Onset of penetrative convection in a ferrofluid-saturated porous layer

The criterion for the onset of penetrative convection in a horizontal layer of ferrofluid-saturated porous medium in the presence of an applied vertical magnetic field via the internal heating model is studied. The impermeable isothermal boundaries are considered to be either paramagnetic or ferromagnetic, and the eigenvalue problem is solved numerically using the Galerkin method. It is noted that the paramagnetic boundaries with large magnetic susceptibility  delay the onset of penetrative ferroconvection the most when compared to the low-susceptibility ones, and the system is least stable if the boundaries are ferromagnetic. Increase in the value of magnetic Darcy-Rayleigh number Rm, heat source strength NS, and nonlinearity of magnetization parameter M3 is to hasten the onset of ferroconvection. Further, increase in the value of  and NS as well as decrease in M3 is to contract the dimension of convection cells.