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: $\mathrm{Re}=O({10}^{-4}–{10}^{-3})$ and $\mathrm{Re}=O(1–10)$. 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, $\varphi =0.69\pm 0.02$). 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.

Figure 1

Experimental apparatus. (a) Photograph and (b) schematic of the experimental setup. An Instron universal testing machine was used to drive disks with radius $r$ and thickness $t$, with or without voids (Fig. 2), through an oil bath. This oil had viscosity $\mu$ and was contained in a pipe with cylindrical radius $R$. The disks were driven at a range of speeds, $U$, and the force the system experienced was recorded by a load cell on the linear stage. The disks were supported by two slender vertical rods, melted to them and press fit into a holder attached to the UTS. (c) Representative photograph of the detail of the joint between the support rod and the disk specimen.

Figure 2

Scanned images of the set of disks used in the experiments (thickness $t=0.80\pm 0.01\mathrm{mm}$). Disk (i) has no voids; disks (ii)–(ix) have constant void fractions ($\varphi =0.69\pm 0.02$), and the size and number of voids are varied. The effective void radius $a$ is the idealized radius calculated for $n$ equally sized voids whose collective area is equal to the void area.

Figure 3

The raw force signal from the load cell is plotted as a function of wetted length $z$, measured by the UTS (black circles). Once the transient forces—highlighted region I in (a)—have subsided, the disk experiences a steady force and the force on the rods increases: highlighted region II in (a), enlarged in (b). The steady-state disk force signal $F_{\mathrm{ss}}$ is isolated from the total force signal (see Fig. 12) and plotted as blue crosses. The horizontal red line is the average $F_d$ of the $F_{\mathrm{ss}}$ signal. The procedure to obtain these force signals is detailed in Appendix pp2.

Figure 4

Experiments were conducted with reference (unperforated) disks of varying radii to explore the wall effect. The characteristic force derived from the analytical prediction for the force on a disk by Sampson [39], shown with the green dashed line, does not account for the presence of the fluid boundary. The wall effect is accounted for in the solution given by Wakiya [48], shown with the blue solid line. Uncertainty in the radius measurements and the characteristic forces are much smaller than the size of the data points (red asterisks) and are not shown here. The results of our numerical simulations are reported with the solid black circles. [Wakiya eqn. = Eq. (12).]

Figure 5

(a) The drag ratio is plotted as a function of thickness for different effective void radii (see legend). Open and closed symbols correspond to experiments conducted in the lower Reynolds number $[\mathrm{Re}=O({10}^{-4}–{10}^{-3}\left)\right]$ and the higher Reynolds number $[\mathrm{Re}=O(1–10\left)\right]$ conditions, respectively. (b) The drag ratio $\mathrm{\Omega }$ decreases as the effective void radius $a$ increases for disks with radius $r=3.25\mathrm{cm}$ (fixed porosity $\varphi =0.69\pm 0.02$). The roman numerals correspond to the scanned disks in Fig. 2. Open and closed symbols correspond to experiments conducted in the lower Reynolds number $[\mathrm{Re}=O({10}^{-4}–{10}^{-3}\left)\right]$ and the higher Reynolds number $[\mathrm{Re}=O(1–10\left)\right]$ conditions, respectively. The analytical result ($+$ symbol) for the annular disk is obtained from Eq. (7). In both (a) and (b), the error bars reflect uncertainty in $\mathrm{\Omega }$ which propagates due to uncertainty in ${\tilde{F}}_c$. When the wall effects for porous disks are appropriately taken into account towards the end of the paper, the drag ratio as a function of the void size is corrected.

Figure 6

(a) Sketch of the numerical flow configuration (not to scale); (b) representative detail of the mesh used to simulate the Stokes flow past the disk (v).

Figure 7

The colored solid lines represent the wall-correction factors for different confinement ratios, here numerically evaluated for the disks considered in the experiments, from the case of the solid disk (i) to the annular disk (ix). The dashed green line is the characteristic force derived from the analytical prediction for the force on a disk by Sampson [39]. The vertical dashed black line corresponds to the confinement ratio used during the experiments, i.e., $\alpha (r/R_p=0.4643)$. The red asterisks are the experimental values obtained for the reference disks, as already reported in Fig. 4.

Figure 8

(a) Wall-correction factors for the perforated disks at the experimental confinement ratio $\alpha (r/R_p=0.464)$ as function of the void radii. The solid blue line represents the linear interpolation between the numerical correction factors. (b) Drag ratio $\mathrm{\Omega }$ as a function of the effective void radius $a$, taking into account the wall effects for porous disks obtained from the numerical simulations.

Figure 9

Experimentally determined permeability $k_V$ computed through Eq. (18) vs (a) $k_1\sim a^2$ and (b) $k_2\sim at$. The solid symbols correspond to experiments done at $\mathrm{Re}=O(1–10)$, for the mineral oil, and open symbols correspond to experiments conducted at $\mathrm{Re}=O({10}^{-4}–{10}^{-3})$, for the silicone oil. Error bars in the vertical direction represent the experimental uncertainty in $\mathrm{\Omega }$ propagating from the the rod force. Error bars are smaller than the solid data points.

Figure 10

Drag ratio $\mathrm{\Omega }$ as a function of the Brinkman parameter $\beta$. The solid symbols correspond to experiments conducted at $\mathrm{Re}=O(1–10)$ (with the mineral oil) and open symbols correspond to experiments conducted at $\mathrm{Re}=O({10}^{-4}–{10}^{-3})$ (with the silicone oil). Error bars of the data points correspond to uncertainty in the rod force. Uncertainty in the prediction (shown shaded in gray) is due to uncertainty in the prediction of $k$. [Vainshtein eqn. = Eq. (9).]

Figure 11

Temperature dependence of the dynamic viscosity of both fluids used: silicone oil and mineral oil (see legend). The dashed and solid lines correspond to fits to Eq. (A1) for the two oils, respectively.

Figure 12

The rod force-speed coefficient $M$, defined in Eq. (B3), is plotted as a function of the speed $U$ for (a) the mineral oil and (b) the silicone oil. In both plots, the solid line is the fitted $M\left(U\right)$ and the gray region represents the $68\%$ confidence interval in both fitted parameters. The outlier data point in (b) was preserved because we could not recall any abnormalities regarding the particular experiment from which it came. The inset in (b) shows the air entrainment that occurred at the free surface in experiments using the silicone oil. Error bars corresponding to variation in the steady-state force signal $F_{\mathrm{ss}}$ and the disk speed $U$ are smaller than the data points and are not shown.