# Thermal Stability and Entropy Generation Analysis for
## 4 Method of Solution
### Code Validation  
The numerical solution from SCCM is then compared with the Shooting-Runge-Kutta method using NDSolve code since the validation of results is a major step in confirming the accuracy of numerical results. In this way, a simple Shooting-Runge-Kutta scheme is implemented by using the NDSolve algorithm in Wolfram Mathematica 13.3. The result is tabled against the SCCM results in Tables 1 and 2 for the parameter values \(R=0.1,\alpha=0.1\), \(\Omega=0.1,\delta=0.1,Gr=0.1,m=0.5,\gamma=0.1,\beta_{1}=0.1,\beta_{2}=0.1, \psi=0.1,Pr=1\), \(\lambda=0.1,\epsilon=0.1\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(y\) & \(\theta(y)_{SCCM}\) & \(\theta(y)_{RK4}\) & \(|\theta(y)_{SCCM}-\theta(y)_{RK4}|\) \\ \hline \(-1.0\) & \(1.1006285586\times 10^{-19}\) & \(0.0\) & \(1.100628558680762\times 10^{-19}\) \\ \hline \(-0.75\) & \(0.02272084331322944\) & \(0.02272084285202028\) & \(4.612091611400082\times 10^{-10}\) \\ \hline \(-0.5\) & \(0.039238086997582716\) & \(0.0392380865496249\) & \(4.479578166738385\times 10^{-10}\) \\ \hline \(-0.25\) & \(0.049427524412263134\) & \(0.049427523946807654\) & \(4.654554797145494\times 10^{-10}\) \\ \hline \(0.0\) & \(0.05315385368892104\) & \(0.053153853296836195\) & \(3.920848437588553\times 10^{-10}\) \\ \hline \(0.25\) & \(0.05026408527117134\) & \(0.050264085011262935\) & \(2.599084072962831\times 10^{-10}\) \\ \hline \(0.5\) & \(0.04058133464454575\) & \(0.04058133447864745\) & \(1.658982951013854\times 10^{-10}\) \\ \hline \(0.75\) & \(0.023902559552278567\) & \(0.023902559431809665\) & \(1.204689022149541\times 10^{-10}\) \\ \hline \(1.0\) & \(1.4543862558\times 10^{-18}\) & \(-1.412512538815998\times 10^{-11}\) & \(1.412512684254624\times 10^{-11}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Validation for the solution of \(\theta(y)\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(y\) & \(f(y)_{SCCM}\) & \(f(y)_{RK4}\) & \(|f(y)_{SCCM}-f(y)_{RK4}|\) \\ \hline \(-1.0\) & \(0.09340042731013098\) & \(0.0934004275457099\) & \(2.355789174846023\times 10^{-10}\) \\ \hline \(-0.75\) & \(0.2782702518827663\) & \(0.2782702628881371\) & \(1.100537083686958\times 10^{-8}\) \\ \hline \(-0.5\) & \(0.41810875686198196\) & \(0.4181087874111443\) & \(3.054916247746675\times 10^{-8}\) \\ \hline \(-0.25\) & \(0.5073637119780844\) & \(0.5073637713586425\) & \(5.93805581372564\times 10^{-8}\) \\ \hline \(0.0\) & \(0.5408032211708456\) & \(0.5408032769405969\) & \(5.576975137255857\times 10^{-8}\) \\ \hline \(0.25\) & \(0.515175556370712\) & \(0.5151755848661947\) & \(2.849548275118962\times 10^{-8}\) \\ \hline \(0.5\) & \(0.4305925030480927\) & \(0.4305925270588788\) & \(2.401078608160745\times 10^{-8}\) \\ \hline \(0.75\) & \(0.29025067501405133\) & \(0.2902507313095687\) & \(5.629551735886196\times 10^{-8}\) \\ \hline \(1.0\) & \(0.09886799791001163\) & \(0.09886808703206998\) & \(8.91220583443264\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Validation for the solution of \(f(y)\). parameter on the flow velocity. As the plot shows, when the viscosity parameter increases, the fluid's dynamic viscosity increases. This is explained by the fact that increasing the parameter \(\alpha\) means that the fluid's viscosity is reduced, leading to diminished resistance to flow. This inevitably leads to increased fluid velocity. Figure 3 shows the influence of the viscous heating parameter on the flow velocity. From the plot, when the viscous dissipation parameter rises, the magnitude of the heat generated within the flow domain increases accordingly; hence, the flow velocity is enhanced, as seen in the figure. Figure 4 shows that the activation energy parameter \(\epsilon\) has little or no significant effect on the fluid velocity. This is explained by the fact that the activation energy parameter implicitly enters the velocity equation through the temperature/viscosity coupling. In this way, its effects on the velocity can, at best, be marginal. An increase in the Grashof number \(Gr\), Figure 5, is observed to increase the fluid velocity.