The eddies nobody stirs

Stir a jar of honey with a wide paddle and the corner between the paddle and the wall appears to do nothing. The obvious reading is that the flow cannot get in there and the fluid is at rest.

It is not at rest. There is an exact answer for what a slow flow does near a corner, it was worked out by asking a question with no particular flow in it at all, and what it says is that the corner contains an infinite sequence of eddies, each turning opposite to its neighbours, each smaller and very much weaker than the last, converging on the vertex.

A question with no flow in it #

The usual way to solve a flow is to state what drives it and work out what happens. Moffatt’s question in 1964 was the other one: what solutions are possible near the corner, whatever is driving them far away?

That is answerable because a corner has no length in it. Two straight walls meeting at an angle look the same at every magnification, so a solution near the vertex can depend on the radius only through a power, and the ansatz

$$\psi = r^\lambda f(\theta)$$

is not a guess so much as the only thing the geometry permits. Putting it into the creeping-flow equations — where the stream function satisfies the biharmonic equation $\nabla^4\psi = 0$, because inertia has been dropped entirely — leaves

$$f(\theta) = A\cos\lambda\theta + B\cos(\lambda-2)\theta$$

for the symmetric case, and the two no-slip conditions at $\theta = \pm\alpha$ leave a two-by-two determinant. Setting it to zero gives the whole subject:

$$\sin 2(\lambda-1)\alpha + (\lambda-1)\sin 2\alpha = 0.$$

This is the same style of argument as the similarity solutions of the boundary layer: a problem with no length scale of its own has solutions that are functions of one variable, and the equation that decides which ones is an eigenvalue problem.

Why a complex exponent is an infinite sequence of eddies #

For a corner narrower than about 146 degrees that equation has no real root. Its first root is complex, and for a 45-degree corner the solver here returns

$$\lambda = 6.3905 + 2.7204\,i.$$

Write $\lambda = p + iq$ and look at what $r^\lambda$ means:

$$r^{p+iq} = r^p\,e^{\,iq\ln r}.$$

The first factor is an amplitude that dies towards the vertex. The second is an oscillation in the logarithm of the radius — and that is the entire phenomenon. The stream function changes sign every time $q\ln r$ advances by $\pi$, which happens at radii in geometric progression, and between two consecutive sign changes sits one closed eddy turning the opposite way to its neighbours.

Two ratios follow immediately, and neither contains anything about what is driving the flow:

$$\frac{r_n}{r_{n+1}} = e^{\pi/q}, \qquad \frac{\psi_n}{\psi_{n+1}} = e^{\pi p/q}.$$

For the 45-degree corner those are 3.17 and 1,603. The solver computes them from the exponent and then measures them again on the constructed field — locating each eddy’s centre by golden section and taking the ratio of the stream function there — and requires the two to agree to a part in ten thousand.

The exponent, and the angle where it stops being complex #

The critical angle is the strongest check available on the whole calculation, and it is worth being explicit about why. It is not an input. The solver walks the root by continuation from a narrow corner, and a separate bisection asks where the imaginary part vanishes; the answer is 146.3085 degrees, and Moffatt’s published figure is 146.3. An equation with a sign wrong somewhere in it would have a critical angle somewhere else, and nothing whatever in a picture of a corner would reveal that — the streamlines would still curl, the eddies would still nest, and the picture would be wrong in a way no reader could see. This is the site’s standing problem with this subject: a wrong flow field is beautiful.

The dependence on angle is fierce. At an opening of 28.5 degrees consecutive eddies are 2.03 times smaller and 829 times weaker; at 45 degrees, 3.17 and 1,603; at 90 degrees, 16.6 times smaller and 36,000 times weaker. A right-angled corner therefore has, for practical purposes, one eddy.

Why nobody has seen the third #

The sequence is infinite and the observable part of it is two. Taneda photographed a second corner eddy in 1979 in a slow flow of glycerine, and that measurement is drawn on this site in the colour kept for a borrowed claim, because nothing here computes what a camera can resolve.

The arithmetic explains the difficulty rather than excusing it. In a 45-degree corner the second eddy carries velocities about 500 times smaller than the first; the third, 250,000 times smaller. A velocity that is a quarter of a millionth of the driving flow is not a hard measurement, it is a different kind of measurement: if the first eddy turns over in ten seconds, the third takes three days.

This is the sense in which “stagnant” is a statement about the instrument. The motion in the corner is not absent, it is exponentially small, and the difference matters for anything that has time on its side: contamination that has to be flushed out of a pipe fitting, heat that has to leave the corner of a channel, a reagent that has to reach the vertex of a mixing vessel. The corner is not sealed, it is slow, and how slow is a number this essay can produce.

What actually clears a corner, which is not the eddies #

The section above leaves a practical question open: if the corner is slow rather than sealed, how long does something trapped in it take to leave? Answering that turns the sequence from a picture into a number, and the answer is that the eddies stop being the mechanism after one or two of them.

The quantity that decides is the Péclet number of each eddy — its own advective transport against molecular diffusion. For eddy $n$ that is $u_n r_n/D$, and since $\psi_n$ is of order $u_n r_n$ by construction, the whole thing collapses to

$$\mathrm{Pe}_n \sim \frac{\psi_n}{D}.$$

An eddy’s Péclet number is its own stream function divided by the diffusivity, which means it falls by the intensity ratio at every step — 1,603 per eddy in a 45-degree corner, thirty-six thousand in a right-angled one. That is a far steeper fall than the velocity’s or the size’s, and it runs out almost immediately.

Put a plausible arrangement into it. A corner a millimetre across in water, driven at a millimetre a second, gives the outermost eddy a stream function of about $10^{-6}\,\mathrm{m^2/s}$; against a solute’s diffusivity of $10^{-9}$ that is a Péclet number of a thousand, which is a genuinely advective eddy. The second one is at 0.6, and the third at $4\times10^{-4}$. For heat rather than solute the fall starts lower and the first eddy is barely advective at all.

So the sequence is real, and the transport it does is over after the first member. Everything inside that is a diffusive pocket — a region where the fluid is moving in nested rings and where nothing carried by the fluid cares, because the rings return it to where it started while diffusion carries it out.

Which turns out to be good news, and the arithmetic reverses the intuition the previous section leaves. Diffusion across a region of size $r$ takes $r^2/D$, and the eddy radii fall geometrically — so the deeper the eddy, the faster it clears. In the arrangement above the second eddy diffuses out in about a hundred seconds and the third in about ten, against an advective turnover for that third eddy of some days. The corner is cleared from the inside outwards, and the slowest step is the first eddy rather than the last.

The design consequence is worth stating plainly, because it is the opposite of what the picture suggests. A dead volume in a pipe fitting or a mixing vessel is not made worse by being sharper. A sharp corner has a small first eddy, and a small first eddy is a short diffusion length; a generously radiused one has a large recirculation, a high Péclet number throughout it, and fluid that genuinely does circulate rather than diffuse. The volume that is hardest to flush is the biggest slow eddy, not the infinite sequence of tiny ones, and the sequence’s practical contribution to residence time is one term.

None of that is visible in a streamline plot, which draws every eddy with the same care and gives the seventh the same visual weight as the first. It is the same failure the rescaled contour levels in the first figure had to work around, arriving in a different currency: a picture of this flow is a picture of its geometry, and its transport is a different problem with a different small number in it.

The check that discriminates, and the one that does not #

There are two things to verify about a field like this, and only one of them can catch a wrong eigenvalue.

The biharmonic equation is satisfied by $A\cos\lambda\theta + B\cos(\lambda-2)\theta$ for every $\lambda$. A wrong exponent gives a perfectly good creeping flow of something; it nests eddies, it alternates their sense, it spaces them geometrically, and it satisfies the governing equation to the precision of the arithmetic. Checking $\nabla^4\psi = 0$ therefore tests the assembly of the solution and says nothing at all about whether the exponent is right.

What the eigenvalue is for is the boundary. Only at an eigenvalue does the fluid come to rest on both walls.

That figure is the rejection test drawn. flowcheck runs the same comparison and requires the perturbed exponent to be refused, on the site’s rule that an assertion that has never rejected anything proves nothing. It is worth stating plainly what the alternative would have been: a figure of nested corner eddies, drawn from a number that was wrong in the third digit, published under a caption about no-slip.

The other corner: what it says about separation #

There is a connection to the rest of the site that took a while to be noticed in the subject itself. A separated flow has a dividing streamline meeting the wall at a point, and downstream of it the recirculating region is bounded by that streamline and by the wall — which is a corner, of whatever angle those two curves make.

So the eddy sequence is not a curiosity confined to sharp machined vees. Wherever the flow lets go of a surface, the region behind it has corners in it, and each of those corners has its own sequence at whatever angle it makes. The same holds inside a lid-driven cavity, at the junction between a tube and a header, and at the trailing edge of any body thin enough for the two surfaces to meet at a sharp angle.

The one requirement is that the local flow be slow, which near a stagnation point it always is. A corner sequence is a local solution: the far field only fixes the amplitude of the outermost eddy, and every ratio inside is set by the angle.

Two families, and only one of them is drawn here #

The eigenvalue equation above is the symmetric one: the eddies it describes are mirror images across the bisector, which is what a corner sees when the flow outside it is symmetric — a stirrer directly above the vertex, a lid driving both walls equally.

There is a second family, whose eddies reverse across the bisector, and it differs by exactly one sign:

$$\sin 2(\lambda-1)\alpha - (\lambda-1)\sin 2\alpha = 0.$$

Its roots are complex too, and its numbers are its own. At 45 degrees the antisymmetric exponent is 10.56 + 3.39i, giving a size ratio of 2.53 and an intensity ratio of 18,000 — eddies packed more closely together and dying much faster than the symmetric family’s. And it stops oscillating at a different angle: the solver puts its critical opening at 159.11 degrees, thirteen degrees wider than the symmetric one.

That gap has a consequence worth stating. Between 146.31 and 159.11 degrees a corner has no symmetric sequence and an antisymmetric one, so whether a corner of, say, 150 degrees contains eddies at all is decided not by its angle but by what is driving it from outside. A corner is not simply eddying or not eddying; it is eddying in a particular way, and which way is a boundary condition rather than a geometry.

Both families are computed here and both critical angles are asserted, because an eigenvalue solver that silently returns the wrong branch is exactly the kind of error this site exists to catch: the picture is a sequence of nested eddies either way.

What the model does not contain #

No inertia anywhere. The whole calculation is the creeping-flow limit, and it is a limit the corner enforces on itself: the local Reynolds number of the $n$-th eddy falls by the size ratio times the velocity ratio each step, so even a briskly stirred vessel is solving Stokes’ equations by the second eddy. What the solution cannot describe is the outermost region, where the flow that drives everything may well be inertial.

The corner is perfectly sharp. A real corner has a fillet radius, and every eddy smaller than that radius is a fiction — the geometry it is a similarity solution of does not exist below the fillet. A machined internal corner with a 0.2 mm radius in a 20 mm channel truncates the 45-degree sequence after about four eddies, which is more than anyone can measure and much less than infinity.

The sequence ends at the continuum. Even for a mathematically sharp corner the fluid stops being a fluid eventually. Starting from a centimetre in air, where the mean free path is 68 nanometres, the 45-degree sequence reaches a hundred mean free paths at the seventh eddy; a 90-degree corner gets there in three. An infinite sequence in an equation is a finite sequence in a fluid, and the number that cuts it off is not in the equation.

Two dimensions. Everything here is a plane flow in a wedge. The three-dimensional versions — a cone, a corner where three faces meet — have their own eigenvalue problems with their own critical angles, and none of them is solved here.

Nothing says how strong the first eddy is. A similarity solution has an arbitrary amplitude: matching it to the flow outside is a separate problem, and every strength quoted in these figures is relative to the outermost eddy drawn.

Who found it, and when #

The complex roots were in the literature fifteen years before anybody believed them. Dean and Montagnon found them in 1949 while studying flow in a corner and reported that the result was difficult to interpret physically, which it is: an exponent with an imaginary part in it looks like a solution that oscillates without any obvious reason to.

Moffatt’s 1964 paper is the interpretation. Its contribution is not the eigenvalue equation but the reading of it — that an oscillation in $\ln r$ is a geometric sequence of eddies, that the ratios are properties of the angle alone, and that the same structure appears wherever two surfaces meet at less than 146 degrees. The paper also treats the antisymmetric case and the three-dimensional cone, and it is unusually easy to read.

Taneda’s photographs of 1979 are the observation, and the honest summary of the experimental situation is that the theory predicts an infinite sequence and the record stands at two.

Where the ladder goes next #

Both exact solutions so far have had walls in them — a film between two surfaces, a flow in a wedge. The next one has no boundary at all: a jet issuing into fluid at rest, which spreads by viscosity alone, keeps its momentum exactly, and gains mass all the way downstream. Its profile is a hyperbolic secant squared, and the exponents in it come out of one conservation law and a dimensional argument.