The sub-barrier fusion data of 32,34S + 89Y colliding systems are analyzed using the coupled-channels (CC) calculations, including couplings to the low-lying 2+ and 3− states in reacting nuclei as well as mutual and multi-phonon excitations of these states, based on the proximity potential model. In this scheme, we focus upon the the effect of surface energy coefficient γ and also thermal properties of liquids and hot nuclei on the inner part of the nuclear potential and thus on the fusion cross sections at bombarding energies far below the Coulomb barrier. As expected, our primary calculations using the original proximity potential 1977 show that this model underestimates the measured fusion cross sections at the whole energy region. It is shown that the mentioned physical effects bring significant cross sections enhancements for both reactions. The indications of the fusion hindrance phenomenon in the fusion cross sections at energies far below the Coulomb barrier show up. The observed S factors for the above-mentioned systems develop a maximum at low energies which can also be predicted by the phenomenological extrapolations. Our results within the framework of the modified form of the proximity potential reveal that the calculated fusion cross sections, S factors, and logarithmic slopes for the two systems are in good agreement with the corresponding experimental data even at the lowest energies

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