We consider the numerical simulation of the combined drawing and radial expansion of a thick tube, which is a possible model for continuous tube-making. Both Newtonian and visco-elastic cases are considered. Mesh refinement studies show convergence; to stabilize the computations we have introduced relaxation factors. The main difficulty here is in finding the region of contact with the sizing die, as well as the form of the free surface. It was found that the extra stress component, τ θθ , is most helpful in considering the characteristics of tube expansion. The maximum value of τ θθ occurs near the outer wall of the exit of the annular die for Newtonian flow and near the interface for K-BKZ liquids. Both for Newtonian and K-BKZ liquids the extra stress component τ θθ in the expansion region, not including the vicinity of the exit of the die, develops from the inner free surface toward the outer free surface. The pressure value needed for tube expansion increases and then decreases as the expansion ratio R 0 /R W increases.