## 3 Solution Procedure  
The governing partial differential system (PDE) is solved numerically using the similarity method. The similarity approach identifies equations for which solutions depend on a certain group of independent variables rather than each one. This technique helps to transform the PDE's (partial differential equations) into ODE's (ordinary differential equations). Transformed ODE's are then solved using the Finite difference method together with the Richardson extrapolation-based doslove routine. Now consider the following [PERSON] transformations [45]:  
\[u=r\Omega F(\xi),\ \ w=\sqrt{\Omega v}H(\xi),\ \ T=T_{\infty}+A\big{(}\frac{r}{R} \big{)}^{2}\theta(\xi), \tag{6}\]  
\[C=C_{\infty}+B\big{(}\frac{r}{R}\big{)}^{2}\phi(\xi),\ \xi=\frac{z}{r}\sqrt{Re}\]  
The reduced ODE system is,  
\[2F+H^{\prime}=0 \tag{7}\]  
\[\bigg{(}1+\frac{1}{\beta_{c}}\bigg{)}F^{\prime}-F(Ha+F)-F^{\prime}H=0 \tag{8}\]  
\[\begin{array}{c}\frac{1}{T}\theta^{\
u}+Nb\theta^{\prime}\phi^{\prime}+Nt( \theta^{\prime})^{2}+Q_{E}exp\{-n\xi_{t}^{*}\}+Q_{T}\theta-2F\theta-H\theta^{ \prime}\\ -\alpha\big{[}4F^{2}\theta+H\theta^{\prime}(4F+H^{\prime})+H(H\theta^{\prime }+2\theta F^{\prime})\big{]}=0\end{array} \tag{9}\]  
\[\begin{array}{c}\phi^{\prime\prime}+\frac{Nt}{Nb}\theta^{\prime\prime}-LePr(2 F\phi+H\phi^{\prime})\\ -\beta LePr\big{[}4F^{2}\phi+H\phi^{\prime}(4F+H^{\prime})+H(H\phi^{\prime\prime }+2\phi F^{\prime})\big{]}=0\end{array} \tag{10}\]  
\[\begin{array}{c}\Big{(}1+\frac{1}{\beta_{c}}\Big{)}F^{\prime}(0)=-2 Ma(1+R), \ H(0)=0,\ \theta(0)=1,\ \phi(0)=1,\\ F(\infty)\to 0,\ \theta(\infty)\to 0,\ \phi(\infty)\to 0\end{array} \tag{11}\]  
The following are the non-dimensional parameters:  
\(Pr=\frac{v\big{(}\rho c_{p}\big{)}}{k}\) denotes Prandtl number, \(Nt=\frac{\tau D_{T}\big{(}T_{\infty}-T_{\infty}\big{)}}{vT_{\infty}}\) denotes thermophoresis number, \(Ha=\frac{\delta B_{0}^{2}}{\rho\Omega}\) denotes Hartmann number, \(Nb=\frac{\tau D_{B}\big{(}C_{\infty}-C_{\infty}\big{)}}{v}\) denotes Brownian motion number, \(Q_{E}=\frac{Q_{t}}{\rho c_{p}\Omega}\) denotes exponential space-dependent heat source number (ESHS), \(Q_{T}=\frac{Q_{t}}{\rho c_{p}\Omega}\) denotes thermal-dependent heat source number (THS), \(Le=\frac{k}{D_{B}\big{(}c_{p}\big{)}}\) denotes [PERSON] number, \(R=\frac{\gamma_{C}B}{\gamma_{T}A}\) denotes Marangoni ratio number, \(Ma=\frac{\alpha_{0}\gamma_{T}A}{N^{2}\mu\Omega}\sqrt{\frac{v}{\Omega}}\)denotes [PERSON] number, \(\alpha=\lambda_{T}\Omega\) denotes the thermal relaxation parameter, and \(\beta=\lambda_{C}\Omega\) denotes the solutal relaxation parameter.  
The local Sherwood and Nusselt numbers are given below:  
\[Sh=\frac{-r\left(\frac{\partial C}{\partial z}\right)_{z=0}}{C_{w}-C_{\infty} }\;, \tag{12}\]  
\[Nu=\frac{-r\left(\frac{\partial T}{\partial z}\right)_{z=0}}{T_{w}-T_{\infty} }\;. \tag{13}\]  
The non-dimensional form of (12)-(13) are:  
\[Sh_{r}=\frac{Sh}{\sqrt{Re}}=-\phi^{\prime}(0), \tag{14}\]  
\[Nu_{r}=\frac{Nu}{\sqrt{Re}}=-\theta^{\prime}(0). \tag{15}\]  
where \(Re=\frac{r^{2}\Omega}{\
u}\) is the local Reynolds number.  
The Equations (7)-(11) are nonlinear and coupled in nature. The analytical solution of these equations can't be guaranteed and, hence, they are solved by FDM along with Richardson extrapolation using dsolve command in MAPLE. Further, obtained FDM results are compared with those estimated by bvp5c method. The comparison values of \(F^{\prime}(1)\) are tabulated in Table 1 and an excellent agreement is established.