The Saffman-Taylor instability, characterized by the branchlike fingers formed when a less viscous fluid is injected into a more viscous one, is typically demonstrated between two rigid walls, as in part (a) of the figure above. But what happens if one of the rigid walls forming the Hele-Shaw cell is replaced with an elastic wall? This is the case for (b) and (c) in the figure. The flexibility of the wall causes the expansion of the air-fluid interface to slow down relative to the rigid wall case and causes the interface to move toward a narrowing fluid-filled gap (as opposed to a constant thickness one). Both of these effects reduce the viscous instability mechanism that drives the fingering instability. With a high enough mass flow rate as in ©, there is still some instability in the interface, but it is dramatically reduced. (Photo credit: D. Pihler-Puzovic et al.)
Tag: Hele-Shaw cell
Protruding Fingers
Instability is a common feature of fluid flows and can generate a near infinite set of patterns. The video above shows the Saffman-Taylor instability, an interface instability that occurs when a fluid of lower viscosity is injected into a higher viscosity fluid. In this case, the fluids inhabit a thin space between two glass plates. The less viscous fluid displaces the more viscous one in a series of branching finger-like shapes. If the situation were reversed, with a more viscous fluid injected into a less viscous one, the interface would be stable and expand radially without any pattern formation. (Video credit: William Jewell College)
Saffman-Taylor Demo
In this video, a thin film of viscous glycerin sits between two glass plates. As the plates are forced apart, air gets entrained from either side, causing finger-like instabilities to form between the two fluids. This is a result of the Saffman-Taylor mechanism. The final dendritic pattern depends on the fluid viscosities, surface tension, and any non-uniformities in the apparatus. (Video credit and submission by M. Goodman)
Labyrinth
A labyrinthine pattern forms in this timelapse video of a multiphase flow in a Hele-Shaw cell. Initially glass beads are suspended in a glycerol-water solution between parallel glass plates with a central hole. Then the fluid is slowly drained over the course of 3 days at a rate so slow that viscous forces in the fluid are negligible. As the fluid drains, fingers of air invade the disk, pushing the beads together. The system is governed by competition between two main forces: surface tension and friction. Narrow fingers gather fewer grains and therefore encounter less friction, but the higher curvature at their tips produces larger capillary forces. The opposite is true of broader fingers. Also interesting to note is the similarity of the final pattern to those seen in confined ferrofluids. (Video credit and submission: B. Sandnes et al. For more, see B. Sandes et al.)
Stick-Slip Bubbles
Varying the rate of injection of air into a wet granular mixture contained in a Hele Shaw cell results in very different flow patterns. At low injection rates, stick-slip bubbles form. As the injection rate increases, patterns are affected by “temporal intermittency” where continuous motion is occasionally interrupted by jamming. Increasing the injection rate still further results in Saffman-Taylor-like fingering. #
Viscous Fingers
A Hele Shaw cell is little more than two glass plates separated by a thin layer of viscous fluid. The cell serves as a good test bed for viscous, low Reynolds number flows such as those found in microfluidics. Here a less viscous fluid is injected into the center of the cell, causing the finger-like protrusions of the less viscous fluid into the more viscous one via the Saffman-Taylor instability.
Flow Vis
Place a viscous fluid in the gap between two plates of glass and you have created a Hele Shaw cell. If a less viscous fluid is then injected between the plates, a fascinating pattern of finger-like protrusions results. This is known as the Saffman-Taylor instability. Because of the relative simplicity of the set-up, it’s possible to create such experiments at home using common household fluids like glycerin, dish soap, dyed water, or laundry detergent. (Photo credits: Jessica Rosencranz, Jessica Todd, Laurel Swift et al, Andrea Fabri et al, Tanner Ladtkow et al, Mike Demmons et al, Trisha Harrison, Justin Cohee, and Erik Hansen)
Viscous Fingers
The Saffman-Taylor instability occurs when a less viscous fluid is injected into a more viscous one, usually in a Hele-Shaw cell. Here oil paint and mineral spirits were painted onto flat surfaces that were pressed together before being pulled apart. The result is viscous fingering of the fluids. #
Ferrofluid Labyrinths
Here’s a different take on ferrofluids. Instead of spikes, we get 2D patterns reminiscent of these ones. Most likely the ferrofluid is trapped under glass as part of a Hele-Shaw cell. The results remind me some of chaotic Rayleigh-Benard convection cells, actually.
Viscous Fingers
This photo shows the Saffman-Taylor instability in a Hele-Shaw cell. Here a viscous fluid has been placed between two glass plates and a second less viscous fluid inserted, resulting in a finger-like instability as the less viscous fluid displaces the more viscous one. This is an effect that can be easily explored at home using common liquids like glycerin, water, dish soap, or laundry detergent. #
