  Spatially homogeneous but totally anisotropic and non-flat Bianchi type II
cosmological model has been studied in general relativity in the presence of
two minimally interacting fluids; a perfect fluid as the matter fluid and a
hypothetical anisotropic fluid as the dark energy fluid. The Einstein's field
equations have been solved by applying two kinematical ans\"{a}tze: we have
assumed the variation law for the mean Hubble parameter that yields a constant
value of deceleration parameter, and one of the components of the shear tensor
has been considered proportional to the mean Hubble parameter. We have
particularly dwelled on the accelerating models with non-divergent expansion
anisotropy as the Universe evolves. Yielding anisotropic pressure, the fluid we
consider in the context of dark energy, can produce results that can be
produced in the presence of isotropic fluid in accordance with the \Lambda CDM
cosmology. However, the derived model gives additional opportunities by being
able to allow kinematics that cannot be produced in the presence of fluids that
yield only isotropic pressure. We have obtained well behaving cases where the
anisotropy of the expansion and the anisotropy of the fluid converge to finite
values (include zero) in the late Universe. We have also showed that although
the metric we consider is totally anisotropic, the anisotropy of the dark
energy is constrained to be axially symmetric, as long as the overall energy
momentum tensor possesses zero shear stress.
