  We consider the stability of clouds surrounded by a hotter confining medium
with respect to which they are in motion, against Kelvin-Helmholtz
instabilities (KHI). In the presence of cooling, sound waves are damped by
dissipation. Whenever cooling times are shorter than sound crossing times, as
they are in the normal interstellar medium, this implies that the instability
generated at the interface of the two media cannot propagate far from the
interface itself. To study how this influences the overall stability, first we
derive an analytic dispersion relation for cooling media, separated by a shear
layer. The inclusion of dissipation does not heal the instability, but it is
shown that only a small volume around the interface is affected, the
perturbation decaying exponentially with distance from the surface; this is
confirmed by numerical simulations. Numerical simulations of spherical clouds
moving in a surrounding intercloud medium by which they are pressure confined
show that these clouds develop a core/halo structure, with a turbulent halo,
and a core in laminar flow nearly unscathed by the KHI. The related and
previously reported ``champagne effect'', whereby clouds seem to explode from
their top sides, is cured by the inclusion of radiative losses.
