## 2 Mathematical Formulation  
If the unsteady magnetohydrodynamic flow is assumed, heat and mass transfer nanofluids flow in a thin finite liquid film over a stretching rotating disk. The thin elastic disk emerges radially from a narrow slit at the origin of the cylindrical coordinate system \((r,\varphi,z)\). The scheme of the formulated problem is shown in Figure 1.  
The time dependent stretching velocity is written as  
\[u=cr(1-bt)^{-1}, \tag{1}\]  
Here \(c\) and \(b\) represent the stretching parameter and positive constant respectively. The time dependent disk rotating velocity is written as  
\[v=r\Omega(1-bt)^{-1}, \tag{2}\]  
where \(\Omega\) is the rotating rate of the disk. The stretchable rotating disk surface temperature \(T_{s}\) is defined as  
\[T_{s}=T_{0}-T_{ref}\frac{\Omega r^{2}}{v_{nf}(1-bt)^{1.5}}, \tag{3}\]  
Here \(T_{0}\) and \(T_{ref}\) represent temperature at the origin and a constant reference temperature, respectively. The magnetic field is applied along the z-direction which is defined as  
\[B(t)=B_{0}(1-bt)^{-0.5}, \tag{4}\]  
The concentration at the disk surface \(C_{s}\) varying with distance \(r\) is written as  
Figure 1: Physical sketch of the problem. \[C_{s}=C_{0}-C_{ref}\frac{\Omega r^{2}}{v_{nf}(1-bt)^{1.5}}, \tag{5}\]  
Here, \(C_{0}\) and \(C_{ref}\) represent the concentration at the origin and reference concentration, respectively. Using the above suppositions, and boundary conditions, the governing equations of the flow problem are  
\[\frac{\partial u}{\partial t}+\frac{u}{r}+\frac{\partial w}{\partial z}=0, \tag{6}\]  
\[\frac{\partial u}{\partial t}+\frac{\partial u}{\partial r}+w\frac{\partial u }{\partial z}-\frac{v^{2}}{r}=v_{nf}\bigg{(}\frac{\partial^{2}u}{\partial z^{ 2}}\bigg{)}-\frac{\sigma_{nf}B_{0}^{2}}{\rho_{nf}(1-bt)}u, \tag{7}\]  
\[\frac{\partial v}{\partial t}+\frac{\partial v}{\partial r}+w\frac{\partial v }{\partial z}+\frac{uv}{r}=v_{nf}\bigg{(}\frac{\partial^{2}v}{\partial z^{2}} \bigg{)}-\frac{\sigma_{nf}B_{0}^{2}}{\rho_{nf}(1-bt)}v, \tag{8}\]  
\[u=cr(1-bt)^{-1}, \tag{1}\]0  
Here \(u,~{}v\) and \(w\) are the velocity components in their respective directions as shown in \[u=cr(1-bt)^{-1}, \tag{1}\]1 Also \(\sigma^{*},k^{*},v_{nf}\), \(\sigma_{nf}\), \(\rho_{nf}\), \(\Big{(}\rho C_{p}\Big{)}_{nf^{\prime}}\), \(\alpha_{nf}\) are the Stephan-Boltzmann constant, mean absorption coefficient, kinematic viscosity, electrical conductivity, density, heat capacity and thermal diffusivity which are defined as  
\[ \begin{array}{ll}\sigma_{nf}=1+\frac{3\big{(}\sigma_{s}/\sigma_{f}-1\big{)} \phi}{\big{(}\sigma_{s}/\sigma_{f}+2\big{)}-\big{(}\sigma_{s}/\sigma_{f}-1 \big{)}\phi}\sigma_{f},&\rho_{nf}=(1-\phi)\rho_{f}+\phi\rho_{s},\\ \Big{(}\rho C_{p}\Big{)}_{nf}=\big{(}1-\phi\big{)}\Big{(}\rho C_{p}\Big{)}_{f} +\phi\Big{(}\rho C_{p}\Big{)},&\frac{k_{nf}}{k_{f}}=\frac{k_{s}+2k_{f}-2\phi \big{(}k_{f}-k_{s}\big{)}}{k_{s}+2k_{f}+2\phi\big{(}k_{f}-k_{s}\big{)}},\\ \alpha_{nf}=\frac{k_{nf}}{\big{(}\rho c_{p}\big{)}_{nf}}.\end{array}  \tag{10}\]  
The respective boundary conditions are  
\[ \begin{array}{ll}u=cr(1-bt)^{-0.5},v=r\Omega(1-bt)^{-0.5},w=0,T=T_{s}~{}~{} \text{at}~{}~{}z=0,\\ \frac{\partial u}{\partial z}=\frac{\partial z}{\partial z}=0,~{}~{}w=\frac{ \partial h}{\partial t}+u\frac{\partial h}{\partial t},~{}~{}\frac{\partial T}{ \partial z}=0~{}~{}\text{at}~{}~{}z=h.\end{array}  \tag{11}\]  
The correspondence variables are defined as  
\[ \begin{array}{ll}u=\frac{r\Omega}{(1-bt)^{0.5}}f^{\prime}(\eta),v=\frac{r \Omega}{(1-bt)^{0.5}}g(\eta),w=-2\big{(}v\Omega/(1-bt)\big{)}^{0.5}f(\eta),\\ h(t)=\Big{(}\frac{v}{\Omega}(1-bt)\Big{)}^{0.5}\beta,T=T_{0}-T_{ref}\frac{ \Omega r^{2}}{v(1-bt)^{1.5}}\theta(\eta),\eta=\big{(}\Omega/\
u(1-bt)\big{)}^{ 0.5}z.\end{array}