  When the universe was about 10 $\mu $seconds old, a first order cosmological
quark - hadron phase transition occurred at a critical temperature of around
200 MeV. In this work, we study the quark-hadron phase transition in the
context of brane-world cosmologies, in which our Universe is a three-brane
embedded in a five-dimensional bulk, and within an effective model of QCD. We
analyze the evolution of the physical quantities, relevant for the physical
description of the early universe, namely, the energy density, temperature and
scale factor, before, during, and after the phase transition. To study the
cosmological dynamics and evolution we use both analytical and numerical
methods. In particular, due to the high energy density in the early Universe,
we consider in detail the specific brane world model case of neglecting the
terms linearly proportional to the energy density with respect to the quadratic
terms. A small brane tension and a high value of the dark radiation term tend
to decrease the effective temperature of the quark-gluon plasma and of the
hadronic fluid, respectively, and to significantly accelerate the transition to
a pure hadronic phase. By assuming that the phase transition may be described
by an effective nucleation theory, we also consider the case where the Universe
evolved through a mixed phase with a small initial supercooling and
monotonically growing hadronic bubbles.
