An important aspect while designing an “R2 z = constant” convergent channel as an extensional rheometer is the appropriate choice of the geometrical parameters and of the Reynolds number range of operation. The higher is the Reynolds number value, the thinner will be the boundary layer where the undesirable no-slip effect is confined, as discussed in the literature. However, if the Reynolds number, Re, is too large, then shear-related pressure losses become important, which is also undesirable in rheometry. Therefore, one design task is to find a range of Re within which the boundary layer is thin enough, and the velocity field in most of the domain is reasonably close to the desired kinematics. In this work we obtained numerical solutions for the flow of Newtonian and viscoelastic fluids through a convergent channel, for representative ranges of Re, dimensionless channel length, L, and dimensionless axial coordinate of inlet section, z0. For all cases, we determined fields of flow type, where regions of shear and of extension can be visualized. Among other findings, it is shown that, depending on the geometrical and flow characteristics, most of the mechanical energy dissipated can be due to shear effects, so that the extensional viscosity cannot be determined via pressure drop measurements.