## 2 Problem Analysis
 \tag{10}\]  
The above transformed model with prescribed controlling parameters is converted into unsteady and steady part separately by using oscillatory Stoke's conditions given in Equation (11) to find the oscillatory behavior of the mechanism by following [11; 28]. Then the oscillating part is converted into real and imaginary parts to compute the oscillating quantities. \[ \begin{split}\tilde{u}=u_{s}+\epsilon u_{t}e^{i\omega\tau},\quad \tilde{v}=v_{s}+\epsilon v_{t}e^{i\omega\tau},\quad\tilde{h}_{x}=\mathrm{h}_{ xs}+\epsilon\mathrm{h}_{xt}e^{i\omega\tau}\\ \tilde{h}_{y}=\mathrm{h}_{ys}+\epsilon\mathrm{h}_{yt}e^{i\omega \tau},\quad\tilde{\theta}=\theta_{\mathrm{s}}+\epsilon\theta_{t}e^{i\omega \tau}\end{split}  \tag{11}\]  
**Steady Part:**  
\[\frac{\partial u_{s}}{\partial x}+\frac{\partial v_{s}}{\partial y}=0 \tag{12}\]  
\[u_{s}\frac{\partial u_{s}}{\partial x}+v_{s}\frac{\partial u_{s}}{\partial y}= \frac{\partial^{2}u_{s}}{\partial y^{2}}+\xi\Big{(}h_{xs}\frac{\partial h_{ xs}}{\partial x}+h_{ys}\frac{\partial h_{xs}}{\partial y}\Big{)}+R_{g}\Big{(}2 \mathrm{o}_{s}-\mathrm{o}_{s}{}^{2}\Big{)}Asin\alpha \tag{13}\]  
\[\frac{\partial h_{xs}}{\partial x}+\frac{\partial h_{ys}}{\partial y}=0 \tag{14}\]  
\[u_{s}\frac{\partial h_{s}}{\partial x}+v_{s}\frac{\partial h_{s}}{\partial y}- h_{xs}\frac{\partial u_{s}}{\partial x}-h_{ys}\frac{\partial u_{s}}{\partial y}= \frac{1}{\gamma}\frac{\partial^{2}h_{s}}{\partial y^{2}} \tag{15}\]  
\[u_{s}\frac{\partial\theta_{s}}{\partial x}+v_{s}\frac{\partial\theta_{s}}{ \partial y}=\frac{1}{P_{r}}\frac{\partial^{2}\theta_{s}}{\partial y^{2}} \tag{16}\]  
with appropriate boundary conditions:  
\[ \begin{split} u_{s}=v_{s}=0,\quad h_{ys}=h_{xs}=0,\quad\theta_{s }=1\quad at\quad y=0\\ u_{s}\to 1,\quad\theta_{s}\to 0,\quad h_{ys}\to 1\quad as \quad y\to\infty\end{split}  \tag{17}\]  
**Oscillating Part:**  
\[\frac{\partial u_{t}}{\partial x}+\frac{\partial v_{t}}{\partial y}=0 \tag{18}\]  
\[ \begin{split} i\omega(u_{t}-1)+& u_{s}\frac{\partial u _{t}}{\partial x}+u_{t}\frac{\partial u_{s}}{\partial x}+v_{s}\frac{\partial u _{t}}{\partial y}+v_{t}\frac{\partial u_{s}}{\partial y}\\ &=\frac{\partial^{2}u_{t}}{\partial y^{2}}+\xi\Big{(}h_{xs}\frac{ \partial h_{xt}}{\partial x}+h_{xt}\frac{\partial h_{xs}}{\partial x}+h_{ys} \frac{\partial h_{xt}}{\partial y}+h_{yt}\frac{\partial h_{xs}}{\partial y} \Big{)}+R_{g}(2\mathrm{o}_{t}\\ &-2\mathrm{o}_{s}\mathrm{o}_{t})\lambda sin\alpha\end{split}  \tag{19}\]\[\frac{\partial h_{xt}}{\partial x}+\frac{\partial h_{yt}}{\partial y}=0 \tag{20}\]  
\[i\omega h_{xt}+u_{s}\frac{\partial h_{t}}{\partial x} +u_{t}\frac{\partial h_{s}}{\partial x}+v_{s}\frac{\partial h_{t}}{ \partial y}+v_{t}\frac{\partial h_{s}}{\partial y}-h_{xs}\frac{\partial u_{t}} {\partial x}-h_{xt}\frac{\partial u_{s}}{\partial x}-h_{ys}\frac{\partial u_{t }}{\partial y}-h_{yt}\frac{\partial u_{s}}{\partial y} \tag{21}\] \[=\frac{1}{\gamma}\frac{\partial^{2}h_{xt}}{\partial y^{2}}\] \[i\omega\theta_{t}+u_{s}\frac{\partial\theta_{t}}{\partial x}+u_{t} \frac{\partial\theta_{s}}{\partial x}+v_{s}\frac{\partial\theta_{t}}{\partial y }+v_{t}\frac{\partial\theta_{s}}{\partial y}=\frac{1}{P_{r}}\frac{\partial^{2 }\theta_{t}}{\partial y^{2}} \tag{22}\]  
with appropriate boundary conditions;  
\[ \begin{array}{ccccc}u_{t}=v_{t}=0,&h_{yt}=h_{xt}=0,&\theta_{t}=1&at&y=0\\ \\ u_{t}\to 1,&\theta_{t}\to 0,&h_{yt}\to 1&as&y\to\infty\end{array}  \tag{23}\]  
Again the unsteady equations are separated into the real and imaginary parts by using the oscillatory Stoke's conditions given in Equation (24) and the primitive formed equations for real, imaginary and steady are given below by following [11; 28]. \[u_{t}=u_{1}+iu_{2},\quad v_{t}=v_{1}+iv_{2},\quad\theta_{t}=\theta_{1}+i \theta_{2},\quad h_{xt}=h_{x1}+ih_{x2},\quad h_{yt}=h_{y1}+ih_{y2} \tag{24}\]  
The steady, real and imaginary are transformed into a convenient form by using the primitive variable formulation for smooth algorithm.