We investigate the nature of dissipative instability appearing in a prominence planar thread filled with partially ionised plasma in the incompressible limit. The importance of partial ionisation is investigated in terms of the ionisation factor and wavelength of waves propagating in the slab. To highlight the role of partial ionisation, we have constructed models describing various situations we can meet in solar prominence fine structure. Matching the solutions for the transversal component of the velocity and total pressure at the interfaces between the prominence slab and surrounding plasmas, we derived a dispersion relation whose imaginary part describes the evolution of the instability. Results are obtained in the limit of weak dissipation. We have investigated the appearance of instabilities in prominence dark plumes using single and two-fluid approximations. We show that dissipative instabilities appear for flow speeds that are less than the Kelvin-Helmholtz instability threshold. The onset of instability is determined by the equilibrium flow strength, the ionisation factor of the plasma, the wavelength of waves and the ion-neutral collisional rate. For a given wavelength and for ionisation degrees closer to a neutral gas, the propagating waves become unstable for a narrow band of flow speeds, meaning that neutrals have a stabilising effect. Our results show that the partially ionised plasma describing prominence dark plumes becomes unstable only in a two-fluid (charged particles-neutrals) model, that is for periods that are smaller than the ion-neutral collision time. The present study improves our understanding of stability of solar prominences and the role of partial ionisation in destabilising the plasma. We show the necessity of two-fluid approximation when discussing the nature of instabilities: waves in a single fluid approximation show a great deal of stability.
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