The drag a distant wall decides

In 1851 George Stokes published the drag on a small sphere moving slowly through a viscous fluid, $F = 6\pi\mu a U$, and it has been one of the most used formulae in physics ever since: it weighed the electron’s charge in Millikan’s oil drops, it sets how fast fog settles and how fast red cells sediment in a test tube, and it underlies every measurement of a molecule’s size from how fast it diffuses. In the same paper Stokes tried to do the same calculation for a long cylinder moving sideways, and found that it could not be done. The equations had no solution. However the constants were chosen, the flow near the cylinder could not be joined to a uniform stream far away.

That failure is Stokes’s paradox, and it is not a quirk of mathematics. A thin fibre drifting through air, a hair falling in water, a flagellum beating beside a bacterium are all long compared with their thickness, and every one of them is in the regime where the calculation fails. The resolution tells something about viscosity that the sphere never reveals: in two dimensions, a slow body’s drag is decided by something far away, and which far thing it is — a wall, the body’s own end or the fluid’s inertia — changes the answer.

A force spread over a circle never finishes spreading #

Momentum going sideways describes viscosity as the diffusion of momentum. A body pushing on a fluid with a force $F$ feeds momentum into it at a rate $F$, and in steady slow flow that momentum has to be carried outward through every surface surrounding the body — the same bookkeeping as counting what comes out of a closed surface around a charge — by the fluid shearing across that surface. The shear stress on a surface is the force divided by the surface’s area, and the velocity gradient is the stress divided by the viscosity.

This is a scaling argument rather than the exact solution — the real stress is not spread evenly over each surface — but it contains the whole of the paradox. Around a sphere the surfaces are spheres whose area grows as $r^2$. The stress falls as $1/r^2$, the velocity gradient falls as $1/r^2$, and adding up the velocity change from the body outward gives a total that converges: most of it is accumulated within a few radii, and the fluid beyond does not need to move. The body can be moving at a finite speed $U$ relative to fluid at rest at infinity, and $U$ is proportional to $F$.

Around an infinitely long cylinder the surfaces are cylinders whose area per unit length grows only as $r$. The stress falls as $1/r$, and the velocity change adds up to a logarithm, which grows without bound. The fluid at a million radii has to be moving relative to the fluid at a thousand radii by as much as the fluid at a thousand radii is moving relative to the cylinder. There is no finite speed at which the cylinder can move relative to still fluid at infinity, and no finite force that achieves it. The slow-flow equations are asked to put a body in motion relative to an infinitely distant reference, and in two dimensions they cannot.

The logarithm is the same one that appears for the field of a line charge, which falls as $1/r$ rather than $1/r^2$ because its flux spreads over cylinders rather than spheres. The electrostatic potential of an infinite wire has no natural zero at infinity for exactly the same reason — the potential difference between the wire and infinity is infinite — and electrostatics handles it by measuring potentials between two finite places. A viscous fluid has no such freedom, because the fluid at infinity is where the stream is specified.

The flow past a thread takes thousands of widths to recover #

What rescues the cylinder is that the slow-flow equations are not the whole story. They drop the fluid’s inertia, which is fair near the body, where viscous stresses dominate. But viscous stresses fall with distance and the inertia of the oncoming stream does not, so at some distance they become comparable. The ratio of the two at distance $r$ is $Ur/\nu$, so they balance at $r \sim \nu/U$ — a distance that, for a slowly moving thin fibre, is very many radii away. Beyond it the flow obeys different equations, which Carl Oseen wrote down in 1910 by keeping the inertia of the uniform stream, and Horace Lamb joined the two regions the following year.

Near the cylinder, Lamb’s solution is the slow-flow solution after all, with one constant left free — the coefficient of the term that grows as $r\ln r$ — and that constant is fixed by requiring the logarithmic growth to reach the stream speed at about the distance where inertia takes over. The result is

$$\varepsilon = \frac{1}{\tfrac12 - \gamma - \ln(Re/8)}, \qquad Re = \frac{2aU}{\nu},$$

with $\gamma = 0.577$ Euler’s constant, and the drag per unit length is $4\pi\mu U\varepsilon$.

The drawing shows what that means for the flow. A sphere’s flow recovers in a handful of radii, and the Reynolds number does not enter; its dashed curve is the same for a raindrop and a bacterium. A cylinder’s flow creeps back towards the stream speed along straight lines on a logarithmic axis, and the slope of each line is set by the Reynolds number. At a Reynolds number of $10^{-4}$ — a hair a tenth of a millimetre thick moving through water at a millimetre a second is about $10^{-1}$; a micrometre-thick fibre drifting through air at a centimetre a second is below $10^{-3}$ — the disturbance is still ten per cent of the stream at fifteen thousand radii. The cylinder is dragging along a region of fluid whose size is set not by the cylinder but by $\nu/U$.

A cylinder’s drag is not proportional to its speed #

Divide Stokes’s drag by $\mu U$ and what is left, $6\pi a$, depends only on the sphere. Divide Lamb’s drag by $\mu U$ and what is left depends on the speed through the Reynolds number: the slower the cylinder moves, the smaller its drag per unit speed. The dependence is weak — a logarithm — so that slowing a cylinder a thousandfold, from a Reynolds number of $10^{-4}$ to $10^{-7}$, reduces its drag per unit speed only from 1.12 to 0.69. But it never stops. The drag is not proportional to the speed at any speed, however low, and the statement “at low Reynolds number drag is linear in velocity”, which is true for every compact body, is false for a long one.

The reason is visible in the previous drawing. Slowing the cylinder enlarges the region of fluid it disturbs in proportion, and the drag depends on that region’s size logarithmically. A larger region means a gentler velocity gradient across most of it, which is a smaller stress at the body for the same speed. The sphere’s disturbed region does not grow as it slows, because it never depended on inertia to cut it off.

Lamb’s result is the first term of an expansion in $\varepsilon$, which is itself one over a logarithm and therefore small only very slowly. Saul Kaplun computed the next term in 1957, reducing the drag by $0.87\varepsilon^3$, and the dashed curve shows it: a correction of a fifth at a Reynolds number of 1, about five per cent at 0.1 and under one per cent at $10^{-4}$. Measurements on fine fibres and wires, made in the 1950s at Reynolds numbers well below one, agree with the corrected formula to within their scatter. Expansions in inverse logarithms are notorious for converging this slowly, and the cylinder was the problem that made them respectable: the method of matched asymptotic expansions, now standard across physics, was largely developed to get it right. The sphere had its own version of the difficulty — in 1889 Alfred Whitehead found that trying to improve Stokes’s solution by adding inertia as a small correction fails at the next order — and it was resolved the same way.

The nearest wall sets the drag #

Inertia is one way to supply a largest length. A wall is another, and it is often the one that matters in practice.

Inside a closed container, the slow-flow equations have a perfectly good solution for a cylinder, because the fluid at the wall is at rest and the logarithm stops growing there. The drag is exact and elementary, and it depends on the container’s radius through the same logarithm that the unbounded problem could not handle: a container ten radii across gives a drag per unit speed of 9.50, a container a million radii across still gives 0.98, and there is no container large enough for the answer to stop changing.

Comparing the two formulae says exactly how far away inertia’s “wall” is: unbounded fluid at Reynolds number $Re$ drags like a container of radius $8e^{3/2-\gamma}/Re$ radii, which works out at $10.07\,\nu/U$ whatever the cylinder’s size. The dots on the curve mark those equivalent walls. For a fibre drifting through water at a micrometre a second, inertia’s wall is ten metres away, and any beaker, channel or cell membrane it is actually moving in is nearer. That fibre’s drag is set by the beaker. It is the reverse of the fourth power in a pipe, where the wall is the whole of the answer because it is close; here it is the whole of the answer although it is far. The drag on a slow thin body in a real apparatus is therefore not a property of the body and the fluid, as a sphere’s is, but of the body, the fluid and the geometry of whatever surrounds them, and it is the most distant relevant boundary — not the nearest surface of the body — that decides it.

The same structure appears whenever the motion itself supplies a length. A cylinder oscillated sideways at frequency $\omega$ disturbs the fluid only within the depth to which shear reaches, $\sqrt{2\nu/\omega}$, and that depth plays the role of the wall: the oscillating cylinder has a finite drag, with the logarithm of the ratio of that depth to its radius in it. This is why vibrating-wire viscometers work, and why their calibration contains a logarithm that a vibrating sphere’s does not.

A thread is dragged twice as hard sideways as lengthwise #

The last way to supply a largest length is for the body to have one. A real fibre is finite, and at distances much greater than its length it looks like a point, so its disturbance falls off in the three-dimensional way and the logarithm stops at about the fibre’s length. The drag per unit length of a thin rod of length $\ell$ is then, to leading order in $1/\ln(\ell/a)$, the cylinder’s drag with the rod’s length in the logarithm.

The drawing shows two curves in the same logarithm, one twice the other. A rod moved broadside is dragged twice as hard per unit length as the same rod moved along its length, to this order, whatever its length. The factor of two is one of the most consequential numbers in biology. Edward Purcell pointed out, in a lecture of 1976 titled Life at low Reynolds number, that a swimmer in this regime cannot coast — its motion stops the instant its stroke stops, because there is no inertia to carry it — and that a stroke which retraces itself in reverse gets it nowhere, however fast or slow the two halves are. A waving filament escapes that trap only because its segments move partly broadside and partly lengthwise and are dragged differently in the two directions. Each segment pushes the fluid backwards harder than it drags it forwards, and the difference is thrust.

The consequence is that a bacterium’s flagellum and a sperm’s tail work because of Stokes’s paradox. A sphere has one drag for every direction; only a slender body, whose drag depends on the logarithm of its own length and so distinguishes its two directions, can make propulsion out of a reciprocal flow. The resistive-force theory that biophysicists use to model swimming is exactly this drawing applied segment by segment along a wave.

The fibre also settles in its own way. Dropped in a viscous fluid at an angle, a long rod does not fall straight down: its broadside and lengthwise drags differ, so the drag is not parallel to its velocity, and it glides sideways as it sinks, at an angle that can reach about twenty degrees from vertical. A sphere cannot do this, because its drag always points straight back along its motion.

A protein in a membrane, and the size that hardly matters #

The most striking consequence of the paradox is in a living cell. A cell membrane is a sheet of lipid about four nanometres thick, and on the scale of a protein embedded in it the sheet is a two-dimensional fluid, a hundred or more times more viscous than the water on either side. A protein diffusing in the membrane is a short cylinder moving sideways in a two-dimensional liquid — exactly the problem Stokes could not solve. If the membrane were the only fluid, the protein’s drag would be undefined and the jiggling that diffusion is would have no rate.

Philip Saffman and Max Delbrück pointed out in 1975 that the water is the cut-off. Momentum put into the membrane leaks out into the water above and below it, so beyond a certain distance — the membrane’s viscosity times its thickness, divided by the water’s viscosity, between a fraction of a micrometre and a few micrometres — the disturbance is no longer two-dimensional, and the logarithm stops. The protein’s diffusion coefficient is then Einstein’s $kT$ over a drag of the form in these drawings, with the logarithm of that length over the protein’s radius in it. The prediction is that doubling a membrane protein’s radius changes its diffusion coefficient by only ten to fifteen per cent, where a sphere in water would slow by half. Measurements of membrane proteins of very different sizes, since the 1980s, have found diffusion coefficients spread over a far narrower range than their sizes, as the logarithm says; for the smallest proteins and for membranes crowded with other proteins the agreement is worse, and how much of the disagreement is crowding and how much is the membrane being thin compared with the protein is still argued.

Nothing about this is special to biology. Any two-dimensional liquid bounded by a three-dimensional one — a soap film, a monolayer of surfactant on water, a sheet of colloids at an interface — has the same structure, a viscous paradox resolved by the neighbouring fluid, and a drag that depends on the size of the moving object only through a logarithm.

Where the slow-flow picture stops #

Small Reynolds number throughout the inner region. Lamb’s solution and everything built on it assume the Reynolds number is small, and its corrections are powers of $1/\ln Re$, which is small only slowly. At a Reynolds number of 1 the first correction is already a fifth, and not far above that the whole scheme fails and the wake begins to separate.

Rigid, straight, smooth bodies. The rod curves use the leading term of slender-body theory and ignore the ends, where the rod’s radius changes; the next term changes the numbers by a constant inside the logarithm, which for a rod a hundred radii long is a noticeable fraction of it. Flexible fibres bend under their own drag and change shape as they sink, which changes their drag in turn.

A Newtonian fluid. Every figure uses a single constant viscosity. A fluid that answers back, with a viscosity that depends on the shear rate, has no single $\nu/U$ at which inertia takes over, and in a polymer solution that can be stretched a fibre’s wake stores elastic stress that changes its drag at the lowest speeds.

A continuum. For a fibre in a gas, the fibre itself must be thicker than the distance a molecule travels between collisions for any of this to apply. A fibre thinner than about a tenth of a micrometre in air sees the gas as molecules, and its drag has a different form entirely.

What the drawings cannot show #

None of the figures shows the flow field itself — the streamlines that bend round a cylinder near it and, far downstream, gather into a slow wake that Oseen’s equations predict and Stokes’s cannot. Past a sphere or a cylinder in Stokes flow the streamlines are symmetric fore and aft; in Oseen’s flow they are not, and that asymmetry, which appears only at distances of order $\nu/U$, is where the drag is ultimately decided. The drawings give its consequence and not its shape.

They also cannot show that the same difficulty returns for any two-dimensional problem in which something spreads by diffusion — heat from a line source in a flow, or a solute released from a long thread — where the answer carries the logarithm of whatever cuts the spreading off. The cylinder is the momentum version of a very general fact about two dimensions.

Still open: how a suspension of fibres settles #

A single fibre’s drag is a solved problem. A suspension of many, sedimenting together, is not. Each fibre’s disturbance extends a long way, falling off only slowly, and neighbours hundreds of radii apart interact through it; experiments and simulations find that fibres in a settling suspension cluster into streamers and align with gravity, that the suspension’s mean settling speed can exceed the speed of an isolated fibre, and that the velocity fluctuations do not settle to a steady level in the way theories of well-mixed suspensions predict. Whether the fluctuations are set by the size of the container, by the stratification the settling itself creates, or by something intrinsic to long-range hydrodynamic interaction, is disputed, and the answer matters for how fibre-laden fluids — from paper pulp to the mineral fibres whose settling in lungs is a health question — behave.

The habit worth carrying away is to ask, of any slow diffusive process, how its disturbance falls off with distance before trusting that the local answer is local. In three dimensions a slow body’s effect dies away and its drag belongs to it; in two it falls off only as a logarithm, and the answer belongs to whatever is far enough away to stop it — a wall, an end, or the distance $\nu/U$ at which the fluid’s inertia finally takes over.