  Based on the strongly lensed gravitational waves (GWs) from compact binary
coalescence, we propose a new strategy to examine the fluid shear viscosity of
dark matter (DM) in the gravitational wave domain, i.e., whether a GW
experiences the damping effect when it propagates in DM fluid with nonzero
shear viscosity. By assuming that the dark matter self-scatterings are
efficient enough for the hydrodynamic description to be valid, our results
demonstrate that future ground-based Einstein Telescope (ET) and satellite GW
observatory (Big Bang Observer; BBO) may succeed in detecting any dark matter
self-interactions at the scales of galaxies and clusters.
