## Asymptotic analysis of mass-dominated convection in a nanofluid

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The threshold conditions for the onset of convection in colloidal suspensions is investigated using the particulate medium formulation. We consider a dilute liquid suspension of solid spherical particles that is confined between two horizontal plates of infinite extent placed at the vertical coordinates Z=0 and at Z=H. The plates are assumed to be rigid, perfectly conducting and impermeable to mass flow. The suspension is heated from below. A quasi-Boussinesq approximation has been adopted i.e. the density will be assumed constant except in the gravity term where it depends on both temperature and concentration. But both the fluid viscosity and the coefficient of the particle diffusion are allowed to depend on the particle concentration through the Einstein formula for the dilute case and through the general empirical formula of suspension viscosity μ=μ_{0}(1-C/C_{M})-2, where μ_{0} is the dynamic viscosity of the base fluid and C_{M} is the maximum packing volume fraction of hard-sphere particles suspension for the moderately concentrated case. An experimental parameter, β, is introduced to depict the coupled effects of thermophoresis, sedimentation and particle diffusion. For a given experimental setup, β is a function of the particle size. The graph of β as function of the particle radius is an inverted parabola with two zero crossings. The first zero crossing occurs near zero particle radius. The second zero crossing occurs at larger size particle radius, although still in the nanosize range.