3.3. Finite- element formulation
The finite-element model may be obtained from above equations by substituting finite-element approximations of the form
\(f=\sum_{j=1}^{2}{f_{j}\psi_{j}}\),\(\ \ \ \ h=\sum_{j=1}^{2}{h_{j}\psi_{j}}\), \(\theta=\sum_{j=1}^{2}{\theta_{j}\psi_{j}}\) ,\(\phi=\sum_{j=1}^{2}{\phi_{j}\psi_{j}}\). (26)
With,\(\text{\ w}_{1}=w_{2}=w_{3}=w_{4}=\psi_{i},\ \ \ \ \ \ \ \ \ (i=1,2,3)\).
Where \(\psi_{i}\) are the shape functions for a typical element\((\eta_{e},\ \eta_{e+1})\) and are defined as
\(\psi_{1}^{e}=\frac{\left(\eta_{e+1}+\eta_{e}-2\ \eta\right)\ (\eta_{e+1}-\eta)}{{{(\eta}_{e+1}-\eta_{e})}^{2}}\),\(\psi_{2}^{e}=\frac{4\left(\eta-\eta_{e}\right)(\eta_{e+1}-\eta)}{{{(\eta}_{e+1}-\eta_{e})}^{2}}\),
\(\text{\ \ ψ}_{3}^{e}=\frac{\left(\eta_{e+1}+\eta_{e}-2\ \eta\right)\ (\eta-\eta_{e})}{{{(\eta}_{e+1}-\eta_{e})}^{2}}\),\(\eta_{e}\leq\eta\leq\eta_{e+1}.\) (27)
The finite element model of the equations thus formed is given by
\(\par \begin{bmatrix}\par \begin{matrix}\left[K^{11}\right]&\left[K^{12}\right]\text{\ \ \ \ \ \ }\left[K^{13}\right]&\left[K^{14}\right]\\ \left[K^{21}\right]&\left[K^{22}\right]\text{\ \ \ \ \ \ }\left[K^{23}\right]&\left[K^{24}\right]\\ \left[K^{31}\right]&\left[K^{32}\right]\text{\ \ \ \ \ \ }\left[K^{33}\right]&\left[K^{34}\right]\\ \end{matrix}\\ \left[K^{41}\right]\text{\ \ \ \ \ \ }\left[K^{42}\right]\text{\ \ \ \ \ }\left[K^{43}\right]\text{\ \ \ \ \ }\left[K^{44}\right]\\ \end{bmatrix}\text{\ \ \ \ \ }\par \begin{bmatrix}\par \begin{matrix}f\\ h\\ \theta\\ \end{matrix}\\ \phi\\ \end{bmatrix}\ =\ \ \ \par \begin{bmatrix}\par \begin{matrix}\left\{r^{1}\right\}\\ \left\{r^{2}\right\}\\ \left\{r^{3}\right\}\\ \end{matrix}\\ \left\{r^{4}\right\}\\ \end{bmatrix}\)
Where \([K^{\text{mn}}]\) and \([r^{m}]\)(m, n = 1, 2, 3, 4) are defined as
\({K_{\text{ij}}}^{11}=\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\frac{\partial\psi_{j}}{\partial\eta}\text{dη}}\),\({K_{\text{ij}}}^{12}=-\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{j}\text{\ dη}}\), \({K_{\text{ij}}}^{13}={K_{\text{ij}}}^{14}=0\),\({\text{\ K}_{\text{ij}}}^{21}=0,\ \)
\({K_{\text{ij}}}^{22}=\int_{\eta_{e}}^{\eta_{e+1}}{\frac{\partial\psi_{i}}{\partial\eta}\frac{\partial\psi_{j}}{\partial\eta}d\eta}+\frac{A_{2}}{A_{1}}\ \overset{\overline{}}{f_{1}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{1}\psi_{j}d\eta}+\frac{A_{2}}{A_{1}}\overset{\overline{}}{f_{2}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{2}\psi_{j}d\eta}-\ \ \frac{A_{2}}{A_{1}}\overset{\overline{}}{h_{1}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{1}\psi_{j}d\eta}-\frac{A_{2}}{A_{1}}\overset{\overline{}}{h_{2}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}{\psi_{2}\psi}_{j}d\eta-\ \beta\ A_{1}\ \int_{\eta_{e}}^{\eta_{e+1}}{\ {(\frac{\partial\psi_{i}}{\partial\eta})}^{2}\frac{\partial\psi_{i}}{\partial\eta}\frac{\partial\psi_{j}}{\partial\eta}d\eta+2\ \beta\ A_{1\ }\overset{\overline{}}{h_{1}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{1}\psi_{j}d\eta}+2\ \beta\text{\ A}_{2\ }\overset{\overline{}}{h_{2}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{2}\psi_{j}d\eta}-\ A_{1\ }\text{M\ }\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\ \psi_{j}d\eta}\text{\ .}}}\)
\({K_{\text{ij}}}^{23}=A_{1\ }\)\(\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\ \psi_{j}d\eta}\),\({K_{\text{ij}}}^{24}={-A}_{1\ \ }\text{Nr}\)\(\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\ \psi_{j}d\eta},\ \ {K_{\text{ij}}}^{31}=0\),\(\text{\ \ \ \ \ \ \ }{K_{\text{ij}}}^{32}=0\ \),
\({K_{\text{ij}}}^{33}=\left(1+A_{4\ }R\right)\int_{\eta_{e}}^{\eta_{e+1}}{\frac{\partial\psi_{i}}{\partial\eta}\frac{\partial\psi_{j}}{\partial\eta}\text{dη}}-Pr\frac{A_{3}}{A_{4}}\ \overset{\overline{}}{f_{1}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{1}\frac{\partial\psi_{j}}{\partial\eta}\text{dη}}-\text{\ Pr}\frac{A_{3}}{A_{4}}\ \overset{\overline{}}{f_{2}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{\text{j\ }}\frac{\partial\psi_{j}}{\partial\eta}\text{dη}}-Pr\ \gamma\ A_{3}\ \overset{\overline{}}{h_{1}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{1}\frac{\partial\psi_{j}}{\partial\eta}\text{dη}}\ -Pr\ \gamma\ A_{3}\ \overset{\overline{}}{h_{2}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{2}\frac{\partial\psi_{j}}{\partial\eta}d\eta,\ }\ {K_{\text{ij}}}^{34}=0.\)
\({K_{\text{ij}}}^{41}=0\),\({\text{\ \ K}_{\text{ij}}}^{42}=0,{\text{\ K}_{\text{ij}}}^{43}=0\ \),
\({K_{\text{ij}}}^{44}=\int_{\eta_{e}}^{\eta_{e+1}}{\frac{\partial\psi_{i}}{\partial\eta}\frac{\partial\psi_{j}}{\partial\eta}\text{dη}}-Sc\ \overset{\overline{}}{f_{1}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{1}\frac{\partial\psi_{j}}{\partial\eta}\text{dη}}-Sc\ \overset{\overline{}}{f_{2}}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{i}\psi_{2}\frac{\partial\psi_{j}}{\partial\eta}d\eta-Sc\ \ C_{r}\int_{\eta_{e}}^{\eta_{e+1}}{\psi_{\text{i\ }}\psi_{j}\text{dη}}}.\)
\(r_{i}^{2}=0\),\(r_{i}^{2}=-\left(\psi_{i}\frac{d\psi_{i}}{\text{dη}}\right)_{\eta_{e}}^{\eta_{e+1}}\),\(r_{i}^{3}=-\left(\psi_{i}\frac{d\psi_{i}}{\text{dη}}\right)_{\eta_{e}}^{\eta_{e+1}}\),\(r_{i}^{4}=-\left(\psi_{i}\frac{d\psi_{i}}{\text{dη}}\right)_{\eta_{e}}^{\eta_{e+1}}\).