# Double-Diffusive Effects on the Onset of Rayleigh-Benard Convection of Water-Based Nanofluids
## 1 Introduction
[8], using theoretical approaches according to which a little addition of nanoparticles stabilized the fluid layer and simultaneously increased the heat transfer rate. Completely opposite results have been obtained by [PERSON] [9; 10], who performed a thermal instability analysis using a two-phase model including the nanoparticle diffusion due to Brownian motion and thermophoresis. Although the nanofluid properties were assumed to be constant with temperature, and the thermophoretic diffusion coefficient was extrapolated by the McNab-Meisen equation [11], valid only for particles greater than 1 \(\upmu\)m, the resulting critical Rayleigh number was found to be smaller by about two orders of magnitude with respect to the base liquid, and as a consequence of the destabilizing effect due to the nanoparticle diffusion, the heat transfer rate further increased. Smaller values of the critical Rayleigh number for a binary fluid mixture heated from below were also reported in the work of [PERSON] et al. [12], in which convective perturbations were found to grow at an effective Rayleigh number well below the critical Rayleigh number for the pure fluid. The stability of a horizontal nanofluid layer heated from below has been widely investigated too by [PERSON] and [PERSON] [13; 14; 15; 16] with a double-diffusive perturbation technique, which takes in account the Brownian diffusion and thermophoresis. Their results show that the instability of the nanofluid layer increased as the Lewis number increases, being related to a major concentration of heavy nanoparticles in the upper part of the layer due to the opposite temperature gradient. Similar results have been pointed out by [PERSON] et al. [17], though the value of the Lewis number considered in their investigation is too small to be representative of the behavior of a nanofluid. Finally, the few numerical works available in the literature on the thermal convection of nanofluids in cavities including the Brownian diffusion and thermophoresis show that the presence of nanoparticles in the base fluid affects the flow features and the heat transfer but has a negligible effect on the onset conditions [18; 19; 20; 21; 22]. In addition to these, many other works numerically analyzed the Rayleigh-Benard convection of nanofluids in cavities with a single-phase approach, yet they are not cited here, being out of the scope of the present work. However, for a comprehensive review on the Rayleigh-Benard instability in nanofluids, the article of [PERSON] and [PERSON] [23] is recommended. In all cited works, results have been presented in the classical Nu-Ra distributions, in which the dimensionless parameters have been calculate at average values of fluid thermophysical properties. This kind of representation is very exhaustive for describing the thermal performance of a fluid when its properties are not significantly affected by other variables of the system. In the case of nanofluids, in which the effective thermal conductivity and dynamic viscosity are extremely dependent on temperature and volume fraction, the dimensionless comparison between base fluid and nanofluid could lead to erroneous findings, so an analysis in dimension form could be more appropriate. In the following sections, a numerical study on the Rayleigh-Benard convection in a shallow cavity filled with different metal-oxide water-based nanofluids is presented through a two-phase model, which accounts for the effects of the Brownian diffusion and thermophoresis, at variable properties with temperature. The study is conducted using the average volume fraction and the diameter of the nanoparticles, the temperature difference imposed between the bottom and top surfaces of the cavity, the average nanofluid temperature, and the aspect ratio of the cavity as controlling parameters.