  We consider the evolution of arbitrarily large perturbations of a prescribed
pure hydrodynamical flow of an electrically conducting fluid. We study whether
the flow perturbations as well as the generated magnetic fields decay or grow
with time and constitute a dynamo process. For that purpose we derive a
generalized Reynolds-Orr equation for the sum of the kinetic energy of the
hydrodynamic perturbation and the magnetic energy. The flow is confined in a
finite volume so the normal component of the velocity at the boundary is zero.
The tangential component is left arbitrary in contrast with previous works. For
the magnetic field we mostly employ the classical boundary conditions where the
field extends in the whole space. We establish critical values of hydrodynamic
and magnetic Reynolds numbers below which arbitrarily large initial
perturbations of the hydrodynamic flow decay. This involves generalization of
the Rayleigh-Faber-Krahn inequality for the smallest eigenvalue of an elliptic
operator. For high Reynolds number turbulence we provide an estimate of
critical magnetic Reynolds number below which arbitrarily large fluctuations of
the magnetic field decay.
