  We construct anisotropic black brane solutions and analyse the behaviour of
some of their metric perturbations. These solutions correspond to field theory
duals in which rotational symmetry is broken due an externally applied,
spatially constant, force. We find, in several examples, that when the
anisotropy is sufficiently big compared to the temperature, some components of
the viscosity tensor can become very small in units of the entropy density,
parametrically violating the KSS bound. We obtain an expression relating these
components of the viscosity, in units of the entropy density, to a ratio of
metric components at the horizon of the black brane. This relation is generally
valid, as long as the forcing function is translationally invariant, and it
directly connects the parametric violation of the bound to the anisotropy in
the metric at the horizon. Our results suggest the possibility that such small
components of the viscosity tensor might also arise in anisotropic strongly
coupled fluids found in nature.
