  In this paper we show that there are circumstances in which the damping of
gravitational waves (GWs) propagating through a viscous fluid can be highly
significant; in particular, this applies to Core Collapse Supernovae (CCSNe).
In previous work, we used linearized perturbations on a fixed background within
the Bondi-Sachs formalism, to determine the effect of a dust shell on GW
propagation. Here, we start with the (previously found) velocity field of the
matter, and use it to determine the shear tensor of the fluid flow. Then, for a
viscous fluid, the energy dissipated is calculated, leading to an equation for
GW damping. It is found that the damping effect agrees with previous results
when the wavelength $\lambda$ is much smaller than the radius $r_i$ of the
matter shell; but if $\lambda\gg r_i$, then the damping effect is greatly
increased.
  Next, the paper discusses an astrophysical application, CCSNe. There are
several different physical processes that generate GWs, and many models have
been presented in the literature. The damping effect thus needs to be evaluated
with each of the parameters $\lambda,r_i$ and the coefficient of shear
viscosity $\eta$, having a range of values. It is found that in most cases
there will be significant damping, and in some cases that it is almost
complete.
  We also consider the effect of viscous damping on primordial gravitational
waves (pGWs) generated during inflation in the early Universe. Two cases are
investigated where the wavelength is either much shorter than the shell radii
or much longer; we find that there are conditions that will produce significant
damping, to the extent that the waves would not be detectable.
