## 5 Conclusions  
A numerical investigation has been carried out to explore the magnetic field effects on 3-D double diffusive natural convection in a cavity filled with an Al\({}_{2}\)O\({}_{3}\)/water micropolar nanofluid. The principal conclusions are listed as follows:  
* For a micropolar nanofluid model, both heat and mass transfer rates are smaller than those of a pure nanofluid model.
* Taking in to account particle microrotation, the three-dimensional flow character is weakened.  
Figure 18: Effect of Rayleigh number Ra and vortex viscosity parameter K on \(\overline{\mathrm{Nu}}\) and \(\overline{\mathrm{Sh}}\) without (Ha = 0) and with (Ha = 40) magnetic field for, N = \(-0.5\) (Top) and N = \(-2\) (Bottom).  
For a fixed value of nanoparticle volume fraction with a wide range Rayleigh number, the heat and mass transfer rates are reduced due to increase in the Hartman number.  
For various Rayleigh numbers, the maximum of the transverse velocity enhances for Hartmann numbers lower than a critical value and diminishes as the Hartmann number increases. This critical value moves towards the highest value of the Ha number as increasing the Ra number.  
In the absence of magnetic field, the increase in nanoparticles volume fraction deteriorates the heat and mass transfer. However, for Ha greater than 30, the enhancement of nanoparticle volume fraction increases the heat and mass transfer rates.  
During non-existence of a magnetic field, an increase in vortex viscosity parameter diminishes the average heat and mass transfer rates and is more pronounced when the magnetic field is enforced. However, for Ha greater than 30, the enhancement of nanoparticle volume fraction increases heat and mass transfer rates.  
[PERSON] and [PERSON] developed the problem and its FORTRAN code. [PERSON] and [PERSON] did the literature review, developed and implemented the computer code, and interpreted the subsequently obtained results. [PERSON] did the editing and removed the grammatical mistakes. [PERSON] in consultation of rest of the Authors reconfirmed the credibility of obtained solutions. All authors have read and approved the final manuscript.  
This research received a funding from the Deanship of Scientific Research at King Khalid University of King Khaled University (G.R.P-176-38).  
The authors declare no conflict of interest.  
The microrotation vector, m/s  
Hatmann number, dimensionless  
K. \(\mathrm{Vortex}\) viscosity, kg. m\({}^{-1}\).s\({}^{-1}\)  
L. \(\mathrm{Enclosure}\) height, m  
Le. \(\mathrm{Lewis}\) number, \(\mathrm{Le}=\alpha/\mathrm{D}\)  
N. \(\mathrm{Buoyancy}\) ratio, \(\mathrm{N}=\frac{\beta_{\mathrm{c}}(\mathrm{Ca}_{-}\mathrm{Cl}_{\mathrm{i}})}{ \beta_{\mathrm{T}}(\mathrm{H}_{-}\mathrm{T}_{\mathrm{c}})}\)  
\(\mathrm{N\overline{U}}\) Average Nusselt number  
Pr. \(\mathrm{Prandtl}\) number, \(\mathrm{Pr}=\
u/\alpha\)  
Ra. \(\mathrm{Thermal}\) Rayleigh number, \(\mathrm{Ra}=\beta_{\mathrm{T}\mathrm{g}}(\mathrm{T}_{h}^{\prime}-\mathrm{T}_{ \mathrm{c}}^{\prime})\mathrm{L}^{3}/(\
u\alpha)\)  
\(\mathrm{\overline{Sh}}\) Average Sherwood number  
t. \(\mathrm{Dimensionless}\) time, \(\mathrm{t}=\frac{\alpha\mathrm{r}^{\prime}}{\mathrm{L}^{2}}\)  
T. \(\mathrm{Dimensionless}\) temperature, \(\mathrm{T}=\frac{\mathrm{T}^{\prime}-\mathrm{T}^{\prime}\mathrm{C}}{\mathrm{H }_{\mathrm{u}}-\mathrm{T}_{\mathrm{c}}}\)  
\(\mathrm{\overline{U}}\) Dimensionless velocity \(\mathrm{\overline{U}}=\frac{\mathrm{U}^{\prime}\mathrm{L}}{\alpha}\)  
x, y, z. \(\mathrm{Dimensionless}\) Cartesian coordinates, x = x\({}^{\prime}\)/L, y = y\({}^{\prime}\)/L, z = z\({}^{\prime}\)/L  
**Greek symbols**  
\(\alpha\) \(\mathrm{Thermal}\) diffusivity, m\({}^{2}\)s\({}^{-1}\)  
\(\mathrm{\beta C}\) Coefficient of compositional expansion, m\({}^{3}\)kg\({}^{-1}\)  
\(\mathrm{\beta T}\) Coefficient of thermal expansion, K\({}^{-1}\)  
\(\mathrm{\varphi}\) Solid volume fraction  
\(\lambda\) Thermal conductivity, Wm\({}^{-1}\)K\({}^{-1}\)  
\(\mu\) Dynamic viscosity of the fluid, kg m\({}^{-1}\) s