Particle-induced viscous fingering: Review and outlook

Changes in bulk rheology of suspensions with the particle concentration are well documented, ranging from normal stress differences to shear-thinning and shear-thickening behaviors. However, relatively little is known about the impact this bulk rheology can have on the interfacial dynamics of free-surface, particle-laden flows. Recently, it has been demonstrated that particles suspended in liquid may cause viscous fingering by modifying the local rheology near the fluid-fluid interface, when a mixture of noncolloidal particles and viscous oil displaces air inside a Hele-Shaw cell. As the same flow configuration sans particles is inherently stable to viscous fingering, the simplicity of the baseline flow renders this a perfect model problem to study to gain insight into the coupling between the bulk rheology and fluid-fluid interface. To that end, we first summarize the recent experimental findings of this particle-induced viscous fingering, which highlights the effect of the particle volume fraction and the particle diameter relative to the gap size on the emergent finger geometries. We also present experimental measurements of the critical viscosity gradient at the fingering onset, as well as the surprising time-independent finger growth regime for sufficiently large volume fractions. These results—both old and new—together highlight the similarities as well as important differences between fingering in suspensions versus pure liquid, suggesting a direction for continued research at the intersection between suspension rheology and interfacial dynamics.

This article appears in the following collection:

Figure 1

Time-sequential images of two representative experiments with corresponding schematics to highlight the key physical picture. (a) ${\varphi }_0=0.28$ and $h/d=10.2$ (weak fingering): Particles first collect on the interface, causing miscible fingering. Miscible fingering is evidenced by the formation of particle clusters normal to the growing interface. Then, the interfacial fingers form between them. (b) ${\varphi }_0=0.30$ and $h/d=2.2$ (band fingering): Particles are shown to clearly form a sharp band on the interface, which first grows radially but starts to break around $7\mathrm{s}$; this causes larger interfacial deformations especially compared to weak fingering in panel (a).

Figure 2

(a) The schematic of the experimental setup shows the injection of suspension into a Hele-Shaw cell from the bottom center of the channel. The side-view schematic below includes the radial $r$ and normal $z$ coordinates, as well as the location of the evolving suspension-air interface $R\left(t\right)$. (b) The phase diagram summarizes all the experimental results of Refs. [49, 50] to show “no fingering,” “weak fingering,” and ‘band fingering” regimes as a function of ${\varphi }_0$ and $h/d$.

Figure 3

(a) The suspension-air interface is described following a curvilinear coordinate $s$. The instantaneous distance from the center to any arbitrary point on the interface is denoted as $R_b\left(s\right)$, while $R$ is the radius of the best fitted circle out of all points along the interface. We identify local $max\left(R_b\right)$ and $min\left(R_b\right)$ inside the small “wedge” defined by ${\theta }_d$; then, the characteristic finger length $\overline{\lambda }$ is computed by averaging $max\left(R_b\right)-min\left(R_b\right)$ over the entire interface. (b) The plot of $\overline{\lambda }$ over time $t$ quantitatively illustrates the difference between band fingering (${\varphi }_0=0.3,h/d=2.2$) and weak fingering (${\varphi }_0=0.28,h/d=10.2$). In addition to being larger in magnitude, there is a peak in $\overline{\lambda }$ for band fingering that corresponds to the moment of band breakup; $\overline{\lambda }$ increases gradually in the weak fingering case.

Figure 4

(a) Even when clear oil is injected ahead of the suspension, particles catch up to the interface, demonstrating that particles move faster than the mixture on average, or ${\overline{u}}_r^{\mathrm{p}}>{\overline{u}}_r$. (b) The plot of the ratio of average velocities $\beta$ as a function of ${\varphi }_0$ includes the calculation result of the suspension balance model (SBM) in a solid line, as well as the experimental measurement for $h=1.40$ and $1.27\mathrm{mm}$. (c) The particle accumulation is evidenced in the plot of $\overline{\varphi }$ over $r/R$ for ${\varphi }_0=0.28$ and $h/d=10.2$; the dotted line indicates the depth-averaged concentration in the upstream regime, ${\overline{\varphi }}_{\mathrm{up}}$.

Figure 5

Processed experimental images at the final time ($t=18\mathrm{s}$) reveal the extent of particle accumulation at varying ${\varphi }_0$ at $h/d=10.2$. The clear rise in $\overline{\varphi }$ relative to ${\varphi }_0$ becomes evident for ${\varphi }_0>0.14$, while miscible fingering and interfacial deformations are visible at ${\varphi }_0=0.25$.

Figure 6

(a) The plot of sample variance $S^2$ for ${\varphi }_0=0.28,h/d=10.2$ (weak fingering) and ${\varphi }_0=0.20,h/d=10.2$ (no fingering) at $t=16\mathrm{s}$. Here, $S^2$ corresponds to the deviation of the particle concentration in each pixel ${\overline{\varphi }}_i$ at given $r$ from the average value $\overline{\varphi }\left(r\right)$ and is a measure of axisymmetry breaking due to miscible fingering. For ${\varphi }_0=0.28, S^2$ rises substantially close to the interface where miscible fingering occurs, while $S^2$ for ${\varphi }_0=0.20$ remains small. (b) Corresponding experimental images for ${\varphi }_0=0.28$ (left) and ${\varphi }_0=0.20$ (right) confirm the effectiveness of $S^2$ in determining the onset of miscible fingering.

Figure 7

The plot of the normalized effective viscosity ${\mu }_s\left(\overline{\varphi }\right)/{\mu }_s\left({\varphi }_0\right)$ over $r/R$ for (a) ${\varphi }_0=0.20,h/d=10.2$ (no fingering) and (b) ${\varphi }_0=0.28,h/d=10.2$ (weak fingering). Because of particle accumulation, ${\mu }_s\left(\overline{\varphi }\right)/{\mu }_s\left({\varphi }_0\right)$ is shown to grow more steeply over time for both cases. In particular, we focus on the slope of the viscosity near the interface $k_c\approx 3$ at the onset of miscible fingering ($t=8.3\mathrm{s}$). On average, this critical slope is measured to be about 2.7 for all ${\varphi }_0\ge 0.23$ for which miscible fingering is observed [inset of panel (b)].

Figure 8

(a) Once the interfacial finger forms between particle clusters (see the inset image), the number of fingers $n$ may increase over time. This increase in $n$ with time is evident in the plot of $n$ vs $t$ for ${\varphi }_0=0.26$. However, for ${\varphi }_0=0.29$ and 0.31, $n$ exhibits an initial increase in time, until it plateaus to a constant value at the saturation time $t_s=14$ and $8\mathrm{s}$, respectively. (b) For ${\varphi }_0=0.29$ and 0.31 at which the finger number saturates, the finger length $\overline{\lambda }$ is also shown to plateau after $t_s$; $\overline{\lambda }$ gradually increases on the timescale of the experiment for ${\varphi }_0=0.26$.