  We study the hydrodynamical transition from an hadronic star into a quark or
a hybrid star. We discuss the possible mode of burning, using a fully
relativistic formalism and realistic Equations of State in which hyperons can
be present. We take into account the possibility that quarks form a diquark
condensate. We also discuss the formation of a mixed phase of hadrons and
quarks, and we indicate which region of the star can rapidly convert in various
possible scenarios. An estimate of the final temperature of the system is
provided. We find that the conversion process always corresponds to a
deflagration and never to a detonation. Hydrodynamical instabilities can
develop on the front. We estimate the increase in the conversion's velocity due
to the formation of wrinkles and we find that, although the increase is
significant, it is not sufficient to transform the deflagration into a
detonation in essentially all realistic scenarios. Concerning convection, it
does not always develop. In particular the system does not develop convection
if hyperons are not present in the initial phase and if the newly formed quark
phase is made of ungapped (or weakly gapped) quarks. At the contrary, the
process of conversion from ungapped quark matter to gapped quarks always allows
the formation of a convective layer. Finally, we discuss possible astrophysical
implications of our results.
