  The quest for quantum degenerate Fermi gases interacting through the
anisotropic and long-range dipole-dipole interaction is an exciting and fast
developing branch within the cold-atoms research program. Recent experimental
progress in trapping, cooling, and controlling polar molecules with large
electric dipole moments has, therefore, motivated much theoretical effort. In a
recent letter, we have briefly discussed the application of a variational
time-dependent Hartree-Fock approach to study theoretically both the static and
the dynamic properties of such a system in a cylinder-symmetric harmonic trap.
We focused on the hydrodynamic regime, where collisions assure the equilibrium
locally. Here, we present a detailed theory, extended to encompass the general
case of a harmonic trap geometry without any symmetry. After deriving the
equations of motion for the gas, we explore their static solutions to
investigate key properties like the aspect ratios in both real and momentum
space as well as the stability diagram. We find that, despite the lack of
symmetry of the trap, the momentum distribution remains cylinder symmetric. The
equations of motion are then used to study the low-lying hydrodynamic
excitations, where, apart from the quadrupole and monopole modes, also the
radial quadrupole mode is investigated. Furthermore, we study the
time-of-flight dynamics as it represents an important diagnostic tool for
quantum gases. We find that the real-space aspect ratios are inverted during
the expansion, while the one in momentum space becomes asymptotically unity. In
addition, anisotropic features of the dipole-dipole interaction are discussed
in detail. These results could be particularly useful for future investigations
of strongly dipolar heteronuclear polar molecules deep in the quantum
degenerate regime.
