One group, three exponents

Four theories in this collection are the same theory and are never presented as one.

Lubrication theory — a bearing, a squeeze film, a coating flow — assumes the gap is small beside the length along it. Hele-Shaw flow assumes the same thing about a cell. Shallow water assumes the depth is small beside the wavelength. Slender-body theory assumes the thickness is small beside the span. Each is an expansion in

$$\varepsilon = \frac{\text{a transverse length}}{\text{a longitudinal length}},$$

and that group contains no speed, no viscosity, no density and no gravity. It is a property of the shape. It is the only group in this collection that could be measured with a ruler and no fluid present at all — which is a stronger statement than it sounds, because every other number here needs an experiment running to have a value.

What each of those theories claims is the same: that the error is $O(\varepsilon^2)$, because the transverse derivatives are down by ε and they enter the equations squared. That is worth checking, and in two of the three cases below it is wrong in different ways.

The duct, whose error is first order #

Take Poiseuille flow in a rectangular duct and compare it with the infinite parallel plates it is supposed to become. The exact answer is a Fourier series and converges in a handful of terms:

$$\frac{Q}{Q_{\text{plates}}} = 1 - \frac{192}{\pi^5}\,\varepsilon \sum_{n\ \text{odd}}\frac{\tanh(n\pi/2\varepsilon)}{n^5},\qquad \varepsilon = h/w.$$

The measured exponent over three decades is $1.0000$, and the coefficient is $0.630249$. Both are worth a moment.

First order, not second. The reason is not that the expansion is done badly: it is that what the reduced problem left out is not a gradient at all. Two side walls are missing, and their effect is proportional to their area, which is proportional to the height, which is proportional to ε. An expansion whose neglected term is a boundary rather than a derivative has no reason to be even in ε, and this one is not.

And the coefficient is not the one people quote. As ε → 0 every tanh goes to one, so what multiplies ε is $(192/\pi^5)\sum_{\text{odd}} n^{-5} = (192/\pi^5)(31/32)\zeta(5)$, computed here as 0.630249. The prefactor alone, $192/\pi^5 = 0.627411$, is what gets printed, and it is half a per cent low — a small error, and exactly the error of reading a coefficient off the front of a series without summing it.

A square duct, incidentally, carries 42.2 per cent of what the parallel plates would. At ε = 1 the “correction” is nearly half the answer, which is what a first-order error does when the small parameter is not small.

The wave, which behaves as advertised #

The second case is the one the folklore is built on.

$$\frac{c}{\sqrt{gh}} = \sqrt{\frac{\tanh kh}{kh}},$$

the exact linear dispersion relation over the shallow-water limit, with $\varepsilon = kh$ the depth over the wavelength.

The measured exponent is $2.0000$ and the coefficient is $0.1666666$, which is a sixth. Both are exactly what the expansion of tanh predicts, and this is the case where the transverse derivative really is the neglected term: the vertical structure of a long wave is a correction to a depth-uniform flow, and it enters the phase speed squared.

So one group and two problems, and the exponents are 1 and 2. Neither is wrong. They are answers to different questions about what got left out.

The body, whose exponent does not exist #

The third case is the interesting one and it is not a larger error — it is a different kind of object.

Lamb’s prolate spheroid has an exact longitudinal added-mass coefficient,

$$\alpha_0 = \frac{2(1-e^2)}{e^3}\left(\tfrac12\ln\frac{1+e}{1-e} - e\right),\qquad k_1 = \frac{\alpha_0}{2 - \alpha_0},$$

and the slender-body limit of it is $k_1 \approx \varepsilon^2(\ln(2/\varepsilon) - 1)$. That logarithm is not an artefact: a line distribution of sources has a logarithmic near field, and what cuts the logarithm off is the body’s own thickness.

On log axes it is not a straight line. Its local slope, measured over one decade at a time, is 1.88 at $\varepsilon = 2\times10^{-4}$ and 1.73 at $3\times10^{-2}$, and it is nowhere near either of the reference slopes drawn beside it. Dividing the error by $\varepsilon^2$ alone leaves a spread of 2.75 across the range; dividing by $\varepsilon^2(\ln(2/\varepsilon)-1)$ leaves 1.37, which is what says the logarithm is there.

How slender is slender enough #

The local slope of the third curve is exactly

$$\frac{\mathrm d\ln(\text{error})}{\mathrm d\ln\varepsilon} = 2 - \frac{1}{\ln(2/\varepsilon) - 1},$$

so it reaches 2 only as ε → 0, and it approaches it like the reciprocal of a logarithm — which is the slowest approach to a limit that occurs anywhere in this collection.

Getting within a tenth needs $\varepsilon = 3\times10^{-6}$. Getting within a hundredth needs $\varepsilon = 2e^{-101} = 2.8\times10^{-44}$: a body 10⁴³ times longer than it is thick, which is roughly the ratio of a proton’s diameter to the observable universe. That is the honest content of “the error is $O(\varepsilon^2\ln 1/\varepsilon)$”: the exponent exists as a limit and no experiment, no computation and no plot will ever see it.

What each theory buys, in dimensions #

The three steps above are worth pricing, because the reason anybody puts up with an expansion whose exponent is uncertain is that it removes a whole dimension from the problem.

Lubrication. A journal bearing is a three-dimensional Stokes problem in a domain with a moving boundary. The reduced problem is Reynolds’ equation — a two-dimensional elliptic equation for the pressure in the gap — and its unknown is a scalar field on a surface rather than a vector field in a volume. The load capacity of a bearing follows from a single integral of it, which is why bearing design was a solved subject decades before anybody could solve the Stokes equations numerically.

Hele-Shaw. Two closely spaced plates make a viscous flow whose depth-averaged velocity is proportional to the pressure gradient, which is Darcy’s law, which is a potential flow. So a Hele-Shaw cell is a physical analogue computer for two-dimensional potential flow, and the streamline photographs in every textbook of this subject are made in one. The slenderness expansion is why an inviscid streamline pattern can be produced by an intensely viscous apparatus.

Slender bodies. The three-dimensional Laplace problem for an airship becomes a line of sources with strength proportional to the rate of change of cross-sectional area, so the whole of the flow is determined by one function of one variable — which is the result that the area rule and Sears–Haack wave drag rest on.

Each of those is a dimension removed, and each is bought with an expansion whose error is one of the three kinds measured above.

Why this matters for reading a plot #

The practical consequence is about method rather than about slender bodies.

A log–log plot with a straight line through it is the standard evidence for a power law, and the third curve above has one available at every scale: measured over a decade it is a straight line, to within the thickness of the ink, with a slope that is a perfectly reportable 1.8. Nothing in the plot says the slope is not the answer. Whether it is depends on whether the underlying function is a power, and a finite range of data cannot answer that.

This collection has hit the same difficulty twice already. The Sears–Haack body’s drag has a length exponent that is exactly −4 and is reported to four decimals because it really is a power law. The Prandtl exponent of a flat plate drifts from a half to a third and is reported as two exponents with a range for each, because it is not.

The discipline that follows is to divide, not to fit. If the suspicion is a logarithm, divide the data by $\varepsilon^2\ln(1/\varepsilon)$ and see whether the result flattens more than dividing by $\varepsilon^2$ does. That is a comparative test with an answer, where “how straight is straight” has none.

What the four theories have in common #

It is worth writing out what the shared expansion actually does, because it is the same three steps in every case.

Non-dimensionalise with different lengths in the two directions. That is the whole trick: $x$ on the long scale, $y$ on the short one, so that $\partial/\partial y$ is $1/\varepsilon$ times $\partial/\partial x$ and the two are not comparable.

Drop the terms that are down by ε². In lubrication that leaves $\partial p/\partial y = 0$, so the pressure is constant across the gap and the flow is locally Poiseuille. In shallow water it leaves the pressure hydrostatic. In slender-body theory it leaves the cross-flow problem two-dimensional at each station.

And solve the reduced problem, which is one dimension smaller. That is the payoff: a three-dimensional Stokes problem becomes an ordinary differential equation for the pressure, and a three-dimensional potential problem becomes a line integral.

Every one of those steps is the same in all four theories, and the difference between them is entirely in what is at the ends. Lubrication has a leading and a trailing edge, shallow water has a shoreline, slender-body theory has a nose and a tail. The expansion is uniformly valid in the middle and fails at the ends, and how badly it fails there is what sets the exponent.

The ends, which is where the exponent comes from #

That gives a rule of thumb worth stating, because it predicts all three of the results above.

If the end region contributes an area, the error is first order. The duct’s side walls are a boundary of the domain and they carry a shear stress over an area proportional to $h$. First order.

If the end region contributes nothing and the neglected term is a derivative, the error is second order. A long wave in an unbounded channel has no ends, and its correction is the vertical structure alone. Second order.

And if the end region contributes a logarithm, there is no exponent. A slender body’s cross-flow problem is two-dimensional at each station, and a two-dimensional potential flow has a logarithmic far field that must be matched to the three-dimensional outer solution. The matching produces $\ln(1/\varepsilon)$, and everything downstream of it inherits the drift.

The same logarithm shows up in Oseen’s correction to Stokes drag, for the same reason and by the same mechanism, which is why it is the subject of the next essay.

What is left out #

The duct comparison is laminar and fully developed. A real duct has an entrance region, and for a high-aspect-ratio duct that region is long — the same development length problem, and one whose own small parameter is different.

The wave is linear and inviscid. Finite amplitude brings in a second small parameter, the wave steepness, and the interesting shallow-water physics is the balance between the two — which is where solitary waves come from and is a different subject.

The spheroid is inviscid and unseparated. Real slender bodies at incidence shed vortices from their lee side, and slender-body theory’s linear normal force is then wrong by a factor rather than by ε².

And none of this is about how accurate the theories are. Lubrication theory at ε = 0.01 is excellent whatever its exponent, because ε² and ε ln(1/ε) are both small at 10⁻². The exponent matters when somebody wants to extrapolate — to say what happens at half the aspect ratio, or to estimate an error bar from a convergence rate — and that is exactly when a drifting exponent gives an answer that is wrong by a factor.

How a drifting exponent gets reported as a constant #

The measurement worth dwelling on is the one that produced no clean number, because it is the case a convergence study is least equipped to notice.

A local exponent is read from two points: $\log(\text{error})$ against $\log\varepsilon$, differenced. For the duct it comes out at 1.0000 everywhere and for the wave at 2.0000 everywhere, and both are true exponents. For the spheroid it moves from 1.88 at $\varepsilon = 2\times10^{-4}$ to 1.73 at $3\times10^{-2}$ and it is not converging to anything — the error is $O(\varepsilon^2\ln(1/\varepsilon))$, whose local slope is $2 - 1/\ln(1/\varepsilon)$ and therefore drifts to 2 only as $\ln(1/\varepsilon) \to \infty$.

How slowly is the number worth having. To bring the local exponent within one per cent of 2 needs $\ln(1/\varepsilon) = 100$, which is $\varepsilon = 2.7 \times 10^{-44}$. No computation and no experiment will ever be run there. The logarithm is unobservable in the exponent and entirely observable in the error.

So a careful convergence study over any realistic range reports 1.8, calls it “close to second order”, and is wrong about the functional form in a way that only matters when it is used — to extrapolate, to set a tolerance, or to claim a method is second-order accurate. The exponent is not measured; it is assumed, and then confirmed by a measurement that could not have refuted it.

The fourth case, which is not measured here #

There is a fourth member of the family worth naming because it fails in a fifth way.

Hele-Shaw flow is the slenderness expansion applied to the gap between two plates, and its reduced problem is a potential flow — which is why the cell is used as an analogue computer for streamline patterns. The error in that identification is not $O(\varepsilon)$ or $O(\varepsilon^2)$: it is $O(\varepsilon^2\mathrm{Re})$, because what the reduction neglects is the inertia, and the inertia has its own parameter.

So the small quantity governing a Hele-Shaw cell’s fidelity is a product of two groups, one geometric and one dynamic, and neither on its own says whether the analogue is any good. That is why the cells in textbook photographs are run in glycerine at speeds of millimetres per second rather than in water: the geometric ratio is fixed by the apparatus, and the only remaining freedom is the Reynolds number.

It is also why the analogy breaks exactly where it would be most useful — round a sharp corner, where the local velocity is large and the local Reynolds number with it.

Two more members, and one of them is not geometric #

The family is larger than four, and the two additions test the end-region rule against evidence it was not fitted to.

Thin-aerofoil theory is the same expansion with $\varepsilon = t/c$, and this collection has already measured its error: the lift departs from the exact conformal map with a fitted exponent of 1.012 — first order, like the duct and not like the wave. The rule above predicts it. What the reduction neglected is not a gradient but a boundary: the tangency condition was applied on the chord line rather than on the surface, and the displacement between the two is proportional to the thickness. Same structure as the duct’s missing side walls, same exponent, arrived at from a different subject.

The boundary layer is the same three steps again — different scales in the two directions, drop what is down by $\varepsilon^2$, solve a problem one dimension smaller — and it differs from every other member in a way worth naming. Its slenderness is not measurable with a ruler. The layer’s thickness over the body’s length is $1/\sqrt{Re}$: a quantity the flow produces rather than one the geometry supplies.

So the family splits. In lubrication, shallow water, slender bodies and thin aerofoils, ε is an input — chosen before the experiment starts, and independent of every dynamic group. In the boundary layer it is an output, and the expansion’s validity is therefore a function of the Reynolds number rather than of the shape, which is why its end regions are a leading edge and a separation point rather than a nose and a tail.

Where the theories came from #

Reynolds wrote lubrication theory in 1886, from Beauchamp Tower’s measurements of a railway journal bearing; Hele-Shaw’s cell is 1898; Airy’s shallow-water theory is 1841; and Munk’s slender-body theory for airships is 1924. None of the four cites any of the others, and there is no reason it should have — they were written for a bearing, a visualisation, a tide and a Zeppelin.

That they are one expansion was understood only with the language of matched asymptotics in the 1950s, which is when the logarithm in the slender-body error stopped being an oddity and became an instance. The unification is worth having for exactly the reason the three exponents above are worth measuring: it says which questions transfer between them and which do not.

What this leaves #

The group collapses four theories onto one small parameter and discards what is at the ends — which is worth nothing, one power of ε, or an entire exponent, depending on the problem.

The next essay is about the machinery for joining two limits when neither holds everywhere, and about what the joining costs: one formula for both ends.