# Double-Diffusive Effects on the Onset of Rayleigh-Benard Convection of Water-Based Nanofluids
## 3 Results
Selected local results to describe the time evolution from the onset of convection up to the steady-state solution are displayed in the sequence reported in Figure 6, in which the streamlines, the isotherm, and the isoconcentration contours are plotted for \(A=0.25\), \(\varphi_{av}=0.04\), and \(\Delta T=0.5\). As expected, the conductive solution is rapidly reached in the initial stages, whereas a longer time is needed before a number of weak convective solutal plumes start evolving from the heated bottom wall. Once the concentration field is sufficiently strong to penetrate the stratified thermal field, the upward movement of the nanoparticles by thermophoresis diffusion determines the thermal stratification breakdown and the growth of large circulatory roll-cells with the formation of hot and cold nanofluid jets traveling upwards and downwards within the cavity. This effect being strongly related to the buoyancy ratio, a pronounced reduction in the critical condition for the onset of convection has to be expected in comparison with the case of the single-phase approach, as shown in Figure 7.  
Figure 4: Distribution of \(Nu\) vs. \(Ra_{n}\) for \(A=1\), using \(\varphi_{av}\) as a parameter. Figure 5: Distribution of \(h_{av}\) vs. \(\Delta T\) for \(A=1\), using \(\varphi_{av}\) as a parameter. Figure 6: Time evolution of streamlines, isotherms and isoconcentration contours for Al\({}_{2}\)O\({}_{3}\) + H\({}_{2}\)O with \(A=0.25\), \(\varphi_{av}=0.4\) and \(\Delta T=0.15\). Figure 7: Distribution of \(h_{av}\) vs. \(\Delta T\) for \(A=0.25\), obtained using single-phase approach and two-phase approach. Distributions of the heat transfer coefficient \(h_{av}\) for the base fluid and the nanofluid using the aspect ratio \(A\) as a parameter are plotted in Figures 8 and 9. It can be noticed that the \(\Delta T\) corresponding to the onset of convection decreases with decreasing the aspect ratio of the cavity. This is due to the progressively more negligible influence of the fluid confinement at the sidewalls for effect of the viscosity. The smaller is the aspect ratio of the enclosure, the larger is the movement capacity of the fluid with consequent enhancement of the heat transfer rate. Moreover, the difference between nanofluid and base fluid tends to reduce due to the cooperating downward solutal driving force, which has the beneficial effect to compensate the increased viscosity, as shown in Figure 10.  
Finally, as far as the role of the nanoparticle material is concerned, its effect on the onset conditions is displayed in Figure 10, where the values of \(h_{av}\) for three different nanofluids, \(\text{Al}_{2}\text{O}_{3}+\text{H}_{2}\text{O}\), \(\text{CuO}+\text{H}_{2}\text{O}\), and \(\text{TiO}_{2}+\text{H}_{2}\text{O}\), have been reported for \(A=1\) and \(\varphi_{av}=0.01\). It is worth pointing out that the onset of convection is anticipated for \(\text{TiO}_{2}+\text{H}_{2}\text{O}\) than for \(\text{CuO}+\text{H}_{2}\text{O}\) and \(\text{Al}_{2}\text{O}_{3}+\text{H}_{2}\text{O}\) due to the higher value of \(S_{T}\). Figure 8: Distribution of \(h_{av}\) vs. \(\Delta T\) for water, using \(A\) as a parameter.