Cessation of pipe flow of a Bingham fluid

The aim of this paper is to examine closely the cessation of the flow of Bingham fluids in a circular pipe. The constitutive law, which has a singularity for zero shear rate, has been regularised using four different relations: that of Papanastasiou, a bi-viscous one and two new models proposed by the authors, based on the error function erf and the hyperbolic function tanh respectively. The results have been obtained with an implicit finite difference method and show that for the cessation of motion there are appreciable differences, particularly for the values of the stopping time, which for the proposed models are intermediate between the Papanastasiou and the bi-viscous ones. The stopping time as function of the Bingham number has been evaluated for the Erf model, emphasizing the very strong agreement between the numerical results and the theoretical upper limit given by Glowinski.