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Microswimming · A course with experiments

L3 · Chapter 7

Flexible tails and turning corkscrews escape the scallop theorem

There are two ways. A corkscrew that keeps turning one way never reverses its motion. A flexible tail waved at one end bends differently on the way out and on the way back, so its shape travels along it as a wave. Both produce thrust because a thin filament resists sideways motion about twice as strongly as motion along its length. Resistive-force theory turns that factor of two into a swimming speed, and a wave pushes its swimmer opposite to the direction the wave travels.

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Purcell’s two answers

After showing that a scallop gets nowhere, Purcell offered two ways out in his Figure 9. A flexible oar waved back and forth bends one way during the first half of the stroke and the other way during the second half. The shapes on the way back differ from the shapes on the way out, so the theorem no longer applies. A stiff oar simply reciprocates.

The second way is a corkscrew. Keep it turning in one direction and its motion never runs backward, so it escapes the theorem entirely. Purcell remarked that the direction such a swimmer goes is not obvious. His Figure 10 sketches E. coli, about two micrometres long, together with its flagella.

Sideways drag is twice lengthwise drag

A thin rod moving through a viscous fluid feels a drag that depends on its direction of motion. Lauga and Powers build a rod from a chain of small spheres and add up their point-force flows. To leading order in one over the logarithm of the aspect ratio, a segment moving sideways feels twice the drag per unit length of one sliding along its length. Each component of the velocity has its own drag coefficient.

This anisotropy makes drag-based thrust possible. A segment moving obliquely feels a drag that points neither straight against its velocity nor straight across the rod, and part of that drag pushes along the filament. Lauga and Powers show that net thrust needs both the segment velocity and the segment orientation to vary along the filament, and a wave provides both.

The theory comes with a caveat that the review states plainly. Because it expands in one over the logarithm of length over radius, it is accurate only for filaments so slender that the radius is exponentially small compared with the length. Slender-body theory improves on it.

f=−ξ∥u∥−ξ⊥u⊥,ξ⊥=2ξ∥=4πηln⁡(L/a)\mathbf{f}=-\xi_{\parallel}\mathbf{u}_{\parallel}-\xi_{\perp}\mathbf{u}_{\perp},\qquad\xi_{\perp}=2\xi_{\parallel}=\frac{4\pi\eta}{\ln(L/a)}

Worked example

Speed from a travelling wave

Take a filament ten micrometres long with a radius of ten nanometres, about the radius Lauga and Powers give for a bacterial flagellum. The logarithm of length over radius is about 6.9, so in water the sideways drag coefficient is 1.82 millipascal seconds and the lengthwise one 0.91.

Send a small sine wave along it: amplitude half a micrometre, wavelength ten micrometres, thirty waves per second. The wave travels at 300 micrometres per second. Lauga and Powers’ equation 36 gives the mean swimming speed. With a drag ratio of two and no head, it reduces to half the wave speed times the square of wavenumber times amplitude. Here wavenumber times amplitude is 0.314, so the filament swims at about 14.8 micrometres per second, opposite to the direction the wave travels.

Two details matter. The speed grows as the square of the amplitude, so doubling the wave height quadruples the speed. A head slows the swimmer, because equation 36 divides by one plus the ratio of the head’s drag to the filament’s lengthwise drag. A spherical head one micrometre in radius has about twice the lengthwise drag of this filament, which cuts the speed to roughly a third, near 4.8 micrometres per second.

⟨U⟩=−ξ⊥−ξ∥2ξ∥ ωkb21+ξ0a/(ξ∥L)ξ0=0, ξ⊥=2ξ∥:⟨U⟩=−12 c (kb)2=−12 (300 μm s−1) (0.314)2≈−14.8 μm s−1\begin{aligned}\langle U\rangle&=-\frac{\xi_{\perp}-\xi_{\parallel}}{2\xi_{\parallel}}\,\frac{\omega kb^{2}}{1+\xi_{0}a/(\xi_{\parallel}L)}\\\xi_{0}=0,\ \xi_{\perp}=2\xi_{\parallel}&:\qquad\langle U\rangle=-\tfrac{1}{2}\,c\,(kb)^{2}=-\tfrac{1}{2}\,(300\ \mu\mathrm{m\,s^{-1}})\,(0.314)^{2}\approx-14.8\ \mu\mathrm{m\,s^{-1}}\end{aligned}

A flexible tail makes its own wave

A sperm cell drives its wave with motors along the whole tail. Machin asked a simpler question: what if only one end is waved? Lauga and Powers balance the filament’s elastic bending force against sideways drag and obtain a hyperdiffusion equation, their equation 48. Its time derivative breaks kinematic reversibility. Even a purely back-and-forth motion at the base sends waves down the filament, because distant parts lag behind the base.

The wavelength and decay length of those waves are set by a penetration length: the fourth root of bending stiffness over sideways drag times frequency. Waved slowly, the filament stays effectively rigid, the stroke is reciprocal and the swimmer goes nowhere. Waved fast, the motion dies within a short distance and most of the filament only adds drag. Lauga and Powers expect the best propulsion when the penetration length is about the filament length; the ratio of the two is called the sperm number.

Rotation follows the same logic. A straight rod spun about its own axis produces no thrust, and Lauga and Powers report that mutant E. coli with straight flagella do not swim. The helix supplies the shape that turns rotation into thrust, while the cell body counter-rotates so that the total torque on the swimmer stays zero.

ξ⊥∂h∂t=−A ∂4h∂x4,ℓ=(Aξ⊥ω)1/4,Sp=Lℓ\xi_{\perp}\frac{\partial h}{\partial t}=-A\,\frac{\partial^{4}h}{\partial x^{4}},\qquad\ell=\left(\frac{A}{\xi_{\perp}\omega}\right)^{1/4},\qquad\mathrm{Sp}=\frac{L}{\ell}

The same trick with three spheres

The three-sphere swimmer escapes the theorem the same way, in miniature. Its four strokes pass a change of shape from one arm to the other in a fixed order, and its loop in shape space encloses area. A travelling wave on a tail does this with many degrees of freedom: each segment runs through the same shapes as its neighbour, a little later.

Reverse the order and the swimmer reverses. In the Lab’s default stroke the left arm moves first and the swimmer travels to the right, toward the arm that moves second. The filament swims against its wave. Purcell’s warning about direction holds for both, and for the three spheres the sign of the loop area in Chapter 5 settles it.

An experiment to think through

Predict, change, explain

  1. In “Four strokes swim forward” the left arm moves first and the swimmer travels to the right. Predict what happens when the right arm leads instead.
  2. Run “Reverse the loop, reverse the swim”, then compare its Distance per cycle and Energy dissipated per cycle with the forward preset.
  3. Explain the result through the order in which the shape change passes from one arm to the other, and compare it with a wave running along a flexible tail.
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 rod segment moves at 45 degrees to its own axis, with sideways drag twice the lengthwise drag. Which way does the drag point?

Read an explained answer

Its sideways component is twice its lengthwise component, so the drag points about 63 degrees from the axis, 18 degrees away from the line of motion. That tilt is the origin of drag-based thrust.

2. Double the amplitude of the travelling wave in the worked example. What happens to the swimming speed?

Read an explained answer

It quadruples to about 59 micrometres per second, provided the small-amplitude formula still holds. With wavenumber times amplitude near 0.63, that assumption is already strained.

3. A filament is made sixteen times stiffer. How does its penetration length change?

Read an explained answer

It doubles, because the penetration length goes as the fourth root of the bending stiffness.

4. Why can a straight, stiff flagellum spun about its axis leave a bacterium in place?

Read an explained answer

Lauga and Powers report that mutant E. coli with straight flagella do not swim. Rotating a straight rod about its own axis leaves its centreline where it is, so no segment sweeps sideways through the fluid and the drag anisotropy yields zero thrust.

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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