Particle-induced miscible fingering: Continuum limit

We experimentally inject silicone oil into the mixture of oil and noncolloidal particles inside a Hele-Shaw cell, to investigate the connection between miscible fingering and the interfacial structure that develops inside the thin gap. Previous studies with pure fluids have demonstrated that the onset of miscible fingering coincides with the transition from a smooth tonguelike structure to a sharp front between invading and defending fluids inside the gap. Our current experiments with suspensions reveal the same general behavior at the onset of miscible fingering, which we capture qualitatively using a lubrication model. However, distinct from the pure liquid counterpart, we observe changes in the interfacial structures inside the gap as the ratio of the gap thickness to particle diameter is systematically varied. We demonstrate that the particle dynamics inside the thin gap may alter interfacial structures even within the continuum limit, which lead to changes in morphologies of miscible fingering. The results of our study suggest a potential use of the wall confinement to control hydrodynamic instabilities in suspensions.

Figure 1

(a) A schematic of the experimental setup (top). A schematic of the interfacial structure from the side view. (b) Bottom-up view of silicone oil injected into suspension of ${\varphi }_0=0.25$. Here, the particle diameter $d=100$–$150\mu \mathrm{m}$, gap thickness $b=2.31\mathrm{mm}$, and flow rate $Q=9.1\mathrm{ml}/\min$.

Figure 2

(a) The depth-averaged concentrations, $\overline{\varphi }\left(r\right)$, is measured at two adjacent times, $t$ and $t+\delta t$ for ${\varphi }_0=0.1$ and $b=1.40$ mm. (b) The schematic illustrates the three-layer structure when oil penetrates the suspension in the $r\text{−}z$ plane. ${\overline{\varphi }}_t$ and ${\overline{\varphi }}_b$ denote the local concentrations of suspension above and below the tip of invading oil, respectively, while $h$ is the thickness of the middle oil layer.

Figure 3

Sketch of the flow system: radial miscible displacement of fluid 1 into fluid 2 with a constant flow rate $Q$ inside a Hele-Shaw cell of gap thickness $b$. The two miscible fluids are separated by a sharp interface of thickness $h(r,t)$.

Figure 4

(a) The plot of the flux function $F\left(h\right)$ versus $h/b$ for the mobility ratios: $M=0.1$, $M=1.5$, and $M=22.5$. (b) The characteristic speed $F^{\prime }\left(h\right)$ plotted as a function of $h/b$ for the same values of $M$.

Figure 5

The plot of similarity solutions, $h\left(\xi \right)$, for $M=1.33$ (solid line) and $M=10$ (dashed line).

Figure 6

(a) State diagram: quarter images of experiments with varying ${\varphi }_0$ and $b/d$. Images were taken after 105 s of the injection of pure oil. (b) The schematic illustrates the averaged interfacial deformation $\overline{l}$ (left panel), which is plotted as a function of $M$ on the right-side panel for $b=1.40\mathrm{mm}$ and $2.31\mathrm{mm}$.

Figure 7

(a)–(f) Interfacial structures of experiments (dashed lines) with $b=2.31\mathrm{mm}$ and mobility ratios: $M=1.33,1.59,1.95,2.48,3.31,4.72$. The profiles of light to dark colors correspond to different instants in time from 35 to 105 s (increment of 10 s) in the experiment (see the color bar for the reference). The error bars shown in panels (c)–(f) correspond to the estimated error in $h/b$ associated with the uniform concentration assumption at $h/b=0.75$ and $t=105$ s, which is denoted with a solid dot on the interfacial shape. The solid lines correspond to the theoretical results for the same values of $M$.

Figure 8

Interfacial structures of experiments (dashed lines) with $b=1.4\mathrm{mm}$ and mobility ratios: $M=1.33,1.59,1.95,2.48,3.31,4.72$. The light to dark colors correspond to increasing times from 35 s to 105 s (in the increment of 10 s) in the experiment. The error bars shown in panels (c)–(f) are the estimated errors in $h/b$ associated with the uniform concentration assumption at $h/b=0.75$ and $t=105$ s, denoted with a solid dot on the interfacial shape. The solid lines correspond to the theoretical results for the same values of $M$.

Figure 9

(a) At $M=1.33$, interfacial structures feature a parabolic shape and are largely independent of the gap thickness: $b=2.31\mathrm{mm}$ (solid lines) and $1.4\mathrm{mm}$ (dashed lines). The profiles of light to dark colors correspond to different times from 15 s to 105 s (every 10 s) in the experiment. (b) Bottom-up images from the experiments show that there is no miscible fingering.

Figure 10

Interfacial structures from experiments with ${\varphi }_0=0.35$ are shown in panels (a), (c), and (e), at three gap thicknesses, $b=2.31,1.8,1.4\mathrm{mm}$, respectively. The profiles of light to dark colors correspond to different instants in time from 15 to 105 s (every 10 s) in the experiment. The corresponding bottom-up images (denoted by a triangle and a star) and a zoomed-in image show the details of the interface. In particular, the zoomed-in images in panels (b), (d), and (f) show the disappearance of stripes as the gap is lowered.

Figure 11

The plot of $t_r$ over ${(b/d)}^2$ shows an approximately linear relationship, for two particle diameters: $d=125\mu \mathrm{m}$ and $325\mu \mathrm{m}$.

Figure 12

(a) Interfacial structures at $t=105\mathrm{s}$ at $\theta$ that corresponds to a peak and a valley of a finger, as well as the $\theta$-averaged value (solid line), for the initial concentration ${\varphi }_0=0.35$ and gap thickness $b=2.31\mathrm{mm}.$ (b) A schematic shows the positions of the two $3^{\mathrm{o}}$ slices corresponding to the peak and the valley of a finger.