# Double-Diffusive Effects on the Onset of Rayleigh-Benard Convection of Water-Based Nanofluids
## 1 Introduction  
The thermal instability of a horizontal nanofluid layer, confined between rigid boundaries and heated from below, has been the subject of several theoretical and numerical studies, for its importance in particular engineering applications, as the cooling of the micro-electronic devices and the solar collectors. However, a number of discrepancies can be found mainly due to the different numerical models used, which basically means single-phase or two-phase approaches assuming either physical properties that are variable or constant with temperature. The common evidence is that the addition of nanoparticles to a base fluid has the effect of increasing the dynamic viscosity of the suspension and, consequently, making the layer more stable. This could be reasonable should the nanofluid behave as a single-phase fluid. On the contrary, since the nanofluid actually behaves as a two-phase fluid, the nanoparticle's motion due to the Brownian diffusion and thermophoresis may induce a concentration gradient which enhances the nanofluid layer's instability, even if the Rayleigh number does not exceed the critical Rayleigh number \(Ra_{c}\) for the onset of convection in a pure fluid. Nanofluid instability due to the aggregation and sedimentation of nanoparticles was observed experimentally by [PERSON] and [PERSON] [1] in a water layer heated from below with the addition of different amounts of TiO\({}_{2}\) nanoparticles. Moreover, although the extra-buoyancy force consequent to the large density difference existing between the solid and liquid phases should lead to a convection enhancement, they measured a heat transfer rate which decreased as the volume fraction was increased, probably as a result of the increase in the effective dynamic viscosity. A totally different result was found by [PERSON] and [PERSON] [2], who conducted experiments using a rectangular cavity filled with Al\({}_{2}\)O\({}_{3}\) + H\({}_{2}\)O, in which significant enhancements of the heat transfer coefficient up to 36% were detected with respect to the base fluid. In their experimental work, [PERSON] et al. [3] suggested that the critical Rayleigh number for the onset of convection of a horizontal colloidal suspension layer heated from below may be strictly related to the size of the suspended particles, whose increase above 100 nm should play a stabilizing effect. On the other hand, when the particle size is smaller than 100 nm, Brownian motion and thermophoresis effects aresufficient to destabilize the layer. However, the experimental measurement of the critical Rayleigh number represents a very difficult task for the extremely long waiting time needed for perturbations for growing and breaking the nanofluid stratification; thus, an accurate investigation can be obtained only by means of a theoretical stability analysis, in the case of an indefinite horizontal layer, or in a finite-volume numerical simulation if the nanofluid is confined within a cavity, provided that the effective properties are correctly evaluated and the concentration boundary conditions are properly imposed. The heat transfer features inside a horizontal rectangular cavity filled with aluminawater nanofluids was investigated analytically by [PERSON] et al. [4] under the hypothesis that the nanofluid behaves as a single-phase fluid. In their model, the Jang-Choi correlation for the thermal conductivity [5] and the Pak-Cho correlation for the dynamic viscosity [6] were used to evaluate the Rayleigh and Prandtl numbers as functions of the mean temperature, the nanofluid volume fraction, and the size of the nanoparticles. They found that the heat transfer rate decreased as the volume fraction was increased and that the nanofluid is more stable than the base fluid. The same results were also obtained by [PERSON] [7] using a finite volume method to simulate a cavity filled with Al\({}_{2}\)O\({}_{3}\) + H\({}_{2}\)O, and by [PERSON] at al.