We numerically solve the cessation of the annular Poiseuille flow of Bingham plastics for various values of the diameter ratio, using the regularized constitutive equation proposed by Papanastasiou and employing finite elements in space and a fully implicit scheme in time. When the yield stress is not zero, the calculated stopping times are finite and just below the theoretical upper bounds provided by Glowinski [R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984]. (c) 2006 Elsevier B.V. All rights reserved.