The experimental data on the $^{16}$O$+^{12}$C and $^{18}$O$+^{12}$C elastic scatterings and their optical model analysis are presented. Detailed and complete elastic angular distributions have been measured at the Strasbourg Vivitron accelerator at several energies covering the energy range between 5 and 10 MeV per nucleon. The elastic scattering angular distributions show the usual diffraction pattern and also, at larger angles, refractive effects in the form of nuclear rainbow and associated Airy structures. The optical model analysis unambiguously shows the evolution of the refractive scattering pattern. The observed structure, namely the Airy minima, can be consistently described by a nucleus-nucleus potential with a deep real part and a weakly absorptive imaginary part. The difference in absorption in the two systems is explained by an increased imaginary (mostly surface) part of the potential in the $^{18}$O$+^{12}$C system. The relation between the obtained potentials and those reported for the symmetrical $^{16}$O$+^{16}$O and $^{12}$C$+^{12}$C systems is drawn.
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