Suppose that it is desired to measure the Darcy permeability k of certain gels that consist of crosslinked polymers with water-filled interstices. These hydrogels are routinely fabricated as disks of diameter D = 5 mm, thickness h = 0.5 mm, and density ρ_o = 1030 kg/m³. Someone suggests that k be inferred from the terminal velocity U of such a disk in water. The presumption is that there will be enough flow through the gel to make U noticeably different from that of an equivalent solid disk. You are asked to evaluate this idea before any equipment is purchased.

(a) For a solid disk that stays horizontal as it settles, relate Re to Ar and C_D. How should Ar be defined for this object?

(b) Determine U for a solid disk with the given properties.

(c) Assuming there is only form drag, calculate the pressure difference ΔP across the disk. This is the average pressure at the bottom (leading) surface minus that at the top.

(d) Suppose that the polymer chains are thought to bundle into fibers of approximately 2 nm diameter and that the volume fraction of solids (polymer) ranges from 0.03 to 0.10. Approximately how large might k and the superficial velocity v_s be? Are the suggested experiments promising?