A finite difference scheme for the unsteady planar motion of a Bingham fluid

We present a numerical scheme based on finite differences to study the unsteady motion of a Bingham fluid in a channel (planar Poiseuille flow). The mathematical problem consists in a parabolic free boundary problem in which the yield surface is the free boundary. This problem has been extensively studied analytically in Comparini (1992). The numerical scheme proposed here exploits the Stefan-type structure of the free boundary problem. We show that the numerical results are in accordance with the analytical findings of Comparini (1992). To the best of our knowledge a comparison between the analytical results of Comparini (1992) and numerical results has never been performed. We also study two particular types of flow: (i) cessation flow; (ii) start-up flow. For both, we perform numerical simulations and we make comparisons between available analytical solutions and numerical solutions obtained with other methods.