  We present the first calculation of the shear viscosity for two-flavor plane
wave (FF) color superconducting quark matter. This is a member of the family of
crystalline color superconducting phases of dense quark matter that may be
present in the cores of neutron stars. The paired quarks in the FF phase
feature gapless excitations on surfaces of crescent shaped blocking regions in
momentum space and participate in transport. We calculate their contribution to
the shear viscosity. We note that they also lead to dynamic screening of
transverse $t^1$, $t^2$, $t^3$ gluons which are undamped in the 2SC phase. The
exchange of these gluons is the most important mechanism of the scattering of
the paired quarks. We find that the shear viscosity of the paired quarks is
roughly a factor of $100$ smaller compared to the shear viscosity of unpaired
quark matter. Our results may have implications for the damping of $r-$modes in
rapidly rotating, cold neutron stars.
