Magnetic field effect on the onset of Soret-driven convection of a nanofluid confined within a Hele-Shaw cell

Abstract

The effect of a magnetic field on the early stages of Soret-driven convection of a nanoparticle suspension with large negative separation ratio χ confined within a Hele-Shaw cell, heated from above, was analyzed. Taking the Lorentz force into account, new stability equations were formulated in a similar (τ, ζ)-domain as well as in a global (τ, z)-domain by introducing the Hele-Shaw Rayleigh number based on the Soret flux (RsH) and the Hele-Shaw Hartmann number (HaH). With and without the quasi-steadiness assumptions, the resulting stability equations were solved analytically by expanding the disturbances as a series of orthogonal functions, and also the numerical shooting method was used. The critical time of the onset of convection and the corresponding wave number were obtained as a function of RsH and HaH. It was found that the magnetic field plays a critical role in the onset of convective instability. The onset time increases with increasing HaH and decreasing RsH. The linear stability limits are independent of the solution methods, if the trial functions for the disturbance quantities are properly chosen. Based on the results of the linear stability analysis, a non-linear analysis was conducted using direct numerical simulations. The non-linear analysis revealed that the convective motion can be apparent far after the linear stability limit.