  We discuss the isentropic trajectories on the QCD phase diagram in the
temperature and the quark chemical potential plane using the
Nambu--Jona-Lasinio model with the Polyakov loop coupling (PNJL model). We
impose a constraint on the strange quark chemical potential so that the strange
quark density is zero, which is the case in the ultra relativistic heavy-ion
collisions. We compare our numerical results with the truncated estimates by
the Taylor expansion in terms of the chemical potential to quantify the
reliability of the expansion used in the lattice QCD simulation. We finally
discuss the strange quark chemical potential induced by the strangeness
neutrality condition and relate it to the ratio of the Polyakov loop and the
anti-Polyakov loop.
