When they reach the bottom of a soap dispenser, frugal handwashers might try adding water to the bottle to push out the last bit of soap. But usually, the water drills right through the soap and jets out an only slightly sudsy splash.
This happens because when you push a less viscous fluid like water into a more viscous fluid like soap in a confined space, the place where the two fluids meet can be unstable, and the runnier liquid might find a path of least resistance.
If you looked very closely, you might see tiny protuberances form at the place where the fluids touch, in a phenomenon physicists call “viscous fingering.” In certain types of confined spaces, the fingers form a branching pattern.
“The viscous fingering instability is one of the most-studied examples of pattern formation, consistently yielding new insights and variations into the formation of branched structures in the natural world, such as rivers splitting into smaller streams,” said Sidney Nagel, Stein-Freiler Distinguished Service Professor of Physics.
In a new study published in Science Advances, Nagel’s team discovered that changing the shape of the interface where the fluids touch can delay onset and slow the growth of the branches—promising improved efficiency for industrial and environmental processes.
Shapeshifting
When one fluid meets another in a confined space, the stability of the interface depends on a few factors: how easily the fluids mix, the difference in their viscosity and how fast the fluids are moving. If the interface becomes unstable, it gets wavy and fingers form.
This reduces efficiency in a lot of scenarios. For example, companies use carbon dioxide to push oil out of reservoirs—but if the interface becomes unstable and forms fingers, the gas can shoot straight through the oil to the extraction well. Engineers are then pumping up gas, leaving oil in the ground.
To better understand the problem, Nagel’s team wanted to delve deeper into the fundamental rules that underscore finger formation.
For fluids that don’t readily mix, such as oil and water, surface tension serves as a sort of skin, helping to stabilize the edge between them. On the other hand, for miscible fluids—which can dissolve together into a uniform solution—there is little to no surface tension. This would suggest greater instability, yet sometimes fingers never develop. Why?
For fingers to form, the interface between runny and thick has to be sharp and abrupt; if the fluids are too similar in viscosity, the interface won’t be sharp enough. Fingers can also be avoided if the runnier fluid is injected slowly enough that it has time to seep into the thicker fluid.
But was there a way to affect finger formation without changing those factors?
“We wanted to know if we could physically change the shape of the interface without altering the viscosity ratio, and whether there’s a direct correlation between its shape and the stability,” said Zhaoning Liu, a graduate student in the Nagel lab and first author on the paper.
Smoothing motion
Viscous fingering instability is often studied using an apparatus consisting of two flat, parallel plates separated by an extremely thin gap.
The team filled the gap with a high viscosity solution. Then they injected a low viscosity solution through a small hole in the top plate. As the thinner liquid spread out from the center, pushing the thicker liquid outward, the advancing edge between them formed a blunt curve, with a fairly flat (sharp and abrupt) face. Fingers eventually formed.
Then they repeated the technique, sliding the bottom plate side to side, a process called shearing, varying how fast and how far, to see how the interface changed.
The motion altered the shape where the two liquids met. The interface bulged outward, forming a pointier curve, and the sharp edge smoothed out.
The team found that the farther and the faster they slid the plates, the longer it took for fingers to start forming, and once they did, they grew more slowly—indicating that there is a direct correlation between the interface shape and its stability.
This breakthrough could have ramifications for industrial processes and environmental applications, the scientists said.
“This study demonstrates a new way to control and delay the instability onset,” said Nagel, “which plays a role in so many industrial processes involving fluids, from oil extraction from the earth to carbon sequestration.”
For example, one climate change mitigation effort has been to lock carbon dioxide inside saltwater aquifers, and the ability to control viscous fingering could be the key to trapping more of the greenhouse gas deep underground.
“There is a long road ahead to take this research and apply it to such problems, but this is a start,” Nagel said.
Citation: Zhaoning Liu et al., Effect of translational shear on interfacial structure in the viscous fingering instability. Sci. Adv. 12, eaeb2907 (2026).
Funding: National Science Foundation
—Adapted from an article first published by the Physical Sciences Division.
