  In this article, we investigate the observed cosmic acceleration in the
framework of a cosmological $f(R,L_m)$ model dominated by bulk viscous matter
in an anisotropic background. We consider the LRS Bianchi type I metric and
derive the Friedmann equations that drive the gravitational interactions in
$f(R,L_m)$ gravity. Further, we assume the functional form
$f(R,L_m)=\frac{R}{2}+L_m^\alpha $, where $\alpha$ is a free model parameter,
and then find the exact solutions of fields equations corresponding to our
viscous matter dominated model. We incorporate the updated H(z) data and the
Pantheon data to acquire the best-fit values of parameters of our model by
utilizing the $\chi^2$ minimization technique along with the Markov Chain Monte
Carlo (MCMC) random sampling method. Further, we present the behavior of
physical parameters that describe the universe's evolution phase, such as
density, effective pressure and EoS parameters, skewness parameter, and the
statefinder diagnostic parameters. We found that the energy density indicates
expected positive behavior, whereas the negative behavior of bulk viscous
pressure contributes to the universe's expansion. The effective EoS parameter
favors the accelerating phase of the universe's expansion. Moreover, the
skewness parameter shows the anisotropic nature of spacetime during the entire
evolution phase of the universe. Finally, from statefinder diagnostic test, we
found that our cosmological $f(R,L_m)$ model lies in the quintessence region,
and it behaves like a de-Sitter universe in the far future. We analyze
different energy conditions in order to test the consistency of the obtained
solution. We found that all energy conditions except SEC show positive
behavior, while the violation of SEC favors the recently observed acceleration
with the transition from decelerated to an accelerated epoch of the universe's
expansion in the recent past.
