## 5 Conclusions  
The Williamson fluid has been taken from the class of pseudoplastic fluids in the presence of magnetic field and heat transfer. The modelled problems of momentum and energy have been solved by using the Optimal Homotopy Asymptotic Method. The strong convergence of OHAM compared to ADM has been discussed in this work. The effects of various embedded parameters have been observed. The physical and numerical comparison of the present work and published work has been achieved in close agreement to each other, and the absolute error has been shown.  
The main points of the work have been observed as:  
* Initially, the liquid film oscillates jointly with the plate for a selected domain \(y\in[0,1]\) and this oscillation rises slowly towards the free surface.
* The gravitational effect near the belt is smaller due to the friction force, and this effect is more clear and rapid at the free surface.
* The magnetic effect on the flow field has been observed, which opposes the fluid motion.
* The thermal boundary layer thickness increases with larger values of Eckert number and the inter molecular forces among the fluid particles decrease and, as a result, the velocity of fluid film increases.
* The fast convergence of OHAM has been observed by comparing its results with ADM.  
The authors greatly acknowledge with thanks the Deanship of Scientific Research (DSR) at King Abdul Aziz University, Jeddah, Saudi Arabia, for technical and financial support. Moreover, all the authors declare that they have no competing interests.  
[PERSON], A.S.K. and S.I. modeled the problem and solved; A.S.A., A.M.A., I.K., participated in the physical discussion of the problem. [PERSON], A.K.A. contributed in the ADM solution of the problem. A.S.A., A.M.A. also supported this work technically and financially; All authors read and approved the final manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{Y}\) & \(\mathbf{ADM}\) & \(\mathbf{OHAM}\) & \(\mathbf{Absolute\ Error}\) \\ \hline
0 & 0.0199375363 & \(-0.0004475377\) & 0.020385074 \\
0.2 & 0.21851290935 & 1.53087 & 0.016596 \\
0.4 & 0.415049659642 & 1.10485 & 0.0291587 \\
0.6 & 0.610432176595 & 0.710142 & 0.033036 \\
0.8 & 0.8052896112883 & 0.342897 & 0.0243532 \\
1 & 1.00000000000 & 1 & \(1.04\times 10^{-17}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: The comparison of ADM and OHAM methods for the temperature profile, when \(W_{e}=0.1\), \(m=0.01;M=0.1,\omega=0.02,t=0.6\), \(Ec=10\), \(\Pr=4.0,\mathrm{C_{1}}=-0.0009350828,\mathrm{C_{2}}=-0.0024496023\), \(\mathrm{C_{3}}=-2.597660913881,\mathrm{C_{4}}=1.59769431544815\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{Y}\) & \(\mathbf{ADM}\) & \(\mathbf{OHAM}\) & \(\mathbf{Absolute\ Error}\) \\ \hline
0 & 1.91632177045 & 1.91632177045 & 0 \\
0.2 & 0.24289103594 & 0.3191514078391 & 0.076260371897 \\
0.4 & \(-0.96869527228\) & \(-0.8959030788252\) & 0.072792193456 \\
0.6 & \(-1.73745720196\) & \(-1.7459175340616\) & 0.008460332096 \\
0.8 & \(-2.08139220298\) & \(-2.2461895367629\) & 0.164797333778 \\
1 & \(-2.01747533731\) & \(-2.4102387658252\) & 0.392763428507 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The comparison of ADM and OHAM methods for the velocity profile, when \(We=0.01\), \(m=8.3;M=0.01,\omega=0.02,t=1.6,\mathrm{C_{1}}=1.3196149901566254\), \(\mathrm{C_{2}}=-0.7108355490082049\).