Hydrodynamic loading of perforated disks in creeping flows

We present results from an experimental investigation of a viscous fluid driven through and around porous disks at low and moderate Reynolds number conditions: Re=O(10−4–10−3) and Re=O(1–10). Specifically, we quantify the hydrodynamic drag that these thin circular disks exhibit as a function of their size and the shape of their voids, while keeping their porosity fixed (void fraction, ϕ=0.69±0.02). We characterize the hydrodynamic loading using the drag ratio, which compares the magnitudes of drag experienced by a porous disk versus that of an impermeable, but otherwise equivalent, reference disk. We find that this drag ratio depends on the effective void radius, but not on the thickness of the disk. During this analysis, great attention has been dedicated to properly account for the effect of the wall confinement on the experimental data. Through scaling analysis, we rationalize our results by comparing them to an existing analytical solution for flow through and around porous disks. In particular, we find that an existing model based on Darcy flow within the porous disk and on Stokes flow outside the disk can be used in conjunction with a permeability model based on aperture flow to predict the forces that porous disks experience, even though the disks have finite thickness. Ultimately, we are able to combine these existing models to successfully predict the dependence of our experimentally measured drag ratio as a function of the Brinkman parameter of the perforated disks, at a fixed level of porosity. In contrast to the sedimentation experiments that are typically employed to evaluate the geometrical effects on the drag forces experienced by objects at low Re, our experiments were displacement controlled.