The world with no inertia

One number decides which physics applies, and this site spends most of its time at the end of the axis where inertia dominates. At that end the useful simplification is to throw viscosity away, which gives an exact theory that is wrong about drag.

At the other end the simplification is the reverse: throw the inertia away. That gives another exact theory, and it fails in a stranger way — not by getting the answer wrong, but by not having one.

What is left when inertia goes #

The Navier–Stokes equations balance three things: the acceleration of the fluid, the pressure gradient pushing it, and the viscous stress resisting it. The Reynolds number is the ratio of the first to the third.

When it is very small — a bacterium, a settling dust grain, honey — the acceleration term is negligible, and dropping it leaves

$$\nabla p = \mu \nabla^2 \mathbf{u}$$

which is linear. That is the same gift ideal flow receives from the opposite direction, and it brings the same benefits: solutions superpose, and closed forms exist.

It also brings a property ideal flow does not have, and which is much stranger.

Reversibility, measured #

The Stokes equations contain no time derivative at all. Reverse the pressure gradient and every velocity reverses; run the driving backwards and every particle retraces its path exactly. A creeping flow has no memory and no arrow of time.

The visible consequence is symmetry. The pattern in front of a body and the pattern behind it are mirror images, because there is nothing to distinguish upstream from downstream.

That is a claim a figure can be made to test rather than illustrate. The measure used here is the mismatch between $\psi(x, y)$ and $\psi(-x, y)$ about the body’s centre, normalised by the size of $\psi$ itself: zero for a perfectly symmetric flow, order one for a completely asymmetric one.

The first row is not “small”. It is the last bit of the arithmetic, which is what a symmetry that holds identically looks like when it is measured numerically. The rows below it are the same measurement on fields the site solved on a grid, with inertia included, and they climb steadily as the term that breaks the symmetry grows.

This is the cleanest demonstration on the site of what the Reynolds number decides. It is not the magnitude of anything: it is whether the flow has a direction of time.

What the solver computed, and how it was checked #

Creeping flow past a cylinder is solved here in closed form, between the body at radius $a$ and an outer boundary at radius $R$ that moves with the stream. The Stokes streamfunction has the form

$$\psi = \sin\theta \left(A r \ln r + B r^3 + C r + \frac{D}{r}\right)$$

and four boundary conditions — no slip at the cylinder, the stream at the outer wall — fix the four coefficients through a four-by-four linear system. The drag on the cylinder is $4\pi\mu A$.

The reason for solving the annulus rather than the unbounded problem is that the unbounded problem has no solution, and the annulus is how that shows itself.

Three things are asserted about that sweep, and the third is what makes it an argument rather than an anecdote.

The drag must fall as the room grows. It must keep falling by more than three per cent at every doubling, so that no amount of enlargement looks like convergence. And $1/\text{drag}$ must be linear in $\ln R$ with a slope of $1/4\pi\mu U$ — because the divergence is logarithmic, and a logarithm is what makes this paradox so easy to miss. The measured slope is 0.079464 against a predicted 0.079577, which is 0.14%.

The rejection test is a set of drags that settle down: handed a table converging on a value, the check refuses it, because that would be no paradox at all.

Why there is no answer #

The failure is not numerical and it is not a defect of the method. It is a genuine property of the two-dimensional problem.

The Stokes equations are a balance between pressure and viscous stress, valid where inertia is negligible. Around a cylinder, the disturbance the body makes decays slowly — logarithmically — so however small the Reynolds number based on the body’s size, there is always a distance far enough away that the neglected inertia term is comparable with the viscous one. The approximation that is excellent near the body fails far from it, always, and no choice of Reynolds number avoids it.

So the far field is not described by the equations being solved, and the outer boundary condition — the thing that would pin down the solution — is being applied in a region where the model is invalid. That is why the answer depends on where the boundary is put.

A sphere does not suffer from this. In three dimensions the disturbance decays as $1/r$ rather than logarithmically, and the inertia term stays negligible everywhere, so Stokes’ law $6\pi\mu a U$ is finite, unambiguous, and correct. Two dimensions is the awkward case, and it is awkward because of a logarithm.

How it was fixed, and what the fix costs #

Oseen’s resolution, in 1910, keeps a linearised version of the inertia term — approximating the convective acceleration by $U \partial\mathbf{u}/\partial x$, which is right far from the body where the disturbance is small. That restores a well-posed problem and gives a drag which, for a cylinder, is

$$D = \frac{4\pi\mu U}{\tfrac{1}{2} - \gamma - \ln(Re/4)}$$

with $\gamma$ Euler’s constant. It depends on the Reynolds number, logarithmically, and it does not depend on the size of the room.

The shape of that fix is worth noticing, because it recurs throughout fluid mechanics. A simplification is excellent in one region and invalid in another; the resolution is not a better approximation everywhere but two approximations, each valid in its own region, matched where they overlap. That is the method of matched asymptotic expansions, and Prandtl’s boundary layer is the same idea applied at the opposite end of the Reynolds axis — inviscid outside, viscous in a thin layer, matched at the edge.

The two great simplifications of the subject fail for the same structural reason and are repaired by the same structural device.

The drag that does have an answer #

Since so much of this essay is about a missing solution, it is worth having the one that is not missing, because it is the most-used result in the whole low-Reynolds-number literature.

For a sphere of radius $a$ moving at $U$ through a fluid of viscosity $\mu$, Stokes’ law gives

$$D = 6\pi\mu a U$$

with no logarithm, no room size and no Reynolds number in it. Two features are worth reading. The drag is linear in the speed, not quadratic — doubling the speed doubles the drag rather than quadrupling it — and it is proportional to the radius rather than to the frontal area, so halving the size halves the drag rather than quartering it.

Balancing that against the weight of a sphere denser than the fluid gives a terminal velocity of

$$V = \frac{2 a^2 g (\rho_s - \rho)}{9\mu}$$

which goes as the square of the size. That single exponent explains a surprising amount: why a river drops its sand in seconds and holds its clay for days, why a fog droplet of ten micrometres falls at about a centimetre per second and effectively hangs in the air, and why sedimentation sorts particles by size so cleanly that it is the basis of a laboratory technique.

It is also the reason the two-dimensional failure is so easy to overlook. The sphere result is taught first, it is clean, and nothing about it hints that the cylinder — the simpler-looking problem, in fewer dimensions — has no answer at all.

What reversibility is good for #

The absence of an arrow of time is not merely a curiosity; it is a hard constraint on what can be done at small scale.

A swimming stroke that is the same forwards and backwards gets nowhere. A scallop opening and closing its shell moves forward on one stroke and back on the other, and at low Reynolds number the two cancel exactly — the scallop theorem. Anything that swims at this scale must therefore have a stroke that is not its own reverse, which is why bacteria use rotating helical flagella and why sperm use travelling waves rather than oars.

The same reversibility produces the classic demonstration in which dye is stirred into glycerine between two cylinders, apparently mixed beyond recovery, and then unstirred by turning the handle backwards. Nothing was mixed. The dye was sheared, and shearing is reversible when inertia is absent.

Where everyday life sits on this axis #

The creeping regime is not exotic. It is where most of the fluid mechanics on Earth happens, by count of events rather than by mass of fluid.

A bacterium swimming at ten micrometres per second, one micrometre long, in water: the Reynolds number is about $10^{-5}$. Its experience of water is, in the standard comparison, roughly a human’s experience of tar — except that the comparison fails, because tar has inertia and the bacterium’s world does not. When its flagellum stops, it coasts for a distance of order a tenth of an atomic diameter and then it is stationary.

A grain of silt settling in a river reaches its terminal velocity almost instantly, and that velocity comes from balancing Stokes drag against weight: for a sphere, $V = 2 a^2 g (\rho_s - \rho)/9\mu$. The square of the size is why silt takes days to clear and sand takes seconds, and why the sorting of sediment by size happens at all.

A lubricating film in a bearing has a Reynolds number in the tens at most, and the whole theory of lubrication is the creeping equations in a thin gap — one of the very few places where an exactly solvable limit is also the engineering model of choice, rather than a stepping stone to one.

What all three share is that the ratio, not the size, decides. A bacterium and a whale are not doing the same thing at different scales; they are in different regimes of the same equations, and the number that separates them is the subject of the rung below.

The limit that is also an engineering theory #

One case deserves more than a bullet, because it is the only place in the subject where an exactly solvable limit is the working model rather than a stepping stone towards one — and because it appears to contradict everything above.

Squeeze a creeping flow into a thin gap and the far field, which is what wrecked the cylinder, is gone: the gap has walls a few micrometres apart and there is nowhere for a slowly-decaying disturbance to go. What is left is a balance between the pressure gradient along the gap and the viscous stress across it, which reduces the equations to one for the pressure alone. That is Reynolds’ lubrication equation, and every plain bearing, thrust pad and piston ring in the world is designed on it.

The result it gives is worth stating because it is unexpected in the same way the scallop theorem is. A parallel gap carries no load at all. Drag fluid through a channel of constant height and the pressure is whatever is imposed at the ends; nothing is generated. Pressure appears only when the film converges in the direction of motion, because then the fluid being dragged in must be squeezed into a smaller passage and can only do so by building a pressure to push some of it back.

So a bearing is a wedge, and it is a wedge for the same reason a swimmer must have a non-reciprocal stroke: a symmetric geometry in a reversible flow cannot produce a one-directional result. The asymmetry has to be put somewhere, and a bearing puts it in the shape of the gap.

The load a wedge carries goes as the viscosity, the speed, and the inverse square of the film thickness, so halving the gap quadruples the pressure. Beau Tower measured such pressures in a railway journal bearing in 1883 and could not account for them; Reynolds explained them three years later.

What the picture cannot show #

The creeping figures are drawn inside a finite room, because that is the only problem that has a solution — so every streamline pattern on this page is a picture of a particular room as much as of a particular body. Enlarging it changes the drag and barely changes the picture, which is precisely the difficulty being described and is very hard to see.

Nor can the figures show the failure’s location. The Stokes solution is excellent near the cylinder and invalid far from it, and there is no visible mark at the crossover, because nothing goes wrong locally. The approximation degrades smoothly into nonsense.

Where the model stops #

The creeping solution is exact for the equations it solves, and those equations describe a real fluid only where the Reynolds number based on the local length scale is small — which, as above, fails far enough away in two dimensions and does not in three.

The site’s grid solver, which supplies the comparison rows, has its own limits recorded elsewhere: it does not resolve the internal structure of a boundary layer, it does not shed a periodic vortex street at these Reynolds numbers, and drag cannot honestly be read out of its wake. Nothing on this page relies on any of those; the symmetry measure is an integral over the streamfunction, which is the quantity the solver advances.

Two exact theories, and the space between them #

It is worth setting the two solvable limits side by side, because between them they define what the rest of this site is doing.

Ideal flow keeps inertia and drops viscosity. It is exact, linear, and predicts no drag on anything — an error of infinite relative size, on the quantity engineers most want. Its saving grace is that the error is localised: the flow is very nearly right everywhere except in a thin layer and a wake, which is what makes the boundary-layer decomposition work.

Creeping flow keeps viscosity and drops inertia. It is exact, linear, and in two dimensions gives no answer at all. Its error is not localised — the neglected term matters far away rather than nearby — which is why the repair had to come from matching to an outer solution rather than from patching a layer.

Everything in between is where the equations are nonlinear and nothing is exactly solvable, which is where every flow anybody flies, sails or swims in actually lives. The two limits are useful because they bracket that region and because each fails in a way that says which term was thrown away.

That symmetry of failure is, in the end, the argument for organising a site about fluids around the Reynolds number rather than around shapes. Shapes do not decide anything. The ratio decides which term can be neglected, neglecting a term is what makes a problem solvable, and every exact result in the subject is a statement about a limit somebody took.

Who found it, and when #

Stokes derived the creeping-flow solution for a sphere in 1851 and noticed in the same work that the cylinder had no solution. The observation sat as a curiosity for half a century, with several attempts to argue it away, until Oseen explained it in 1910 by identifying the far field as the place where the approximation fails.

Lamb produced the practical drag formula in 1911, and the full matched-asymptotic treatment — which makes the logic explicit and generates the higher-order terms — is Kaplun’s and Proudman and Pearson’s, in 1957. That is a hundred and six years from the paradox to its complete resolution, on a problem that is linear.

Where the ladder goes next #

Below this rung, one number decides introduces the Reynolds number as the thing that separates regimes, and the length in it is about how easily that number is quoted meaninglessly.

This rung is the far end of the axis, and its lesson is the site’s central one arriving from an unexpected direction: an exact theory failed, and the failure was informative. Ideal flow’s failure is a wrong answer, which is embarrassing and useful. Creeping flow’s is no answer at all, which is worse and more interesting, because a missing solution says exactly where the model’s hypotheses stopped being true.