  The phonon contribution to the shear viscosity $\eta$ in superfluid neutron
stars is calculated by assuming neutron pairing in a $^1S_0$ channel. The shear
viscosity is obtained by means of variational methods for the solution of the
Boltzmann equation amended by a collision term which takes into account the
binary collisions of phonons. Effective field theory techniques are used to
extract the phonon scattering rates in terms of the equation of state (EoS) of
the system. We find that $\eta \propto 1/T^5$, the proportionality factor
depending on the EoS of the system. Our results indicate that the phonon
contribution to $\eta$ might have important effects for the different
oscillation modes of the star.
