Linear and nonlinear analysis of thermal instability in a porous saturated by a nanofluid

In this present article, the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is investigated analytically using linear and weakly nonlinear analysis. The model used for the nanofluid incorporates the effect of Brownian motion and thermophoresis. The effect of Raleigh-Darcy number, Lewis Number, modified diffusivity ratio, Vadasz number and normalized porosity parameter on the stability of the system is investigated. The analysis reveals that for a typical nanofluid (with large Lewis number) the prime effect of the nanofluids is via buoyancy effect coupled with the conservation of nanoparticles to the thermal energy equation being a second-order effect. Stationary and oscillatory modes of convections have been studied. It is found that the critical thermal Raleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution. The linear stability analysis is based on normal mode technique, while for nonlinear theory is based on the truncated representation of Fourier series method. A weakly nonlinear analysis is used to obtain the concentration and thermal Nusselt number. The behavior of the concentration and thermal Nusselt numbers is investigated by a solving the finite amplitude equations. Obtained results have been presented graphically and discussed in details.