  Previous work has shown that the interaction between gravitational waves
(GWs) and a shell of viscous matter leads to damping of the GWs and heating of
the matter, and that these effects may be astrophysically significant. This
result was derived using the theory of linear perturbations about a Minkowki
background, and in this work the model is extended to be more physically
realistic by allowing the background geometry to be Schwarzschild. It is found
that the difference between using a Schwarzschild or Minkowski background is
minimal when either $r\gg M$ or $\lambda< M$, where $r$ is the radius of the
shell, $\lambda$ is the GW wavelength and $M$ is the mass of the system in
geometric units (so that $1M_\odot=1.48$km). However, when $r\sim 6M$ and
$\lambda\sim 25M$, then the damping and heating effects are about 9 times
larger on a Schwarzschild background than on Minkowski, and such situations
occur astrophysically.
