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Sandbox Physics
Microswimming · A course with experiments

L2 · Chapter 3

Opening and closing a shell gets you nowhere

In Stokes flow, the distance a swimmer moves depends only on which shapes it passes through, never on how fast. A reciprocal stroke runs through a sequence of shapes and then back through the same sequence, so every bit of progress on the way out is undone on the way back. Purcell called this the scallop theorem. It holds near walls and in narrow channels, and it fails once the fluid remembers its past, as a polymer solution does.

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Purcell’s scallop

Purcell drew a scallop with a single hinge. It can only open and close. At high Reynolds number it could open slowly and snap shut quickly, and the quick snap would push it along. Real scallops are big enough for that, and Lauga and Powers note in a footnote that they swim at high Reynolds number.

At low Reynolds number the trick fails. Purcell’s caption to the drawing says that time does not matter, only the sequence of configurations. Snap the shell shut quickly or slowly and the body moves the same distance; open it again and the body moves back by exactly that distance.

Any swimmer with a single way to change shape has the same problem. The only periodic motion open to one hinge is to go out and come back along the same path. Purcell called such a motion reciprocal.

The pace drops out

Describe the swimmer’s shape with a few numbers, such as a hinge angle or the lengths of two arms. Because Stokes flow is linear and instantaneous, the swimmer’s velocity at each moment is a linear combination of how fast those numbers are changing. The coefficients depend only on the current shape.

Integrate the velocity over time to get the distance. The rate of change of the shape cancels against the time step, and what remains is an integral along the path in shape space. Lauga and Powers go through this argument in their equations 14 to 17. Run the same path twice as fast and the integral comes out the same.

The argument uses two facts only: the flow equations contain no time derivative, and the velocity is linear in the shape rates. It says nothing about the swimmer’s geometry, so it covers every reciprocal swimmer at once.

U(t)=∑kAk(s) s˙k(t)⟹ΔX=∫0T∑kAk(s) dskdt dt=∫s(0)s(T)∑kAk(s) dskU(t)=\sum_{k}A_{k}(\mathbf{s})\,\dot{s}_{k}(t)\quad\Longrightarrow\quad\Delta X=\int_{0}^{T}\sum_{k}A_{k}(\mathbf{s})\,\frac{ds_{k}}{dt}\,dt=\int_{\mathbf{s}(0)}^{\mathbf{s}(T)}\sum_{k}A_{k}(\mathbf{s})\,ds_{k}

Coming back undoes going out

A reciprocal stroke runs along a path from one shape to another and then back along the same path. The return integral is the outward integral with its limits swapped, so the two cancel exactly. Lauga and Powers write this as their equation 18. The displacement over the full stroke is zero, however unevenly the two halves are paced.

The swimmer still moves during each half. The theorem says only that the second half brings it back. To get anywhere, the shape must return by a different route, so that the path in shape space encloses some area. With one degree of freedom no such route exists, and a swimmer needs at least two independent ways to change shape.

ΔXcycle=∫s0s1A⋅ds+∫s1s0A⋅ds=0\Delta X_{\mathrm{cycle}}=\int_{\mathbf{s}_{0}}^{\mathbf{s}_{1}}\mathbf{A}\cdot d\mathbf{s}+\int_{\mathbf{s}_{1}}^{\mathbf{s}_{0}}\mathbf{A}\cdot d\mathbf{s}=0

Worked example

A reciprocal stroke of the three-sphere swimmer

The three-sphere swimmer of the next chapter has two arms, so it can make strokes of either kind. Take the default spheres, one micrometre in radius in water, with both arms starting at ten micrometres.

First shorten only the left arm to six micrometres, then lengthen it again. While it shortens, the middle sphere moves 1.353 micrometres to the left; while it lengthens, the middle sphere moves 1.353 micrometres back. Doing the shortening ten times faster changes the time each half takes and leaves both distances as they were.

The Lab’s reciprocal preset moves both arms together from ten to six micrometres and back. The computed displacement per cycle is zero to the precision of the arithmetic, below a ten-millionth of a nanometre. The cycle still costs about 7.08 attojoules of dissipated energy, because viscous friction is paid in both halves while the distances cancel.

Where the theorem holds and where it ends

Lauga and Powers stress that the proof never used the shape of the surroundings. A reciprocal swimmer near a wall or inside a narrow channel also gets nowhere, as long as the flow obeys the Stokes equations. Strictly, the theorem holds only when every Reynolds number in the problem is zero; the review cites work on how it breaks down once inertia enters.

Breaking the symmetry of the stroke is necessary for swimming, and it is still not enough. Lauga and Powers give a counterexample: two swimmers that are mirror images of each other, placed head to head. Together their motion is non-reciprocal, yet the mirror symmetry forbids their centre of mass from going anywhere.

The fluid itself matters most. In a polymer solution the stress depends on how the fluid was deformed a moment earlier, so time returns to the equations. The review describes a sheet in a model viscoelastic fluid, the Oldroyd-B fluid, that sends a wave rightward for one third of a period and leftward for two thirds. The motion is reciprocal, and the sheet gains a net displacement because its speed now depends nonlinearly on the wave frequency. Groups of swimmers escape the theorem too: Lauga and Powers point out that two reciprocal non-swimmers can use their hydrodynamic interaction to swim together.

An experiment to think through

Predict, change, explain

  1. Before you run “Back-and-forth goes nowhere”, predict its Distance per cycle and whether its Energy dissipated per cycle is zero.
  2. While the reciprocal preset runs, change Arm speed. Then drag one corner of the loop below the stage so the path opens up and encloses some area.
  3. Explain why Arm speed changes the time per cycle and the energy while the distance stays at zero, and why the distance appears as soon as the path encloses an area.
Stroke a three-sphere swimmer

The three-sphere Lab couples three equal spheres through point-force Stokes flow in unbounded water. It leaves out walls, neighbours and thermal noise.

Try explaining it yourself

1. A single-hinge scallop at low Reynolds number closes ten times faster than it opens. What is its net displacement per cycle?

Read an explained answer

Zero. Both halves pass through the same shapes in opposite order, and the pace of each half drops out of the integral.

2. A three-sphere swimmer keeps its two arms equal at every moment while they grow and shrink together. Does it swim?

Read an explained answer

It stays put on average. Its path in the plane of the two arm lengths runs back and forth along the diagonal line, so it encloses zero area and the outward and return integrals cancel.

3. Does a reciprocal stroke cost energy even though it goes nowhere?

Read an explained answer

Yes. The dissipated power is positive at every instant the spheres move, so each half costs energy. For the Lab’s reciprocal preset the cost is about 7.08 attojoules per cycle.

4. In a polymer solution, can a scallop-like swimmer with one hinge make progress?

Read an explained answer

It can. The stress now depends on the recent history of the flow, so the displacement depends on the pace of each half. Lauga and Powers describe a reciprocal sheet in an Oldroyd-B fluid that gains a net displacement this way.

Sources to read alongside

  1. Purcell, Life at low Reynolds number, American Journal of Physics 45, 3 (1977)

    Pages 3 to 6 were read: the Reynolds-number estimates of Figure 3, the coasting example of Figure 4, the scallop theorem of Figure 6, the two-hinge swimmer of Figure 7 and the flexible oar and corkscrew of Figure 9. Pages 7 to 11 were not read for this course.

  2. Lauga & Powers, The hydrodynamics of swimming microorganisms, arXiv:0812.2887v1 (2008); Reports on Progress in Physics 72, 096601 (2009)

    Preprint version read. Sections 3 and 4 supply the Stokes equations, the point-force flow and the proofs of the scallop theorem; Sections 5 to 7 cover resistive-force theory, flexible filaments, walls and synchronisation.

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