  We have constructed a spherically symmetric structure model in a cosmological
background filled with perfect fluid with non-vanishing pressure and studied
its quasi-local characteristics. This is done by using the Lema\^{i}tre
solution of the Einstein equations and suggesting an algorithm to integrate it
numerically. The result shows intriguing effects of the pressure inside the
structure. The evolution of the central black hole within the FRW universe, its
decoupling from the expanding parts of the model, the structure of its
space-like apparent horizon, the limiting case of the dynamical horizon tending
to a slowly evolving horizon, and the decreasing mass in-fall to the black hole
is also studied. The quasi-local features of this cosmological black hole may
not be inferred from the weak field approximation although the gravity outside
the structure is very weak.
