3. Numerical solution of the problem
The variational Finite element process [39 – 42] is implemented to
evaluate numerically above equations (10) - (12) with boundary
conditions (13) - (14). The procedure of this method is as follows.
For the solution of system of non-linear ordinary differential equation
(10) – (12) together with boundary conditions (13) - (14), first we
assume that
\(\frac{\text{df}}{\text{dη}}=h\) (16)
The equations (10) to (12) then reduces to
\(h^{{}^{\prime\prime}}+\frac{A_{2}}{A_{1}}f\text{\ h}^{{}^{\prime}}-\frac{A_{2}}{A_{1}}h^{2}-\alpha\ A_{1}\left[f^{2}h^{{}^{\prime\prime}}-2\ f\ h\ h^{{}^{\prime}}\right]-A_{1}Mh+A_{1}[\theta-NrS]=0\)(17)
\(\left(1+A_{4}R\right)\theta^{{}^{\prime\prime}}-Pr\ \frac{A_{3}}{A_{4}}\text{\ f\ }\theta^{{}^{\prime}}-Pr\ \beta\ A_{3}[f^{2}\theta^{{}^{\prime\prime}}+f\ h\ \theta^{{}^{\prime}}]=\ 0\ \)(18)
\(S^{{}^{\prime\prime}}-Sc\ f\ S^{{}^{\prime}}-\ Sc\ C_{r}\ S=0\) (19)
The boundary conditions take the form
\(\eta=0,\ \ f=V_{0},\ \ h=1,\ \ \theta^{{}^{\prime}}=-\text{Bi}\left(1-\theta\right),\ \ S=1\)(20)
\(\eta\rightarrow\infty,\ \ \ \ \ \ \ h=0,\ \ \ \ \ \ \ \ \ \ \ \ \theta=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ S=0\)(21)