Pulsatile Poiseuille flow of a viscoplastic fluid in the gap between coaxial cylinders

Several materials that are of interest in engineering present a yield stress and behave as viscoplastic fluids. This paper investigates numerically the motion of a Bingham fluid between two coaxial cylinders due to a periodic pressure gradient and/or to the periodic displacement of the internal cylinder. The constitutive equation presents a discontinuity at the zero shear rate. To overcome the difficulty, the rheologic law has been regularized using a smooth function based on the error function. The velocity fields have been calculated using an implicit finite difference method. The procedure has been validated, comparing the numerical results with the analytical solution of the same problem for a Newtonian fluid. The nonlinear behavior of the fluid is emphasized, comparing the effects due to the simultaneous action of the pressure gradient and the displacement of the internal wall with the sum of the effects due to the single actions. In all cases, the mean discharge in a period increases. The comparison between the effects of the forcing agents shows that if the dimensionless frequency is less than 10 the increases of the discharge obtained by applying the pulsatile pressure gradient or moving the internal wall are similar. At low frequencies the action of the gradient exceeds that of the moving wall, whereas for higher frequencies the effect of the moving wall increases rapidly because a fixed displacement of the internal cylinder leads to very great values for the velocity of the internal wall