Brinkmann viscosity action in porous MHD convection

• The paper is devoted to MHD convection in a porous horizontal layer with large pores.
• The stabilizing action of the Brinkmann viscosity is analyzed.
• Steady and oscillatory instability are analyzed.
• Unconditional asymptotic nonlinear stability is obtained by linear stability.
• The definitive boundedness of solutions is obtained by looking for L2-absorbing sets.

The stabilizing effect of Brinkmann viscosity (BV) in MHD convection in a horizontal porous layer L – filled by an electrically conducting fluid, heated from below and imbedded in a transverse magnetic field – is analyzed. The critical Rayleigh number of linear stability is found and – in closed forms – conditions for the onset of steady or oscillatory convection are obtained. Via the linearization principle given in [16] it is shown that unconditional nonlinear stability of thermal magnetic conduction solution is guaranteed by linear stability. The long-time behavior is characterized via the existence of L2-absorbingL2-absorbing sets.