Computations of the flow of non-Newtonian fluids in the presence of a reentrant corner have a long history of convergence problems, which are believed to originate from a nonsquare-integrable stress singularity. Local flow analyses near such a corner have been inconclusive, due to the nonlinearity and the model dependence of the governing equations. We have used molecular dynamics simulations to compute the flow of both a Newtonian liquid and a model polymer melt through a channel with a reentrant corner, providing an unbiased and convergent calculation. The fluids interact via Lennard-Jones potentials, and for the polymer case we employ FENE chains of length up to 30. For the Newtonian fluid, the shear stress near the corner is found to agree with the Stokes flow prediction of Moffatt. In the non-Newtonian case, the shear stress has a stronger apparent divergence, increasing with velocity but not with chain length, which appears to saturate at an integrable value of approximately 0.8. The molecular origin of the stress enhancement is the additional elongation and rotation of the molecules near the reentrant corner.