Creeping flows of Bingham fluids through arrays of aligned cylinders

Numerical simulations are presented for flows of Bingham fluids through periodic square arrays of aligned cylinders, for cases in which fluid inertia can be neglected. The aim is to quantify the dependence of the drag coefficient of the cylinders on the Bingham number. The results for large Bingham numbers, and also for dilute arrays of cylinders (low solid area fraction) are shown to approach previous analytical results for a single cylinder. The results for concentrated arrays are shown to agree with a lubrication theory. Although the rheology is strongly nonlinear and significant unyielded regions are shown to develop, the drag coefficient is approximately a linear function of the Bingham number. This is shown to be the case for flows along a principal axis of the array and also seems to hold for flow at 45° (in the plane perpendicular to the cylinders). It is shown that the drag force on a cylinder in the array immersed in a Bingham fluid is approximately equal to the sum of the drag forces in the corresponding cases of Newtonian and perfectly plastic fluids. This result is used to derive a criterion for the critical pressure gradient, required for flow. Implications for large-scale modelling of flow through fibrous media are discussed.