The onset of Soret-driven convection of a nanoparticles suspension confined within a Hele-Shaw cell or in a porous medium

In this study, the early stages of a Soret-driven convection of a nanoparticles suspension with large negative separation ratio ψ confined within a Hele-Shaw cell or in a porous media, heated from above is investigated using linear and nonlinear analyses. The new stability equations are formulated in a similarity transformation (τ,ζ)-domain as well as a (τ,z)-domain by introducing a new characteristic dimensionless parameter RsH. With and without the quasi-steadiness assumptions, the resulting stability equations are solved analytically by expanding the disturbances as a series of orthogonal functions, and in addition, the numerical shooting method is used. The critical time of the onset of convection and the corresponding wave number are obtained as a function of RsH. It is found that the onset time of convective instability decreases with increasing RsH. The linear stability limits are independent of the solution methods and the coordinate system, if the trial functions for the disturbance quantities are properly chosen. Based on the results of the linear stability analysis, the nonlinear analyses are also tried by employing a finite volume method. The present nonlinear theory explained the linear stability theory well and also predicted the existing experimental results reasonably.