On the penetrative Benard–Marangoni convection in a ferromagnetic fluid layer

The onset of penetrative Benard–Marangoni convection in a horizontal ferromagnetic fluid layer in the presence of a uniform vertical magnetic field via an internal heating model is investigated by performing the linear stability analysis. It is observed that the internal heating alters the basic temperature, magnetic field intensity as well as magnetization distributions from linear to parabolic with respect to fluid layer depth and turns out the eigenvalue equations with variable coefficients. The eigenvalue problem is solved numerically using the Galerkin technique by considering either the Rayleigh number or the Marangoni number as the eigenvalue. The influence of buoyancy, magnetic and surface tension forces acting together as well as in isolation on the stability characteristics of the system is analyzed in detail. It is observed that the effect of internal heating NsNs, magnetic Rayleigh number RmRm, nonlinearity of the magnetization M3M3 is to hasten, while increasing the Biot number Bi   is to delay the onset of Benard–Marangoni ferroconvection. Moreover, increasing NsNs, RmRm, M3M3, and decreasing Bi is to decrease the size of the convection cells.