  We investigate the behavior of shear viscosity in the presence of small
anisotropy and a finite chemical potential. First, we construct an anisotropic
Reissner Nordstr{\"o}m blackbrane in 5 dimensions in a simple Einstein-Maxwell
theory with a small linear dilaton. This solution is characterized by three
mass scales : anisotropy $\rho$, temperature T and chemical potential $\mu$. We
find this solution upto second order in the dilaton anisotropy parameter
$\rho$. This blackbrane solution corresponds to an anisotropic phase where the
anisotropy is small compared to the temperature and chemical potential. We find
that in this anisotropic phase, some components of the anisotropic shear
viscosity tensor, which are spin one with respect to the residual symmetry
after breaking rotational invariance, violates the KSS bound (${\eta \over
s}\ge {1 \over 4 \pi} $) proposed by Kovtun, Son and Starinets. We identify the
regions of the parameter space where these violations are significant. We carry
out a similar analysis in 4 dimensions and find similar violation of the KSS
bound for the spin one components to demonstrate the generality of the result.
Our results are particularly relevant in the context of strongly coupled
systems found in nature. We also provide an intuitive understanding of the
results using dimensional reduction and a Boltzmann calculation in a weakly
coupled version of a similar system. The Boltzmann analysis performed in a
system of weakly interacting particles in a linear potential also shows that
components of the viscosity tensor may be reduced. It is intriguing that the
Boltzmann analysis also predicts the corrections to be negative and that too in
a manner similar to the anisotropic strongly coupled theories with smooth
gravity duals.
