Unsteady axial Poiseuille flow of a Bingham fluid in an annulus

Several fluids which are used in civil engineering, as bentonite, montmorillonite, slurries, behave as Bingham fluids, i.e. fluids which present a yield stress. This paper investigates numerically the start-up, the cessation and some pulsating flows, of a Bingham plastic between two coaxial cylinders. Its constitutive law presents a knee for zero shear rate, which represents a severe difficulty for solving any problem of unsteady motion, both analytically and numerically. A suitable way to avoid the obstacle is to regularize the constitutive equation using a smooth function which approximate the Bingham law. The approximating function depends on a parameter, and as it tends to infinity the model tends (in distribution theory sense) to the true Bingham law. The calculation is carried out with a implicit finite difference method. The results show that the time required to reach the steady state is infinite for the start-up whereas is finite for stopping.