# Double-Diffusive Effects on the Onset of Rayleigh-Benard Convection of Water-Based Nanofluids
## 2 Materials and Methods  
### Mathematical Formulation  
Consider a two-dimensional rectangular enclosure of height \(H\) and width \(W\), filled with a nanofluid. The bottom of the enclosure is heated at a uniform temperature \(T_{h}\), while the top is cooled at a uniform temperature \(T_{c}\), both sidewalls of the cavity being adiabatic, as depicted in Figure 1. The resulting flow is laminar, with negligible pressure work and viscous dissipation. It is assumed that Brownian and thermophoresis diffusion are the only significant slip mechanisms acting between nanoparticles and base fluid, as already discussed by [PERSON] [24]. The thermophysical properties of base fluid vary with temperature and are dependent on the local mass fraction of the suspended solid phase. Radiative heat transfer, as well as the Dufour effect, are neglected. With these hypotheses, the system of governing equations reduce to:  
\[\frac{\partial\rho_{n}}{\partial t}+\
abla\cdot(\rho_{n}\mathbf{V}) =0 \tag{1}\] \[\frac{\partial(\rho_{n}\mathbf{V})}{\partial t}+\
abla\cdot(\rho _{n}\mathbf{V}\mathbf{V}) =\
abla\cdot\boldsymbol{\tau}+\rho_{n}\mathbf{g}\] (2) \[\frac{\partial(\rho_{n}c_{n}T)}{\partial t}+\
abla\cdot(\rho_{n} \mathbf{V}c_{n}T) =\
abla\cdot(k_{n}\
abla T)\] (3) \[\frac{\partial(\rho_{n}m)}{\partial t}+\
abla\cdot(\rho_{n} \mathbf{V}m) =-\
abla\cdot\mathbf{J}_{p} \tag{4}\]  
where \(t\) is the time, \(\mathbf{V}\) is the velocity vector, \(\mathbf{J}_{p}\) is the nanoparticle diffusion mass flux, \(m\) is the mass fraction of the solid phase, and \(k_{n}\) is the nanofluid thermal conductivity. As  
Figure 1: Geometry and coordinate system. demonstrated by several researchers [25; 26; 27; 28; 29; 30], the nanofluid is considered to behave as a Newtonian fluid, and thus the stress tensor can be expressed as:  
\[\mathbf{\tau}=-(p+\frac{2}{3}\mu_{n}\
abla\cdot\mathbf{\mathrm{V}})\mathbf{\mathrm{I}}+\mu_{ n}[\
abla\mathbf{\mathrm{V}}+(\
abla\mathbf{\mathrm{V}}^{t})] \tag{5}\]  
where \(\mu_{n}\) is the effective dynamic viscosity. The nanoparticle diffusion mass flux is given by the sum of Brownian and thermophoretic contributions, and it is expressed by:  
\[\mathbf{\mathrm{J}}_{p}=-\rho_{n}(D_{B}\
abla m+D_{T}\frac{\
abla T}{T}) \tag{6}\]  
in which \(D_{B}\) and \(D_{T}\) represent the Brownian and thermophoretic diffusion coefficients. The Brownian diffusion coefficient, \(D_{B}\), is evaluated through the well-known Stokes-Einstein equation [31]:  
\[D_{B}=\frac{\kappa_{b}T}{3\pi\mu_{f}d_{p}} \tag{7}\]  
where \(\mu_{f}\) and \(d_{p}\) are, respectively, the dynamic viscosity of the base fluid and the diameter of the suspended nanoparticles, and \(\kappa_{b}=1.38066\times 10^{-23}\) J/K is the Boltzmann constant. As derived by [PERSON] and [PERSON] [32], the thermophoretic diffusion coefficient, \(D_{T}\), is expressed as:  
\[D_{T}=S_{T}\frac{\mu_{f}}{\rho_{f}}m \tag{8}\]  
where the value of the thermophoresis parameter, \(S_{T}\), is calculated by the following correlation proposed by [PERSON] et al. [33]:  
\[\frac{\partial\rho_{n}}{\partial t}+\
abla\cdot(\rho_{n}\mathbf{V}) =0 \tag{1}\]0  
where \(k_{s}\), \(k_{f}\), and \(\varphi_{av}\) represent, respectively, the thermal conductivities of the nanoparticles and of the base fluid and the volume fraction of the suspended solid phase.