  This work is concerned with advancing multi-fluid models in General
Relativity, and in particular focuses on the modelling of dissipative fluids
and turbulent flows. Such models are required for an accurate description of
neutron star phenomenology, and binary neutron star mergers in particular. In
fact, the advent of multi-messenger astronomy offers exciting prospects for
exploring the extreme physics at play during such cosmic fireworks. We first
focus on modelling dissipative fluids in relativity, and explore the arguably
unique model that is ideally suited for describing dissipative multi-fluids in
General Relativity. Modelling single fluids in relativity is already a hard
task, but for neutron stars it is easy to argue that we need to understand even
more complicated settings: the presence of superfluid/superconducting mixtures,
for example, means that we need to go beyond single-fluid descriptions. We then
consider turbulent flows and focus on how to perform "filtering" in a curved
spacetime setting. We do so as most recent turbulent models in a Newtonian
setting are based on the notion of spatial filtering. As the same strategy is
beginning to be applied in numerical relativity, we focus on the foundational
underpinnings and propose a novel scheme for carrying out filtering, ensuring
consistency with the tenets of General Relativity. Finally, we discuss two
applications of relevance for binary neutron star mergers. We focus on the
modelling of ($\beta$-)reactions in neutron star simulations, and provide a
discussion of the magneto-rotational instability that is suited to highly
dynamical environments like mergers. We focus on these two problems as
reactions are expected to source the dominant dissipative contribution to the
overall dynamics, while the magneto-rotational instability is considered
crucial for sustaining the development of turbulence in mergers.
