**3. Numerical Methodology**

Various numerical methods can be exploited in order to find solutions of the differential equations whether in parallel or in sequential (see, e.g., [45–48]). Here, the main equations and accompanying initial boundary conditions are addressed using the Galerkin finite element method. The non-linear partial differential equations are transformed into linear equations using the weighted residual technique [42].

For the validation, the (Nu) on the hot surface at (Re = 500, Ha = 0, φ = 4% and N = 4) is utilized. Table 2 displays the results of the mesh independence study. The findings illustrate that the grid size of 21,999 is the ideal option.

**Table 2.** Grid independence test for Re = 100, Ha = 0, φ = 4% and N = 4.


By using numerical research, the existing results were validated. The results from the model used in the current study are compared to those presented in Ghalambaz et al. [44], as seen in Figure 2.

**Figure 2.** Comparison of current work with that of Mohammad Ghalambaz et al reprinted/adapted with permission from Ref. [44]. 2022, Elsevier . **Figure 2.** Comparison of current work with that of Mohammad Ghalambaz et al. reprinted/adapted with permission from Ref. [44]. 2022, Elsevier.
