onset of convection in a nanofluid layer confinedwithin a hele-shaw cell

In this study, the onset of nanofluid convection confined within a hele-shaw cell is investigated byperforming a classical linear stability analysis. the model used for nanofluid combines the effects ofbrownian motion and thermophoresis, while for hele-shaw cell brinkman model are employed. the newstability equations are formulated by introducing new characteristic dimensionless parameters such as thehele-shaw number, the hele-shaw rayleigh number and the nanoparticle concentration hele-shaw rayleigh number. the resulting stability equations are solved numerically by using higher order galerkin method. it is found that the nanoparticle concentration hele-shaw rayleigh number, the lewis number and the modified diffusivity ratio hasten the onset of convection, while the hele-shaw number delays the onset of convection. a comparison is also made between the existing boundary conditions for nanoparticle and obtained that the zero nanoparticle flux boundary conditions under the thermophoretic effects has more destabilizing effect than the fixed nanoparticle boundary conditions.