Transp Porous Med DOI 10.1007/s11242-017-0868-2 Porous MHD Convection: Effect of Vadasz Inertia Term Florinda Capone 1 · Roberta De Luca 1 Received: 8 March 2017 / Accepted: 27 April 2017 © Springer Science+Business Media Dordrecht 2017 Abstract The onset of porous convection in an electrically conducting fluid uniformly heated from below and embedded in an external transverse constant magnetic field is analysed. In particular the effect of Vadasz inertia term, measured through the Vadasz number V a , on the instability threshold, is investigated. For the three-dimensional perturbations and full non- linear problem it is shown that sub-critical instabilities do not exist and the global nonlinear stability is guaranteed by the linear stability. The long-time behaviour is characterized via the existence of L 2 -absorbing sets. Keywords Porous convection · Magnetic field · Vadasz inertia term · Routh–Hurwitz conditions · Stability 1 Introduction Thermal convection in electrically conducting fluids embedded in porous media under the action of an external transverse magnetic field is the subject of Porous Magneto- Hydrodynamic (MHD) convection. In the past as nowadays this problem has been widely studied by many researchers because of its relevant applications in a lot of research fields like geophysics (geothermal energy recovery, subterraneans flows of plasma in the Earth’s magnetic field) and engineering (cooling of electronic devices, metallurgy, aerospace engi- neering, use of heat pipe in PC). Since the inhibitory effect of the magnetic field on the onset of convection is used in many practical situations, both in clear fluids and in porous media, many efforts have been made in order to prove (rigorously) the stabilizing effect of magnetic field (see [1–7, 9–21] and the references therein). Furthermore, due to the increasing interest B Florinda Capone fcapone@unina.it Roberta De Luca roberta.deluca@unina.it 1 Department of Mathematics and Applications “Renato Caccioppoli”, University of Naples Federico II, Via Cinzia, 80126 Naples, Italy 123