The exact theory, drawn by viscosity

The exact theory of fluid flow is built on throwing viscosity away. It gives closed-form answers to everything and it predicts that nothing has any drag, and the standard teaching about it is that it applies at high Reynolds number, away from walls, where the viscous term is genuinely small.

There is one apparatus in which it is exactly right, and its Reynolds number is a hundredth.

Why it works #

Put two plates a distance $h$ apart and push a viscous liquid between them. The gap is small compared with everything else, so the dominant velocity gradients are across it, and the momentum equation reduces to a balance between the pressure gradient and one viscous term:

$$\frac{\partial p}{\partial x} = \mu\frac{\partial^2 u}{\partial z^2}.$$

Integrating across the gap with no-slip at both plates gives a parabola, and averaging the parabola gives

$$\bar{\mathbf{u}} = -\frac{h^2}{12\mu}\nabla p .$$

The averaged velocity is proportional to the gradient of a scalar. Continuity then forces $\nabla^2 p = 0$, so the depth-averaged flow is a potential flow with $\phi = -h^2 p/12\mu$, and an obstacle spanning the gap is a body the averaged velocity cannot cross.

Every consequence follows. The streamlines are ideal-flow streamlines. The stagnation points are where ideal flow puts them. The speed at the shoulder of a circular obstacle is exactly twice the free stream. And the flow is a Stokes flow, dominated by viscosity, containing no inertia worth the name.

The paradox it steps around #

There is something stronger to say than “it happens to look the same”, and it involves a problem this collection has already computed to have no answer.

A genuinely two-dimensional creeping flow past a cylinder does not exist. Stokes’ paradox: every candidate solution grows logarithmically with distance and cannot be matched to a uniform stream, so the problem of a cylinder in an unbounded Stokes flow has no solution at all. This site computes that by watching the drag of an annulus fail to converge as the outer radius grows.

So the two-dimensional creeping problem is insoluble, and the nearly two-dimensional one is soluble in closed form. What rescues it is the third dimension: the plates supply a drag proportional to $\bar u/h^2$, which is a term the unbounded problem does not have and which cuts off the logarithm at a distance of order $h$.

Squeezing a flow into a plane makes it harder, not easier, and the Hele-Shaw cell is the standard demonstration.

Three things it gets wrong #

None of them is the pattern, and all of them are about speed.

The no-slip condition on the obstacle is violated. The averaged model has fluid sliding along the cylinder at twice the free stream, which a viscous fluid never does. The true flow satisfies no-slip in a region about $h/\pi$ wide around the obstacle — invisible in a cell whose gap is a few per cent of the obstacle, and not invisible in one where it is not.

A particle does not travel at the streamline’s speed. The profile across the gap is a parabola, so a tracer at mid-gap moves at 1.5 times the average and one near a plate barely moves at all. A dye streak in a cell is therefore spread across the gap into a shape that has nothing to do with the flow’s history: the pattern is right and the timing is not.

And the analogy fails when inertia returns, at a threshold that is not the Reynolds number. What has to be small is $Re\,(h/L)$, the reduced Reynolds number, which is why a cell with a millimetre gap works at speeds where a plain shear flow of the same fluid would already have inertia in it.

What the demonstration is really demonstrating #

Hele-Shaw’s cell arrived in 1898 as a way of seeing streamlines, and it was received as a vindication of potential flow — here at last was a photograph of the thing the theory described. It is worth being careful about what it vindicates, because the received reading has it backwards.

It is not evidence that potential flow describes real flows at high Reynolds number. The cell is at the other end of the parameter range entirely. What it shows is that Laplace’s equation has more than one physical realisation, which is a mathematical fact rather than an aerodynamic one.

It is evidence about which features of a flow the equation controls. The pattern of streamlines past a body is decided by the equation; the pressure recovery behind the body is too, and the cell shows that as well — the pressure at the rear stagnation point in a cell really does return to the front value, and there really is no pressure drag on the obstacle. There is drag, and all of it comes from the two plates.

And it is a demonstration of separation’s absence, which is the useful part. The cell shows the flow potential theory predicts including the part a real high-Reynolds-number flow refuses to perform: the closing-up behind the body. Photographs of a cell next to photographs of a wind tunnel are the fastest way to see what separation costs.

The number the analogy is good to #

The agreement between the two fields in the first figure is quoted at eight parts in a hundred billion, which is a suspicious number and deserves explanation: it is a statement about arithmetic rather than about physics.

The two fields are computed from different starting points — one from a harmonic pressure and a mobility, the other from a velocity potential — and they agree to the accuracy of the differencing, because the depth-averaging is exact for this geometry. There is no error term. The averaged equation is not an approximation to the Stokes equations; it is what the Stokes equations become after an integration that throws nothing away, provided the gap is uniform and the flow is slow.

The physical error is elsewhere, and it has a size: the analogy holds to order $(h/L)^2$ in the geometry and to order $Re\,(h/L)$ in the dynamics. For a two-centimetre obstacle in a one-millimetre gap those are $2.5\times10^{-3}$ and, at a centimetre per second in glycerine, $5\times10^{-3}$. A per cent, from two independent directions, which is what the apparatus is actually good for and is far better than any wind tunnel manages against the same theory.

What it is used for #

Two things, and the second is why the apparatus is still built.

Teaching and visualisation. A cell with dye ports along the inlet edge shows a streamline pattern directly, at a speed slow enough to photograph, with no unsteadiness and no wake. Prandtl used one; so does everybody who wants to show a flow net without computing it.

And porous media. The averaged equation $\bar{\mathbf{u}} = -(h^2/12\mu)\nabla p$ is Darcy’s law with permeability $h^2/12$, exactly. So a Hele-Shaw cell is an analogue computer for flow through a porous medium, and the analogy is not a resemblance but an identity of the governing equation. Petroleum engineers used cells to study waterflooding, and the Saffman–Taylor instability — the branching fingers a less viscous fluid makes as it invades a more viscous one — was discovered in a cell in 1958 and is a genuine phenomenon of oil reservoirs.

The cell stopped illustrating and started asking #

An apparatus built to display a settled theory ended up posing one of applied mathematics’ more stubborn questions, and the route from one to the other is short enough to follow.

Put two fluids in the cell instead of one, with the less viscous fluid pushing the more viscous. The averaged law says each moves at a mobility times a pressure gradient, and the mobilities differ. Perturb the interface: a bump that gets ahead is now in fluid that resists less, so it feels a steeper local gradient and gets further ahead. The interface is unstable, at every wavelength, with a growth rate rising with the wavenumber — which means the fastest-growing disturbance is the smallest one, and the problem is ill-posed until something is added to stop it.

What stops it is surface tension, which penalises curvature and therefore penalises short wavelengths as $k^3$. Balancing the two gives a most-unstable wavelength, and what actually happens in the cell is that the interface breaks into fingers of about that spacing, they compete, and one wins. A single finger runs the length of the channel with a steady shape.

Then the question: how wide is the finger?

Saffman and Taylor answered it in closed form in 1958, with surface tension omitted, and got a family — an exact steady finger of width $\lambda W$ for every $\lambda$ between zero and one, all of them satisfying the equation and both boundary conditions. Experiment gives one answer: the finger occupies very nearly half the channel, and approaches a half from above as the speed rises.

That is this collection’s recurring shape arriving in a new subject. The equations are exact, the boundary conditions are exact, and they leave a free parameter that the physics evidently does not leave — the same position as an ideal flow with any circulation at all, or a closed wake with a mode that costs nothing. And the resolution has the same character: the missing number is supplied by a term that was dropped.

What is unusual is how it is supplied. Restore surface tension as a small parameter and the family does not simply narrow towards a half — it collapses to a discrete set, of which the physical finger is the first member, and the selection happens at a size that is exponentially small in the parameter. No power series in the surface tension sees it: expand to any order and the whole family survives, because the term that selects is smaller than every term in the expansion. It took until the mid-1980s to work out, it required borrowing techniques from beyond-all-orders asymptotics, and the same structure turned out to govern the selection of a dendrite’s tip in a solidifying alloy.

A singular perturbation that acts beyond every order is the hardest kind there is, and the apparatus that produced it was built to photograph streamlines. There is a moral in that about demonstrations. The cell’s advertised virtue is that it makes a known answer visible; its actual contribution was to make a question visible that nobody solving the equations had noticed was there, because at zero surface tension the equations answer confidently and answer with a family.

How far the family of analogies goes #

It is worth listing them, because the list makes clear what is being shared and what is not.

Every one of these is $\nabla^2\phi = 0$ with a no-flux boundary: ideal flow, with $\phi$ a velocity potential; Hele-Shaw flow, with $\phi$ proportional to pressure; Darcy flow in a uniform porous medium, the same; steady heat conduction, with $\phi$ the temperature; electrostatics in a charge-free region, with $\phi$ the electric potential; and steady current flow in a uniform conductor.

What they share is the equation and the boundary condition, which between them fix the field completely. What they do not share is anything about how the field relates to the material. In ideal flow the fluid can support no shear at all; in a conductor there is no fluid; in a Hele-Shaw cell the shear is the entire mechanism.

So a conducting-paper analogue was a legitimate computer for ideal flow — cut the body’s shape out of a sheet, pass a current, probe the potential — and this collection has already noted that it was used that way for thirty years. The Hele-Shaw cell is the same device with the current replaced by syrup, and its advantage is that the streamlines can be seen rather than reconstructed from a probe.

The cell that is not uniform #

One extension is worth a paragraph because it changes what the apparatus is for.

Let the gap vary: $h = h(x,y)$. The averaged law becomes $\bar{\mathbf{u}} = -(h^2/12\mu)\nabla p$ with $h$ a function of position, and continuity now gives

$$\nabla\cdot\left(h^3\nabla p\right) = 0,$$

which is not Laplace’s equation. It is the equation for flow through a non-uniform porous medium, and it is also Reynolds’ lubrication equation — the same equation that decides whether a bearing carries a load.

So a cell with a machined gap is an analogue for a heterogeneous reservoir, and a cell with a moving plate and a converging gap is a bearing. Three subjects, one equation, and the only thing that changes between them is which quantity is being called the pressure.

What the picture cannot show #

The third dimension. Every figure here is a plan view of a flow whose entire mechanism is across the gap, and the mechanism is invisible in the plan. The one figure that shows it is the profile, which is a picture of the direction the plan discards.

The thin region round the obstacle. No-slip is satisfied in a layer of thickness $h/\pi$ against the obstacle’s side wall, where the flow turns from the averaged model’s slip to the true zero. At the scale a figure is drawn at, that layer is a line.

And the drag. There is a drag on the obstacle in a cell, and it is not the drag the ideal solution gives — which is zero. It comes from the plates, it is proportional to the flow rate and to $\mu/h^2$, and it is not a property of the obstacle’s shape in any way potential flow would recognise.

There is a general lesson in that pair and it is not about fluids. An equation does not know what it is about. Laplace’s equation arrived here from a velocity potential in one case and from a mobility law in the other, and having arrived it has no memory of which. Everything the equation and the boundary condition determine will be the same in both; everything they do not — the speeds, the stresses, the no-slip condition, the drag — is free to differ completely, and does. Knowing which quantities fall on which side of that line is most of what it means to understand a model.

Who found it, and when #

Henry Selby Hele-Shaw described the cell in 1898, as an apparatus for making stream-line motion visible; Stokes commented on it immediately, identifying why the pattern comes out potential, and the theory was tidied within a year. The paradox it evades is Stokes’ own, from 1851. Saffman and Taylor found the fingering instability in 1958, and the mathematics of that problem — the shape of a free boundary advancing in a harmonic field — turned out to be one of the deepest in applied analysis, and is still open in parts.

The surprising connection is that the cell and the aerofoil theory it was built to illustrate have exchanged places in usefulness. Potential flow as a model of air is a first approximation that needs correcting; potential flow as a model of a Hele-Shaw cell is exact to the order of $(h/L)^2$ and needs no correction at all. The apparatus built to demonstrate a theory about air turns out to be one of the very few places the theory is not an approximation, and the reason is that the plates supply, by brute force, the one thing an ideal fluid cannot: a mechanism for a flow to be brought to rest.

Where the ladder goes next #

Below this rung are the exact theory it realises and the creeping-flow regime it lives in.

Beside it are what a photograph of a flow actually shows, which is the general form of the caution above, and Darcy’s law, which is the same equation with the gap replaced by a pore.

And above it is the other direction the analogy runs: what happens when the viscous layer is thin rather than everything, and has to be handed back to the outer problem as a change of shape.