  In this article, we attempt to describe cosmic late-time acceleration of the
universe in the framework of $f(R,L_m)$ gravity by using an effective equation
of state when the account is taken of bulk viscosity. We presume a non-linear
$f(R,L_m)$ functional form, specifically, $f(R,L_m)=\frac{R}{2}+L_m^\alpha $,
where $\alpha$ is free model parameter. We obtain the exact solution of our
bulk viscous matter dominated $f(R,L_m)$ model, and then we utilize the
combined $H(z)+Pantheon+Analysis$ data sets to estimate the best fit values of
the free parameters of our model. Then we characterize the behaviour of the
matter-energy density, effective pressure, and the equation of state (EoS)
parameter incorporating the viscous type fluid. The evolution profile of the
effective EoS parameter depicts an acceleration phase of the cosmic expansion
whereas the pressure with the effect of viscosity exhibits negative behaviour
that can lead to the accelerating expansion of the universe. Moreover, the
cosmic matter-energy density shows expected positive behaviour. Further, we
investigate the behaviour of statefinder parameters for the assumed $f(R,L_m)$
model. We find that the evolutionary trajectory of the given model lies in the
quintessence region. In addition, we employ the Om diagnostic test that
indicates our model exhibits quintessence behavior. Lastly, we check the energy
condition criteria and find that violation of SEC occurs in the past, whereas
NEC and DEC satisfies the positivity criteria. We find that our $f(R,L_m)$
cosmological model with the effect of bulk viscosity provides a good fit of the
recent observational data and can efficiently describe the cosmic expansion
scenario.
