  We consider spherically symmetric inhomogeneous pressure Stephani universes,
the center of symmetry being our location. The main feature of these models is
that comoving observers do not follow geodesics. In particular, comoving
perfect fluids have necessarily a radially dependent pressure. We consider a
subclass of these models characterized by some inhomogeneity parameter $\beta$.
We show that also the velocity of sound, like the (effective) equation of state
parameter, of comoving perfect fluids acquire away from the origin a time and
radial dependent change proportional to $\beta$. In order to produce a
realistic universe accelerating at late times without dark energy component one
must take $\beta < 0$. The redshift gets a modified dependence on the scale
factor $a(t)$ with a relative modification of $-9\%$ peaking at $z\sim 4$ and
vanishing at the big-bang and today on our past lightcone. The equation of
state parameter and the speed of sound of dustlike matter (corresponding to a
vanishing pressure at the center of symmetry $r=0$) behave in a similar way and
away from the center of symmetry they become negative -- a property usually
encountered for the dark energy component only. In order to mimic the observed
late-time accelerated expansion, the matter component must significantly depart
from standard dust, presumably ruling this subclass of Stephani models out as a
realistic cosmology. The only way to accept these models is to keep all
standard matter components of the universe including dark energy and take an
inhomogeneity parameter $\beta$ small enough.
