Formation and Destabilization of the Particle Band on the Fluid-Fluid Interface

An inclusion of particles in a Newtonian liquid can fundamentally change the interfacial dynamics and even cause interfacial instabilities. For instance, viscous fingering can arise even in the absence of the destabilizing viscosity ratio between invading and defending phases, when particles are added to the viscous invading fluid inside a Hele-Shaw cell. In the same flow configuration, the formation and breakup of a dense particle band are observed on the interface, only when the particle diameter $d$ becomes comparable to the channel gap thickness $h$. We experimentally characterize the evolution of the fluid-fluid interface in this new physical regime and propose a simple model for the particle band that successfully captures the fingering onset as a function of the particle concentration and $h/d$.

Figure 1

(a) The sequence of images shows the particles “chasing” and “catching up” to the fluid-fluid interface when the pure oil is injected ahead of the particle-oil mixture; the dotted line indicates the location of the liquid-air interface. (b) Schematic of the experimental setup [45]: suspension is injected into the Hele-Shaw cell from the center, and experiments are recorded with the camera from above. (c) An analytical expression for the velocity ratio $\beta$ as a function of $h/d$ is plotted with experimental data for different values of ${\varphi }_0$ and $h/d$; the inset shows the schematic of a single particle moving with ${\overline{u}}^p$ at the channel centerline.

Figure 2

(a) Fingering phenomena depend on the initial particle volume fraction, ${\varphi }_0$, at given $h/d=3.5$: At small ${\varphi }_0$, the distribution of suspension remains uniform (no fingering regime); interfacial deformations are first observed at ${\varphi }_0=0.12$ (weak fingering) and become more pronounced as ${\varphi }_0$ increases in the band fingering regime. The black arrows in the pure oil case indicate the flow direction. (b) Schematic of the particle band of width $w(t)$ and the concentration ${\varphi }_b\approx 0.5$, and the upstream regime in which the particle concentration is given by ${\overline{\varphi }}_{\text{up}}={\varphi }_0/\beta$.

Figure 3

(a) The plot of the interfacial deviation from a circle $\mathrm{\Lambda }$ over dimensionless time $tQ/(\pi R_0^{2}h)$, where $R_0=10\mathrm{cm}$. Two rows of time-sequential images correspond to $h/d=3.5$ (top) and 8.2 (bottom), respectively, while all other experimental values remain the same (i.e., ${\varphi }_0=0.3$). The peak in $\mathrm{\Lambda }$ (circle) is only observed for the large particle case and coincides with the moment at which the particle band breaks. For $h/d=8.2$, particles gradually accumulate on the interface, which leads to weak fingering; the value of $\mathrm{\Lambda }$ (triangle) is much lower than the large particle counterpart. (b) Schematic illustrating the sequence of events that lead to fingering: the jump in particle concentration between the displacing suspension and particle band leads to miscible fingering, resulting in the band break-up and pronounced fingers.

Figure 4

(a) The experimental data of the change in the particle band volume $\mathrm{\Delta }{V}_b$ are shown to follow Eq. (2). (b) The inverse of the wave number $n^{-1}$ linearly scales with the ratio of the critical band width to the critical radius $w_f/R_f$. The inset consists of the plot of $n$ versus $X\equiv 1-\sqrt{[1-(R_i/R_f{)}^2](1-\zeta )}$, where $\zeta \equiv ({\varphi }_0/{\varphi }_b)(1-{\beta }^{-1})$ (symbols listed in the Supplemental Material [47]).

Figure 5

All experimental runs are organized into a phase diagram that shows the dependence of fingering instability on ${\varphi }_0$ and $h/d$: no fingering (square), weak fingering (triangle), and band fingering (circle). The function of $\mathrm{\Pi }=0.63$ correctly captures the transition between weak and band fingering, especially in the limit of $h/d\sim 1$.