A swimmer that cannot go backwards

A bacterium is about as long as a wavelength of green light — small enough that Stokes’ law is the right drag law for it — and swims at about thirty micrometres a second. Its Reynolds number is around $10^{-4}$, which means the fluid it is in has no memory and no inertia: stop beating and it stops, in about a microsecond, having coasted about a tenth of an atomic diameter.

That regime has a consequence that is easy to state and startling the first time: a stroke that looks the same played backwards achieves nothing. The Stokes equations contain no time derivative, so reversing the boundary motion reverses the flow exactly, and any body that retraces its own shape retraces its own displacement. A scallop, which has one hinge and can only open and close, cannot swim at low Reynolds number at all.

So a microscopic swimmer needs a stroke with a direction in it, and the simplest one there is is a travelling wave.

The flow, in six lines #

Let the sheet be at $y = b\sin k(x - ct)$, small enough that $kb \ll 1$. To first order in that parameter, the material points of an inextensible sheet move vertically only, so at $y = 0$ the boundary conditions are $u_x = 0$ and $u_y = \partial h/\partial t$.

The biharmonic stream function that satisfies them is

$$\psi = bc\,(1 + ky)\,e^{-ky}\,\sin k(x - ct),$$

and every quantity in this essay comes from differentiating it.

The velocity along the sheet works out as $u_x = -bck^2 y\,e^{-ky}\sin k(x-ct)$, which is zero at the sheet as required, rises to a maximum one radian of wavelength above it, and is gone within about three.

A dissipation with nothing in it #

Carrying the derivatives into the dissipation function gives

$$\Phi = 4\mu\,b^2c^2k^6\,y^2 e^{-2ky},$$

and the notable thing is what is absent. There is no $x$ and no $t$. The sheet destroys energy at the same rate under the crest of the wave, under the trough, and everywhere between, at every instant of the cycle.

That is not obvious and it is not a general property of swimmers. It happens because the two contributions to Φ — the normal strains and the shear — carry $\cos^2$ and $\sin^2$ of the phase respectively, with the same coefficient, and they sum to one.

Integrating across the layer gives the rate of working per unit area:

$$W = \mu k^3 b^2 c^2.$$

Checked here two ways, as every result on this site is: by integrating the closed-form Φ, and by differencing the velocity field and integrating that. They agree to a part in a million, and the phase-independence is checked separately, because a sign error would leave a plausible total and a varying distribution.

The speed, from one line of the boundary condition #

The swimming is second order in the amplitude and does not need a second solve.

The no-slip condition is imposed on the sheet’s actual position, not at $y = 0$. Expanding it to second order leaves a mean tangential slip at the plane:

$$\langle U \rangle = \left\langle h\,\frac{\partial u_x}{\partial y}\Big|_{0}\right\rangle = -\tfrac12 k^2b^2c.$$

That is Taylor’s result. The fluid far from the sheet drifts backwards relative to it at $\tfrac12k^2b^2c$, which is the sheet swimming forwards at that speed — against the direction the wave travels.

The number is small. At $kb = 0.2$ the sheet moves at two per cent of its own wave speed. A real flagellum has $kb$ of order one and moves at a substantial fraction, which is beyond what this expansion can say.

The result that reads like a mistake #

Now put the two together. The power goes as $b^2$. The speed goes as $b^2$. So the work per unit distance travelled is

$$\frac{W}{U} = 2\mu k c,$$

with no amplitude in it whatever.

Doubling the amplitude quadruples the power, quadruples the speed, and leaves the cost of the journey alone. A swimmer choosing an amplitude is choosing how fast to get there and not how much it will cost.

What does change the cost is the wave speed, linearly. So a large slow undulation is cheap and a small fast ripple is expensive, at the same swimming speed — which is a real design statement, and it is the direction real flagella have gone: amplitudes comparable with the wavelength, frequencies as low as the required speed allows.

Written at a fixed swimming speed $U$ rather than a fixed wave speed, the power is $4\mu U^2/kb^2$, which falls as the square of the amplitude. Same statement, and in this form it is the one an organism would care about.

Where the heat is, which is a layer of its own #

The dissipation profile has a shape and the shape is worth reading, because it says how much fluid a swimmer disturbs.

$\Phi \propto y^2 e^{-2ky}$ peaks at $y = 1/k$ — one radian of wavelength above the sheet — and is down by a factor of a hundred by $y = 4/k$. So essentially all of the destruction happens within about half a wavelength of the surface, and the fluid beyond that is being moved and not sheared.

That is a layer, and it is the same kind of object as a Stokes layer at an oscillating wall or a boundary layer on a plate, with one important difference: its thickness has nothing to do with the viscosity. It is set by the wavelength, and only by the wavelength, because there is no other length in the problem and no time for a diffusive one to develop — the fluid responds instantaneously.

A swimmer’s hydrodynamic reach is therefore its own wavelength, and that has a consequence for populations. Two bacteria a wavelength apart are strongly coupled; ten wavelengths apart they are barely coupled at all through this mechanism, though the far field of a finite swimmer decays algebraically and reaches much further.

Why a reciprocal stroke gets nothing #

The scallop theorem can be seen directly in the same arithmetic.

A standing wave — $h = b\sin kx\cos\omega t$ — is the sum of two travelling waves of equal amplitude going opposite ways. Each contributes a mean slip of $\pm\tfrac12k^2(b/2)^2c$, and they cancel exactly. The sheet oscillates and goes nowhere.

That cancellation is not a coincidence of this geometry; it is the theorem. In a Stokes flow, reversing the boundary motion reverses everything, so the displacement produced by the second half of a time-reversible stroke is exactly minus the displacement produced by the first. What a low-Reynolds swimmer needs is not force but a broken time symmetry, and the travelling wave is the cheapest way to break it.

What it is worth as an efficiency #

The natural next question is how good a swimmer this is, and the honest answer needs the question sharpened, because an infinite sheet has no drag and therefore no obvious denominator.

The convention that is usable is Lighthill’s: compare the power the swimmer spends with the power it would take to tow the same body, held rigid, at the same speed. For a waving sheet that ratio comes out at order $(kb)^2$ — so at $kb = 0.2$ the efficiency is a few per cent, and at $kb = 1$ it is a few tens of per cent.

Those are terrible numbers by the standards of anything that swims at a large Reynolds number, and they are unimprovable rather than a failure of design. At low Reynolds number, propulsion and drag use the same mechanism, so there is no separating them: everything a swimmer does to push on the fluid also drags on it. A fish separates them by leaving a jet behind, which requires inertia to carry the jet away, which is exactly what is absent here.

The comparison with drifting, which is the alternative #

A bacterium that stopped swimming would still move, because everything at that scale is being battered by thermal motion. It is worth putting the two on the same axis, because the comparison is what decides whether swimming is worth doing at all.

A one-micron sphere in water diffuses with a coefficient of about $4\times10^{-13}$ m²/s, so it covers a micron in about two and a half milliseconds and a millimetre in about a month. Swimming at thirty microns a second covers a micron in a thirtieth of a second — slower than diffusion — and a millimetre in half a minute.

So swimming is worse than diffusion at short range and enormously better at long range, and the crossover is at about thirty microns, which is thirty body lengths. That number is the reason motility exists: below it there is nothing to gain, and above it the gain grows without limit, because diffusion’s time goes as the square of the distance and swimming’s goes as the first power.

It is also why the cost result above matters less than it looks. The bill for a journey is $2\mu k c$ per unit area per unit distance, and at these scales it is a fraction of a per cent of what the organism is spending anyway. The constraint on a bacterium’s swimming is not energetic; it is that the Péclet number formed on its own size is about one, so it cannot outrun the diffusion of the thing it is chasing over any distance shorter than its own sensing range.

What the picture cannot show #

The expansion is in $kb$ and real swimmers are not small in it. A bacterial flagellum has an amplitude comparable with its wavelength. The second-order result above is the beginning of a series whose next term is $O((kb)^4)$, and at $kb = 1$ the series is not obviously converging.

The sheet is infinite and flat. A real swimmer is a finite body, and finiteness matters enormously: it is what makes a drag and therefore an efficiency definable, and it introduces the rotation that a helical flagellum uses and a plane wave cannot.

Nothing here is a flagellum’s mechanics. The sheet’s shape is imposed. A real flagellum’s shape is the result of internal motors working against the fluid’s resistance and against the filament’s own elasticity, and the wave that emerges is a solution rather than an input.

The sheet is inextensible and its material points move vertically only. A sheet that also stretches along itself swims at a different speed, and the boundary condition that distinguishes them is exactly the kind this collection counts carefully.

And there is only one swimmer. A suspension of them interacts through disturbance fields that decay slowly, which is why bacterial suspensions do collective things — bioconvection, active turbulence — that no single-swimmer calculation predicts.

A crowd of them, and a viscosity that goes the wrong way #

The list above ends by naming the absent case — a suspension of swimmers rather than one — and it is worth following, because the answer contradicts one of the most secure results in this collection.

Start from what a swimmer looks like from far away. It is force-free: nothing is holding it, so the thrust its flagellum makes and the drag its body suffers are equal and opposite. The leading term of its far field is therefore not a point force but a stresslet — a pair of opposed forces a body-length apart — which decays as $1/r^2$ rather than $1/r$, and whose sign depends on which end the propulsion is at.

A pusher — a bacterium with its flagellum behind, or a spermatozoon — drives fluid outward along its own axis and draws it inward from the sides. A puller — an alga rowing with flagella in front — does the reverse. The two have stresslets of opposite sign, and everything below follows from that.

Now put a suspension of them in a shear flow. The shear tends to align elongated bodies, and an aligned swimmer’s stresslet adds a stress of its own. For a puller the added stress opposes the shear and the suspension is stiffer. For a pusher it acts with the shear — the swimmer’s own disturbance is doing part of the work the shearing apparatus would otherwise have to do — and the apparent viscosity falls.

That is a startling thing to be able to say, because it contradicts the one result about suspensions everybody knows. Einstein’s calculation says a dilute suspension of rigid passive particles has a viscosity raised by five halves of the volume fraction, and the sign of that is not negotiable: passive particles resist deformation, so they always add. An active suspension is not bound by it, because its particles carry their own stress and are not merely being deformed.

Measurements bear it out and go further than anybody expected. Suspensions of swimming E. coli have been driven at low shear rates with an apparent viscosity falling steadily with concentration and, at a few per cent by volume, reaching essentially zero — a suspension of bacteria in water flowing under no measurable stress at all. The energy is not coming from nowhere; it is coming from the bacteria’s metabolism, delivered into the shear through the stresslet, and the apparatus that would have supplied it is spared.

Push the concentration higher and the same stresslets destabilise the suspension outright: aligned pushers reinforce one another’s disturbance, an ordered state is unstable, and the result is the swirling, jetting, continually reorganising motion called active turbulence — a flow with a broad range of scales, at a Reynolds number of $10^{-4}$, where the energy is injected at the scale of an organism rather than cascaded from anywhere.

Which is the sharpest possible statement of what this essay’s regime allows. A fluid with no inertia cannot support a cascade, cannot support an instability driven by momentum, and cannot be stirred by anything reciprocal — and a crowd of swimmers produces something that looks like turbulence anyway, out of the one ingredient a Stokes flow has left: a stress that the particles carry themselves.

What a real organism does with this #

Three things about actual swimmers that the sheet gets right, and one it does not.

Right: the wave has to travel. Every flagellated organism propagates a wave along its flagellum rather than beating it back and forth, and the direction of propagation determines the direction of swimming. A spermatozoon’s wave travels from head to tail and it swims head first.

Right: the amplitude is large. Real flagellar amplitudes are a fifth to a half of the wavelength, which is exactly where the sheet’s arithmetic says the swimming stops being negligible. A swimmer that operated at $kb = 0.05$ would move at a thousandth of its wave speed and would be outrun by diffusion.

Right: efficiency is terrible and nobody minds. The power a bacterium spends swimming is a fraction of a per cent of its metabolic rate. At that scale propulsion is nearly free and what is scarce is information — which is why bacterial motility is organised around sensing gradients rather than around going fast.

Wrong: the sheet is planar and most flagella are helical. A plane wave produces no torque; a helical one does, and a free swimmer must counter-rotate its body to balance it. That counter-rotation is a substantial fraction of a bacterium’s expenditure and has no counterpart here at all.

Who found it, and when #

Taylor published in 1951, in the first of two papers that founded the subject; the second, the following year, did the finite cylindrical filament. Purcell’s lecture Life at Low Reynolds Number, which named the scallop theorem and made the reversibility argument famous, was given in 1976 and is still the best short account of why the regime is strange.

The surprising connection is with a completely different way of getting motion out of an oscillation. A waving sheet swims because the mean of a product of two oscillating quantities is not zero even though each has zero mean — the displacement and the velocity gradient are in phase. That is exactly the mechanism of steady streaming, where a flow with no mean anywhere produces a steady circulation, and of Stokes drift in a water wave, where a particle in an orbit that does not quite close drifts forwards. Three completely different phenomena, all of them second-order means of first-order oscillations, and in all three the effect is invisible to any calculation that stops at first order.

Where the ladder goes next #

Below this rung is the world with no inertia, which is the regime everything here depends on, and the reciprocal theorem, which gives the swimming speed in general without the expansion.

Beside it is the surface that moves with the flow, which is the other essay here about a mobile boundary, and an oscillation with somewhere to go, which is the same second-order mean arriving from a completely different problem.