Universal stability of hydromagnetic flows in a porous medium

Abstract

The linear and non-linear (energy theory) stability of viscous incompressible homogeneous conducting How of a liquid in a porous medium is investigated with an applied uniform magnetic field in the Z-direction. Our analysis is restricted to Darcy's law and arbitrary magnetic Reynolds number. In the case of linear theory the stability of equilibrium flows is discussed and is shown that the motion is unconditionally stable, whereas in the case of energy theory the motion is stable only if, 0 ≤ N' ≤ (1+∈/R_m)(1-R_e) where N' is the magnetic parameter, R_m the magnetic Reynolds number and R_e the Reynolds number. It is shown that for small magnetic Reynolds number the stability region increases whereas for large magnetic Reynolds number the stability region decreases. It is also shown that, since stability conditions obtained from linear and energy theories are quite different, the region of possible sub-critical instability may be quite large.