Eqs. (8) – (10) are then reduced to the following system
\begin{equation}
x_{3}^{{}^{\prime}}=\ \left(\frac{1}{1+\ \beta}\right)\left[-x_{1}x_{3}-\ {x_{2}}^{2}-\text{Gr}x_{4}+M\left(x_{2}-1\right)-1\right]\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ (15)\nonumber \\
\end{equation}\begin{equation}
x_{5}^{{}^{\prime}}=\ -\Pr x_{1}x_{5}-\left(1+\ \frac{1}{\beta}\right)\text{PrEc}{x_{3}}^{2}\ -\text{MPrEc}\left(x_{2}-1\right)^{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(16)\nonumber \\
\end{equation}subject to the following initial conditions,
\begin{equation}
x_{1}\left(0\right)=0,\ \ x_{2}\left(0\right)=\lambda,\ \ x_{5}\left(0\right)=\ \text{βi}\left[\theta\left(0\right)-1\right],\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(17)\nonumber \\
\end{equation}\begin{equation}
x_{2}\left(0\right)=s_{1},\ \ x_{5}\left(0\right)=\ s_{2}\text{\ \ }\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\nonumber \\
\end{equation}Using the unknown initial conditions \(s_{1}\) and \(s_{2}\) in Eq.
(17), Eq. (15) and Eq. (16) are integrated numerically. The accuracy of
the assumed missing initial conditions was checked by comparing the
calculated value of the dependent variable at the terminal point with
its given value there. If differences exist, improved values of the
missing initial conditions are obtained and the process repeated. The
accuracy and robustness for solving the boundary value problems have
been repeatedly confirmed previously \(\mathbf{[23]}\).
From the process of numerical computation, the fluid velocity\(f^{\prime}(\eta)\) and temperature \(\theta(\eta)\) are compared with the
given boundary conditions.
Results and Discussion
The numerical computations are carried out for the various values of the
physical parameter with Runge-Kutta Fehlberg integration scheme. The
effects of the varying physical parameters – magnetic field parameter\((M)\), Casson parameter \((\beta)\), velocity ratio parameter\((\lambda)\), Grashof number \((Gr)\), Biot number \((Bi)\), Eckert
number \((Ec)\) and Prandtl number \((Pr)\) on velocity an temperature
profiles has been analyzed. The obtained computation results are
presented graphically in Figures (1) – (8) and discussed.
The effects of various values of magnetic field parameter \(M\) on the
flow field velocity and temperature profiles are displayed in Figures
(1) and (2). As \(M\) increases, the flow field velocity decreases and
also increases with decreasing values in \(M\). Due to the Lorentz force
induced by the dual actions of electric and magnetic fields, the
velocity boundary layer thickness decreases. Similarly, for\(\lambda=0.2\), the temperature profiles increases with increasing
values of \(M\). The obtained result is in agreement with\(\mathbf{[}\mathbf{22}\mathbf{,26]}\).