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Tiny symmetric swimmer evades basic rule of fluid dynamics

Play a video of swimmer Michael Phelps in reverse, and you’ll notice right away that his stroke looks dramatically different running forward or backward in time. Such time asymmetry typifies the swimming motion of the vast majority of animals—and it’s absolutely essential for bacteria and other microbes. But now, a team of physicists has developed a tiny mechanical swimmer that can inch along, even though its stroke is symmetric in time. The result opens a new conceptual lane in fluid dynamics and could aid in efforts to develop tiny swimming robots for drug delivery and other purposes.

“It’s a significant step forward and it’s done in a very clean and elegant way,” says Ramin Golestanian, a living-systems physicist at the Max Planck Institute for Dynamics and Self-Organization, who was not involved in the new work.

Imagine a clam shell that opens and closes in a cycle that looks the same, whether it is run forward or in reverse. The shell will push itself forward when it closes and expels water, but it will pull itself backward by the same distance when it reopens and draws in water. Or so says the 4-decade-old scallop theorem, which states that a swimmer that cycles through a motion that is symmetrical in time, or reciprocal, won’t go anywhere.

Strictly speaking, this rule applies only when the surrounding liquid is so thick that it stops moving as soon as the swimmer stops pushing on it. A sufficiently strong clam could close its shell and create currents that continue to flow behind it, keeping it moving forward even as it reopens. (A human swimmer depends on similar “inertial” flows.) However, on the microscopic scale, even water behaves like wet concrete, damping out such lingering flows and stopping any swimmer with a symmetrical stroke. That’s why bacteria propel themselves in more complicated ways, such as twirling, corkscrew-shaped flagella.

Guided by the scallop theorem, physicists have been trying to devise the simplest mechanical swimmers. In 2008, Golestanian and a colleague showed that, in theory, a gizmo consisting of just three identical beads connected by two springs can swim. While the inner bead sits there, the outer beads oscillate in and out, slightly out of sync. That makes the motion asymmetric in time, and the swimmer inches in the direction of the bead that’s ahead in its cycle.

Now, Maxime Hubert, a biophysicist at the Friedrich Alexander University (FAU) of Erlangen Nuremberg, and colleagues have developed a swimmer so simple that it is only capable of reciprocal motion. It consists of a single spring connecting two beads, which, by basic physics, must move in and out in sync and symmetrically in time. The scallop theorem would seem to rule out such a swimmer, and yet it moves, the researchers report in a paper in print at Physical Review Letters.

To make the thing go, the researchers relied on the inertia, not of the fluid, but of the beads themselves. If the beads are big enough and heavy enough, they will coast through the liquid for some time. The bigger the bead, the longer it will coast and, conversely, the slower it will react to any shove. So the new swimmer consists of one larger bead and one smaller one. When the spring extends, the smaller bead reacts more quickly but slows down faster, acting like an anchor that repeatedly shoots out and drags the larger bead along, Hubert explains.

After confirming the theory in computer simulations, the researchers clinched the case in an experiment. Exploiting surface tension, they suspended a pair of mismatched steel beads a few hundred micrometers in diameter on the surface of water in a petri dish. Surface tension tended to pull the beads together. At the same time, a magnetic field magnetized the beads and made them repel each other, keeping them a certain distance apart, just as if they were connected by a spring. Wiggling the direction of the magnetic field made the beads move in and out. And the pair swam in the direction of the smaller bead, by about one-thousandth of a body length per cycle.

So does the advance sink the scallop theorem? Not quite, Golestanian says. The motion of the beads alone is symmetrical in time, but their motion relative to the water is not, once they get moving. Because of that overall motion, the small bead starts each cycle lurching forward across the surface before the larger bead starts to move backward. Like a person walking the wrong way on a moving sidewalk, the larger bead must pick up enough speed before it actually moves backward relative to the stationary water. That makes the beads’ motion relative to the water slightly asymmetrical in time. “No rules of physics are broken,” Golestanian says.

The next step is to explore controlling the swimmer by changing the bead sizes and other variables, says Daphne Klotsa, a soft-matter physicist at the University of North Carolina, Chapel Hill, who had previously shown in computer simulations that a two-bead swimmer can move if it can whip up lingering flows in the fluid. “It’s such a simple system, and yet is has all these layers” of physics, she says. “It opens up a lot of questions.”

Co-author Ana-Suncana Smith, a theoretical physicist at FAU, wonders whether nature hasn’t already evolved its own way to exploit the newly identified effect. “We haven’t looked for a biological example,” she says, but, “I would be surprised if nature didn’t use it at some [size] scale.”