ABSTRACT A tidal disruption event (TDE) occurs when a star passes too close to a supermassive black hole and gets torn apart by its gravitational tidal field. After the disruption, the stellar debris form an expanding gaseous stream. The morphology and evolution of this stream are particularly interesting as it ultimately determines the observational properties of the event itself. In this work, we perform 3D hydrodynamical simulations of the TDE of a star modelled as a polytropic sphere of index γ = 5/3 and study the gravitational stability of the resulting gas stream. We provide an analytical solution for the evolution of the stream in the bound, unbound, and marginally bound cases, which allows us to describe the stream properties and analyse the time-scales of the physical processes involved, applying a formalism developed in star formation context. Our results are that, when fragmentation occurs, it is fuelled by the failure of pressure in supporting the gas against its self-gravity. We also show that a stability criterion that includes also the stream gas pressure proves to be far more accurate than one that only considers the black hole tidal forces, giving analytical predictions of the time evolution of the various forces associated with the stream. Our results point out that fragmentation occurs on time-scales longer compared with the observational windows of these events and is thus not expected to give rise to significant observational features.