## 2 Problem Formulation
[PERSON] [43] defined the dynamic viscosity of nano-liquid in term of base liquid as:  
\[\mu_{nf}=\frac{\mu_{f}}{(1-\varphi)^{2.5}}. \tag{8}\]  
The density, specific heat capacity, and rate of thermal conductivities of the nanofluid [7; 8] are expressed as:  
\[ \begin{split}&\rho_{nf}=\rho_{f}(1-\varphi)+\varphi\,\rho_{CNT},\;\mu_{nf}=\frac{\mu_{f}}{(1-\varphi)^{2.5}},\\ &\alpha_{nf}=\frac{\mu_{f}}{\left(\rho\;c_{p}\right)_{nf}},\; \big{(}\rho\;C_{p}\big{)}_{nf}=(1-\varphi)\big{(}\rho\;C_{p}\big{)}_{f}+\varphi \big{(}\rho\;C_{p}\big{)}_{CNT},\\ &\sigma_{nf}=\sigma_{f}\;\Big{(}\frac{1+3(\sigma-1)\;\varphi}{( \sigma+2)-(\sigma-1)\varphi}\Big{)}.\end{split}  \tag{9}\]  
Here, the model by [PERSON] [44] is implemented to calculate the thermal conductivities  
\[\frac{k_{nf}}{k_{f}}=\frac{1-\varphi+2\varphi\Big{(}\frac{k_{CNT}}{k_{CNT}-k_ {f}}\;ln\;\Big{(}\frac{k_{CNT}+k_{f}}{2\;k_{f}}\Big{)}\Big{)}}{1-\varphi+2 \varphi\Big{(}\frac{k_{f}}{k_{CNT}-k_{f}}\;ln\;\Big{(}\frac{k_{CNT}+k_{f}}{2 \;k_{f}}\Big{)}\Big{)}}. \tag{10}\]  
where \(\rho_{CNT}\) shows the density of carbon nanotube, \(\big{(}\rho c_{p}\big{)}_{CNT}\) denotes the specific heat of the carbon nanotube, and \(\rho_{f},\;\big{(}\rho c_{f}\big{)}_{f}\) represent the density and specific heat of base fluid, respectively. \(q^{rad}=-\frac{4\sigma^{*}}{3k^{*}}\frac{qT^{4}}{\sigma^{2}}\) has been achieved using the Roseland approximation [43], where \(\sigma^{*}\) is the [PERSON] constant, and \(k^{*}\) is the absorption coefficient. We can observe that temperature difference in liquid motion is suitably negligible, so \(T^{4}\) can be expressed in linear form using the Taylor's series, neglecting the higher terms: \(T^{4}=4T_{\infty}^{3}-3T_{\infty}^{4}\). Therefore, we have \(q^{rad}=-\frac{16\sigma^{*}}{3k^{*}}T^{3}\frac{qT}{\sigma^{2}}\). Using the following suitable similarities variable [13],  
\[ \begin{split}& u=c\,x\;f^{\prime}(\eta),\;v=c\,x\;g(\eta),\;w=- \sqrt{c\;v_{f}}\,f(\eta),\\ &\Theta(\eta)=T_{\infty}+\Big{(}T_{f}-T_{\infty}\Big{)}\Theta( \eta),\quad\eta=\Big{[}\frac{c}{v_{f}}\Big{]}^{\frac{1}{2}}.\end{split}  \tag{11}\]  
Equation (3) is verified identically and Equations (4)-(10) take the following system when we use the similarities variables:  
\[\Big{(}1+\frac{1}{\beta}\Big{)}f^{\prime\prime}(\eta)-\zeta_{1}\zeta_{2}\Big{[} f(\eta)f^{\prime\prime}(\eta)+\big{(}f^{\prime}(\eta)\big{)}^{2}+2\,K\,g(\eta) \;-\zeta_{4}Mf^{\prime}(\eta)\big{]}=0, \tag{12}\]  
\[\Big{(}1+\frac{1}{\beta}\Big{)}\;g^{\prime\prime}(\eta)-\zeta_{1}\zeta_{2} \big{[}f(\eta)\,g^{\prime}(\eta)+f^{\prime}(\eta)\,g(\eta)\;-2\,K\,f^{\prime}( \eta)-\zeta_{4}Mg(\eta)\big{]}=0, \tag{13}\]  
\[\frac{d}{d\eta}\Big{(}\zeta_{5}+\frac{4}{3}Rad[1+(\Theta(\eta)-1)\Theta(\eta)] ^{3}\Big{)}\Theta^{\prime}(\eta)+\zeta_{3}Pr\big{[}f(\eta)\Theta^{\prime}( \eta)+Q_{h}\,\Theta(\eta)\big{]}, \tag{14}\]  
\[f(\eta)=g(\eta)=0,\;f^{\prime}(\eta)=1,\;\Theta^{\prime}(\eta)=\frac{k_{f}}{k _{nf}}(1-\Theta(\eta))\lambda,\;at\;\eta=0, \tag{15}\]  
\[f^{\prime}(\eta)=g(\eta)=\Theta(\eta)=0,\;at\;\eta=\infty.\]Here \(K\) represent the rotation parameter, \(M\) represents the magnetic parameter, \(Pr\) represents the Prandtl number, \(Q_{h}\) denotes the heat generation or absorption parameter, \(Rd\) represents radiation parameter, and \(\lambda\) is the Biot number, and are expressed as follows:  
\[K=\frac{\Omega}{c},M=\frac{\sigma_{f}B_{0}{}^{2}g}{c},\ Pr=\frac{\left(\rho\ c_ {p}\right)_{f}}{k_{f}},Q_{h}=\frac{Q}{c\left(\rho c_{p}\right)_{f}},\ Rd=\frac{4\ \sigma^{*}\ \mathrm{T}_{\infty}^{3}}{3k^{*}k_{f}},\lambda=\frac{h_{f}}{k_{f}}\sqrt{\frac{ v_{f}}{c}}. \tag{16}\]  
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