A stroke is worth the area it encloses

The standard account of swimming in a fluid with no inertia is a proof of impossibility. The Stokes equations contain no time derivative, so reversing every boundary motion reverses the flow exactly; a body that retraces its own sequence of shapes therefore retraces its own displacement, and a scallop — one hinge, open and shut — goes nowhere however violently it beats. The waving sheet is built to evade that theorem, and it does, by carrying a wave that has a direction in it.

What the argument does not do is anything constructive. It sorts strokes into two classes and says that one of them is useless. It offers no way to compare two strokes that are both non-reciprocal, no way to ask whether a stroke is a good one, and no reason to expect that the answer should be simple. The impression it leaves is that swimming at low Reynolds number is about breaking a symmetry, and that once the symmetry is broken the rest is detail.

The rest is not detail, and the symmetry is not the point.

The velocity is linear in the rates, and does not know the rates #

Take any swimmer whose shape is described by a few numbers — for a swimmer with two hinges, two numbers, call them $s_1$ and $s_2$. Its shape at any instant is a point in a plane, and a stroke is a closed curve in that plane.

Because the Stokes equations are linear, the flow is linear in the boundary velocities, and the boundary velocities are linear in the rates $\dot s_1$ and $\dot s_2$ at which the shape is being changed. Because the equations have no time in them, the coefficients depend on the shape and not on the rates. So the swimmer’s velocity is

$$V = A(s_1, s_2)\,\dot s_1 + B(s_1, s_2)\,\dot s_2,$$

with $A$ and $B$ functions of position in shape space alone. That is the whole of the structure, and everything below is a consequence of it.

Integrate over one cycle. The rates appear only as $\dot s_i\,dt = ds_i$, so the time drops out entirely and the displacement is a line integral round the closed curve:

$$\Delta X = \oint\!\left(A\,ds_1 + B\,ds_2\right) = \iint\!\left(\frac{\partial B}{\partial s_1} - \frac{\partial A}{\partial s_2}\right)ds_1\,ds_2,$$

the second form by Green’s theorem. A quantity that looked as though it needed a solved time-dependent flow turns out to be a property of a closed curve and of a field defined on the plane that curve lives in. Nothing about how quickly the curve is traversed survives.

This is the same structure as a geometric phase, and the language of gauge fields is not decoration here — $A$ and $B$ transform as a connection under a change of what “the swimmer’s position” means, and the curl in the second integral does not. Which is the mathematical statement of something a reader can check physically: two people who disagree about where to put the mark on a swimmer will disagree about $A$ and $B$ everywhere and agree about the distance travelled in a cycle.

Three spheres, because two is not enough #

To put numbers on that, the swimmer has to be simple enough to solve at every shape and complicated enough to have a shape space with an inside.

Najafi and Golestanian’s swimmer is three spheres in a line, joined by two arms whose lengths are prescribed. Two spheres would give a one-dimensional shape space, in which every closed path is an out-and-back, and the scallop theorem would be the only thing there was to say. Three gives a plane.

The hydrodynamics is the simplest that couples the spheres at all: each sphere’s velocity is its own force divided by $6\pi\mu a$, plus the flow the other two make at its position, $F/4\pi\mu d$. With the arm rates prescribed and the swimmer force-free — nothing is holding it — that is three equations for the three forces, and the swimmer’s velocity follows. Solving it at one shape gives one value of $A$ and one of $B$; solving it on a grid gives the field.

The viscosity cancels out of $A$ and $B$ entirely, which it must: a force-free body in a linear flow has a velocity that is a ratio of two things both proportional to $\mu$. What is left is pure geometry, and the numbers below are pure numbers.

At the symmetric shape, with both arms equal and the spheres a tenth of an arm across, the field comes out as $A = 0.0101$ and $B = -0.0101$. The equality of magnitude is a symmetry — shortening the front arm and shortening the rear one move the swimmer the same distance in opposite directions, because at that shape the swimmer is its own mirror image — and it is the clearest possible statement of why one arm is not enough. A swimmer with a single degree of freedom has only $A$, its displacement is $\oint A\,ds_1$ round a path that goes out and comes back, and that integral is zero whatever $A$ is.

The curl at that shape is 0.063, in units of displacement per unit of shape-space area, both measured in arm lengths. Those units are worth pausing on because they are what makes the number sayable at all: a stroke that varies each arm by three tenths of its length encloses 0.09 square arm lengths, so it should be worth about 0.0057 arm lengths a cycle, and the computed value is 0.0058. A swimmer of this design moves a little over half a per cent of its own size per stroke, and would need two hundred strokes to advance by one body length.

That is a poor performance and it is characteristic rather than accidental. Nothing about three spheres is efficient; what they are is solvable at every shape, which is what the argument needs.

Pacing is free, which is not a small statement #

The most immediately useful consequence is the one that sounds least like a result.

A stroke played at twice the rate covers exactly the same ground per cycle, and therefore swims twice as fast — which is the same rate-independence that makes a force in Stokes flow a statement about the instantaneous geometry rather than about its history. A stroke played with one arm moving slowly and the other quickly covers the same ground as the same loop played evenly. There is no such thing as a badly timed stroke, only a badly chosen loop.

That is not obvious and it is not true of swimming with inertia, where the phasing between the two degrees of freedom is most of what a designer has to get right. A fish’s stroke is a shape and a timing; a bacterium’s is a shape alone. The reason is the same one that makes the whole subject strange: with no time in the equations, there is nothing for a timing to interact with.

It also means that a swimmer’s speed and its stroke rate are related by a constant that the swimmer’s geometry fixes once and for all. Doubling a flagellar motor’s speed doubles the swimming speed, exactly, with no saturation and no optimum — a statement that is true only until the motor’s own torque-speed curve is the thing that bends, which is a fact about motors rather than about fluids.

The practical form of that, for anybody reading a measurement rather than making one, is that a plot of swimming speed against beat frequency for a low-Reynolds-number organism should be a straight line through the origin, and any curvature in it is evidence about the organism rather than about the fluid. A flagellum that bends more at high frequency traces a different loop and therefore lies on a different line; a motor that loses torque traces the same loop more slowly. The two are distinguishable, and the shape-space picture is what says they are: one changes the loop, the other changes only the pace, and the pace does not appear in the yield.

The scallop theorem, with no symmetry in it #

The reciprocal stroke now needs no argument at all.

A stroke is reciprocal when the swimmer retraces its shapes, which means the path in shape space goes out along a curve and comes back along the same curve. Such a path encloses no area. Its line integral is therefore zero, term by term, because every contribution on the way out is cancelled by the identical contribution on the way back with $ds$ reversed.

Computed, the displacement of an out-and-back stroke is exactly zero — not small, not at the level of round-off, but identically zero, because the quadrature sums a set of numbers with their own negatives. A square stroke of the same width displaces the swimmer by 0.0058 of an arm length per cycle.

So the scallop theorem is a statement about a degenerate loop rather than about time reversal. Both derivations are correct, and the second one says more: it tells a designer that the enemy is not symmetry but flatness, and that a stroke can fail for reasons that have nothing to do with being its own reverse. A path that wanders over a large region of shape space and returns along a route that crosses itself can enclose very little net area while looking nothing like a reciprocal stroke, and it will swim correspondingly badly.

It also makes the other direction obvious. Any two-dimensional shape space admits loops with interiors, so any swimmer with two independent degrees of freedom can swim, provided the curl does not happen to vanish. Purcell’s three-link swimmer, the three spheres here, a two-hinged rod, a pair of rotating flaps: none of them needs a special mechanism, and the only thing a designer has to avoid is an apparatus whose two controls are not independent.

The area is the measure, and small loops know it #

For a loop small enough that the curl is nearly constant across it, the displacement is simply the curl times the area. That gives the scaling directly: a stroke twice as large in every shape coordinate carries the swimmer four times as far per cycle, which is a strong return and is why real strokes are large rather than delicate.

It also explains why the amplitude result for a swimming sheet has the form it does. Taylor’s sheet swims at $\tfrac12 k^2 b^2 c$ — quadratic in the amplitude — and that quadratic is this area law seen in a different set of coordinates: the sheet’s shape space is parameterised by the amplitude and the phase of its wave, a travelling wave traces a circle of radius proportional to $b$ in that plane, and the area of that circle goes as $b^2$. The sheet’s amplitude-squared and the square stroke’s side-squared are the same statement, which is not visible from either calculation alone.

For larger loops the curl varies and the simple product fails, which is where the choice of where to put the loop starts to matter as much as how big to make it. The field above is largest at short arms, so a swimmer that operates with its spheres close together does better per unit of shape-space area than one that operates stretched out — at the price of a smaller region to loop in, since the arms cannot be shorter than the spheres are wide.

Those two pressures point in opposite directions and their balance is a real design question rather than a rhetorical one. Shrinking the operating point towards short arms raises the curl but shrinks the largest loop that fits, and since the yield is curl times area, and area goes as the square of the room available, the second effect wins wherever the first is merely algebraic. What that predicts is that a good swimmer of this kind should use the whole of its available shape space rather than a comfortable patch in the middle of it — arms swinging between nearly closed and fully extended — which is what organisms with a small number of joints are observed to do, and which looks like extravagance until the area law is written down.

Which way it goes, and what decides #

Nothing so far has said which direction the swimmer travels, and the answer is the one piece of the structure that is genuinely a matter of orientation rather than of magnitude.

Traversing the loop the other way round reverses the displacement exactly. That is arithmetic — a line integral changes sign with the direction of the path — and it is checked here to one part in $10^{12}$ against the forward stroke. So a swimmer reverses by running its stroke backwards, which sounds like a contradiction of the reversibility argument and is not: running a stroke backwards is not the same as making it reciprocal. The first traverses the same loop the other way and goes the other way; the second traverses out and back within one cycle and goes nowhere.

The sign of the curl is what decides the direction for a given sense of traversal, and the curl can change sign. Over the region drawn above it does not — the field is one-signed, so every anticlockwise stroke anywhere in it swims the same way — but there is no principle requiring that, and a swimmer whose shape space contained a line of zero curl would have strokes on either side of it that swam in opposite directions while looking identical to an observer watching the arms.

That is not a hypothetical. It is how a swimmer with more than two degrees of freedom steers: the curl becomes a matrix of components, different two-dimensional sections of shape space have different signs, and a change of which section the stroke loops in turns the organism without any of its parts doing anything a naive observer would call steering.

The loop does not have to be a square #

Nothing above used the shape of the path, and it is worth checking that, because it is the part a reader is most entitled to disbelieve.

A circular stroke of the same width as the square carries the swimmer a distance that agrees with the curl times $\pi r^2$ to one per cent, the residual being the variation of the curl across the loop rather than any failure of the rule. Green’s theorem itself is verified directly, by computing the line integral round the boundary and the area integral over the inside with no arithmetic in common: they agree to three parts in ten million on a small loop and to six parts in a million on a loop four times the size, the drift being the grid’s.

That check is worth more than it looks. The line integral uses the gauge field; the area integral uses its curl, which is computed by differencing the gauge field at four neighbouring shapes. A sign error in the mobility, a missing factor in the force-free condition, or a mistake in what “the swimmer’s position” means would move one of those two and not the other.

What the swimming sheet was measuring instead #

The sheet’s work-per-distance result — a cost with no amplitude in it — now reads as a statement about two different quadratic forms. The displacement per cycle is the enclosed area, which goes as the square of the stroke’s size. The dissipation per cycle is a quadratic form in the shape rates, which also goes as the square of the stroke’s size at fixed pacing. Their ratio is therefore independent of size, exactly, and the cancellation that looked like a coincidence is two areas dividing out.

What does not cancel is the pacing, and this is the one place where the two accounts differ in what they permit. The displacement has no time in it; the dissipation does, because a rate appears squared in it. Going twice as fast is free in distance and costs four times as much power, which is why a swimmer’s optimum is set by its power supply rather than by its geometry, and why the question “what is the best stroke” only becomes well posed once the efficiency is written as the enclosed area divided by a dissipation.

What the picture cannot show #

The mobility here is the leading term in the sphere radius over the arm length. Each sphere sees the others as point forces, which is the first term of a series in $a/s$. At the arm lengths drawn the ratio is a tenth and the correction is of order one per cent; at arms twice the sphere diameter it is not small, and the field above is wrong in exactly the region where it is largest.

The swimmer is a line and the shape space is a plane. A real organism has many degrees of freedom, the shape space is high-dimensional, and the curl becomes a two-form. Everything above survives that generalisation formally, and the picture does not: there is no “enclosed area” to point at in six dimensions, and the useful statement becomes a statement about which two-dimensional sections of the shape space have large curl.

The arms are massless, rigid and prescribed. A real flagellum’s shape is the outcome of internal motors working against the fluid and against the filament’s own elasticity, and what that filament converts into thrust is the anisotropy of its own drag; its stroke is a solution rather than an input, and it need not close.

And nothing here is an efficiency. The area law says what a stroke wins and says nothing about what it costs, and the cost is dissipation in the fluid rather than work against inertia. Two strokes enclosing the same area can dissipate very differently, and choosing between them needs the second quadratic form.

Who found it, and when #

Shapere and Wilczek wrote the gauge-theoretic account of swimming in 1989, and it is theirs that this essay follows: the swimmer’s displacement as a holonomy, the shape space as the base manifold, and the connection whose curvature decides what a stroke is worth. Purcell had given the three-link swimmer and the scallop theorem in his 1976 lecture, which is where the impossibility argument comes from. Najafi and Golestanian’s three spheres, in 2004, are the version simple enough to solve at every shape.

The surprising connection is with a mechanism that has no fluid in it at all. A falling cat, which is not rotating and lands feet-down, does it by changing its shape through a cycle: its angular momentum is conserved and zero throughout, its orientation at the end of the cycle is not the orientation at the start, and the rotation it achieves is a line integral of a connection over the loop its shape traces. The cat and the bacterium are the same theorem. What a fluid with no inertia supplies — a velocity determined by the shape rates alone, with no memory — conservation of angular momentum supplies for the cat, and in both cases the yield of a cycle is an area rather than a duration.

Still open: whether the best stroke is the largest one #

The area law makes the yield of a stroke computable and leaves the optimisation undone, because the cost has not been written in the same terms.

The displacement is $\iint\Omega\,ds_1\,ds_2$ with $\Omega$ the curl. The dissipation over a cycle is $\oint \dot s^{\mathsf T} G(s)\,\dot s\,dt$ with $G$ a positive-definite metric on shape space — which does depend on the pacing, so for a fixed loop the cheapest traversal is the one that minimises that integral at fixed period — the same shape of question as the section that carries a given flow at least cost, and that is a geodesic-like problem with a known answer. The efficiency of a loop is then the ratio of an area integral to a boundary integral of a metric, and the best stroke is the loop that maximises it.

That is an isoperimetric problem, and it is the shape of problem whose answers are usually circles — but the metric here is not flat and the curl is not uniform, so the optimum need not be a circle and need not be centred where the curl is largest. Computing it for the three spheres, with the same mobility used above, would give a stroke that is optimal rather than merely chosen. The interesting question is whether the answer is close to what organisms do, or whether the real constraint is something this calculation does not contain — the arms’ maximum extension, the motor’s torque-speed curve, or the fact that a real swimmer must also steer.