beta \right)}_f\right]\right]\theta \left(\eta \right)cos\gamma \end{array}\right\}-\left(M^2+\frac{1}{Da}\right)f\left(\eta \right)=0$   (11)

$\left(1+\frac{4}{3}R\right){\theta }^{″}+3\left[\frac{1}{\left[\left(1-\varphi \right)+\varphi \left[{\left(\rho C_p\right)}_s/{\left(\rho C_p\right)}_f\right]\right]}\left[\frac{k_s+\left(m-1\right)k_f-\left(m-1\right)\varphi \left(k_f-k_s\right)}{k_s+\left(m-1\right)k_f+\varphi \left(k_f-k_s\right)}\right]\right]Prf{\theta }^{\prime }=0$   (12)

and the boundary conditions as

$f=0,f^{\prime }=0,\theta =1,when\eta =0$   (13)

$f^{\prime }=0,\theta =0,when\eta =\infty$   (14)

Basic concepts of differential transform method

The relatively new semi-analytical method, differential transformation method introduced by Zhou¹⁹ has proven proved very effective in providing highly accurate solutions to differential equations, difference equation, differential-difference equations, fractional differential equation, pantograph equation and integro-differential equation. Therefore, this method is applied to the present study. The basic definitions and the operational properties of the method are as follows

If $u(t)$  is analytic in the domain T, then the function $u(t)$  will be differentiated continuously with respect to time t.

$\frac{d^{p}u(t)}{d{t}^p}=\phi (t,p)$  for all  $t\in T$ (15)

for $t=t_i$ , then $\phi (t,p)=\phi (t_i,p)$  , where p belongs to the set of non-negative integers, denoted as the p-domain. We can therefore write Eq. (15) as

$U(p)=\phi (t_i,p)={\left[\frac{d^{p}u(t)}{d{t}^p}\right]}_{t=t_i}$   (16)

where $U_p$  is called the spectrum of $u(t)$  at

Expressing $u(t)$ in Taylor’s series as $t=t_i$

$u(t)={\displaystyle \sum _p^{\infty }\left[\frac{{\left(t-t_i\right)}^p}{p!}\right]}U(p)$   (17)

where Equ. (15) is the inverse of $U(k)$  us symbol ‘D’ denoting the differential transformation process and combining (16) and (17), we have

$u(t)={\displaystyle \sum _{p=0}^{\infty }\left[\frac{{\left(t-t_i\right)}^p}{p!}\right]}U(p)=D^{-1}U(p)$   (18)

Basic concepts of multi-step differential transform method

The limitation of classical DTM is shown when is being used for solving differential equations with the boundary conditions at infinity i.e. the obtained series solution through the DTM for such equation with the boundary condition become divergent. Besides that, generally, power series solutions are not useful for large values of the independent variable. In order to overcome this shortcoming, the multi-step DTM is developed (Table 4). The basic concepts of the multi-step DTM for solving non-linear initial-value problem is presented as follows;

$u\left(t,f,f^{\prime },\mathrm{...}{f}^{\left(h\right)}\right)=0,$   (19)

subject to the initial conditions

$f^{\left(k\right)}\left(0\right)=c_k,k=0,1,\mathrm{...}h-1.$   (20)

Let $\left[0,T\right]$ be the interval over which we want to find the solution of the initial value problem of Eq. (19). In actual application of the DTM, the approximate solution of the initial value problem of Eq. (19) can be expressed by the following finite series:

$f\left(t\right)={\displaystyle \sum _{m=0}^M{a}_m{t}^{m}t\in \left[0,T\right]}.$   (21)

The multi-step approach introduces a new idea for constructing the approximate solution. Assume that the interval $\left[0,T\right]$ is divided into N subintervals $\left[t_{i-1},t_i\right],i=1,2,\mathrm{...},N$ of equal step size $H=T/N$ by using the nodes $t=iH$ . The main idea of the multi-step DTM is as follows. First, we apply the DTM to Eq. (19) over the interval $\left[0,t_1\right],$ we will obtain the following approximate solution,

$f_1\left(t\right)={\displaystyle \sum _{m=0}^K{a}_{1m}{t}^{m}t\in \left[0,t_1\right],}$   (22)

Using the initial conditions $f^{\left(k\right)}\left(0\right)=c_k.$ For $i\ge 2$ and at each subinterval $\left[t_{i-1},t_i\right]$ we will use the initial conditions $f_i^{\left(k\right)}\left(t_{i-1}\right)=f_{i-1}^{\left(k\right)}\left(t_{i-1}\right)$ and apply the DTM to Eq. (19) over the interval $\left[t_{i-1},t_i\right],$ where $t_0$ in Eq. (15) is replaced by $t_{i-1}$ . The process is repeated and generates a sequence of approximate solution, $f_i\left(t\right),i=1,2,\dots ,N$ for the solution $f\left(t\right)$ ,

$f_i\left(t\right)={\displaystyle \sum _{m=0}^K{a}_{im}{\left(t-t_{i-1}\right)}^m,t\in \left[t_i,t_{i+1}\right],}$   (23)

where $M=K\cdot N$ . In fact, the multi-step DTM assumes the following solution:

$f\left(t\right)\{\begin{array}{l}{f}_1\left(t\right),t\in \left[0,t_1\right]\\ .\\ .\\ .\\ f_i\left(t\right),t\in \left[t_i,t_{i+1}\right]\\ .\\ .\\ .\\ f_N\left(t\right),t\in \left[t_{N-1},t_N\right]\end{array}$   (24)

which shows that there is a separate function for every sub domain.

Following the above definition, it could be stated that the multi-step DTM for every sub-domain is defined as

$F_i\left(m\right)=\frac{H^m}{m!}{\left[\frac{d^{m}u(t)}{d{t}^m}\right]}_{t=t_i}$   (25)

The inverse multi-step DTM is

$f_i(t)={\displaystyle \sum _{m=0}^{\infty }\left[\frac{{\left(t-t_i\right)}^m}{m!}\right]}{F}_i(m)$   (26)

The new algorithm, multi-step DTM is simple for computational performance for all values of H. It is easily observed that if the step size $H=T$ , then the multi-step DTM reduces to the classical DTM. Using the operational properties of the differential transformation method, the differential transformation of the governing differential Eq. (10) is given as

$\begin{array}{l}\left(p+1\right)\left(p+2\right)\left(p+3\right)F\left(p+3\right)\\ +{\left(1-\varphi \right)}^{2.5}\left\{\begin{array}{l}{\alpha }_1\left[3{\displaystyle \sum _{l=0}^p\left(p-l+1\right)\left(p-l+2\right)F\left(l\right)F\left(p-l+2\right)}-2{\displaystyle \sum _{l=0}^p\left(l+1\right)\left(p-l+1\right)F\left(l+1\right)F\left(p-l+1\right)}\right]\\ +{\alpha }_2\Theta \left(p\right)cos\gamma -\left(M^2+D{a}^{-1}\right)F\left(p\right)\end{array}\right\}=0\end{array}$   (27)

Eq. (27) can be written as

$F\left(p+3\right)=\frac{{\left(1-\varphi \right)}^{2.5}}{\left(p+1\right)\left(p+2\right)\left(p+3\right)}\left\{{\alpha }_1\left[\begin{array}{l}2{\displaystyle \sum _{l=0}^p\left(l+1\right)\left(p-l+1\right)F\left(l+1\right)F\left(p-l+1\right)}\\ -3{\displaystyle \sum _{l=0}^p\left(p-l+1\right)\left(p-l+2\right)F\left(l\right)F\left(p-l+2\right)}\end{array}\right]-{\alpha }_2\Theta \left(p\right)cos\gamma +\left(M^2+D{a}^{-1}\right)F\left(p\right)\right\}$   (28)

For the Eq.(11), we have the recursive relation in differential transform domain as

$\left(p+1\right)\left(p+2\right)\Theta \left(p+2\right)+\left\{3{\alpha }_{3}Pr{\displaystyle \sum _{l=0}^p\left(l+1\right)\Theta \left(l+1\right)F\left(p-l\right)}\right\}=0$   (29)

which can be written as

$\Theta \left(p+2\right)=\frac{-3{\alpha }_{3}Pr}{\left(p+1\right)\left(p+2\right)}\left\{{\displaystyle \sum _{l=0}^p\left(l+1\right)\Theta \left(l+1\right)F\left(p-l\right)}\right\}$   (30)

where

$\begin{array}{l}{\alpha }_1=\left(1-\varphi \right)+\varphi \left(\frac{{\rho }_s}{{\rho }_f}\right)\\ {\alpha }_2=\left[\left(1-\varphi \right)+\varphi \left[{\left(\rho \beta \right)}_s/{\left(\rho \beta \right)}_f\right]\right]\\ {\alpha }_3=\left[\frac{1}{\left[\left(1-\varphi \right)+\varphi \left[{\left(\rho C_p\right)}_s/{\left(\rho C_p\right)}_f\right]\right]}\left[\frac{k_s+\left(m-1\right)k_f-\left(m-1\right)\varphi \left(k_f-k_s\right)}{k_s+\left(m-1\right)k_f+\varphi \left(k_f-k_s\right)}\right]\right]\end{array}$

Also, recursive relation for the boundary conditions in Eq. (13) are

$\begin{array}{l}F\left(p\right)=0\Rightarrow F\left(0\right)=0,\left(p+1\right)F\left(p+1\right)=0\Rightarrow F(1)=0,\theta \left(p\right)=1\Rightarrow \theta \left(0\right)=1,\\ F\left(2\right)=a,\theta \left(1\right)=b,\end{array}$   (31)

where a and b are unknown constants which will be found later.

It should be noted that the transformations which included “a” and “b” are established from values of $f^{″}\left(0\right)\text{=}a\text{ and }{\theta }^{\prime }\left(0\right)=b$

From Eq. (29), we have the following boundary conditions in differential transform domain

$F\left[0\right]=0,F\left[1\right]=0,\theta \left[0\right]=1,F\left[2\right]=a,\theta \left[1\right]=b$   (32)

Using p=0, 1, 2, 3, 4, 5, 6, 7… in the above recursive relations in Eq. (29), we arrived at

$F\left[3\right]=-\frac{1}{6}{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma$

$F\left[4\right]=-\frac{1}{24}{\left(1-\varphi \right)}^{5/2}b{\alpha }_{2}cos\gamma$

$F\left[5\right]=\frac{a\left(2a{\left(1-\varphi \right)}^{5/2}{\alpha }_{1}Da+M^{2}Da+1\right)}{60Da}$

$F\left[6\right]=-\frac{\left(Da{M}^2+1\right){\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma }{720Da}$

$F\left[7\right]=-\frac{{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}bcos\gamma \left(\begin{array}{l}-40Da{\left(1-\varphi \right)}^{5/2}Rda{\alpha }_1-30{\left(1-\varphi \right)}^{5/2}{\alpha }_{1}aDa\\ +4Da{M}^{2}Rd-18Da\mathrm{Pr}a{\alpha }_3+3Da{M}^2+4Rd+3\end{array}\right)}{\left(15120+20160Rd\right)Da}$

$F\left[8\right]=\frac{1}{\left(120960+161280Rd\right)D{a}^2}\left(\begin{array}{l}6a+8D{a}^2{M}^{4}Rda+16Da{M}^{2}Rda-144{\left(1-\varphi \right)}^{5/2}{\alpha }_1{a}^{2}Da+312D{a}^2{a}^3{\varphi }^5{\alpha }_1{}^2\\ -1560D{a}^2{a}^3{\varphi }^4{\alpha }_1{}^2+3120D{a}^2{a}^3{\varphi }^3{\alpha }_1{}^2-3120D{a}^2{a}^3{\varphi }^2{\alpha }_1{}^2+1560D{a}^2{a}^3\varphi {\alpha }_1{}^2\\ -416D{a}^{2}Rd{a}^3{\alpha }_1{}^2-192D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}Rd{a}^2{\alpha }_1-15D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\alpha }_1{\alpha }_2{}^2\\ -9D{a}^2\mathrm{Pr}b{(}^c{\alpha }_2{}^2{\alpha }_3+100D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c\varphi {\alpha }_1{\alpha }_2{}^2\\ -200D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\varphi }^2{\alpha }_1{\alpha }_2{}^2-100D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\varphi }^4{\alpha }_1{\alpha }_2{}^2\\ +200D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\varphi }^3{\alpha }_1{\alpha }_2{}^2+20D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\varphi }^5{\alpha }_1{\alpha }_2{}^2\\ -192Da{\left(1-\varphi \right)}^{5/2}Rd{a}^2{\alpha }_1-144D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}{a}^2{\alpha }_1-4160D{a}^{2}Rd{a}^3{\varphi }^2{\alpha }_1{}^2\\ +2080D{a}^{2}Rd{a}^3\varphi {\alpha }_1{}^2+4160D{a}^{2}Rd{a}^3{\varphi }^3{\alpha }_1{}^2+416D{a}^{2}Rd{a}^3{\varphi }^5{\alpha }_1{}^2-2080D{a}^{2}Rd{a}^3{\varphi }^4{\alpha }_1{}^2\\ +8Rda+12Da{M}^{2}a-312D{a}^2{a}^3{\alpha }_1{}^2+6D{a}^2{M}^{4}a-20D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\alpha }_1{\alpha }_2{}^2\\ +75D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c\varphi {\alpha }_1{\alpha }_2{}^2+150D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\varphi }^3{\alpha }_1{\alpha }_2{}^2\\ -150D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\varphi }^2{\alpha }_1{\alpha }_2{}^2+15D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\varphi }^5{\alpha }_1{\alpha }_2{}^2\\ -75D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\varphi }^4{\alpha }_1{\alpha }_2{}^2+9D{a}^2\mathrm{Pr}b{(}^c{\varphi }^5{\alpha }_2{}^2{\alpha }_3-45D{a}^2\mathrm{Pr}b{(}^c{\varphi }^4{\alpha }_2{}^2{\alpha }_3\\ +90D{a}^2\mathrm{Pr}b{(}^c{\varphi }^3{\alpha }_2{}^2{\alpha }_3-90D{a}^2\mathrm{Pr}b{(}^c{\varphi }^2{\alpha }_2{}^2{\alpha }_3+45D{a}^2\mathrm{Pr}b{(}^c\varphi {\alpha }_2{}^2{\alpha }_3\end{array}\right)$

$F\left[9\right]=-\frac{{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma }{\left(1088640+1451520Rd\right)D{a}^2}\left(\begin{array}{l}3-252{\left(1-\varphi \right)}^{5/2}{\alpha }_{1}aDa+216D{a}^2{a}^2{\varphi }^5{\alpha }_1{}^2-1080D{a}^2{a}^2{\varphi }^4{\alpha }_1{}^2+2160D{a}^2{a}^2{\varphi }^3{\alpha }_1{}^2\\ -2160D{a}^2{a}^2{\varphi }^2{\alpha }_1{}^2+1080D{a}^2{a}^2\varphi {\alpha }_1{}^2-288D{a}^{2}Rd{a}^2{\alpha }_1{}^2-15D{a}^2{b}^{2}cos\gamma {\varphi }^5{\alpha }_1{\alpha }_2\\ +75D{a}^2{b}^{2}cos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-150D{a}^2{b}^{2}cos\gamma {\varphi }^3{\alpha }_1{\alpha }_2+150D{a}^2{b}^{2}cos\gamma {\varphi }^2{\alpha }_1{\alpha }_2\\ +20D{a}^{2}Rd{b}^{2}cos\gamma {\alpha }_1{\alpha }_2-75D{a}^2{b}^{2}cos\gamma \varphi {\alpha }_1{\alpha }_2-336D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}Rda{\alpha }_1\\ +8Da{M}^{2}Rd+200D{a}^{2}Rd{b}^{2}cos\gamma {\varphi }^2{\alpha }_1{\alpha }_2+100D{a}^{2}Rd{b}^{2}cos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-1440D{a}^{2}Rd{a}^2{\varphi }^4{\alpha }_1{}^2\\ -200D{a}^{2}Rd{b}^{2}cos\gamma {\varphi }^3{\alpha }_1{\alpha }_2+9D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}{b}^{2}cos\gamma {\alpha }_2{\alpha }_3-20D{a}^{2}Rd{b}^{2}cos\gamma {\varphi }^5{\alpha }_1{\alpha }_2\\ -100D{a}^{2}Rd{b}^{2}cos\gamma \varphi {\alpha }_1{\alpha }_2-336Da{\left(1-\varphi \right)}^{5/2}Rda{\alpha }_1+288D{a}^{2}Rd{a}^2{\varphi }^5{\alpha }_1{}^2\\ +15D{a}^2{b}^{2}cos\gamma {\alpha }_1{\alpha }_2-252D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}a{\alpha }_1-2880D{a}^{2}Rd{a}^2{\varphi }^2{\alpha }_1{}^2+1440D{a}^{2}Rd{a}^2\varphi {\alpha }_1{}^2\\ +2880D{a}^{2}Rd{a}^2{\varphi }^3{\alpha }_1{}^2+6M^{2}Da+3D{a}^2{M}^4+4D{a}^2{M}^{4}Rd-216D{a}^2{a}^2{\alpha }_1{}^2+4Rd\end{array}\right)$

$F\left[10\right]=-\frac{{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cosy}{3628800{\left(3+4Rd\right)}^{2}D{a}^2}\left(\begin{array}{l}-77760D{a}^{2}Rd{a}^{2}b\varphi {\alpha }_1{}^2+9b+5832D{a}^2{a}^{2}b{\alpha }_1{}^2+16D{a}^2{M}^{4}R{d}^{2}b+24D{a}^2{M}^{4}Rdb\\ +32Da{M}^{2}R{d}^{2}b+48Da{M}^{2}Rdb+378Dacos\gamma {\alpha }_1{\alpha }_2-378D{a}^2{M}^{2}cos\gamma {\varphi }^5{\alpha }_1{\alpha }_2\\ +1890D{a}^2{M}^{2}cos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-672DaR{d}^{2}cos\gamma {\varphi }^5{\alpha }_1{\alpha }_2-3780D{a}^2{M}^{2}cos\gamma {\varphi }^3{\alpha }_1{\alpha }_2\\ +3360DaR{d}^{2}cos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-1008DaRdcos\gamma {\varphi }^5{\alpha }_1{\alpha }_2+672D{a}^2{M}^{2}R{d}^{2}cos\gamma {\alpha }_1{\alpha }_2\\ -6720DaR{d}^{2}cos\gamma {\varphi }^3{\alpha }_1{\alpha }_2+5040DaRdcos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-10368D{a}^{2}R{d}^2{a}^{2}b{\varphi }^5{\alpha }_1{}^2\\ +51840D{a}^{2}R{d}^2{a}^{2}b{\varphi }^4{\alpha }_1{}^2-15552D{a}^{2}Rd{a}^{2}b{\varphi }^5{\alpha }_1{}^2-103680D{a}^{2}R{d}^2{a}^{2}b{\varphi }^3{\alpha }_1{}^2\\ +77760D{a}^{2}Rd{a}^{2}b{\varphi }^4{\alpha }_1{}^2+103680D{a}^{2}R{d}^2{a}^{2}b{\varphi }^2{\alpha }_1{}^2-155520D{a}^{2}Rd{a}^{2}b{\varphi }^3{\alpha }_1{}^2\\ -51840D{a}^{2}R{d}^2{a}^{2}b\varphi {\alpha }_1{}^2+155520D{a}^{2}Rd{a}^{2}b{\varphi }^2{\alpha }_1{}^2+5328D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}Rd{a}^{2}b{\alpha }_1{\alpha }_3\\ +3780Dacos\gamma {\varphi }^2{\alpha }_1{\alpha }_2+672DaR{d}^{2}cos\gamma {\alpha }_1{\alpha }_2+378D{a}^2{M}^{2}cos\gamma {\alpha }_1{\alpha }_2+3360D{a}^2{M}^{2}R{d}^{2}cos\gamma {\varphi }^4{\alpha }_1{\alpha }_2\\ -3780Dacos\gamma {\varphi }^3{\alpha }_1{\alpha }_2+1890Dacos\gamma {\varphi }^4{\alpha }_1{\alpha }_2+3240D{a}^2\stackrel{2}{\mathrm{Pr}}{a}^{2}b{\alpha }_3{}^2-378Dacos\gamma {\varphi }^5{\alpha }_1{\alpha }_2\\ +10080D{a}^2{M}^{2}Rdcos\gamma {\varphi }^2{\alpha }_1{\alpha }_2-3360D{a}^2{M}^{2}R{d}^{2}cos\gamma \varphi {\alpha }_1{\alpha }_2-10080D{a}^2{M}^{2}Rdcos\gamma {\varphi }^3{\alpha }_1{\alpha }_2\\ +6720D{a}^2{M}^{2}R{d}^{2}cos\gamma {\varphi }^2{\alpha }_1{\alpha }_2+5040D{a}^2{M}^{2}Rdcos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-6720D{a}^2{M}^{2}R{d}^{2}cos\gamma {\varphi }^3{\alpha }_1{\alpha }_2\\ -1008D{a}^2{M}^{2}Rdcos\gamma {\varphi }^5{\alpha }_1{\alpha }_2-3408Da{\left(1-\varphi \right)}^{5/2}Rdab{\alpha }_1+3780D{a}^2{M}^{2}cos\gamma {\varphi }^2{\alpha }_1{\alpha }_2\\ -2272Da{\left(1-\varphi \right)}^{5/2}R{d}^{2}ab{\alpha }_1-1278D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}ab{\alpha }_1-5040DaRdcos\gamma \varphi {\alpha }_1{\alpha }_2\\ -144Da\mathrm{Pr}Rdab{\alpha }_3+10080DaRdcos\gamma {\varphi }^2{\alpha }_1{\alpha }_2-3360DaR{d}^{2}cos\gamma \varphi {\alpha }_1{\alpha }_2-10080DaRdcos\gamma {\varphi }^3{\alpha }_1{\alpha }_2\\ +6720DaR{d}^{2}cos\gamma {\varphi }^2{\alpha }_1{\alpha }_2-1890D{a}^2{M}^{2}cos\gamma \varphi {\alpha }_1{\alpha }_2+1008D{a}^2{M}^{2}Rdcos\gamma {\alpha }_1{\alpha }_2\\ -672D{a}^2{M}^{2}R{d}^{2}cos\gamma {\varphi }^5{\alpha }_1{\alpha }_2-108D{a}^2{M}^2\mathrm{Pr}ab{\alpha }_3+9D{a}^2{M}^{4}b+18Da{M}^{2}b-1278Da{\left(1-\varphi \right)}^{5/2}ab{\alpha }_1\\ -108Da\mathrm{Pr}ab{\alpha }_3+1008DaRdcos\gamma {\alpha }_1{\alpha }_2-1890Dacos\gamma \varphi {\alpha }_1{\alpha }_2+58320D{a}^2{a}^{2}b{\varphi }^2{\alpha }_1{}^2-29160D{a}^2{a}^{2}b\varphi {\alpha }_1{}^2\\ +29160D{a}^2{a}^{2}b{\varphi }^4{\alpha }_1{}^2-58320D{a}^2{a}^{2}b{\varphi }^3{\alpha }_1{}^2-5832D{a}^2{a}^{2}b{\varphi }^5{\alpha }_1{}^2+10368D{a}^{2}R{d}^2{a}^{2}b{\alpha }_1{}^2\\ +15552D{a}^{2}Rd{a}^{2}b{\alpha }_1{}^2+16R{d}^{2}b+24Rdb-5040D{a}^2{M}^{2}Rdcos\gamma \varphi {\alpha }_1{\alpha }_2-144D{a}^2{M}^2\mathrm{Pr}Rdab{\alpha }_3\\ +3996D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}{a}^{2}b{\alpha }_1{\alpha }_3-3408D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}Rdab{\alpha }_1-2272D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}R{d}^{2}ab{\alpha }_1\end{array}\right)$

In the same manner, the expressions for F [11], F[12], F[13], F[14], F[15] are found but they are too large to be included in this paper.

Also, using p=0, 1, 2, 3… in the above recursive relations in Eq. (30), we arrived at following solutions

$\Theta \left[2\right]=0$

$\Theta \left[3\right]=0$

$\Theta \left[4\right]=-\frac{3}{4}\left(\frac{{\alpha }_3\mathrm{Pr}ab}{3+4Rd}\right)$

$\Theta \left[5\right]=\frac{3{\alpha }_3\mathrm{Pr}{\left(1-\varphi \right)}^{5/2}b{\alpha }_{2}cos\gamma }{120+160Rd}$

$\Theta \left[6\right]=\frac{{\alpha }_3\mathrm{Pr}{b}^2{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma }{240+320Rd}$

$\Theta \left[7\right]=-\frac{{\alpha }_3\mathrm{Pr}ba\left(\begin{array}{l}3+8Da{\left(1-\varphi \right)}^{5/2}Rda{\alpha }_1+6{\left(1-\varphi \right)}^{5/2}{\alpha }_{1}aDa\\ +4Da{M}^{2}Rd-180Da\mathrm{Pr}a{\alpha }_3+3Da{M}^2+4Rd\end{array}\right)}{280{\left(3+4Rd\right)}^{2}Da}$

$\Theta \left[8\right]=\frac{{\alpha }_3\mathrm{Pr}{\left(1-\varphi \right)}^{5/2}b{\alpha }_{2}cos\gamma \left(4Da{M}^{2}Rd-630Da\mathrm{Pr}a{\alpha }_3+3Da{M}^2+4Rd+3\right)}{4480{\left(3+4Rd\right)}^{2}Da}$

$\Theta \left[9\right]=\frac{{\alpha }_3\mathrm{Pr}b{\alpha }_{2}cos\gamma }{40320{\left(3+4Rd\right)}^{2}Da}\left(\begin{array}{l}4Da{M}^2{\left(1-\varphi \right)}^{5/2}Rdb-1026{\alpha }_3\mathrm{Pr}ba{\left(1-\varphi \right)}^{5/2}Da-315Da\mathrm{Pr}cos\gamma {\varphi }^5{\alpha }_2{\alpha }_3\\ +40DaRdab{\varphi }^5{\alpha }_1+3Da{M}^2{\left(1-\varphi \right)}^{5/2}b+1575Da\mathrm{Pr}cos\gamma {\varphi }^4{\alpha }_2{\alpha }_3\\ -200DaRdab{\varphi }^4{\alpha }_1+30Daab{\varphi }^5{\alpha }_1+150Daab\varphi {\alpha }_1-30Daab{\alpha }_1\\ -3150Da\mathrm{Pr}cos\gamma {\varphi }^3{\alpha }_2{\alpha }_3+400DaRdab{\varphi }^3{\alpha }_1-150Daab{\varphi }^4{\alpha }_1\\ +3150Da\mathrm{Pr}cos\gamma {\varphi }^2{\alpha }_2{\alpha }_3-400DaRdab{\varphi }^2{\alpha }_1+300Daab{\varphi }^3{\alpha }_1\\ +4{\left(1-\varphi \right)}^{5/2}Rdb-1575Da\mathrm{Pr}cos\gamma \varphi {\alpha }_2{\alpha }_3+200DaRdab\varphi {\alpha }_1\\ -300Daab{\varphi }^2{\alpha }_1+3b{\left(1-\varphi \right)}^{5/2}+315{\alpha }_3\mathrm{Pr}{\alpha }_{2}cos\gamma Da-40DaRdab{\alpha }_1\end{array}\right)$

$\Theta \left[10\right]=-\frac{{\alpha }_3\mathrm{Pr}b}{403200{\left(3+4Rd\right)}^{3}D{a}^2}\left(\begin{array}{l}-1152D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}Rd{a}^2{\alpha }_1-450D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\varphi }^2{\alpha }_1{\alpha }_2{}^2+18a-9072{\alpha }_3\mathrm{Pr}{a}^{2}Da\\ +48D{a}^2{M}^{4}Rda+96Da{M}^{2}Rda-432{\left(1-\varphi \right)}^{5/2}{\alpha }_1{a}^{2}Da+936D{a}^2{a}^3{\varphi }^5{\alpha }_1{}^2-4680D{a}^2{a}^3{\varphi }^4{\alpha }_1{}^2\\ +9360D{a}^2{a}^3{\varphi }^3{\alpha }_1{}^2-9360D{a}^2{a}^3{\varphi }^2{\alpha }_1{}^2+4680D{a}^2{a}^3\varphi {\alpha }_1{}^2-45D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\alpha }_1{\alpha }_2{}^2\\ -12096D{a}^2{M}^2\mathrm{Pr}Rd{a}^2{\alpha }_3-18144D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}{a}^3{\alpha }_1{\alpha }_3-768D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}R{d}^2{a}^2{\alpha }_1\\ -12096Da\mathrm{Pr}Rd{a}^2{\alpha }_3-9072D{a}^2{M}^2\mathrm{Pr}{a}^2{\alpha }_3+450D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\varphi }^3{\alpha }_1{\alpha }_2{}^2\\ -225D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\varphi }^4{\alpha }_1{\alpha }_2{}^2+45D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c{\varphi }^5{\alpha }_1{\alpha }_2{}^2+3429D{a}^2\mathrm{Pr}b{(}^c{\varphi }^5{\alpha }_2{}^2{\alpha }_3\\ +34290D{a}^2\mathrm{Pr}b{(}^c{\varphi }^3{\alpha }_2{}^2{\alpha }_3-34290D{a}^2\mathrm{Pr}b{(}^c{\varphi }^2{\alpha }_2{}^2{\alpha }_3-80D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^{2}b{(}^c{\alpha }_1{\alpha }_2{}^2\\ -4572D{a}^2\mathrm{Pr}Rdb{(}^c{\alpha }_2{}^2{\alpha }_3-17145D{a}^2\mathrm{Pr}b{(}^c{\varphi }^4{\alpha }_2{}^2{\alpha }_3-120D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\alpha }_1{\alpha }_2{}^2\\ -24192D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}Rd{a}^3{\alpha }_1{\alpha }_3+225D{a}^2{\left(1-\varphi \right)}^{5/2}b{(}^c\varphi {\alpha }_1{\alpha }_2{}^2+600D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c\varphi {\alpha }_1{\alpha }_2{}^2\\ -1200D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\varphi }^2{\alpha }_1{\alpha }_2{}^2+1200D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\varphi }^3{\alpha }_1{\alpha }_2{}^2\\ -600D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\varphi }^4{\alpha }_1{\alpha }_2{}^2+120D{a}^2{\left(1-\varphi \right)}^{5/2}Rdb{(}^c{\varphi }^5{\alpha }_1{\alpha }_2{}^2+32R{d}^{2}a\\ +18D{a}^2{M}^{4}a-1152Da{\left(1-\varphi \right)}^{5/2}Rd{a}^2{\alpha }_1-432D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}{a}^2{\alpha }_1+12480D{a}^{2}Rd{a}^3\varphi {\alpha }_1{}^2\\ -936D{a}^2{a}^3{\alpha }_1{}^2-3429D{a}^2\mathrm{Pr}b{(}^c{\alpha }_2{}^2{\alpha }_3+17145D{a}^2\mathrm{Pr}b{(}^c\varphi {\alpha }_2{}^2{\alpha }_3\\ +400D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^{2}b{(}^c\varphi {\alpha }_1{\alpha }_2{}^2+36Da{M}^{2}a+800D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^{2}b{(}^c{\varphi }^3{\alpha }_1{\alpha }_2{}^2\\ -800D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^{2}b{(}^c{\varphi }^2{\alpha }_1{\alpha }_2{}^2+80D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^{2}b{(}^c{\varphi }^5{\alpha }_1{\alpha }_2{}^2-400D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^{2}b{(}^c{\varphi }^4{\alpha }_1{\alpha }_2{}^2\\ -45720D{a}^2\mathrm{Pr}Rdb{(}^c{\varphi }^2{\alpha }_2{}^2{\alpha }_3+22860D{a}^2\mathrm{Pr}Rdb{(}^c\varphi {\alpha }_2{}^2{\alpha }_3+4572D{a}^2\mathrm{Pr}Rdb{(}^c{\varphi }^5{\alpha }_2{}^2{\alpha }_3\\ -22860D{a}^2\mathrm{Pr}Rdb{(}^c{\varphi }^4{\alpha }_2{}^2{\alpha }_3+45720D{a}^2\mathrm{Pr}Rdb{(}^c{\varphi }^3{\alpha }_2{}^2{\alpha }_3-24960D{a}^{2}Rd{a}^3{\varphi }^2{\alpha }_1{}^2+24960D{a}^{2}Rd{a}^3{\varphi }^3{\alpha }_1{}^2\\ -12480D{a}^{2}Rd{a}^3{\varphi }^4{\alpha }_1{}^2+2496D{a}^{2}Rd{a}^3{\varphi }^5{\alpha }_1{}^2+8320D{a}^{2}R{d}^2{a}^3\varphi {\alpha }_1{}^2+181440D{a}^2\stackrel{2}{\mathrm{Pr}}{a}^3{\alpha }_3{}^2+48Rda\\ +16640D{a}^{2}R{d}^2{a}^3{\varphi }^3{\alpha }_1{}^2-16640D{a}^{2}R{d}^2{a}^3{\varphi }^2{\alpha }_1{}^2-8320D{a}^{2}R{d}^2{a}^3{\varphi }^4{\alpha }_1{}^2-768Da{\left(1-\varphi \right)}^{5/2}R{d}^2{a}^2{\alpha }_1\\ +1664D{a}^{2}R{d}^2{a}^3{\varphi }^5{\alpha }_1{}^2-2496D{a}^{2}Rd{a}^3{\alpha }_1{}^2+64Da{M}^{2}R{d}^{2}a+32D{a}^2{M}^{4}R{d}^{2}a-1664D{a}^{2}R{d}^2{a}^3{\alpha }_1{}^2\end{array}\right)$

In the same manner, the expressions for Θ [11], Θ[12], Θ[13], Θ[14], Θ[15]… are found but they are too large to be included in this paper

From the definition in Eq. (18), the solutions of Eqs. (10) and (11) are given as

$\begin{array}{l}f\left(\eta \right)=F[0]+\eta F[1]+{\eta }^{2}F[2]+{\eta }^{3}F[3]++{\eta }^{4}F[4]+{\eta }^{5}F[5]\\ +{\eta }^{6}F[6]+{\eta }^{7}F[7]+{\eta }^{8}F[8]+{\eta }^{9}F[9]++{\eta }^{10}F[10]+\mathrm{...}\end{array}$   (33)

$\begin{array}{l}\theta \left(\eta \right)=\Theta [0]+\eta \Theta [1]+{\eta }^2\Theta [2]+{\eta }^3\Theta [3]++{\eta }^4\Theta [4]+{\eta }^5\Theta [5]\\ +{\eta }^6\Theta [6]+{\eta }^7\Theta [7]+{\eta }^8\Theta [8]+{\eta }^9\Theta [9]++{\eta }^{10}\Theta [10]+\mathrm{...}\end{array}$   (34)

Now, consider similar fully coupled third and second orders ordinary differential equations presented in Eqs. (11) and (12), but at this time, we take a=1 and b=1

${g^{‴}}_{}\left(\eta \right)+{\left(1-\varphi \right)}^{2.5}\left\{{\alpha }_1\left(3g\left(\eta \right){g^{″}}_{}\left(\eta \right)-2{\left({f^{\prime }}_{}\left(\eta \right)\right)}^2\right)+{\alpha }_2{\vartheta }_{}\left(\eta \right)cos\gamma -\left(M^2+D{a}^{-1}\right)g\left(\eta \right)\right\}=0$   (35)

${{\vartheta }^{″}}_{}\left({\eta }_{}\right)+3{\alpha }_{3}Prg\left({\eta }_{}\right){{\vartheta }^{\prime }}_{}\left({\eta }_{}\right)=0$   (36)

With initial conditions as

&$g=0,g^{\prime }=0,g^{″}=1,\vartheta =1,{\vartheta }^{\prime }=1when\eta =0$ nbsp; (37)

Following the similar solution procedures of Eqs. (10) and (11), the solutions of Eqs. (35) and (36) are

$\begin{array}{l}g\left(\eta \right)=G[0]+\eta G[1]+{\eta }^{2}G[2]+{\eta }^{3}G[3]++{\eta }^{4}G[4]+\eta GF[5]\\ +{\eta }^{6}G[6]+{\eta }^{7}G[7]+{\eta }^{8}G[8]+{\eta }^{9}G[9]++{\eta }^{10}G[10]+\mathrm{...}\end{array}$   (38)

$\begin{array}{l}\vartheta \left(\eta \right)=\Phi [0]+\eta \Phi [1]+{\eta }^2\Phi [2]+{\eta }^3\Phi [3]++{\eta }^4\Phi [4]+{\eta }^5\Phi [5]\\ +{\eta }^6\Phi [6]+{\eta }^7\Phi [7]+{\eta }^8\Phi [8]+{\eta }^9\Phi [9]++{\eta }^{10}\Phi [10]+\mathrm{..}\end{array}$   (39)

where

$G\left[0\right]=0,G\left[1\right]=0,,G\left[2\right]=\frac{1}{2},\theta \left[0\right]=1,\theta \left[1\right]=1$

$G\left[3\right]=-\frac{1}{6}{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma$

$G\left[4\right]=-\frac{1}{24}{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma$

$G\left[5\right]=\frac{\left({\left(1-\varphi \right)}^{5/2}{\alpha }_{1}Da+M^{2}Da+1\right)}{120Da}$

$G\left[6\right]=-\frac{\left(Da{M}^2+1\right){\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma }{720Da}$

$G\left[7\right]=-\frac{{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma \left(\begin{array}{l}-20Da{\left(1-\varphi \right)}^{5/2}Rd{\alpha }_1-15{\left(1-\varphi \right)}^{5/2}{\alpha }_{1}Da\\ +4Da{M}^{2}Rd-9Da\mathrm{Pr}{\alpha }_3+3Da{M}^2+4Rd+3\end{array}\right)}{\left(15120+20160Rd\right)Da}$

$G\left[8\right]=\frac{1}{\left(120960+161280Rd\right)D{a}^2}\left(\begin{array}{l}3+4D{a}^2{M}^{4}Rd+8Da{M}^{2}Rd-36{\left(1-\varphi \right)}^{5/2}{\alpha }_{1}Da+78D{a}^2{a}^3{\varphi }^5{\alpha }_1{}^2\\ -195D{a}^2{\varphi }^4{\alpha }_1{}^2+390D{a}^2{\varphi }^3{\alpha }_1{}^2-390D{a}^2{\varphi }^2{\alpha }_1{}^2+195D{a}^2\varphi {\alpha }_1{}^2\\ -52D{a}^{2}Rd{\alpha }_1{}^2-48D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}Rd{\alpha }_1-15D{a}^2{\left(1-\varphi \right)}^{5/2}{\left(cos\gamma \right)}^2{\alpha }_1{\alpha }_2{}^2\\ -9D{a}^2\mathrm{Pr}{\left(cos\gamma \right)}^2{\alpha }_2{}^2{\alpha }_3+100D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{\left(cos\gamma \right)}^2\varphi {\alpha }_1{\alpha }_2{}^2\\ -200D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{\left(cos\gamma \right)}^2{\varphi }^2{\alpha }_1{\alpha }_2{}^2-100D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{\left(cos\gamma \right)}^2{\varphi }^4{\alpha }_1{\alpha }_2{}^2\\ +200D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{\left(cos\gamma \right)}^2{\varphi }^3{\alpha }_1{\alpha }_2{}^2+20D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{\left(cos\gamma \right)}^2{\varphi }^5{\alpha }_1{\alpha }_2{}^2\\ -48Da{\left(1-\varphi \right)}^{5/2}Rd{\alpha }_1-36D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}{\alpha }_1-520D{a}^{2}Rd{\varphi }^2{\alpha }_1{}^2\\ +260D{a}^{2}Rd\varphi {\alpha }_1{}^2+520D{a}^{2}Rd{\varphi }^3{\alpha }_1{}^2+52D{a}^{2}Rd{\varphi }^5{\alpha }_1{}^2-260D{a}^{2}Rd{\varphi }^4{\alpha }_1{}^2\\ +4Rd+6Da{M}^2-39D{a}^2{\alpha }_1{}^2+3D{a}^2{M}^4-20D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{\left(cos\gamma \right)}^2{\alpha }_1{\alpha }_2{}^2\\ +75D{a}^2{\left(1-\varphi \right)}^{5/2}{\left(cos\gamma \right)}^2\varphi {\alpha }_1{\alpha }_2{}^2+150D{a}^2{\left(1-\varphi \right)}^{5/2}{\left(cos\gamma \right)}^2{\varphi }^3{\alpha }_1{\alpha }_2{}^2\\ -150D{a}^2{\left(1-\varphi \right)}^{5/2}{\left(cos\gamma \right)}^2{\varphi }^2{\alpha }_1{\alpha }_2{}^2+15D{a}^2{\left(1-\varphi \right)}^{5/2}{\left(cos\gamma \right)}^2{\varphi }^5{\alpha }_1{\alpha }_2{}^2\\ -75D{a}^2{\left(1-\varphi \right)}^{5/2}{\left(cos\gamma \right)}^2{\varphi }^4{\alpha }_1{\alpha }_2{}^2+9D{a}^2\mathrm{Pr}{\left(cos\gamma \right)}^2{\varphi }^5{\alpha }_2{}^2{\alpha }_3-45D{a}^2\mathrm{Pr}{\left(cos\gamma \right)}^2{\varphi }^4{\alpha }_2{}^2{\alpha }_3\\ +90D{a}^2\mathrm{Pr}{\left(cos\gamma \right)}^2{\varphi }^3{\alpha }_2{}^2{\alpha }_3-90D{a}^2\mathrm{Pr}{\left(cos\gamma \right)}^2{\varphi }^2{\alpha }_2{}^2{\alpha }_3+45D{a}^2\mathrm{Pr}{\left(cos\gamma \right)}^2\varphi {\alpha }_2{}^2{\alpha }_3\end{array}\right)$

$G\left[9\right]=-\frac{{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma }{\left(1088640+1451520Rd\right)D{a}^2}\left(\begin{array}{l}3-126{\left(1-\varphi \right)}^{5/2}{\alpha }_{1}Da+54D{a}^2{\varphi }^5{\alpha }_1{}^2-270D{a}^2{\varphi }^4{\alpha }_1{}^2+540D{a}^2{\varphi }^3{\alpha }_1{}^2\\ -540D{a}^2{\varphi }^2{\alpha }_1{}^2+270D{a}^2\varphi {\alpha }_1{}^2-72D{a}^{2}Rd{\alpha }_1{}^2-15D{a}^{2}cos\gamma {\varphi }^5{\alpha }_1{\alpha }_2\\ +75D{a}^{2}cos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-150D{a}^{2}cos\gamma {\varphi }^3{\alpha }_1{\alpha }_2+150D{a}^{2}cos\gamma {\varphi }^2{\alpha }_1{\alpha }_2\\ +20D{a}^{2}Rdcos\gamma {\alpha }_1{\alpha }_2-75D{a}^{2}cosy\varphi {\alpha }_1{\alpha }_2-168D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}Rd{a}_1\\ +4Da{M}^{2}Rd+200D{a}^{2}Rdcos\gamma {\varphi }^2{\alpha }_1{\alpha }_2+100D{a}^{2}Rdcos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-360D{a}^{2}Rd{\varphi }^4{\alpha }_1{}^2\\ -200D{a}^{2}Rdcos\gamma {\varphi }^3{\alpha }_1{\alpha }_2+9D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}cos\gamma {\alpha }_2{\alpha }_3-20D{a}^{2}Rdcos\gamma {\varphi }^5{\alpha }_1{\alpha }_2\\ -100D{a}^{2}Rdcos\gamma \varphi {\alpha }_1{\alpha }_2-168Da{\left(1-\varphi \right)}^{5/2}Rd{a}_1+72D{a}^{2}Rd{\varphi }^5{\alpha }_1{}^2\\ +15D{a}^{2}cos\gamma {\alpha }_1{\alpha }_2-126D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}{\alpha }_1-720D{a}^{2}Rd{\varphi }^2{\alpha }_1{}^2+360D{a}^{2}Rd\varphi 2^{}{\alpha }_1{}^2\\ +720D{a}^{2}Rd{\varphi }^3{\alpha }_1{}^2+6M^{2}Da+3D{a}^2{M}^4+4D{a}^2{M}^{4}Rd-54D{a}^2{\alpha }_1{}^2+4Rd\end{array}\right)$

$G\left[10\right]=-\frac{{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cosy}{3628800{\left(3+4Rd\right)}^{2}D{a}^2}\left(\begin{array}{l}-19440D{a}^{2}Rd\varphi {\alpha }_1{}^2+9b+1458D{a}^2{\alpha }_1{}^2+16D{a}^2{M}^{4}R{d}^2+24D{a}^2{M}^{4}Rd\\ +32Da{M}^{2}R{d}^2+48Da{M}^{2}Rd+378Dacos\gamma {\alpha }_1{\alpha }_2-378D{a}^2{M}^{2}cos\gamma {\varphi }^5{\alpha }_1{\alpha }_2\\ +1890D{a}^2{M}^{2}cos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-672DaR{d}^{2}cos\gamma {\varphi }^5{\alpha }_1{\alpha }_2-3780D{a}^2{M}^{2}cos\gamma {\varphi }^3{\alpha }_1{\alpha }_2\\ +3360DaR{d}^{2}cos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-1008DaRdcos\gamma {\varphi }^5{\alpha }_1{\alpha }_2+672D{a}^2{M}^{2}R{d}^{2}cos\gamma {\alpha }_1{\alpha }_2\\ -6720DaR{d}^{2}cos\gamma {\varphi }^3{\alpha }_1{\alpha }_2+5040DaRdcos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-2592D{a}^{2}R{d}^2{\varphi }^5{\alpha }_1{}^2\\ +12960D{a}^{2}R{d}^2{\varphi }^4{\alpha }_1{}^2-3888D{a}^{2}Rd{\varphi }^5{\alpha }_1{}^2-25920D{a}^{2}R{d}^2{\varphi }^3{\alpha }_1{}^2\\ +77760D{a}^{2}Rd{a}^2{\varphi }^4{\alpha }_1{}^2+25920D{a}^{2}R{d}^2{\alpha }_1{}^2-77760D{a}^{2}Rd{a}^2{\alpha }_1{}^2\\ -12960D{a}^{2}R{d}^2\varphi {\alpha }_1{}^2+38880D{a}^{2}Rd{\varphi }^2{\alpha }_1{}^2+1332D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}Rd{\alpha }_1{\alpha }_3\\ +3780Dacos\gamma {\varphi }^2{\alpha }_1{\alpha }_2+672DaR{d}^{2}cos\gamma {\alpha }_1{\alpha }_2+378D{a}^2{M}^{2}cos\gamma {\alpha }_1{\alpha }_2+3360D{a}^2{M}^{2}R{d}^{2}cos\gamma {\varphi }^4{\alpha }_1{\alpha }_2\\ -3780Dacos\gamma {\varphi }^3{\alpha }_1{\alpha }_2+1890Dacos\gamma {\varphi }^4{\alpha }_1{\alpha }_2+810D{a}^2\stackrel{2}{\mathrm{Pr}}b{\alpha }_3{}^2-378Dacos\gamma {\varphi }^5{\alpha }_1{\alpha }_2\\ +10080D{a}^2{M}^{2}Rdcos\gamma {\varphi }^2{\alpha }_1{\alpha }_2-3360D{a}^2{M}^{2}R{d}^{2}cosy\varphi {\alpha }_1{\alpha }_2-10080D{a}^2{M}^{2}Rdcos\gamma {\varphi }^3{\alpha }_1{\alpha }_2\\ +6720D{a}^2{M}^{2}R{d}^{2}cos\gamma {\varphi }^2{\alpha }_1{\alpha }_2+5040D{a}^2{M}^{2}Rdcos\gamma {\varphi }^4{\alpha }_1{\alpha }_2-6720D{a}^2{M}^{2}R{d}^{2}cos\gamma {\varphi }^3{\alpha }_1{\alpha }_2\\ -1008D{a}^2{M}^{2}Rdcos\gamma {\varphi }^5{\alpha }_1{\alpha }_2-1704Da{\left(1-\varphi \right)}^{5/2}Rd{\alpha }_1+3780D{a}^2{M}^{2}cos\gamma {\varphi }^2{\alpha }_1{\alpha }_2\\ -1138Da{\left(1-\varphi \right)}^{5/2}R{d}^2{\alpha }_1-639D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}{\alpha }_1-5040DaRdcos\gamma \varphi {\alpha }_1{\alpha }_2\\ -72Da\mathrm{Pr}Rd{\alpha }_3+10080DaRdcos\gamma {\varphi }^2{\alpha }_1{\alpha }_2-3360DaR{d}^{2}cos\gamma \varphi {\alpha }_1{\alpha }_2-10080DaRdcos\gamma {\varphi }^3{\alpha }_1{\alpha }_2\\ +6720DaR{d}^{2}cos\gamma {\varphi }^2{\alpha }_1{\alpha }_2-1890D{a}^2{M}^{2}cos\gamma \varphi {\alpha }_1{\alpha }_2+1008D{a}^2{M}^{2}Rdcos\gamma {\alpha }_1{\alpha }_2\\ -672D{a}^2{M}^{2}R{d}^{2}cos\gamma {\varphi }^5{\alpha }_1{\alpha }_2-54D{a}^2{M}^2\mathrm{Pr}{\alpha }_3+9D{a}^2{M}^4+18Da{M}^2-639Da{\left(1-\varphi \right)}^{5/2}{\alpha }_1\\ -54Da\mathrm{Pr}{\alpha }_3+1008DaRdcos\gamma {\alpha }_1{\alpha }_2-1890Dacos\gamma \varphi {\alpha }_1{\alpha }_2+14580D{a}^{2}b{\varphi }^2{\alpha }_1{}^2-7290D{a}^2\varphi {\alpha }_1{}^2\\ +7290D{a}^2{\varphi }^4{\alpha }_1{}^2-14580D{a}^{2}b{\varphi }^3{\alpha }_1{}^2-1458D{a}^2{\varphi }^5{\alpha }_1{}^2+2592D{a}^{2}R{d}^2{\alpha }_1{}^2\\ +3888D{a}^{2}Rd{\alpha }_1{}^2+16R{d}^2+24Rd-5040D{a}^2{M}^{2}Rdcos\gamma \varphi {\alpha }_1{\alpha }_2-72D{a}^2{M}^2\mathrm{Pr}Rd{\alpha }_3\\ +999D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}{\alpha }_1{\alpha }_3-1704D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}Rd{\alpha }_1-1388D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}R{d}^2{\alpha }_1\end{array}\right)$

Similarly

$\Phi \left[2\right]=0$

$\Phi \left[3\right]=0$

$\Phi \left[4\right]=-\frac{3}{8}\left(\frac{{\alpha }_3\mathrm{Pr}}{3+4Rd}\right)$

$\Phi \left[5\right]=\frac{3{\alpha }_3\mathrm{Pr}{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma }{120+160Rd}$

$\Phi \left[6\right]=\frac{{\alpha }_3\mathrm{Pr}{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cos\gamma }{240+320Rd}$

$\Phi \left[7\right]=-\frac{{\alpha }_3\mathrm{Pr}\left(\begin{array}{l}3+4Da{\left(1-\varphi \right)}^{5/2}Rd{\alpha }_1+3{\left(1-\varphi \right)}^{5/2}{\alpha }_{1}Da\\ +4Da{M}^{2}Rd-90Da\mathrm{Pr}{\alpha }_3+3Da{M}^2+4Rd\end{array}\right)}{560{\left(3+4Rd\right)}^{2}Da}$

$\Phi \left[8\right]=\frac{{\alpha }_3\mathrm{Pr}{\left(1-\varphi \right)}^{5/2}{\alpha }_{2}cosy\left(4Da{M}^{2}Rd-315Da\mathrm{Pr}{\alpha }_3+3Da{M}^2+4Rd+3\right)}{4480{\left(3+4Rd\right)}^{2}Da}$

$\Phi \left[9\right]=\frac{{\alpha }_3\mathrm{Pr}{\alpha }_{2}cos\gamma }{40320{\left(3+4Rd\right)}^{2}Da}\left(\begin{array}{l}4Da{M}^2{\left(1-\varphi \right)}^{5/2}Rd-516{\alpha }_3\mathrm{Pr}b{\left(1-\varphi \right)}^{5/2}Da-315Da\mathrm{Pr}cos\gamma {\varphi }^5{\alpha }_2{\alpha }_3\\ +20DaRd{\varphi }^5{\alpha }_1+3Da{M}^2{\left(1-\varphi \right)}^{5/2}+1575Da\mathrm{Pr}cos\gamma {\varphi }^4{\alpha }_2{\alpha }_3\\ -100DaRd{\varphi }^4{\alpha }_1+15Da{\varphi }^5{\alpha }_1+75Da\varphi {\alpha }_1-15Da{\alpha }_1\\ -3150Da\mathrm{Pr}cos\gamma {\varphi }^3{\alpha }_2{\alpha }_3+200DaRd{\varphi }^3{\alpha }_1-75Da{\varphi }^4{\alpha }_1\\ +3150Da\mathrm{Pr}cos\gamma {\varphi }^2{\alpha }_2{\alpha }_3-200DaRd{\varphi }^2{\alpha }_1+150Da{\varphi }^3{\alpha }_1\\ +4{\left(1-\varphi \right)}^{5/2}Rd-1575Da\mathrm{Pr}cos\gamma \varphi {\alpha }_2{\alpha }_3+100DaRd\varphi {\alpha }_1\\ -150Da{\varphi }^2{\alpha }_1+3{\left(1-\varphi \right)}^{5/2}+315{\alpha }_3\mathrm{Pr}{\alpha }_{2}cos\gamma Da-20DaRd{\alpha }_1\end{array}\right)$

$\Phi \left[10\right]=-\frac{{\alpha }_3\mathrm{Pr}b}{403200{\left(3+4Rd\right)}^{3}D{a}^2}\left(\begin{array}{l}-1152D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}Rd{a}^2{\alpha }_1-450D{a}^2{\left(1-\varphi \right)}^{5/2}{(}^c{\varphi }^2{\alpha }_1{\alpha }_2{}^2+9-2268{\alpha }_3\mathrm{Pr}Da\\ +24D{a}^2{M}^{4}Rd+48Da{M}^{2}Rd-108{\left(1-\varphi \right)}^{5/2}{\alpha }_{1}Da+117D{a}^2{\varphi }^5{\alpha }_1{}^2-585D{a}^2{\varphi }^4{\alpha }_1{}^2\\ +1170D{a}^2{\varphi }^3{\alpha }_1{}^2-1170D{a}^2{\varphi }^2{\alpha }_1{}^2+585D{a}^2\varphi {\alpha }_1{}^2-45D{a}^2{\left(1-\varphi \right)}^{5/2}{(}^c{\alpha }_1{\alpha }_2{}^2\\ -3024D{a}^2{M}^2\mathrm{Pr}Rd{\alpha }_3-2268D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}{\alpha }_1{\alpha }_3-192D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}R{d}^2{\alpha }_1\\ -3024Da\mathrm{Pr}Rd{\alpha }_3-2268D{a}^2{M}^2\mathrm{Pr}{\alpha }_3+450D{a}^2{\left(1-\varphi \right)}^{5/2}{(}^c{\varphi }^3{\alpha }_1{\alpha }_2{}^2\\ -225D{a}^2{\left(1-\varphi \right)}^{5/2}{(}^c{\varphi }^4{\alpha }_1{\alpha }_2{}^2+45D{a}^2{\left(1-\varphi \right)}^{5/2}{(}^c{\varphi }^5{\alpha }_1{\alpha }_2{}^2+3429D{a}^2\mathrm{Pr}{(}^c{\varphi }^5{\alpha }_2{}^2{\alpha }_3\\ +34290D{a}^2\mathrm{Pr}{(}^c{\varphi }^3{\alpha }_2{}^2{\alpha }_3-34290D{a}^2\mathrm{Pr}{(}^c{\varphi }^2{\alpha }_2{}^2{\alpha }_3-80D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^2{\left(cosy\right)}^2{\alpha }_1{\alpha }_2{}^2\\ -4572D{a}^2\mathrm{Pr}Rd{(}^c{\alpha }_2{}^2{\alpha }_3-17145D{a}^2\mathrm{Pr}{(}^c{\varphi }^4{\alpha }_2{}^2{\alpha }_3-120D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{(}^c{\alpha }_1{\alpha }_2{}^2\\ -3024D{a}^2{\left(1-\varphi \right)}^{5/2}\mathrm{Pr}Rd{\alpha }_1{\alpha }_3+225D{a}^2{\left(1-\varphi \right)}^{5/2}{(}^c\varphi {\alpha }_1{\alpha }_2{}^2+600D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{(}^c\varphi {\alpha }_1{\alpha }_2{}^2\\ -1200D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{(}^c{\varphi }^2{\alpha }_1{\alpha }_2{}^2+1200D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{(}^c{\varphi }^3{\alpha }_1{\alpha }_2{}^2\\ -600D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{(}^c{\varphi }^4{\alpha }_1{\alpha }_2{}^2+120D{a}^2{\left(1-\varphi \right)}^{5/2}Rd{(}^c{\varphi }^5{\alpha }_1{\alpha }_2{}^2+16R{d}^2\\ +9D{a}^2{M}^4-288Da{\left(1-\varphi \right)}^{5/2}Rd{\alpha }_1-108D{a}^2{M}^2{\left(1-\varphi \right)}^{5/2}{\alpha }_1+1560D{a}^{2}Rd\varphi {\alpha }_1{}^2\\ -117D{a}^2{\alpha }_1{}^2-3429D{a}^2\mathrm{Pr}{(}^c{\alpha }_2{}^2{\alpha }_3+17145D{a}^2\mathrm{Pr}{(}^c\varphi {\alpha }_2{}^2{\alpha }_3\\ +400D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^2{(}^c\varphi {\alpha }_1{\alpha }_2{}^2+36Da{M}^{2}a+800D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^2{(}^c{\varphi }^3{\alpha }_1{\alpha }_2{}^2\\ -800D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^2{(}^c{\varphi }^2{\alpha }_1{\alpha }_2{}^2+80D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^2{(}^c{\varphi }^5{\alpha }_1{\alpha }_2{}^2-400D{a}^2{\left(1-\varphi \right)}^{5/2}R{d}^2{(}^c{\varphi }^4{\alpha }_1{\alpha }_2{}^2\\ -45720D{a}^2\mathrm{Pr}Rdb{(}^c{\varphi }^2{\alpha }_2{}^2{\alpha }_3+22860D{a}^2\mathrm{Pr}Rdbcos{y}^2\varphi {\alpha }_2{}^2{\alpha }_3+4572D{a}^2\mathrm{Pr}Rdbcos{y}^2{\varphi }^5{\alpha }_2{}^2{\alpha }_3\\ -22860D{a}^2\mathrm{Pr}Rd{(}^c{\varphi }^4{\alpha }_2{}^2{\alpha }_3+45720D{a}^2\mathrm{Pr}Rd{(}^c{\varphi }^3{\alpha }_2{}^2{\alpha }_3-3120D{a}^{2}Rd{\varphi }^2{\alpha }_1{}^2+3120D{a}^{2}Rd{\varphi }^3{\alpha }_1{}^2\\ -1560D{a}^{2}Rd{\varphi }^4{\alpha }_1{}^2+312D{a}^{2}Rd{\varphi }^5{\alpha }_1{}^2+1040D{a}^{2}R{d}^2\varphi {\alpha }_1{}^2+22680D{a}^2{\mathrm{Pr}}^2{a}^3{\alpha }_3{}^2+24Rd\\ +2080D{a}^{2}R{d}^2{\varphi }^3{\alpha }_1{}^2-2080D{a}^{2}R{d}^2{a}^3{\varphi }^2{\alpha }_1{}^2-1040D{a}^{2}R{d}^2{\varphi }^4{\alpha }_1{}^2-192Da{\left(1-\varphi \right)}^{5/2}R{d}^2{\alpha }_1\\ +208D{a}^{2}R{d}^2{\varphi }^5{\alpha }_1{}^2-312D{a}^{2}Rd{\alpha }_1{}^2+32Da{M}^{2}R{d}^2+16D{a}^2{M}^{4}R{d}^2-208D{a}^{2}R{d}^2{\alpha }_1{}^2\end{array}\right)$

The functions in Eq. (33) and (34) and that in Eq. (38) and (39) have relations as follows:

$f\left(\eta \right)=a^{k}g\left(a^q\eta \right)\to f^{\prime }\left(\eta \right)=a^{k+q}{g}^{\prime }\left(a^q\eta \right)\to f^{\prime }\left(\infty \right)=a^{k+q}{g}^{\prime }\left(\infty \right)$   (40)

and

$\theta \left(\eta \right)=b^r\vartheta \left(b^s\eta \right)\to \theta \left(\infty \right)=b^r\vartheta \left(\infty \right)$   (41)

From Eq. (12),  $f^{\prime }\left(\infty \right)\text{=0 and }\theta \left(\infty \right)=0$

Since  $a\ne 0\text{ and }b\ne 0\to g^{\prime }\left(\infty \right)=0\text{ and }\vartheta \left(\infty \right)=0$

Applying multi-step DTM

To solve the boundary layer problems, the domain [0,∞) is replaced by [0, η ͚). But η ͚ should be great enough that the solution is not dependent on. The solution domain should be divided to N equal parts (H= η ͚ /N). So, we have

${g^{‴}}_i\left({\eta }_i\right)+{\left(1-\varphi \right)}^{2.5}\left\{\begin{array}{l}\left[\left(1-\varphi \right)+\varphi \left(\frac{{\rho }_s}{{\rho }_f}\right)\right]\left(3g_i\left({\eta }_i\right){g^{″}}_i\left({\eta }_i\right)-2{\left({g^{\prime }}_i\left({\eta }_i\right)\right)}^2\right)\\ +\left[\left(1-\varphi \right)+\varphi \left[{\left(\rho \beta \right)}_s/{\left(\rho \beta \right)}_f\right]\right]{\vartheta }_i\left({\eta }_i\right)cos\gamma -\left(M^2+D{a}^{-1}\right)g_i\left({\eta }_i\right)\end{array}\right\}=0$   (42)

$\left(i-1\right)H\le {\eta }_i<iH,fori\le i\le N$

${{\vartheta }^{″}}_i\left({\eta }_i\right)+3\left[\frac{1}{\left[\left(1-\varphi \right)+\varphi \left[{\left(\rho C_p\right)}_s/{\left(\rho C_p\right)}_f\right]\right]}\left[\frac{k_s+\left(m-1\right)k_f-\left(m-1\right)\varphi \left(k_f-k_s\right)}{k_s+\left(m-1\right)k_f+\varphi \left(k_f-k_s\right)}\right]\right]Pr{g}_i\left({\eta }_i\right){{\vartheta }^{\prime }}_i\left({\eta }_i\right)=0$   (43)

$\left(i-1\right)H\le {\eta }_i<iH,fori\le i\le N$

Applying multi-step DTM on Eq. (42) and Eq. (43)

$G_i\left(p+3\right)=\frac{{\left(1-\varphi \right)}^{2.5}{H}^3}{\left(p+1\right)\left(p+2\right)\left(p+3\right)}\left\{\begin{array}{l}{\alpha }_1\left[\begin{array}{l}2{\displaystyle \sum _{l=0}^p\frac{\left(l+1\right)}{H}{G}_i\left(l+1\right)\frac{\left(p-l+1\right)}{H}{G}_i\left(p-l+1\right)}\\ -3{\displaystyle \sum _{l=0}^p{G}_i\left(l\right)\frac{\left(p-l+1\right)\left(p-l+2\right)}{H^2}{G}_i\left(p-l+2\right)}\end{array}\right]\\ -{\alpha }_2{\vartheta }_i\left(p\right)cos\gamma -\left(M^2+D{a}^{-1}\right)G_i\left(p\right)\end{array}\right\}$   (44)

$fori\le i\le N$

${\vartheta }_i\left(p+2\right)=\frac{-3H^{2}Pr}{\left(p+1\right)\left(p+2\right)}\left\{{\alpha }_3{\displaystyle \sum _{l=0}^p\frac{\left(l+1\right)}{H}{\vartheta }_i\left(l+1\right)G_i\left(p-l\right)}\right\}$   (45)

$fori\le i\le N$

The initial conditions for the problem are considered for the first sub domain (i =1). Followng Eq. (24), the differential transform for the initial conditions for Eq. (35) and (36) and for Eqs. (44) and (45) are

$\begin{array}{l}{G}_1\left(0\right)=g_1\left(0\right)=0,G_1\left(1\right)=H{g^{\prime }}_1\left(0\right)=0,G_1\left(2\right)=\frac{H^2}{2}{g^{″}}_1=\frac{H^2}{2},\\ {\Psi }_1\left(0\right)={\vartheta }_1\left(0\right)=1,{\Psi }_1\left(1\right)=H{\vartheta }^{\prime }\left(0\right)=H\end{array}$   (46)

The boundary conditions of each subdomain are continuity of the

$g_i\left({\eta }_i\right),{g^{\prime }}_i\left({\eta }_i\right),g^{″}\left({\eta }_i\right),\vartheta \left({\eta }_i\right),and{\vartheta }^{\prime }\left({\eta }_i\right)$   (47)

These boundary conditions can be obtained from Eq. (26):

$\begin{array}{l}{g}_i\left({\eta }_{i+1}\right)={\displaystyle \sum _{k=0}^m{G}_i}\left(k\right)\\ g_{i+1}\left({\eta }_{i+1}\right)=G_{i+1}\left(0\right)\to G_{i+1}\left(0\right)={\displaystyle \sum _{k=0}^m{G}_i}\left(k\right)\end{array}$   (48a)

$\begin{array}{l}{g^{\prime }}_i\left({\eta }_{i+1}\right)={\displaystyle \sum _{k=1}^m\frac{k}{H}{G}_i}\left(k\right)\\ {g^{\prime }}_i\left({\eta }_{i+1}\right)=\frac{G_{i+1}\left(1\right)}{H}\to G_{i+1}\left(1\right)={\displaystyle \sum _{k=1}^{m}k{G}_i}\left(k\right)\end{array}$   (48b)

$\begin{array}{l}{g^{″}}_i\left({\eta }_{i+1}\right)={\displaystyle \sum _{k=2}^m\frac{k\left(k-1\right)}{H^2}{G}_i}\left(k\right)\\ {g^{″}}_i\left({\eta }_{i+1}\right)=\frac{2G_{i+1}\left(2\right)}{H^2}\to G_{i+1}\left(2\right)=\frac{1}{2}{\displaystyle \sum _{k=1}^{m}k\left(k-1\right)G_i}\left(k\right)\end{array}$   (48c)

$\begin{array}{l}{\vartheta }_i\left({\eta }_{i+1}\right)={\displaystyle \sum _{k=0}^m{\Psi }_i}\left(k\right)\\ {\vartheta }_{i+1}\left({\eta }_{i+1}\right)={\Psi }_{i+1}\left(0\right)\to {\Psi }_{i+1}\left(0\right)={\displaystyle \sum _{k=0}^m{\Psi }_i}\left(k\right)\end{array}$   (49a)

$\begin{array}{l}{{\vartheta }^{\prime }}_i\left({\eta }_{i+1}\right)={\displaystyle \sum _{k=1}^m\frac{k}{H}{\Psi }_i}\left(k\right)\\ {{\vartheta }^{\prime }}_i\left({\eta }_{i+1}\right)=\frac{{\Psi }_{i+1}\left(1\right)}{H}\to {\Psi }_{i+1}\left(1\right)={\displaystyle \sum _{k=1}^{m}k{\Psi }_i}\left(k\right)\end{array}$   (49b)

The values of the $g^{\prime }\left({\eta }_{\infty }\right)\text{ and }\vartheta \left({\eta }_{\infty }\right)$ can be calculated by differentiating from Eq. (24)

$\begin{array}{l}{g^{\prime }}_i\left(\infty \right)\simeq {g^{\prime }}_i\left({\eta }_{\infty }\right)={g^{\prime }}_i\left({\eta }_{N+1}\right)={\displaystyle \sum _{k=1}^m\frac{k}{H}{G}_N}\left(k\right)\\ {\vartheta }_i\left(\infty \right)\simeq {\vartheta }_i\left({\eta }_{\infty }\right)={\vartheta }_i\left({\eta }_{N+1}\right)={\displaystyle \sum _{k=0}^m{\Psi }_N}\left(k\right)\end{array}$   (50)

Now, Eq. (11) and (12) are solved with a similar process like Eqs. (35) and (36) using multi-step DTM. The only difference is that the condition $f^{″}\left(0\right)=a$ is replaced by the condition $g^{″}\left(0\right)=1$

It should be noted as mentioned previously that the unknown parameters “a” and “b” in the solutions are unknown constants. The infinite boundary conditions i.e. $\eta \to \infty ,f^{\prime }=0,\theta =0$ are applied. The resulting simultaneous equations are solved to obtain the values of “a” and “b” for the respective values of the physical and thermal properties of the nanofluids under considerations (Figures 1–4). 

Flow and heat transfer parameters

The determinations of other physically important quantities such as shear stress, drag, heat transfer rate and heat transfer coefficient) associated with the free convection flow and heat transfer problem are very important in the design of equipment. Therefore, in this analysis physically important quantities are computed. The local heat transfer coefficient at the surface of the vertical plate can be obtained from

Fluid flow parameter 

Skin friction coefficient

$c_f=\frac{{\tau }_w}{{\rho }_{nf}{u}^2}=\frac{{\mu }_{nf}{\frac{\partial u}{\partial y}|}_{y=0}}{{\rho }_{nf}{u}^2}=\frac{{\mu }_{nf}{\left(\frac{\partial u}{\partial \eta }\cdot \frac{\partial \eta }{\partial y}\right)|}_{y=0}}{{\rho }_{nf}{u}^2}$   (51)

The dimensionless form of the skin friction coefficient,

$\begin{array}{l}{c}_f{\left(R{e}_x\right)}^{1/2}=\frac{f^{″}\left(0\right)}{{\left(1-\varphi \right)}^{2.5}}\\ c_f{\left(R{e}_x\right)}^{1/2}\frac{{\tau }_w}{{\left(4G{r}_x^3\right)}^{1/4}\left(\nu \mu \right)}=f^{″}\left(0\right)\frac{f^{″}\left(0\right)}{{\left(1-\varphi \right)}^{2.5}}\end{array}$   (52)

Heat transfer parameter

$h_x=-\frac{k_{nf}}{T_w-T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}_{y=0}=-k_{nf}{\theta }^{\prime }\left(0\right)\frac{1}{x}{\left(\frac{1}{4}G{r}_x\right)}^{1/4}$   (53)

The local nusselt number

The local Nusselt number is

$\begin{array}{l}N{u}_x=\frac{h_{x}x}{k_{nf}}=-\left(\frac{x}{T_w-T_{\infty }}\right){\left(\frac{\partial T}{\partial y}\right)}_{y=0}=-{\theta }^{\prime }\left(0\right){\left(\frac{1}{4}G{r}_x\right)}^{1/4}\\ N{u}_x=-\frac{{\theta }^{\prime }\left(0\right)}{\sqrt{2}}G{r}_x^{1/4}=f\left(\left(Pr\right)G{r}_x^{1/4}\right)\end{array}$   (54)

where $\varphi \left(Pr\right)=-\frac{{\theta }^{\prime }\left(0\right)}{\sqrt{2}}$ is a function of Prandtl number. The dependence of ϕ on the Prandtl number is evidenced by Eq. (54).

It could also be shown that

$\frac{N{u}_x}{{\left(R{e}_x\right)}^{1/2}}=-\frac{k_{nf}}{k_f}{\theta }^{\prime }\left(0\right)=-\left[\frac{k_s+\left(m-1\right)k_f-\left(m-1\right)\varphi \left(k_f-k_s\right)}{k_s+\left(m-1\right)k_f+\varphi \left(k_f-k_s\right)}\right]{\theta }^{\prime }\left(0\right)$   (55)

where $R{e}_x$  and $G{r}_x$ are the local Reynold and Grashof numbers defined as:

$R{e}_x=\frac{ux}{{\nu }_{nf}}\text{ and }G{r}_x=\frac{g\beta \left(T_w-T_{\infty }\right)x^3}{{\nu }^3}$