# Double-Diffusive Effects on the Onset of Rayleigh-Benard Convection of Water-Based Nanofluids
## 3 Results  
Numerical simulations are performed using \(\rm Al_{2}O_{3}+H_{2}O\) nanofluid, for values of the volume fraction, \(\varphi_{av}\), in the range between 0 and 0.04, and for values of the aspect ratio, \(A=H/W\), in the range between 0.25 and 1. The height of the enclosure, \(H\), the average diameter of the nanoparticles, \(d_{p}\), and the temperature of the bottom wall have been fixed to 0.01 m, 25 nm, and 300 K, respectively, for all simulations. With these values, varying the temperature difference within the cavity, the Rayleigh number based on the height of the plate lies in the range between \(5\times 10^{2}\leq Ra_{n}\leq 10^{4}\), in which the effective physical properties of the nanofluid are calculated at the average temperature and average volume fraction. The more common way to present heat transfer results is the typical dimensionless distributions displayed in Figure 4, where the Nusselt number \(Nu\) is plotted versus \(Ra_{n}\) for a square cavity using the average volume fraction as a parameter. As expected, \(Nu\) increases for values of \(Ra_{n}\) above the critical value, \(Ra_{c}\), corresponding to the departure from the perfectly conducting solution of a nanofluid static layer with an \(Nu\) value equal to unity. What seems important to point out is that the Nu-Ra data do not lie on a unique interpolation curve, which is what would be expected if the nanofluid should behave as a pure fluid, thus illuminating the two-phase behaviou of the nanofluids due to the Brownian motion and thermophoresis diffusion of the suspended nanoparticles, as already experimentally found by [PERSON] et al. [40]  
Figure 3: Comparison between the present numerical values and the corresponding data of [PERSON] et al. [40] for \(\rm Al_{2}O_{3}+H_{2}O\) in a differentially heated vertical enclosure: (**a**) \(\varphi_{av}=0.01\); (**b**) \(\varphi_{av}=0.04\). and [PERSON] et al. [42] using Al\({}_{2}\)O\({}_{3}\) + H\({}_{2}\)O in cavities differentially heated at the sides. As clearly shown, the dispersion of a progressively larger amount of particles in mass flux into the base fluid seems to result in two main effects: an increase in \(Nu\) and a progressive reduction in the nanofluid layer's stability, with the consequent decrease in the critical Rayleigh number for the onset of convection. Actually, if the analysis is conducted considering the distributions of the average heat transfer coefficient \(h_{av}\) plotted versus the temperature difference \(\Delta T\) imposed across the cavity for different values of the average volume fraction \(\varphi_{av}\), as shown in Figure 5, it is evident that the heat transfer coefficient decreases when the average volume fraction is increased, while the value of \(\Delta T\) corresponding to the onset of convection increases as the average volume fraction is increased, which is a direct consequence of the growth in the dynamic viscosity. Thus, it can be concluded that the dispersion of nanoparticles in the base fluid is detrimental for the heat transfer performance and that the nanofluid layer is more stable than the base fluid. The values of \(\Delta T\) corresponding to the onset of convection, for water and alumina-water nanofluids with \(\varphi_{av}=0.01\) and \(\varphi_{av}=0.04\), are, respectively, 0.14 K, 0.16 K, and 0.2 K. Notice that the value assumed by the heat transfer coefficient \(h_{av}\) at the onset of convection corresponds to the theoretical conductance \(k/H\) of the stratified nanofluid layer. On account of the above considerations, it is clear that using a two-phase modeling, which permits us to account for the non-negligible slip effects occurring between suspended nanoparticles and base fluid, i.e., Brownian diffusion and, much more importantly, thermophoresis, makes the representation of the heat transfer performance not meaningful in terms of dimensionless parameters.