## 3 Exact Solution  
Consider a solution of velocity in the form of the following polynomial expression [29]  
\[w(x,y)=K_{1}x^{4}+K_{2}y^{4}+K_{3}x^{2}y^{2}+K_{4}x^{2}+K_{5}y^{2}+K_{6}, \tag{21}\]  
The solution of velocity can be written in this polynomial form, since the following polynomial, when used in momentum equation and corresponding boundary conditions, gives an exact solution of flow profile that exactly satisfies the equation and boundary conditions.  
Substituting Equation (21) in Equation (17) and making comparison of \(x^{2},\,y^{2},\,x^{0},\,y^{0}\) coefficients on both sides of the Equation, we utilize the three Equations  
\[12K_{1}+2K_{3}=0,\] (i)  
\[2K_{3}+12K_{2}=0,\] (ii)  
\[2K_{4}+2K_{5}=\frac{\mathrm{d}\mathrm{p}}{\mathrm{d}z}(1+\lambda_{1}),\] (iii)  
Also, we avail three more Equations by using Equation (21) in (19) and coefficients balancing of \(x^{4},\,x^{2},\,x^{0}\) gives  
\[K_{1}\mathrm{a}^{4}+K_{2}\mathrm{b}^{4}-K_{3}\mathrm{a}^{2}\mathrm{b}^{2}=0,\] (iv)  
\[-2K_{2}\mathrm{b}^{4}+K_{3}\mathrm{a}^{2}\mathrm{b}^{2}+K_{4}\mathrm{a}^{2}- K_{5}\mathrm{b}^{2}=0,\] (v)  
\[K_{2}\mathrm{b}^{4}+K_{5}\mathrm{b}^{2}+K_{6}=-1-\frac{2\pi\mathrm{\Phi} \alpha\beta\mathrm{Cos}(2\pi\mathrm{z})}{1-2\pi\mathrm{\Phi}\alpha\beta \mathrm{Cos}(2\pi\mathrm{z})},\] (vi)We can solve these six Equations (_i_)-(_vi_) and have  
\[K_{1}=0,\ K_{2}=0,\ K_{3}=0,\ K_{4}=\frac{\mathrm{b}^{2}\frac{ \mathrm{d}\varphi}{\mathrm{d}\varphi}(1+\lambda_{1})}{2\left(\mathrm{a}^{2}+ \mathrm{b}^{2}\right)},\] \[K_{5}=\frac{\mathrm{a}^{2}\frac{\mathrm{d}\varphi}{\mathrm{d} \varphi}(1+\lambda_{1})}{2\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)},\ K_{6}=- \frac{\mathrm{a}^{2}\mathrm{b}^{2}\frac{\mathrm{d}\varphi}{\mathrm{d}\varphi} (1+\lambda_{1})}{2\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)}+\frac{1}{2\pi \mathrm{d}\varphi\mathrm{d}\alpha\beta\mathrm{Cos}(2\pi\mathrm{z})-1},\]  
The values of these constants are utilized in Equation (21) and this provides an exact velocity solution given as  
\[\mathrm{w}(\mathrm{x},\mathrm{y})=-1-\frac{2\pi\mathrm{d}\varphi\alpha\beta \mathrm{Cos}(2\pi\mathrm{z})}{1-2\pi\mathrm{d}\varphi\alpha\beta\mathrm{Cos}(2 \pi\mathrm{z})}+\frac{1}{2}\frac{\mathrm{a}^{2}\mathrm{b}^{2}(1+\lambda_{1})}{ \left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)}\frac{\mathrm{d}\varphi}{\mathrm{d }\mathrm{z}}\left(\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{ \mathrm{b}^{2}}-1\right), \tag{22}\]  
The exact value of \(q(z)\) is computed by integration of (22)  
\[\mathrm{q}(\mathrm{z})=-\frac{\mathrm{a}^{3}\mathrm{b}^{3}\frac{\mathrm{d} \mathrm{p}}{\mathrm{d}\mathrm{z}}\pi(1+\lambda_{1})}{4\left(\mathrm{a}^{2}+ \mathrm{b}^{2}\right)}+\frac{\mathrm{a}\mathrm{b}\pi}{-1+2\pi\mathrm{d}\varphi \alpha\beta\mathrm{Cos}(2\pi\mathrm{z})}, \tag{23}\]  
From Equation (23), \(\frac{dp}{dz}\) is computed as  
\[\frac{\mathrm{d}\mathrm{p}}{\mathrm{d}\mathrm{z}}=\frac{4\left(\mathrm{a}^{2} +\mathrm{b}^{2}\right)\left[-\int_{0}^{1}\mathrm{a}\mathrm{b}\mathrm{d} \mathrm{z}+\mathrm{a}\mathrm{b}\pi+\mathrm{Q}+2\pi\left(\int_{0}^{1}\mathrm{a }\mathrm{b}\mathrm{d}\mathrm{z}-\mathrm{Q}\right)\alpha\beta\phi\mathrm{Cos}(2 \pi\mathrm{z})\right]}{\mathrm{a}^{3}\mathrm{b}^{3}\pi(1+\lambda_{1})(-1+2\pi \mathrm{d}\varphi\alpha\beta\mathrm{Cos}(2\pi\mathrm{z}))}, \tag{24}\]  
The mathematical relation that is used to evaluate the pressure rise for a single wavelength is given as  
\[\Delta\mathrm{P}=\int\limits_{0}^{1}\frac{\mathrm{d}\varphi}{\mathrm{d}\mathrm{ z}}\mathrm{d}\mathrm{z}, \tag{25}\]  
The exact mathematical result of temperature is computed as  
\[\theta(x,y)=\frac{-B_{r}\left(\frac{dp}{dz}\right)^{2}\mathrm{a}^{2}b^{2}\left( \frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{b}^{2}}-1 \right)\left[b^{6}x^{2}+a^{2}b^{4}\left(b^{2}+\mathrm{c}x^{2}-y^{2}\right)+a^ {6}\left(b^{2}+y^{2}\right)+a^{4}b^{2}\left(4b^{2}-x^{2}+6y^{2}\right)\right] \left(1+\lambda_{1}\right)}{12\left(\mathrm{a}^{2}+b^{2}\right)^{2}\left( \mathrm{a}^{4}+6a^{2}b^{2}+b^{4}\right)}, \tag{26}\]