## 5 Conclusions  
In this article we have analyzed three-dimensional Williamson fluid flow past a linear porous and stretching sheet with the effect of thermal radiation. The system of equations has been solved by the homotopy analysis method. The impact of implanted parameters has been deliberated through graphs and discussed in detail. The concluding observations are as follows:  
The velocity function \(f^{\prime}(\eta)\) reduces with enhancement of \(f^{\prime}(\eta)\), \((We)\) and \((\gamma)\).  
* The velocity function \(g^{\prime}(\eta)\) improves with enhancement of the stretching parameter, while reduces with an increase in \((\gamma)\) and \((We)\).
* The temperature function \(\theta(\eta)\) improves with an increase in \((Rd)\), \((Nt)\), \((Nb)\) and \((Bi_{1},Bi_{2})\), while reduces with an increase in \((c)\) and \((\Pr)\).
* The concentration function \(\phi(\eta)\) enhances with an increase in \((Nt)\) and \((Bi_{2})\), while reduces with an increase in \((c)\) and \((Nb)\).
* It is observed that increasing \(c\) and \(\gamma\) reduces the surface drag forces \(f^{\prime\prime}(0)\) and \(g^{\prime\prime}(0)\), while increasing \(We\) augmented the surface drag forces \(f^{\prime\prime}(0)\) and \(g^{\prime\prime}(0)\).
* \(Nt\). It is concluded from Table 2, that the higher values of \(c,\Pr\) and \(Rd\) enhance the heat transfer rate, while the augmented values of \(Nb\) and \(Nt\) decline the heat transfer rate.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(c\) & **Pr** & \(Rd\) & **Nt** & **Nb** & **Nu Previous Results [43]** & **_Nt_ Present Results** \\ \hline
1.0 & 0.4 & 0.3 & 0.01 & 0.01 & 0.261896 & 0.119212 \\
1.1 & & & & & 0.318272 & 0.120401 \\
1.2 & & & & & 0.379145 & 0.121608 \\
0.5 & & & & & - & 0.127300 \\
0.6 & & & & & - & 0.134579 \\
0.7 & & & & & 0.140705 \\  & & 0.4 & & & - & 0.155214 \\  & & 0.5 & & - & 0.192457 \\  & & 0.6 & & - & 0.241744 \\  & & 0.02 & & 0.293770 & 0.155120 \\  & & 0.06 & & 0.274299 & 0.154747 \\  & & 0.08 & & 0.255629 & 0.154467 \\  & & & 0.03 & 0.285647 & 0.155044 \\  & & & 0.05 & 0.253739 & 0.154874 \\  & & & 0.07 & 0.222739 & 0.154704 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Numerical values of heat transfer rate for different embedded parameters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(c\) & \(Le\) & **Nt** & **Nb** & **Sh Previous Results [43]** & _Sh_ **Present Results** \\ \hline
1.0 & 0.1 & 0.01 & 0.01 & 0.223924 & 0.291024 \\
1.1 & & & & 0.255507 & 0.293757 \\
1.2 & & & & 0.289266 & 0.296524 \\  & 0.2 & & & - & 0.291022 \\  & 0.3 & & & - & 0.291021 \\  & 0.4 & & & - & 0.291019 \\  & 0.02 & & 0.213208 & 0.200865 \\  & & 0.03 & & 0.144515 & 0.290705 \\  & & 0.04 & & 0.089302 & 0.290545 \\  & & & 0.02 & 0.272275 & 0.290847 \\  & & & 0.03 & 0.286051 & 0.290671 \\  & & & 0.04 & 0.290593 & 0.290494 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Numerical values of mass transfer rate for different embedded parameters.  
**Author Contributions:** [PERSON], [PERSON] and [PERSON] modeled and solved the problem and wrote the manuscript. [PERSON] and [PERSON] thoroughly checked the mathematical modeling and English corrections. [PERSON] contributed in the results and discussions and solution. All the corresponding authors finalized the manuscript after its internal evaluation.  
**Funding:** This research was funded by National Natural Science Foundation of China No. [11471262].  
**Conflicts of Interest:** The authors declare no conflict of interest.