Revisiting the Saffman-Taylor Experiment: Imbibition Patterns and Liquid-Entrainment Transitions

We revisit the Saffman-Taylor experiment focusing on the forced-imbibition regime where the displacing fluid wets the confining walls. We demonstrate a new class of invasion patterns that do not display the canonical fingering shapes. We evidence that these unanticipated patterns stem from the entrainment of thin liquid films from the moving meniscus. We then theoretically explain how the interplay between the fluid flow at the contact line and the interface deformations results in the destabilization of liquid interfaces. In addition, this minimal model conveys a unified framework which consistently accounts for all the liquid-entrainment scenarios that have been hitherto reported.

Figure 1

Viscous fingering pattern, and associated water-thickness fields, ${\eta }_{\text{oil}}={10}^3\mathrm{cp}$. (a) Drainage pattern, $Q=0.5\mu \mathrm{l}/\min$. (b), (c) and (d) Imbibition patterns observed at three different flow rates. See also movies 1, 2, and 3 in the Supplemental Material [12].

Figure 2

Forced imbibition dynamics on a hydrophilic surface: ${\theta }_0=7°$. (a) Spatiotemporal plot of $⟨h(x,t){⟩}_y$ for ${\eta }_{\text{oil}}={10}^3\mathrm{cp}$ and $\mathrm{Q}=1.5\mu \mathrm{l}/\min$. (b) Sketch of the entrained film in the $xz$ plane and probability distribution functions of the water-pattern thickness. Full line: $\mathrm{Q}=1.8\mu \mathrm{l}/\min$, dashed line: $\mathrm{Q}=1.5\mu \mathrm{l}/\min$, dotted line: $\mathrm{Q}=0.5\mu \mathrm{l}/\min$, ${\eta }_{\text{oil}}={10}^3\mathrm{cp}$. The three curves are shifted by a constant offset to facilitate the reading. (c) Mean film thickness plotted versus the imposed water flow rate. (d) Phase diagram. Symbols: local critical capillary number at which entrainment occurs, $C{a}^{\star }$ plotted versus the viscosity ratio between the two fluids, $\eta$. Error bars: one standard deviation.

Figure 3

(a) Computed meniscus profile for increasing values of Ca (from black to red). ${\theta }_0=10°$, $\eta ={10}^3$, $\lambda /H={10}^{-5}$. (b) Variations of the apparent contact angle with Ca. Same parameters as in (a). (c) Computed meniscus profile for increasing values of Ca (from black to blue). ${\theta }_0=10°$, $\eta ={10}^{-2}$, $\lambda /H={10}^{-5}$. (d) Variations of the apparent contact angle with Ca. Same parameters as in (c).

Figure 4

Numerical results. (a) Stability diagram for a liquid interface driven in a Hele-Shaw channel: numerics. ${\theta }_0=10°$, $\lambda /H={10}^{-5}$. Insets: sketches of the interface shape in the $xz$ plane. (b) Stream lines and pressure field in a wedge of angle ${\theta }_0=20°$ for $\eta /{\eta }^{\star }=10$. Note the existence of a depression ahead of the contact line. (c) Stream lines and pressure field in a wedge of angle ${\theta }_0=20°$ for $\eta /{\eta }^{\star }=0.1$. The color coding indicates the local pressure (arbitrary units).