## 3 Numerical Approach and Benchmarking of the Code
The power laws scheme is used to treat the convective terms. To discretize the temporal derivatives, the fully implicit procedure is retained to ensure the stability of computational scheme. The grids are considered uniform in all directions with additional nodes on boundaries. The successive relaxation-iterating scheme is employed in solving the resulting algebraic equations. The solution is considered as acceptable as the next convergence criterion is satisfied for every time step:  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{Pr}\,\mathrm{Ha}^{2}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}}\Big{)}\Big{(} \frac{\partial\tau}{\partial\kappa}\Big{)}\]1  
The superscript n corresponds to the iteration number. The solution is regarded as a steady state if the following criterion is satisfied:  
\[-\mathrm{Pr}\cdot\mathrm{K}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}} \Big{)}\Big{(}\frac{\partial\Psi_{1}}{\partial\kappa^{2}}+\frac{\partial\Psi_ {1}}{\partial\gamma^{2}}+\frac{\partial\Psi_{1}}{\partial\alpha^{2}}\Big{)}+ \mathrm{Pr}\,\mathrm{Ha}^{2}\Big{(}\frac{\rho_{1}}{\rho_{\rm{d}}}\Big{)}\Big{(} \frac{\partial\tau}{\partial\kappa}\Big{)}\]2  
where m denotes the number of the time step. To determine an appropriate grid for numerical simulations, a grid independence study has been undertaken first. A dimensionless time step equal to \(10^{-4}\) is retained. Figure 2 presents the effect of grid size on the average Nusselt and the average Sherwood numbers on the hot wall for N = \(-\)0.5, \(\varphi\) = 0.04 and K =1 for different Rayleigh numbers. Hence, the (51 \(\times\) 51 \(\times\) 51) grid was estimated to be convenient for the present work at least for Ra \(\leq 10^{5}\) because it allows a good compromise between the computational cost (a significant reduction of the execution time) and the accuracy of the results obtained. To prove the precision of the present code, comparisons with previously published results are necessary. But due to the lack of availability of data on the particular problems of three-dimensional double diffusive convection in micropolar nanofluid investigated in this research, the validation of our three-dimensional code was made in different stages:  
First, the 3-D double diffusive convection code for a Newtonian fluid is compared with the numerical results of [PERSON] and [PERSON] ([PERSON] and [PERSON], 2018). The validation which is reported in a previous work ([PERSON] et al. (2018)) shows good agreement. Second, the proposed three-dimensional nanofluid model that takes into account the nanoparticles microrotation is validated against experimental data of [PERSON] et al. (2016). Figure 3 demonstrates that the experimental data and the present numerical results reveal similar trends. Apparently, as the microrotation parameter K augments our numerical results are closer to the experimental data. The proposed model is consistent with the assumptions describing that the nanoparticles microrotation must be taken into account. The third validation concerns the model of micropolar nanofluid under a magnetic field. To that end, a 2-D version of our code is derived and the results are verified against the numerical solutions reported by [PERSON] and [PERSON] (2018). As revealed in Figure 4, a very good agreement is found. The results of the present investigation are in good agreement with the results of the models used ([PERSON] and [PERSON], 2018; [PERSON] and [PERSON], 2018; [PERSON] and [PERSON], 2018). Hence, the developed computer code can be effectively and reliably used to simulate the problem under consideration.