The cell draws a larger cylinder

The exact theory, drawn by viscosity explains why syrup between two close plates draws the streamlines of ideal flow round an obstacle. Averaged across the gap, the slow flow obeys Darcy’s law — the averaged velocity is the gradient of the pressure, times the plates’ permeability $h^2/12\mu$ — so it is the gradient of a harmonic function, and the pattern of streamlines is ideal flow’s to a part in ten billion.

It also names the one place the analogy cannot be right. Ideal flow slides past the obstacle’s side at twice the stream speed. The syrup touching the obstacle’s side does not move at all. Somewhere between the obstacle’s wall and the flow the dye shows, the fluid must be brought to rest, and the averaged law has no way to do it: it is first order in the velocity and can satisfy only one condition at a wall.

This essay puts the missing term back, solves the problem it makes, and finds that the whole correction — seen from any distance larger than a few gaps — is a single number.

The term Darcy’s law leaves out #

Darcy’s law comes from the cross-gap part of the viscous stress: the velocity profile across the gap is a parabola, and its curvature, averaged, is a friction $12\mu\bar u/h^2$ against the plates. The in-plane part of the viscous stress — the shear between neighbouring streamlines in the plane of the cell — is smaller by the square of the gap over the obstacle’s size and is dropped. Keeping it gives Brinkman’s equation for the averaged flow,

$$-\nabla p + \mu\nabla^2\bar{\mathbf u} - \frac{\mu}{\delta^2}\,\bar{\mathbf u} = 0, \qquad \delta = \frac{h}{\sqrt{12}},$$

which is second order again and can meet no-slip at the obstacle. It has one length, the Brinkman length $\delta$, a little under a third of the gap. Where the velocity varies slowly on that scale the Laplacian is negligible and Darcy’s law returns; where it varies on that scale, which is next to a wall, the two terms balance.

This is a stated model, and it is worth being exact about what it assumes. It keeps the parabolic profile across the gap right up to the obstacle, where the true three-dimensional flow is no longer a parabola. The exact three-dimensional side-wall layer — the one the account of the cell quotes — decays over $h/\pi$, which is 9 per cent thicker than $h/\sqrt{12}$. The Brinkman equation gets the layer’s existence, its scale and its consequences right, and its thickness to that accuracy; it is the model used to interpret Hele-Shaw experiments for that reason.

The cylinder, in closed form #

For a cylinder of radius $a$ in a stream $U$, write the stream function as $\psi = U\sin\theta\,f(r)$. The Brinkman equation splits cleanly: any part of $f$ that is harmonic satisfies it with the pressure doing the work, and any part with $\nabla^2 g = g/\delta^2$ satisfies it with no pressure at all. So

$$f(r) = r + \frac{A}{r} + B\,K_1\!\left(\frac{r}{\delta}\right),$$

with $K_1$ the modified Bessel function that decays over one Brinkman length. The first two terms are ideal flow past a cylinder of some radius; the third is the layer. No-slip — both velocity components zero at $r = a$ — fixes $A$ and $B$ by two linear equations, and everything else follows. The pressure comes from the harmonic part alone, $p = -(\mu U/\delta^2)(r - A/r)\cos\theta$, because the Bessel part’s in-plane stress exactly cancels its own friction against the plates.

The solution was checked two ways: the wall conditions hold to machine precision, and the Bessel part, differenced off the computed stream function, satisfies its own equation to $3\times10^{-7}$. The Bessel functions were evaluated from their integral representation and checked against tabulated values and their large-argument series to $6\times10^{-12}$.

The layer, and the fastest fluid #

The speed along the flank is the first figure. It rises from zero at the wall, reaches a maximum a few Brinkman lengths out, and then follows ideal flow — or rather follows a curve slightly above it. How close the maximum gets to ideal flow’s twice the stream speed depends only on the cylinder’s size in Brinkman lengths.

For the cell a demonstration is usually built as — a post ten millimetres in radius in a gap of one — the Brinkman length is 0.29 mm and the cylinder is 35 of them across its radius. The fastest averaged fluid moves at 1.82 times the stream speed, 1.1 mm from the wall, and the layer inside it is where the dye gets smeared. The analogy’s largest single error, twice the stream speed at the wall, is replaced by an error of nine per cent at the fastest point and a smooth fall to zero inside a millimetre — which is invisible in a photograph of the whole cell and is exactly where a velocity measured with particles would be most wrong.

A layer that does not grow #

An ordinary boundary layer thickens as it runs along a body, as the square root of the distance travelled, because the only thing limiting it is how long viscosity has had to spread the wall’s influence outward. That growth is the whole shape of the thin-layer theory, and it is why a boundary layer is thin at a leading edge and thick at a trailing one.

The Brinkman layer does not grow. It is the same thickness on the front of the cylinder, over the shoulder and round the back, and it would be the same thickness on a body a kilometre long. The reason is the plates: they remove momentum from the averaged flow everywhere at the rate $\mu\bar u/\delta^2$, whatever the flow’s history, so the wall’s influence spreads outward only until that sink matches the in-plane diffusion that carries it, and that happens at one Brinkman length regardless of how far downstream the fluid has come.

That makes it a relative of the layer that stops at a depth rather than of the ordinary boundary layer. The Ekman layer under a rotating fluid has a fixed thickness for the same structural reason: a body force — there the Coriolis force, here the plates’ friction — acts everywhere in the fluid and balances the viscous spreading at one fixed distance. Neither layer has a downstream history, and so neither separates. The cell’s flow closes up behind the cylinder as smoothly as it parted in front, which is the same reason the cell shows no wake at all.

What the far field sees #

Now the result that makes the correction a single number. Far from the cylinder the Bessel term has decayed to nothing and

$$\psi \to U\sin\theta\left(r + \frac{A}{r}\right),$$

which is ideal flow past a cylinder of radius $\sqrt{-A}$. Ideal flow past the real cylinder would have $A = -a^2$. The cell’s flow has $-A$ slightly larger: the far field is ideal flow past a larger cylinder, and the amount it is larger by turns out to be, to within a part in ten thousand at forty Brinkman lengths, exactly one Brinkman length.

That is the same idea as the body the outer flow sees in a boundary layer. Fluid slowed near a wall carries less flux than it would have, the streamlines outside are pushed away by the deficit, and the outer flow behaves as if the body were thicker by a displacement thickness. Here the layer does not grow along the body — it has the same thickness everywhere, because it is set by the plates rather than by the history of the flow — so the displacement is a single number all the way round, and the cylinder simply looks one Brinkman length bigger.

The drag, which ideal flow says is zero #

The cell’s pressure is not ideal flow’s. It is the velocity potential times the plates’ resistance, so instead of recovering behind the body as d’Alembert’s argument requires, it falls steadily from the front of the obstacle to the back, the way it falls everywhere else in the cell. An obstacle in a Hele-Shaw cell has a drag, and in Darcy’s law for a cylinder it is

$$F_{\text{Darcy}} = \frac{2\pi\mu U a^2}{\delta^2} = \frac{24\pi\mu U a^2}{h^2}$$

per unit depth of the gap — pure pressure drag, with no viscous stress on the obstacle at all, because the Darcy flow slips.

With the layer included the obstacle carries pressure and wall shear together. The two are integrated round the real cylinder here, numerically, and the total comes out as

$$F = -A\,F_{\text{Darcy}} = \left(\frac{a_{\text{apparent}}}{a}\right)^2 F_{\text{Darcy}},$$

to a part in two million at four sizes. The drag on the real cylinder is exactly Darcy’s drag on the larger cylinder the far field sees. It follows from the wall conditions in three lines — the shear at the wall turns out to equal the Bessel coefficient over $\delta^2$, and it adds to the pressure force in just the proportion that replaces $a^2$ by $-A$ — and the numerical integral is the check that the three lines are right.

Why it has to be true is worth a paragraph, because it is the cell’s version of a theorem that recurs across this subject: a force on a body is written in its far field. In a Darcy medium the pressure far from the body is a dipole, and the strength of that dipole is fixed by how much flux the body displaces — which, seen from far away, is ideal flow past a cylinder of radius $\sqrt{-A}$. The force on the body is the momentum the plates’ friction has to absorb to hold that dipole, and it depends on nothing but the dipole’s strength. Two bodies with the same far field have the same drag, whatever they look like close to. The real cylinder with its layer and the larger cylinder without one are, from far away, the same body.

That identity also corrects the account the analogy is usually given. The cell does not reproduce d’Alembert’s paradox; it reproduces ideal flow’s streamlines with a pressure that has nothing to do with ideal flow’s, and the obstacle feels the pressure the plates’ resistance has built up across it. What the cell demonstrates about drag is that a pattern and a force can be decided by different things: the pattern by Laplace’s equation, which the two flows share, and the force by what the pressure is proportional to, which they do not.

What this means for a cell someone builds #

The numbers give a builder’s rule. The analogy’s error near the body is controlled by the post’s radius in Brinkman lengths, $a\sqrt{12}/h$, not by the Reynolds number and not by the gap alone. To keep the drag within five per cent of Darcy’s needs the post to be forty Brinkman lengths in radius, or about twelve gaps; to keep the fastest flank speed within five per cent of ideal flow’s needs about eighty-five Brinkman lengths, or twenty-five gaps. The near field converges more slowly than the far field, because the far field only feels the layer through a displacement of one Brinkman length while the flank speed is measured inside it.

That ordering matters for what a cell is used for. As a picture of a flow net, where only the pattern at a distance matters, a cell with a gap of a tenth of the obstacle is excellent. As an instrument for the flow next to an obstacle — a porous-medium experiment in which grains are posts a few gaps across — the Brinkman layer is a leading-order effect and the Darcy description of the flow between neighbouring grains is wrong by tens of per cent. That is the regime in which Darcy’s law meets a wall, and the Brinkman equation was written for porous media in 1947 for exactly that reason.

The same arithmetic runs the other way in a microfluidic chip, where the cell is not a demonstration but the device. Arrays of posts etched into a channel are used to sort cells and particles by size, and a typical array has posts ten micrometres in radius in a channel twenty micrometres deep. That is a post under two Brinkman lengths in radius — far to the left of every figure above — and there the Brinkman layer is not a correction to the flow between posts; it is most of that flow. The drag on such a post is two and a half times Darcy’s, and the streamlines between neighbouring posts, which decide where a particle is carried, are not the ideal-flow streamlines a design drawn from potential theory would predict. Devices of that kind are designed with full three-dimensional Stokes solutions for exactly that reason.

And a measurement made in any cell has the cross-gap profile to contend with before it reaches the layer. A particle photographed from above sits somewhere in the gap, and a particle at mid-gap moves at one and a half times the averaged speed plotted here. Near the obstacle the two corrections compound: the profile is no longer a parabola, and the averaged speed is itself below ideal flow’s. A velocity field measured in a cell is therefore compared with the Brinkman solution after averaging over the depth the camera sees, not with ideal flow and not with the averaged solution directly.

What the picture cannot show #

The figures are drawn in the plane of the cell, and the layer they show is a gap-averaged one. Across the gap the flow near the obstacle is not a parabola; the true velocity field in the corner where the obstacle meets a plate is three-dimensional, with no-slip on two walls at once, and the gap-average is a summary of it that the Brinkman model gets right in scale and approximately in shape. A tracer photographed in that corner will not follow any of the curves drawn.

Nor do the figures show the plates’ own contribution. The friction on the two plates is what drives the whole pressure field, and integrated over the cell it is most of the force the syrup exerts; the drag computed here is the part that acts on the obstacle, which is what an experiment that weighs the obstacle measures.

Where the model stops #

The parabolic profile. Brinkman’s equation assumes the cross-gap profile is a parabola everywhere. Within about one gap of the obstacle it is not, and the exact layer decays over $h/\pi$ rather than $h/\sqrt{12}$. The apparent growth, the drag identity and the flank speed are therefore the model’s, accurate to the nine-per-cent difference between the two lengths when the layer is thin and less well when it is not.

No inertia. The reduced Reynolds number, $Re\,h/a$, is assumed negligible. Inertia enters the averaged equations at that order and makes the flow asymmetric front to back, which none of the figures show.

A uniform gap. A cell whose plates are not quite parallel has a permeability that varies from place to place, and that turns the cell into a bearing as much as a flow visualiser.

The convention the numbers depend on #

Speeds are gap-averaged and quoted in units of the stream’s gap-averaged speed; the mid-gap speed is one and a half times as large everywhere the profile is a parabola. Lengths are in cylinder radii unless given in millimetres, and the Brinkman length is $h/\sqrt{12}$ — not $h/\pi$, which is the exact three-dimensional decay length and is sometimes quoted in its place.

Who found it, and when #

Henry Selby Hele-Shaw built the cell in 1898 to photograph streamlines, and Stokes explained within the year why they matched ideal flow. H. C. Brinkman wrote his equation in 1947 for the viscous force a dense swarm of particles exerts on a fluid, and it was carried over to Hele-Shaw cells when experiments began to resolve the flow near obstacles; the Bessel-function solution for the cylinder is standard in the porous-media literature.

Still open: what the layer does when the obstacle is not a circle #

The cylinder is the case in which the displacement is uniform round the body. On an obstacle with corners — a square post, a flat plate across the stream — the layer cannot have the same thickness everywhere, because ideal flow round a corner has an infinite speed there and the Brinkman layer is what caps it. The far field will still be ideal flow past some slightly larger body, but that body is no longer the obstacle inflated uniformly; its shape near the corners is set by how the layer rounds them off.

The calculation that would follow solves the Brinkman equation round a square post, by a boundary-integral method with the modified-Bessel kernel in place of the logarithm, and asks what shape the apparent body is — whether its corners are rounded by one Brinkman length, as a uniform displacement would suggest, or by more, because the layer thickens where the outer flow is fastest. Beside it is the sharp edge that decides an aerofoil’s circulation, which is the same question about a corner asked of a flow with no layer at all.