## 4 Results and Discussion  
The numerical calculations have been performed for the velocity, temperature, solutal, and Casson nano-fluid volume fraction functions for a different assessment of physical parameters, as magnetic parametre \(M\), Prandtl number \(Pr\), Unsteadiness \(\sigma\), permeability \(k_{p}\), Brownian motion \(Nb\), thermophoresis \(Nt\), thermal radiation \(R\), Dufour \(Nd\), Schmidt \(Sc\), buoyancy \(\lambda_{1},\lambda_{2},\lambda_{3}\), Suction/Injection \(f_{w}\), [PERSON] \(Sr\), Lewis number \(Le\), hydro-dynamic slip \(S_{f}\), thermal slip \(S_{\theta}\), solutal slip \(S_{\phi}\), and nano-particles concentration slip \(S_{g}\). The detail of the present results and the appraisal of flow velocity is made with the exact solution that is given by [PERSON] [32] as \(f(\eta)=1-e^{(-\eta)}\) under the special case (\(M=0,\sigma=0,\beta\rightarrow\infty,\lambda_{1}=\lambda_{2}=\lambda_{3}=0,S_{f} =0,f_{w}=0,kp=0\)). The finite element method's outcomes have decent concurrence with the exact solution, which approves the validity of the finite element method. It can be seen clearly in Table 2, and in Table 3, that the skin friction coefficient attained by the finite element method is equivalent to the numerical outcomes of [PERSON] et al. [33] and the exact solution of [PERSON] et al. [34] under special case \(\sigma=0,\beta=0,\lambda_{1}=\lambda_{2}=\lambda_{3}=0,S_{f}=0,f_{w}=0,kp=0\). To confirm the accuracy of the presented numerical results, the results obtained by the finite element method for skin friction co-efficient for steady and unsteady flow have been compared with the numerical results that have previously been reported in studies and shown in Table 4. Regarding our results, there is admirable conformity among our outcomes and previously available research articles, which approves the cogency and the accuracy of the current results that are obtained by the finite element method (FEM). Table 5 describes the results of the heat transfer rate that are acquired by the finite element method, which are compared with the results of earlier studies and with the accurate solution of [PERSON] et al. [35] in a special case (Nt = Nb = 0). We observe that our results are in complete agreement and the grid invariance test has performed to sustain accuracy for 4 decimal points. In Table 6, local skin friction coefficient \(-f^{\prime\prime}(0)\), the rates of heat transfer \(-\theta^{\prime}(0)\), and mass transfer \(-\phi^{\prime}(0)\) that are acquired by FEM, which are also compared with the published research work, show excellent correlation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(M\) & **[PERSON] et al. [33]** & **[PERSON] et al. [34]** & **FEM (Our Results)** & **Error in \%** \\  & \(\beta=0\) & **Exact Solution (a)** & **(b)** & \(|(\frac{b-a}{a})|\times 100\) \\ \hline
0.0 & 1.000 & 1.00000 & 1.0000078 & 0.000078 \\
0.2 & 1.095 & 1.095445 & 1.0954462 & 0.00010 \\
0.5 & 1.224 & 1.224745 & 1.2247452 & 0.00001 \\
1.0 & 1.414 & 1.414214 & 1.4142142 & 0.00001 \\
1.2 & 1.483 & 1.483240 & 1.4832385 & 0.00010 \\
1.5 & 1.581 & 1.581139 & 1.5811392 & 0.00001 \\
2.0 & 1.732 & 1.732051 & 1.7320515 & 0.00002 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Assessment of skin friction coefficient for different values of M. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\eta\) & **[PERSON] [32]** & **FEM** & \(\eta\) & **Crane [32]** & **FEM** \\ \hline
0 & 1 & 1 & 7 & 0.0009 & 0.0009 \\
1 & 0.1379 & 0.1379 & 8 & 0.0003 & 0.0003 \\
2 & 0.1353 & 0.1353 & 9 & 0.0001 & 0.0001 \\
3 & 0.0498 & 0.0498 & 10 & 0.0000 & 0.0000 \\
4 & 0.0183 & 0.0018 & 11 & 0.0000 & 0.0000 \\
5 & 0.0067 & 0.0067 & 12 & 0.0000 & 0.0000 \\
6 & 0.0025 & 0.0025 & 13 & 0.0000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The comparison of the exact solution of Crane and FEM for the flow velocity \(f^{\prime}(\eta)\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(M\) & **[PERSON]** [36]** & **[PERSON] and [PERSON]** [25]** & **FEM (Our Results)** & **[PERSON] et al.