We explore the viability of a bulk viscous matter-dominated Universe to explain the present accelerated expansion of the Universe. The model is composed by a pressureless fluid with bulk viscosity of the form zeta = zeta_0 + zeta_1 * H where zeta_0 and zeta_1 are constants and H is the Hubble parameter. The pressureless fluid characterizes both the baryon and dark matter components. We study the behavior of the Universe according to this model analyzing the scale factor as well as some curvature scalars and the matter density. On the other hand, we compute the best estimated values of zeta_0 and zeta_1 using the type Ia Supernovae (SNe Ia) probe. We find that from all the possible scenarios for the Universe, the preferred one by the best estimated values of (zeta_0, zeta_1) is that of an expanding Universe beginning with a Big- Bang, followed by a decelerated expansion at early times, and with a smooth transition in recent times to an accelerated expansion epoch that is going to continue forever. The predicted age of the Universe is a little smaller than the mean value of the observational constraint coming from the oldest globular clusters but it is still inside of the confidence interval of this constraint. A drawback of the model is the violation of the local second law of thermodynamics in redshifts z >= 1. However, when we assume zeta_1 = 0, the simple model zeta = zeta_0 evaluated at the best estimated value for zeta_0 satisfies the local second law of thermodynamics, the age of the Universe is in perfect agreement with the constraint of globular clusters, and it also has a Big-Bang, followed by a decelerated expansion with the smooth transition to an accelerated expansion epoch in late times, that is going to continue forever.
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