  This present study deals with the dissipative instability that appears in a
compressible partially ionised plasma slab embedded in a uniform magnetic
field, modelling the state of the plasma in solar prominences. In the partially
ionised plasma, the dominant dissipative effect is the Cowling resistivity. The
regions outside the slab (modelling the solar corona) are fully ionised, and
the dominant mechanism of dissipation is viscosity. Analytical solutions to the
extended magnetohydrodynamic (MHD) equations are found inside and outside of
the slab and solutions are matched at the boundaries of the slab. The
dispersion relation is derived and solutions are found analytically in the
slender slab limit, while the conditions necessary for the appearance of the
instability is investigated numerically for the entire parameter space. Our
study is focussed on the effect of the compressibility on the generation and
evolution of instabilities. We find that compressibility reduces the threshold
of the equilibrium flow, where waves can be unstable, to a level that is
comparable to the internal cusp speed, which is of the same order of flow
speeds that are currently observed in solar prominences. Our study addresses
only the slow waves, as these are the most likely perturbations to become
unstable, however the time-scales of the instability are found to be rather
large ranging from $10^5$-$10^7$ seconds. It is determined that the instability
threshold is further influenced by the concentration of neutrals and the
strength of the viscosity of the corona. Interestingly, these two latter
aspects have opposite effects. Our numerical analysis shows that the interplay
between the equilibrium flow, neutrals and dispersion can change considerably
the nature of waves.
