How small is small enough

Creeping flow is the cleanest approximation in the subject. Drop the inertia terms from the Navier–Stokes equations and what is left is linear, so solutions add, uniqueness is guaranteed, and the flow past a sphere can be written down. Stokes did it in 1851 and got

$$F = 6\pi\mu a U,$$

which is exact, contains no fitted constant, and is the reference every small-particle calculation on this site and everywhere else is checked against. The world with no inertia is what that world looks like: reversible, memoryless, and utterly unlike the one large things live in.

The approximation has a hypothesis, and the hypothesis is that the Reynolds number is small. Which raises the only question worth asking about it, and the one that is almost never answered with a number: how small?

Why the obvious answer is wrong, and interestingly so #

The obvious answer is one. The Reynolds number is $\rho U d/\mu$, which is the ratio of a representative inertia term to a representative viscous term, so at $\mathrm{Re} = 1$ the terms are the same size and below it the one that was dropped is the smaller.

The trouble is buried in the word representative. The inertia term is $\rho u\cdot\nabla u$ and the viscous term is $\mu\nabla^2 u$. Estimating both on the scale of the sphere gives $\rho U^2/a$ and $\mu U/a^2$, whose ratio is the Reynolds number, so far from the sphere both estimates are wrong.

Out at a distance $r$ the disturbance velocity from a Stokes sphere falls as $a/r$. The viscous term is then of order $\mu U a/r^3$ and the inertia term, because the disturbance is being convected past at the free-stream speed rather than at its own, is of order $\rho U^2 a/r^2$. Their ratio is $\mathrm{Re}\,r/a$ — which grows with distance, and exceeds one at $r \sim a/\mathrm{Re}$ however small the Reynolds number is.

So Stokes’ approximation is not uniformly valid anywhere. It is good near the sphere and bad far away, and there is no Reynolds number small enough to fix that; making it smaller only pushes the failure further out.

Oseen’s repair, and the number it gives #

Oseen’s fix in 1910 was to keep the convective term but linearise it about the free stream rather than dropping it: $u\cdot\nabla u \to U\,\partial u/\partial x$. That is exact far away, where the disturbance is small compared with the free stream, and wrong near the sphere, where it is not — but near the sphere the term is negligible anyway, so the error is second order everywhere. The result is

$$C_D = \frac{24}{\mathrm{Re}}\left(1 + \frac{3\mathrm{Re}}{16} + \dots\right),$$

with the Reynolds number on the diameter. The relative error in Stokes’ law is therefore $(3\mathrm{Re}/16)/(1 + 3\mathrm{Re}/16)$, which inverts exactly:

$$\mathrm{Re} = \frac{16}{3}\cdot\frac{\varepsilon}{1-\varepsilon}.$$

One per cent gives $16/297 = 0.0539$. Five per cent gives 0.281. Twenty per cent gives $4/3$.

And at $\mathrm{Re} = 1$ the error is $3/19 = 15.8$ per cent. The value where the two terms balance is not a boundary of validity; it is a place where the answer is already wrong by a sixth.

The term after that has a logarithm in it #

Proudman and Pearson pushed the expansion further in 1957 and found the next term is not a power of the Reynolds number at all:

$$C_D = \frac{24}{\mathrm{Re}}\left(1 + \frac{3}{16}\mathrm{Re} + \frac{9}{160}\mathrm{Re}^2\ln\mathrm{Re} + \dots\right).$$

A logarithm in an expansion is a signal, and the signal is precise: the problem has no single length scale in it. The sphere supplies one length, $a$; the balance of inertia and viscosity in the far field supplies another, $a/\mathrm{Re}$, which is called the Oseen length; and an expansion that has to describe both regions cannot be a power series in the ratio.

That is what matched asymptotic expansions were invented for, and this problem is the case they were invented on. The inner region near the sphere and the outer region beyond the Oseen length have different governing balances, each is solved in its own variables, and the two are matched in the overlap — which is the same argument structure as the overlap layer in a turbulent channel, where a logarithm appears for exactly the same reason and is far more famous.

Where this bites, which is almost everywhere small #

Stokes’ law is the workhorse of every calculation involving a small particle in a fluid, and the Reynolds numbers involved are usually quoted as “small” without a value.

A 10 µm cloud droplet settling in air falls at about 3 mm/s, giving $\mathrm{Re} = 0.002$. Comfortably inside the one-per-cent band; Stokes’ law is right to a part in three thousand.

A 100 µm drizzle drop falls at about 0.27 m/s, giving $\mathrm{Re} = 1.8$. Stokes’ law is 25 per cent low, and a settling calculation that uses it will predict a drop reaching the ground far too soon.

A 1 mm sand grain in water settles at about 0.1 m/s, giving $\mathrm{Re} = 100$. Stokes’ law is wrong by a factor of five, and sediment transport uses a fitted correlation instead.

A red blood cell in plasma is at $\mathrm{Re} \sim 10^{-3}$, and a bacterium swimming at $10^{-4}$ — which is the regime where the reversibility of the equations forbids a swimming stroke that undoes itself, and where the whole of low-Reynolds-number locomotion lives.

The gap between the first and the second of those is a factor of ten in size and a factor of a thousand in Reynolds number, and Stokes’ law goes from being exact to being useless across it.

The cases, in a table #

Putting the same statement across the sizes a reader can hold makes the sharpness of it visible in a way the curve does not. Water drops in air, settling at terminal speed:

The whole of the useful range is inside the first four rows, and the boundary of it — the row where the one per cent threshold falls — is a thirty-micron droplet. That is a fog droplet rather than a raindrop, and it is a long way below the size at which anybody’s intuition says inertia has begun to matter.

The same statement runs the other way in a liquid, because water’s kinematic viscosity is fifteen times smaller than air’s while its density is eight hundred times larger. A particle settling in water reaches the threshold at a smaller size than one settling in air, which is why sedimentation analysis of soils works on the clay fraction and stops working on the silt.

The two-dimensional version, where there is no answer to correct #

The sharpest evidence that the non-uniformity is real rather than a technicality is what happens in two dimensions. Repeat Stokes’ calculation for a cylinder and there is no solution at all: no function satisfies the creeping equations, vanishes on the cylinder and approaches a uniform stream at infinity. That is Stokes’ paradox, and it is not a failure of ingenuity.

The reason is the same non-uniformity, worse. In three dimensions the disturbance falls as $a/r$ and the neglected inertia term catches the retained viscous one at $r \sim a/\mathrm{Re}$; in two dimensions the disturbance falls only logarithmically, so the far field is dominated by the neglected term everywhere and the inner problem has no boundary condition to satisfy.

What a numerical experiment shows instead is a drag that never settles: put the cylinder in a circular domain of radius $R$ and the computed drag falls as $1/\ln(R/a)$, with no limit as $R$ grows. The answer depends on the size of the room, which is a solver’s way of reporting that the problem as posed has none.

The same failure, for a particle that is following a flow #

The consequence that matters most on this site is not the drag on a settling sphere. It is that whether a particle follows a flow is decided by a relaxation time built entirely on Stokes’ law:

$$\tau = \frac{\rho_p d^2}{18\mu},$$

and the Stokes number $\tau U/L$ that follows from it. Every threshold in that essay — including the exact critical value of one eighth — assumes the drag on the particle is Stokesian.

The particle Reynolds number is $\rho|u - v|d/\mu$, formed on the slip velocity rather than on the flow speed, and it is small when the particle is following well. So the assumption is self-consistent exactly where it matters least, and fails exactly where the particle is doing something interesting: a droplet being flung out of a bend at $\mathrm{St} = 3$ has a slip velocity comparable with the flow speed, and its particle Reynolds number is whatever the flow’s is, divided by the length ratio.

The one place the Reynolds number is not the criterion at all #

Worth stating, because it is the commonest way this threshold is misapplied. A lubricating film solves the creeping-flow equations exactly while having a Reynolds number in the hundreds.

The reason is that its criterion is not $\mathrm{Re}$ but $\mathrm{Re}\,(h/B)^2$, with $h/B$ the film aspect ratio, which is of order $10^{-3}$ in a bearing. Inertia is negligible not because the Reynolds number is small but because the geometry suppresses it, and a journal turning at two thousand revolutions a minute is genuinely in the creeping regime while being nowhere near $\mathrm{Re} = 0.054$.

So “creeping flow” names a balance of terms, and the Reynolds number is only one of the ways that balance can be reached. Reading the Reynolds number as the criterion attributes to a bearing a hypothesis it does not make, and to a settling drop a validity it does not have.

The other hypothesis nobody quotes a number for #

Stokes’ law has a second assumption in it, and it is the one an experiment breaks first. The solution is written for a sphere in an unbounded fluid, and every measurement of it has ever been made in a container.

The correction has the same shape as everything else in this essay, and it is slow. For a sphere on the axis of a cylindrical tube of radius $R$,

$$F = 6\pi\mu a U\left(1 + 2.104\,\frac{a}{R} + \dots\right),$$

so the walls increase the drag, and they do so at first order in the ratio of the two lengths rather than at third. The reason is the reason for the whole essay: a Stokes disturbance decays as $a/r$, which is slow enough that a boundary at distance $R$ is still felt at order $a/R$. The same long reach that makes the far field the wrong place to drop the inertia term makes the container an active participant.

The arithmetic is unforgiving. A sphere ten radii from the wall carries twenty-one per cent extra drag; a hundred radii still carries two per cent, which is twice the tolerance this essay has been computing Reynolds numbers to. Getting the wall term below one per cent requires $R > 210a$ — a container two hundred sphere-radii wide, which for a millimetre ball is a tube twenty centimetres across.

So the standard instrument for measuring viscosity is dominated by this term. A falling-ball viscometer drops a 2 mm ball down a 20 mm tube, giving $a/R = 0.1$ and a wall correction of a fifth, and every such instrument applies a tabulated wall factor before it reports anything. That correction is larger than the Oseen term over the entire range in which Stokes’ law is otherwise good — at $\mathrm{Re} = 0.054$ the inertial error is one per cent and the wall error in the same apparatus is twenty.

A second boundary does the same thing at a distance. Neighbouring particles are boundaries too, so a suspension settles more slowly than an isolated sphere would: the interaction is again long-ranged, it enters at the cube root of the volume fraction rather than at the volume fraction itself, and a one-per-cent suspension by volume already settles measurably slowly. Hindered settling is described by a fitted correlation for the same reason the intermediate drag range is.

The moral is the essay’s own, applied to a hypothesis it had not questioned. “Small Reynolds number” was found to need a number and a tolerance; “unbounded fluid” and “isolated particle” need them just as much, they are broken by every real experiment, and their errors are larger. A calculation quoting Stokes’ law to three figures at $\mathrm{Re} = 0.01$ has controlled the smallest of its three errors with great care.

What to replace “small Reynolds number” with #

If the threshold is not one, what should be quoted instead? The honest answer has three parts, and all three are in the arithmetic above.

A tolerance. There is no value at which creeping flow becomes false; there is a value at which it becomes wrong by more than a stated amount, and the value moves with the amount. One per cent is $\mathrm{Re} = 0.054$; ten per cent is $\mathrm{Re} = 0.71$. A calculation that needs two figures and a calculation that needs a sign are entitled to different boundaries.

Which observable. The drag, the wake symmetry and the pressure distribution all depart at different rates, and a threshold for one of them is not a threshold for the others.

Which length. The Reynolds number on the radius and on the diameter differ by two, and the factor of nineteen this essay is about is quoted on the diameter. Half of the disagreements about where creeping flow ends are this and nothing else.

Those three together are the general prescription, and it is what a group of order one is worth applied to the oldest exact solution in the subject.

What the picture cannot show #

The Oseen correction is itself asymptotic. It is the first term of an expansion valid as $\mathrm{Re} \to 0$, and using it to compute an error at $\mathrm{Re} = 1$ — as the figures here do, to make a point — is using an asymptotic series outside the range where its own remainder is small. The 15.8 per cent quoted at the balance point is the right order and is not the right number; measurement gives about 13 per cent there.

Nothing here computes the drag. The site’s own solver produces flow fields on a grid and the creeping solution in closed form, and both are drawn; the correction is imported from Oseen’s linearisation, which this collection does not solve.

And the thresholds are for the drag alone. The streamlines of a Stokes flow are fore-and-aft symmetric, and that symmetry is broken by inertia at a rate which is not the same as the drag’s — a visible wake appears well before the drag error reaches ten per cent, and a figure drawn to show reversibility fails earlier than a calculation of the force does.

Who found it, and when #

Stokes solved the sphere in 1851 and immediately found that the same method gives nothing at all for a cylinder — no solution exists that both satisfies no-slip and is bounded at infinity. That is Stokes’ paradox, and it is the two-dimensional face of the same non-uniformity: the neglected term is worse in two dimensions, so badly that there is no answer to correct.

Oseen resolved both in 1910. Proudman and Pearson explained why it worked in 1957, and in doing so wrote one of the founding papers of matched asymptotics — a technique now used everywhere from boundary layers to semiconductor devices, and which began with somebody asking how small a Reynolds number has to be.

The surprising connection is with the far end of the same axis. The overlap layer of a turbulent channel produces a logarithm for the same structural reason this problem does — two regions with different governing balances, matched in between — and the logarithm is celebrated there and almost unknown here. The same mathematics is famous at one end of the Reynolds axis and a footnote at the other, and the difference is only that turbulence is harder to avoid.

Where the ladder goes next #

Above this rung is the drag correlation itself, from the Oseen correction through the intermediate range to the drag crisis, which this collection draws and does not derive: nothing between $\mathrm{Re} = 1$ and $\mathrm{Re} = 10^5$ has a closed form, and every curve through that region is a fit.

Beside it sits the bearing, where creeping flow holds without a small Reynolds number, and the particle, whose threshold rests on the law this essay bounds. Below it is the world with no inertia and, further down, the number itself.