The Simha-Somcynsky (S-S) equation of state (eos) was used to compute the free volume parameter, h, from the pressure-volume-temperature (PVT) dependencies of eight molten polymers. The predicted by eos variation of h with T and P was confirmed by the positron annihilation lifetime spectroscopy; good agreement was found for h(P=constant)=h(T) as well as for h(T=constant)=h(P). Capillary shear viscosity (eta) data of the same polymers (measured at three temperatures and six pressures up to 700 bars), were plotted as log eta vs 1/h, the latter computed for T and P at which eta was measured. In previous works, such a plot for solvents and silicone oils resulted in a "master curve" for the liquid, in a wide range of T and P. However, for molten polymers, no superposition of data onto a "master curve" could be found. The superposition could be obtained allowing the characteristic pressure reducing parameter, P*, to vary. The necessity for using a "rheological" characteristic pressure reducing parameter, P*(R)=kappa P*, with kappa=1 to 2.1 indicates that the free volume parameter extracted from the thermodynamic equilibrium data may not fully describe the dynamic behavior. After eliminating possibility of other sources for the deviation, the most likely culprit seems to be the presence of structures in polymer melts at temperatures above the glass transition, T-g. For example, it was observed that for amorphous polymers at T congruent to 1.52T(g) the factor kappa=1, and the deviation vanish.