  Evolution of the bulk viscous matter dominated universe has been analysed
using the full causal, Israel-Stewart theory for the evolution of bulk viscous
pressure in the context of recent acceleration of the universe. The form of
bulk viscosity is taken as $\xi=\alpha\rho^{1/2}. $ We obtained analytical
solutions for the Hubble parameter and scale factor of the universe. The model
parameters have been computed using the Type Ia supernovae observational data.
The evolution of the prominent cosmological parameters were obtained. The age
of the universe for the best estimated model parameters is found to be less
than observational value. The viscous matter behaves like stiff fluid in the
early evolutionary phase and then evolves to a negative pressure fluid in the
later phase. The equation of state is found to be stabilized with value $\omega
>-1$ and thus, the model is not showing any of the phantom behaviour during the
evolution. The local as well as generalized second law of thermodynamics are
satisfied in this model, while it was shown by many that the local second law
is breaking in the Eckart formalism approach. The statefinder geometric
diagnostic shows that the present model is distinct from the standard
$\Lambda$CDM model of the universe. One of the marked deviation seen in this
model compared to a corresponding model using Eckart approach is that the bulk
viscosity decreases with expansion of the universe, while in it increases from
negative value in the early universe towards positive values values in the
Eckart formalism.
