## 2 Mathematical Model
### Non-Dimensionlization, Lubrication Approach and Boundary Conditions
The relation of velocity and stream function \(\psi\) is defined as  
\[u\ =\ \frac{\partial\psi}{\partial y},\ v\ =\ -\alpha\frac{\partial\psi}{ \partial x}. \tag{18}\]  
After introducing Equations (17) and (18), Equation (2) is satisfied identically and Equations (3)-(6) becomes  
\[R_{e}\alpha\Bigg{(}\frac{\partial\psi}{\partial y}\frac{\partial}{\partial x }-\frac{\partial\psi}{\partial x}\frac{\partial}{\partial y}\Bigg{)}\frac{ \partial\psi}{\partial y}\ =\ -\frac{\partial p}{\partial x}+\Bigg{(}\alpha\frac{\partial S_{xx}}{ \partial x}+\frac{\partial S_{xy}}{\partial y}\Bigg{)}+{m_{e}}^{2}U_{lls}\Phi, \tag{19}\]  
\[R_{e}\alpha\Bigg{(}\frac{\partial\psi}{\partial y}\frac{\partial}{\partial x}- \frac{\partial\psi}{\partial x}\frac{\partial}{\partial y}\Bigg{)}\frac{ \partial\psi}{\partial x}\ =\ -\frac{\partial p}{\partial y}+\Bigg{(}\alpha\frac{\partial S_{xy}}{ \partial x}+\frac{\partial S_{yy}}{\partial y}\Bigg{)}, \tag{20}\]  
\[R_{e}\alpha\Bigg{(}\frac{\partial\psi}{\partial y}\frac{\partial}{\partial x}- \frac{\partial\psi}{\partial x}\frac{\partial}{\partial y}\Bigg{)}\Theta\ =\ \frac{1}{P_{r}}\Bigg{(}\alpha^{2}\frac{\partial^{2}}{ \partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\Bigg{)}\theta+E_{c}\Bigg{[} \alpha^{2}\frac{\partial}{\partial x}\bigg{(}\frac{\partial\psi}{\partial y} \bigg{)}S_{xx}-\alpha\frac{\partial\psi}{\partial x}\Bigg{)}S_{yy}+\Bigg{(} \frac{\partial^{2}\psi}{\partial y^{2}}-\alpha^{2}\frac{\partial^{2}\psi}{ \partial x^{2}}\Bigg{)}S_{xy}\Bigg{]}, \tag{21}\]  
\[R_{e}\alpha\Bigg{(}\frac{\partial\psi}{\partial y}\frac{\partial}{\partial x}- \frac{\partial\psi}{\partial x}\frac{\partial}{\partial y}\Bigg{)}\Omega\ =\ \frac{1}{S_{c}}\Bigg{(}\alpha^{2}\frac{\partial^{2}}{ \partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\Bigg{)}\Omega+S_{r}\Bigg{(} \alpha^{2}\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2} }\Bigg{)}\theta \tag{22}\]  
Now subject to long-wavelength assumption [10] on Equations (19)-(22), ignoring higher powers of \(\alpha\), we get;  
\[\frac{\partial p}{\partial x}\ =\ \frac{\partial}{\partial y}\Bigg{[}\frac{ \partial^{2}\psi}{\partial y^{2}}+2\Gamma\Bigg{(}\frac{\partial^{2}\psi}{ \partial y^{2}}\Bigg{)}^{3}\Bigg{]}+{m_{e}}^{2}U_{lls}\Phi, \tag{23}\]  
\[\frac{\partial p}{\partial y}\ =\ 0 \tag{24}\]\[\frac{\partial^{2}\Theta}{\partial y^{2}}+B_{r}\Bigg{[}\left(\frac{\partial^{2} \Psi}{\partial y^{2}}\right)^{2}+2\Gamma\left(\frac{\partial^{2}\Psi}{\partial y ^{2}}\right)^{4}\Bigg{]}\ =\ 0 \tag{25}\]  
\[\frac{\partial^{2}\Omega}{\partial y^{2}}+S_{c}S_{r}\left(\frac{\partial^{2} \Theta}{\partial y^{2}}\right)\ =\ 0 \tag{26}\]  
where \(S_{xy}\ =\ \frac{\partial^{2}\Psi}{\partial y^{2}}+2\Gamma\left(\frac{\partial^{2} \Psi}{\partial y^{2}}\right)^{3}\). Eliminating pressure from Equations (23) and (24), we get  
\[\frac{\partial^{2}}{\partial y^{2}}\left[\frac{\partial^{2}\psi}{\partial y^{ 2}}+2\Gamma\left(\frac{\partial^{2}\Psi}{\partial y^{2}}\right)^{3}\right]+m _{e}{}^{2}U_{HS}\frac{\partial\Phi}{\partial y}\ =\ 0 \tag{27}\]  
The dimensionless boundary conditions are:  
\[R_{e}\alpha\Bigg{(}\frac{\partial\psi}{\partial y}\frac{\partial}{\partial x }-\frac{\partial\psi}{\partial x}\frac{\partial}{\partial y}\Bigg{)}\frac{ \partial\psi}{\partial y}\ =\ -\frac{\partial p}{\partial x}+\Bigg{(}\alpha\frac{\partial S_{xx}}{ \partial x}+\frac{\partial S_{xy}}{\partial y}\Bigg{)}+{m_{e}}^{2}U_{lls}\Phi, \tag{19}\]0  
\[R_{e}\alpha\Bigg{(}\frac{\partial\psi}{\partial y}\frac{\partial}{\partial x }-\frac{\partial\psi}{\partial x}\frac{\partial}{\partial y}\Bigg{)}\frac{ \partial\psi}{\partial y}\ =\ -\frac{\partial p}{\partial x}+\Bigg{(}\alpha\frac{\partial S_{xx}}{ \partial x}+\frac{\partial S_{xy}}{\partial y}\Bigg{)}+{m_{e}}^{2}U_{lls}\Phi, \tag{19}\]1  
here f is dimensionless mean flow rate in moving frame and \(\Theta\) is dimensionless mean flow rate in fixed frame.