## 2 Governing Equations
Further, \(\rho_{uf}\), \(\mu_{uf}\), \(\sigma_{uf}\), \(\left(\rho C_{{}_{P}}\right)_{uf}\) and \(k_{uf}\) denote the density, dynamic viscosity, electric conductivity, heat capacity and the thermal conductivity of the fluid, respectively [27, 28, 29]; where,  
The Maxwell-Garnett model for the effective thermal conductivity of nanofluids is used in this problem as [14]:  
\[\frac{k_{uf}}{k_{{}_{f}}}=\frac{k_{s}+2k_{f}-2\varphi\left(k_{f}-k_{s}\right) }{k_{s}+2k_{f}+\varphi\left(k_{f}-k_{s}\right)} \tag{6}\]  
As proposed by [PERSON] [18], the effective electrical conductivity of the nanofluid can be written as:  
\[\frac{\sigma_{uf}}{\sigma_{{}_{f}}}=1+\frac{3\left(\frac{\sigma_{s}}{\sigma_{{ }_{f}}}-1\right)\varphi}{\left(\frac{\sigma_{{}_{s}}}{\sigma_{{}_{f}}}+2 \right)-\left(\frac{\sigma_{{}_{s}}}{\sigma_{{}_{f}}}-1\right)\varphi} \tag{7}\]  
Here, \(\mu_{f}\) is the viscosity of the base fluid and \(\varphi\) is the volume fraction for the nanoparticles. \(k_{f}\), \(k_{s}\) are the thermal conductivities, and \(\rho_{f}\), \(\rho_{s}\) are the densities of the base fluid and nanoparticles, respectively. Furthermore  
\[\frac{1}{r}\frac{\partial}{\partial r}(r\overset{\wedge}{u_{r}})=0 \tag{1}\]0 \[\frac{1}{r}\frac{\partial}{\partial r}(r\overset{\wedge}{u_{r}})=0 \tag{1}\]1 \[\Theta_{n+1}(\eta)=\Theta_{n}(\eta)\int_{0}^{\eta}\lambda_{\Theta} \left[ \begin{matrix}\left[\theta^{\prime\prime}(s)+4\alpha^{2}\Theta(s)+ \frac{2K_{2}Pr}{K_{3}}\alpha^{2}\Theta(s)F(s)\right.\\ K_{2}EcPr\\ K_{3}Re(1-\phi)^{2.5}\left[4\alpha^{2}F^{2}(s)+\left(F^{\prime}(s)\right)^{2} \right]\end{matrix} \right]ds \tag{16}\]  
where the Lagrange multipliers for velocity and temperature profile are \(\lambda_{F}\left(s\right)\) and \(\lambda_{\Theta}(s)\), respectively. Approximate Lagrange multipliers for the above problem can be obtained using the variational approach as \(\lambda_{F}\left(s\right)=-\frac{\left(s-\eta\right)^{2}}{2! }\) and \(\lambda_{\Theta}(s)=\left(s-\eta\right)\), so that the above iterative scheme can be reformulated as:  
\[F_{n+1}(\eta)=F_{n}(\eta)\] \[-\int\limits_{0}^{\eta}\frac{\left(s-\eta\right)^{2}}{2!}\left(F^ {\prime}=\left(s\right)+\left(4-\left(1-\varphi\right)^{2.5}A_{1}Ha\right) \alpha^{2}F^{\prime}\left(s\right)+\left(1-\varphi\right)^{2.5}K_{1}2\alpha\, \mathrm{Re}\,F\left(s\right)F^{\prime}\left(s\right)\right)ds\] \[\Theta_{n+1}(\eta)=\Theta_{n}(\eta)\] \[+\int_{0}^{\eta}(s-\eta)\left[ \begin{matrix}\left[\theta^{ \prime\prime}(\eta)+4\alpha^{2}\Theta(\eta)+\frac{2K_{2}Pr}{K_{3}}\alpha^{2} \Theta(\eta)F(\eta)\right.\\ K_{2}EcPr\\ K_{3}Re(1-\phi)^{2.5}\left[4\alpha^{2}F^{2}(\eta)+\left(F^{\prime}(\eta) \right)^{2}\right]\end{matrix} \right]ds\] \[+\int_{0}^{\eta}(s-\eta)\left[ \begin{matrix}\left[\theta^{ \prime\prime}(\eta)+4\alpha^{2}\Theta(\eta)+\frac{2K_{2}Pr}{K_{3}}\alpha^{2} \Theta(\eta)F(\eta)\right.\\ K_{2}EcPr\\ K_{3}Re(1-\phi)^{2.5}\left[4\alpha^{2}F^{2}(\eta)+\left(F^{\prime}(\eta) \right)^{2}\right]\end{matrix} \right]ds\]  
For the initial guess, using boundary conditions for velocity and temperature profiles, we get:  
\[F_{0}\left(\eta\right)=1+\beta\frac{\eta^{2}}{2}\ \,\]  
\[\Theta_{0}(\eta)=\gamma\]  
where \(\beta\) and \(\gamma\) are the constants that will be calculated by using the boundary conditions \(F\left(1\right)=S\) and \(\Theta(1)=1\). Next, a few terms of the solution are given as:  
\[ \begin{split} F_{1}(\eta)=1&+\frac{\beta\eta^{2}}{2}-0.0083 33333333333318 Re\alpha\beta^{2}\eta^{6}(1.-1.\phi)^{2.5}K_{1}\\ &\quad+\eta^{4}(-0.166666666666666674\alpha^{2}\\ &\quad+0.041666666666666685\text{Ha}\alpha^{2}\beta(1.-1.\phi)^{ 2.5}A_{1}\\ &\quad-0.08333333333333337 Re\alpha\beta(1.-1.\phi)^{2.5}K_{1}) \end{split}  \tag{17}\]  
\[