## 4 Further Validation with Variational Iteration Method (VIM)  
The MATLAB code has been validated against limited published results in Table 1. The case benchmarked used the data: \(Me=\delta=\gamma=Nb=Nt=Ec=Sc=Pe=Sb=0\). Therefore melting, slip, nanoscale and bioconvection effects are not validated against. In order to further validate the model, an alternative algorithm was used to solve the boundary value problem. The variational iteration method [56] has been used in several related studies [49; 50; 57; 58] and is now used. The comparison with the benchmark case is shown in the last column of Table 1 and correlation is excellent. Further validations with the general model are illustrated in Figure 2a-d where the influence of temperature dependent viscous parameter, \(h_{2}\), and Eckert number, \(Ec\), on velocity, temperature, nanoparticle volume fraction and micro-organism density number function are visualized.  
\begin{table}
\begin{tabular}{c c c c} \hline \(\delta\) & **[PERSON] [54]** & **[PERSON] et al. [55]** & **Present Results (BVP4C)** \\ \hline
0 & 1.0000 & 1.0000000 & 1.0000000 \\
0.1 & 0.8721 & 0.87208247 & 0.87204247 \\
0.2 & 0.7764 & 0.77637707 & 0.77593307 \\
0.5 & 0.5912 & 0.59119548 & 0.59119589 \\
1.0 & 0.4302 & 0.43015970 & 0.43016000 \\
2.0 & 0.2840 & 0.28397959 & 0.28399832 \\
5.0 & 0.1448 & 0.14484019 & 0.14464015 \\
10.0 & 0.0812 & 0.08124198 & 0.08124091 \\
20.0 & 0.0438 & 0.04378834 & 0.04378790 \\
50.0 & 0.0186 & 0.01859623 & 0.01857868 \\
100.0 & 0.0095 & 0.00954997 & 0.00954677 \\ \hline \end{tabular}
\end{table}
Table 2: Comparison of skin friction \(f^{\prime\prime}\left(0\right)\) for \(\delta\) when \(Me=M=h_{2}=h_{4}=h_{6}=h_{8}=\gamma=Nb=Nt=Ec=Sc=Pe=Sb=0,Pr=1\).  
Figure 2: Influences of \(h_{2}\) and \(Ec\) on the dimensionless: (**a**) velocity, (**b**) temperature, (**c**) nanoparticle volume fraction, (**d**) micro-organism density number function.  
The circles correspond to the VIM solution and the lines to the BVP4C computations. The comparison cases are for \(Ec\) = 0.1 and \(h_{2}\) = 0.4 in Figure 2a-d and very close agreement is attained. Confidence in the MATLAB BVP4C code is therefore justifiably high. Both velocity and temperature profile increase as both \(Ec\) and increase in Figure 2a,b. A significant temperature enhancement is obtained with an increase in both temperature dependent viscous parameter, \(h_{2}\), and Eckert number, \(Ec\), on temperature and nanoparticle volume fraction. However, increasing Eckert number and temperature dependent viscous parameter \(h_{2}\) are observed to suppress the volume fraction of nanoparticles throughout the boundary layer. Thermal boundary layer thickness is therefore elevated with greater viscous heating effect owing to the conversion of kinetic energy into heat (thermal energy) in the boundary layer. With increasing values of temperature dependent viscous parameter \(h_{2}\), nanofluid viscosity increases (based on the formulae in Equation (7)), which exacerbates thermal diffusion in the regime manifesting in temperature elevation. However, this simultaneously reduces the species diffusion of nanoparticles in the boundary layer and results in a decrease in nanoparticle species boundary layer thickness.