  In the violent post-merger of binary neutron-star mergers strong oscillations
are present that impact the emitted gravitational-wave (GW) signal. The
frequencies, temperatures and densities involved in these oscillations allow
for violations of the chemical equilibrium promoted by weak-interactions, thus
leading to a nonzero bulk viscosity that can impact dynamics and GW signals. We
present the first simulations of binary neutron-star mergers employing the
self-consistent and second-order formulation of the equations of relativistic
hydrodynamics for dissipative fluids proposed by M\"uller, Israel and Stewart.
With the spirit of obtaining a first assessment of the impact of bulk viscosity
on the structure and radiative efficiency of the merger remnant we adopt a
simplified approach for the viscosity, which we assume to be constant within
the stars, but which we vary in strength for different binaries, thus exploring
the possible behaviours and obtaining strict upper limits. In this way, we find
that large bulk viscosities are very effective at damping the
collision-and-bounce oscillations that characterize the dynamics of the stellar
cores right after the merger. As a result, the $m=2$ deformations and the
gravitational-radiation efficiency of the remnant are considerably reduced,
with qualitative and quantitative changes in the post-merger spectrum that can
be large in the case of the most extreme configurations. Overall, our crude but
self-consistent results indicate that bulk viscosity reduces the energy
radiated in GWs by $\lesssim 1\%$ in the (realistic) scenario of small
viscosity, and by $\lesssim 15\%$ in the (unrealistic) scenario of large
viscosity.
