  We present a Newtonian multi-fluid formalism for superfluid neutron star
cores, focussing on the additional dissipative terms that arise when one takes
into account the individual dynamical degrees of freedom associated with the
coupled "fluids". The problem is of direct astrophysical interest as the nature
of the dissipative terms can have significant impact on the damping of the
various oscillation modes of the star and the associated gravitational-wave
signatures. A particularly interesting application concerns the
gravitational-wave driven instability of f- and r-modes. We apply the developed
formalism to two specific three-fluid systems: (i) a hyperon core in which both
Lambda and Sigma^- hyperons are present, and (ii) a core of deconfined quarks
in the colour-flavour-locked phase in which a population of neutral K^0 kaons
is present. The formalism is, however, general and can be applied to other
problems in neutron-star dynamics (such as the effect of thermal excitations
close to the superfluid transition temperature) as well as laboratory
multi-fluid systems.
