  Many models of gamma-ray bursts (GRBs) involve accretion onto a compact
object, usually a black hole, at a mass accretion rate of order a fraction of a
solar mass per second. If the accretion disk is larger than a few tens or
hundreds of Schwarzschild radii, the accretion will proceed via a
convection-dominated accretion flow (CDAF) in which most of the matter escapes
to infinity rather than falling onto the black hole. Models involving the
mergers of black hole white dwarf binaries and black hole helium star binaries
fall in this category. These models are unlikely to produce GRBs since very
little mass reaches the black hole. If the accretion disk is smaller, then
accretion will proceed via neutrino cooling in a neutrino-dominated accretion
disk (NDAF) and most of the mass will reach the center. Models involving the
mergers of double neutron star binaries and black hole neutron star binaries
fall in this category and are capable of producing bright GRBs. If the
viscosity parameter $\alpha$ in the NDAF has a standard value $\sim0.1$, these
mergers can explain short GRBs with durations under a second, but they are
unlikely to produce long GRBs with durations of tens or hundred of seconds. If
the accretion disk is fed by fallback of material after a supernova explosion,
as in the collapsar model, then the time scale of the burst is determined by
fallback, not accretion. Such a model can produce long GRBs. Fallback models
again require that the accretion should proceed via an NDAF rather than a CDAF
in order for a significant amount of mass to reach the black hole. This
condition imposes an upper limit on the radius of injection of the gas.
