  We consider a system consisting of a strongly interacting, ultracold unitary
Fermi gas under harmonic confinement. Our analysis suggests the possibility of
experimentally studying, in this system, an anisotropic shear viscosity tensor
driven by the anisotropy in the trapping potential. In particular, we suggest
that this experimental setup could mimic some features of anisotropic
geometries that have recently been studied for strongly coupled field theories
which have a gravitational dual. Results using the AdS/CFT correspondence in
these theories show that in systems with a background linear potential, certain
viscosity components can be made much smaller than the entropy density,
parametrically violating the bound proposed by Kovtun, Son and Starinets (KSS).
This intuition, along with results from a Boltzmann analysis that we perform,
suggests that a violation of the KSS bound can perhaps occur in the unitary
Fermi gas system when it is subjected to a suitable anisotropic trapping
potential which may be approximated to be linear in a suitable range of
parameters.
  We give a concrete proposal for an experimental setup where an anisotropic
shear viscosity tensor may arise. In such situations, it may also be possible
to observe a reduction in the spin one component of the shear viscosity from
its lowest value observed so far in ultracold Fermi gases. In extreme
anisotropic situations, the reduction may be enough to reduce the shear
viscosity to entropy ratio below the proposed KSS bound, although this regime
is difficult to analyze in a theoretically controlled manner.
