## 2 Problem Analysis
The primitive forms of the steady, real and imaginary equations are given below by using Equation (25),  
\[ \begin{array}{ccccc}u_{s}(x,y)=U_{s}(X,Y),v_{s}(x,y)=x^{\frac{-1}{2}}V_{s}( X,Y),h_{ys}(x,y)=x^{\frac{-1}{2}}\varphi_{ys}(X,Y),\\ \\ h_{xs}(x,y)=\varphi_{xs}(X,Y),\quad\quad\theta_{s}(x,y)=\theta_{s}(X,Y),&Y=x^{ \frac{-1}{2}}y,&X=x\end{array}  \tag{25}\]  
**For steady equations:**  
\[X\frac{\partial U_{s}}{\partial X}-\frac{Y}{2}\frac{\partial U_{s}}{\partial Y }+\frac{\partial V_{s}}{\partial Y}=0 \tag{26}\]  
\[XU_{s}\frac{\partial U_{s}}{\partial X}+\left[V_{s}-\frac{Y}{2}U_{s}\right] \frac{\partial U_{s}}{\partial Y}=\frac{\partial^{2}U_{s}}{\partial Y^{2}}+ \xi\bigg{[}X\varphi_{xs}\frac{\partial\varphi_{xs}}{\partial Y}+\bigg{(} \varphi_{ys}-\frac{Y}{2}\varphi_{xs}\bigg{)}\frac{\partial\varphi_{xs}}{ \partial Y}\bigg{]} \tag{27}\]  
\[ \begin{array}{ccccc}X\frac{\partial\varphi_{xs}}{\partial X}+\bigg{[}V_{s}- \frac{Y}{2}U_{s}\bigg{]}\frac{\partial\varphi_{s}}{\partial Y}-X\varphi_{xs} \frac{\partial U_{s}}{\partial Y}-\bigg{(}\varphi_{ys}-\frac{Y}{2}\varphi_{xs }\bigg{)}\frac{\partial U_{s}}{\partial Y}=\frac{1}{\gamma}\frac{\partial^{2} \varphi_{s}}{\partial Y^{2}}\end{array}  \tag{29}\]  
\[XU_{s}\frac{\partial\theta_{s}}{\partial X}+\bigg{[}V_{s}-\frac{Y}{2}U_{s} \bigg{]}\frac{\partial\theta_{s}}{\partial Y}=\frac{1}{P_{r}}\frac{\partial^{2 }\theta_{s}}{\partial Y^{2}} \tag{30}\]  
with boundary conditions as:  
\[ \begin{array}{ccccc}U_{s}=V_{s}=0,&\varphi_{ys}=\varphi_{xs}=0,&\theta_{s}= 1&\text{at}&Y=0\\ \\ U_{s}\to 1,&\theta_{s}\to 0,&\varphi_{ys}\to 1&\text{as}&Y\to\infty\end{array}  \tag{31}\]For real equations:  
\[X\frac{\partial U_{1}}{\partial X}-\frac{Y}{2}\frac{\partial U_{1}}{\partial Y}+ \frac{\partial V_{1}}{\partial Y}=0 \tag{32}\]  
\[X\bigg{[}U_{s}\frac{\partial U_{1}}{\partial X}+U_{1} \frac{\partial U_{s}}{\partial X}\bigg{]}+\bigg{[}V_{s}-\frac{Y}{2}U _{s}\bigg{]}\frac{\partial U_{1}}{\partial Y}+\bigg{[}V_{1}-\frac{Y}{2}U_{1} \bigg{]}\frac{\partial U_{s}}{\partial Y}-\omega X(U_{2}+1\] (33) \[=\frac{\partial^{2}U_{1}}{\partial Y^{2}}\] \[+  
\[X\bigg{[}U_{s}\frac{\partial\varphi_{2}}{\partial X}+U_{2} \frac{\partial\varphi_{s}}{\partial X}\bigg{]}+\bigg{[}V_{s}- \frac{Y}{2}U_{s}\bigg{]}\frac{\partial\varphi_{2}}{\partial Y}+\bigg{[}V_{2}- \frac{Y}{2}U_{2}\bigg{]}\frac{\partial\varphi_{s}}{\partial Y}+\omega X \varphi_{1} \tag{41}\] \[-\bigg{[}X\bigg{(}\varphi_{xs}\frac{\partial U_{2}}{\partial x}+ \varphi_{x2}\frac{\partial U_{s}}{\partial x}\bigg{)}+\bigg{(}\varphi_{ys}- \frac{Y}{2}\varphi_{xs}\bigg{)}\frac{\partial U_{2}}{\partial Y}+\bigg{(} \varphi_{y2}-\frac{Y}{2}\varphi_{xx}\bigg{)}\frac{\partial U_{s}}{\partial Y} \bigg{]}\] \[=\frac{1}{\gamma}\frac{\partial^{2}\varphi_{2}}{\partial Y^{2}}\] \[X\bigg{[}U_{s}\frac{\partial\theta_{2}}{\partial X}+U_{2}\frac{ \partial\theta_{s}}{\partial X}\bigg{]}+\bigg{[}V_{s}-\frac{Y}{2}U_{s}\bigg{]} \frac{\partial\theta_{2}}{\partial Y}+\bigg{[}V_{2}-\frac{Y}{2}U_{2}\bigg{]} \frac{\partial\theta_{s}}{\partial Y}+\omega X\theta_{1}=\frac{1}{P_{r}} \frac{\partial^{2}\theta_{2}}{\partial Y^{2}} \tag{42}\]  
with boundary conditions as:  
\[ \begin{array}{l}U_{2}=V_{2}=0,\quad\varphi_{y2}=\varphi_{x2}=0,\quad\theta_ {2}=0\quad\text{at}\quad Y=0\\ U_{2}\to 0,\quad\quad\theta_{2}\to 0,\quad\quad\varphi_{y2}\to 0\quad\quad\text{ as}\quad\quad Y\to\infty\end{array}  \tag{43}\]