Start-up of axisymmetric Poiseuille flow of a viscoplastic fluid

The start-up of a Bingham fluid in a pipe, due to a suddenly applied constant pressure gradient, has been examined numerically. The constitutive equation of the fluid, which has a discontinuity 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: the first based on the error function erf and the latter on the hyperbolic function tanh.. The results have been obtained with an implicit finite difference method: the numerical procedure has been first validated for a Newtonian fluid, for which the analytical results are well known. To compare the four models, the value of the regularization parameters is assigned in such a way that the tangent viscosity at zero shear stress is the same. The calculation shows that the models are practically equivalent for the start-up problem.