## 2 Description and Formulation of the Governing Equations
[41] and [PERSON] and [PERSON] [42], and based on the inherent assumptions outlined earlier, the relevant initial, wall and far-field conditions can be written as:  
\[t<0,\ u=\mathrm{v}=0\quad\mathrm{any}\ x,\ y\]  
\[t>0,u=u_{\mathrm{slip}},\ \mathrm{v}=0,\ T=T_{w}+T_{\mathrm{slip}},\ D_{B}\frac{ \partial C}{\partial y}+\frac{\eta_{T}}{T_{\infty}}\frac{\partial T}{\partial y }=0\ \mathrm{at}\ y=0, \tag{5}\]  
\[u\to u_{\varepsilon},\ T\to T_{\infty},\ C\to C_{\infty}\ \mathrm{as}\ y\to\infty,\]  
Here \(\alpha=\frac{k}{(\rho c)_{f}}\): Thermal diffusivity, \(\tau_{1}=\frac{(\rho c)_{f}}{(\rho c)_{f}}\): Ratio of heat capacity of the nanoparticle and fluid, \((u,v)\): Components of velocity, \(u_{\mathrm{slip}}=\frac{N_{1}\mu}{\rho}\frac{\partial u}{\partial y}\) velocity slip with velocity slip factor \(N_{1}\), \(T_{\mathrm{slip}}=D_{1}\frac{\partial T}{\partial y}\): Thermal slip with thermal slip factor \(D_{1}\), \(\mu\) kinematic viscosity of the fluid, \(\rho\): Density of the fluid, \(D_{B}\): Diffusion coefficient for Brownian motion, \(D_{T}\): Diffusion coefficient for thermophoresis, \((\beta_{T},\beta_{C})\) coefficients of thermal and mass expansions, \(L=\sqrt{\frac{a}{a}}\): Characteristic length. Note that \(K=0\) implies flow without a pressure force, and \(K=1\) corresponds to flow, due to pressure force. It is assumed that the boundary layer is optically thick, and the nonlinear Rosseland diffusion approximation for radiation is valid. The radiative heat flux (with significant absorption), as reported by [PERSON] and [PERSON] [43], is defined as \(q_{r}=-\frac{4\omega_{0}}{3k_{1}}\frac{\partial T^{4}}{\partial y}\), where \(\sigma_{1}\) (=\(5.67\times 10^{-8}\) W/m\({}^{2}\)-K\({}^{4}\)) is the Stefan-Boltzmann constant and \(k_{1}\) (m\({}^{-1}\)) is the Rosseland mean absorption coefficient. To reduce the number of variables and equations, we invoke the dimensional stream function \(\psi\) which satisfies the equations \(u=\frac{\partial\psi}{\partial y},\ v=-\frac{\partial\psi}{\partial x}\). Note that \(\psi\) satisfies the continuity Equation (1) automatically. Let us invoke the following co-ordinate transformations ([PERSON] et al. [39], [PERSON] et al. [44]):  
\[ \begin{array}{l}\tau=\alpha t,\ \eta=\sqrt{\frac{a}{a}}y,\ u=\alpha\frac{ \partial^{2}(\eta_{1}\tau)}{\partial\eta},\ \mathrm{v}=-\sqrt{a\alpha}f(\eta,\tau),\\ \theta=\frac{T-T_{\infty}}{T_{w}-T_{\infty}}=\theta(\eta,\tau),\ \phi=\frac{C-C_{\infty}}{C_{w}-C_{\infty}}=\phi(\eta,\tau),\ g(\tau)=\frac{ \mathbf{g}^{*}(t)}{\delta_{0}},\end{array} \\[\frac{\partial^{2}\theta}{\partial\eta^{2}}+f\frac{\partial\theta}{\partial\eta}+Nb \frac{\partial\theta}{\partial\eta}\frac{\partial\phi}{\partial\eta}+Nt\Big{(} \frac{\partial\theta}{\partial\eta}\Big{)}^{2}+\frac{4}{3R}\frac{\partial}{ \partial\eta}\Big{[}[1+(T_{r}-1)\theta]^{3}\frac{\partial\theta}{\partial \eta}]=\Omega\frac{\partial\theta}{\partial\eta}, \tag{8}\]  
\[\frac{\partial^{2}\phi}{\partial\eta^{2}}+Le\,f\frac{\partial\phi}{\partial \eta}+\frac{Nb}{Nt}\frac{\partial^{2}\theta}{\partial\eta^{2}}=Le\,\Omega \frac{\partial\phi}{\partial\eta}. \tag{9}\]  
with boundary conditions:  
\[ \begin{array}{l}\frac{\partial f}{\partial\eta}(\tau,0)=\delta_{0}\frac{ \partial^{2}f}{\partial\eta^{2}}(\tau,0),f(\tau,0)=0,\,\,\theta(\tau,0)=1+ \delta_{T}\frac{\partial\theta}{\partial\eta}(\tau,0),\\ Nb\phi^{\prime}(0)+Nt\theta^{\prime}(0)=0,\frac{\partial f}{\partial\eta}( \tau,\infty)=K,\theta(\tau,\infty)=\phi(\tau,\infty)=0.\end{array}