  In this work, we investigate the dynamics of bulk viscous models with
decaying vacuum energy density (VED) in a spatially homogeneous and isotropic
flat Friedmann-Lema\^{i}tre- Robertson-walker (FLRW) spacetime. We particularly
are interested to study the viscous model which considers first order deviation
from equilibrium, i.e., the Eckart theory. In the first part, using the
different forms of the bulk viscous coefficient, we find the main cosmological
parameters, like Hubble parameter, scale factor, deceleration parameter and
equation of state parameter analytically. We discuss some cosmological
consequences of the evolutions and dynamics of the different viscous models
with decaying VED. We examine the linear perturbation growth in the context of
the bulk viscous model with decaying VED to see if it survives this further
level of scrutiny. The second part of the work is devoted to constrain the
viscous model of the form $\zeta \propto H$, where $\zeta$ is the bulk viscous
coefficient and $H$ is the Hubble parameter, using three different combinations
of data from type Ia supernovae (Pantheon), $H(z)$ (cosmic chronometers),
Baryon Acoustic Oscillation and $f(z)\sigma_8(z)$ measurements with Markov
Chain Monte Carlo (MCMC) method. We show that the considered model is
compatible with the cosmological probes, and the $\Lambda$CDM recovered in
late-time of the evolution of the Universe. Finally, we obtain selection
information criteria (AIC and BIC) to study the stability of the models.
