Plots of the temperature dependence of the shear viscosity eta of C12E5/D2O mixtures with compositions in the range 0.60 x 10(-2) < y < 10.08 x 10(-2) (critical composition y(c) = 1.08 x 10(-2); y, mass fraction of C12E5) are S-formed (a shallow minimum at a lower temperature T-eta min and a maximum at a higher temperature T-eta max). The measurements are carried out in the temperature range 5 degrees C < T < T-P;T-P approximate to 30 degrees C approaching the lower part of the binodal curve (Tp, temperature of phase separation of the mixture). In the temperature range (T-P - T-eta max) the viscosity decreases with increasing temperatures. At compositions y < 0.60 x 10(-2) the viscosity decreases with increasing temperature monotonically. The self-diffusion coefficient of C12E5 in C12E5/D2O mixtures is measured in the same temperature and composition range. The findings of both types of experiments are rationalized in terms of a model developed by Nilsson et al. (J. Phys. Chem. 1983, 87, 1377). The results give support to the hypothesis that the characteristic features of the temperature and composition dependence of both transport coefficients in the large aggregate region of the phase diagram have a common cause : The structural dynamics of the large micellar aggregates increases approaching the liquid/liquid coexistence curve at compositions y > y(c). Concentration fluctuations with long range correlations appear to be involved in this process.