We have calculated the viscosities of a variant of the Gay-Berne fluid as a function of the temperature by performing molecular dynamics simulations. We have evaluated the Green-Kubo relations for the various viscosity coefficients. The results have been cross-checked by performing shear flow simulations. At high temperatures there is a nematic phase that is transformed to a smectic A phase as the temperature is decreased. The nematic phase is found to be flow stable. Close to the nematic-smectic transition point the liquid crystal model system becomes flow unstable. This is in agreement with the theoretical predictions by Jahnig and Brochard [F. Jahnig and F. Brochard, J. Phys. 35, 301 (1974)]. In a planar Couette flow one can define the three Miesowicz viscosities or effective viscosities eta(1), eta(2), and eta(3). The coefficient eta(1) is the viscosity when the director is parallel to the streamlines, eta(2) is the viscosity when the director is perpendicular to the shear plane, and eta(3) is the viscosity when the director is perpendicular to the vorticity plane. In the smectic phase eta(1) is undefined because the strain rate field is incommensurate with the smectic layer structure when the director is parallel to the streamlines. The viscosity eta(3) is found to be fairly independent of the temperature. The coefficient eta(2) increases with the temperature. This is unusual because the viscosity of most isotropic Liquids decreases with the temperature. This anomaly is due to the smectic layer structure that is present at low temperatures. This lowers the friction because the layers can slide past each other fairly easily.