Time-dependent thermodynamics was applied to steady-state melt flow of polyethylene. The steady-state viscosity behavior and first normal-stress difference were examined as a couple. The latter was a measure of the energy intensity (energy per volume); the time derivative of that was treated as the rate of energy production. In steady-state flow, viscosity decreased and the rate of the energy production increased with increasing shear stress. From the sign of the rate of production, steady flow was found to be in the stable region. The stress growth leading into steady-state flow was known from our previous work to be in the unstable region. This may be called pseudo-stable region. Under a constant rate of shear deformation, both shear stress and viscosity increased rapidly. A change into steady-state flow was interpreted to be the fracture point. The growth of the normal stress was slower than that of shear stress. The first normal-stress difference was a manifestation of deformation, which is a volume increase caused by shear stress. The volume increase led to fracture.