  We consider a cosmological model dominated by bulk viscous matter with total
bulk viscosity coefficient proportional to the velocity and acceleration of the
expansion of the universe in such a way that
$\zeta=\zeta_{0}+\zeta_{1}\frac{\dot{a}}{a}+\zeta_{2}\frac{\ddot{a}}{\dot{a}}.$
We show that there exist two limiting conditions in the bulk viscous
coefficients, ($\zeta_{0}$, $\zeta_{1}$, $\zeta_{2}$) which corresponds to a
universe having a Big-Bang at the origin, followed by an early decelerated
epoch and then making a smooth transition into an accelerating epoch. We have
constrained the model using the type Ia Supernovae data, evaluated the best
estimated values of all the bulk viscous parameters and the Hubble parameter
corresponding to the two limiting conditions. We found that even though the
evolution of the cosmological parameters are in general different for the two
limiting cases, they show identical behavior for the best estimated values of
the parameters from both the limiting conditions. A recent acceleration would
occur if $\tilde{\zeta}_{0}+\tilde{\zeta}_{1}>1$ for the first limiting
conditions and if $\tilde{\zeta}_{0}+\tilde{\zeta}_{1}<1$ for the second
limiting conditions. The age of the universe predicted by this model is found
to be less than that predicted from the oldest galactic globular clusters. The
total bulk viscosity seems to be negative in the past and becomes positive when
$z\leq0.8$. So the model violates the local second law of thermodynamics.
However, the model satisfies the generalized second law of thermodynamics at
the apparent horizon throughout the evolution of the universe. We also made a
statefinder analysis of the model and found that it is distinguishably
different from the standard $\Lambda$CDM model at present, but shows a de
Sitter type behavior in the far future of the evolution.
