  We present the first multi-fluid analysis of a dense neutron star core with a
deconfined colour-flavour-locked superconducting quark component. Accounting
only for the condensate and (finite temperature) phonons, we make progress by
taking over results for superfluid $^4$He. The resultant two-fluid model
accounts for a number of additional viscosity coefficients (compared to the
Navier-Stokes equations) and we show how they enter the dissipation analysis
for an oscillating star. We provide simple estimates for the gravitational-wave
driven r-mode instability, demonstrating that the various phonon processes that
we consider are not effective damping agents. Even though the results are
likely of little direct astrophysical importance (since we consider an overly
simplistic stellar model) our analysis represents significant technical
progress, laying the foundation for more detailed numerical studies and
preparing the ground for the inclusion of additional aspects (in particular
associated with kaons) of the problem.
