Rotational viscosity coefficient is measured in different nematic phases of a binary liquid crystal system 6OCB-8OCB, as a function of relative concentration of the constituents, and of temperature at each concentration, using a transient NMR technique. Experiments are performed on four such mixtures with relative concentrations chosen such that for two of them there is a re-entrant nematic phase with an intervening smectic A phase (high temperature (HN) and re-entrant (RN) nematic phases), whereas for the other two there is only a single nematic phase (SN). Variation of gamma(1) with concentration and temperature is distinct in the HN and RN phases, while the corresponding behaviour in the SN phase is qualitatively the same as that in the RN phase. The variation of activation energies in all the phases as a function of the relative concentration further confirms this observation. It is suggested that the dynamics in HN phase is influenced by the smectic order fluctuations remnant in the system due to the underlying smectic phase, and hence is dynamically different from the SN and RN phases, which are obtained by direct melting of the solid.