  We study the bulk viscosity of moderately hot and dense, neutrino-transparent
relativistic $npe\mu$ matter arising from weak-interaction direct Urca
processes. This work parallels our recent study of the bulk viscosity of
$npe\mu$ matter with a trapped neutrino component. The nuclear matter is
modeled in a relativistic density functional approach with two different
parametrizations -- DDME2 (which does not allow for the low-temperature
direct-Urca process at any density) and NL3 (which allows for low-temperature
direct-Urca process above a low-density threshold). We compute the
equilibration rates of Urca processes of neutron decay and lepton capture, as
well as the rate of the muon decay, and find that the muon decay process is
subdominant to the Urca processes at temperatures $T\geq 3$MeV in the case of
DDME2 model and $T\geq 1$MeV in the case of NL3 model. Thus, the
Urca-process-driven bulk viscosity is computed with the assumption that pure
leptonic reactions are frozen. As a result the electronic and muonic Urca
channels contribute to the bulk viscosity independently and at certain
densities the bulk viscosity of $npe\mu$ matter shows instead of the standard
one-peak (resonant) form a "flattened" shape. In the final step, we estimate
the damping timescales of density oscillations by the bulk viscosity. We find
that, e.g., at a typical oscillation frequency $f=1$kHz, the damping of
oscillations is most efficient at temperatures $3\leq T\leq 5$MeV and densities
$n_B\leq 2n_0$ where they can affect the evolution of the post-merger object.
