Why the list is this long

Five exact solutions now stand in this sequence, and they fall into two kinds. Stokes’ oscillating wall, the asymptotic suction layer and the started plate are parallel flows: the nonlinear term is identically zero and what remains is the diffusion equation. The rotating disc and the wedge are similarity reductions: the nonlinear term survives and the answer is an ordinary differential equation.

Both kinds appear on every textbook list, and every such list is about a dozen long. The explanation usually offered is that the equations are hard and nobody has found the rest.

That is not the explanation. A similarity reduction is a solution invariant under a subgroup of the equations’ own symmetry group, which means the catalogue of reductions is the catalogue of subgroups. The group has eleven generators. The subgroups of an eleven-generator group are a finite, countable, listable object, and the list of exact solutions is short because that object is small. It was going to be short before anybody started looking.

The eleven, and a test that could reject #

The symmetries are usually just stated. Here they are tested, and the test is the one this collection applies to everything else: transform a known solution, and ask whether the equations still hold for the result — recomputing the residual from the transformed field by finite differences, with nothing differentiated by hand.

Time translation, three space translations, three rotations, three Galilean boosts and one scaling: eleven. Beside them sits the pressure gauge, which is infinite-dimensional — the pressure is defined up to any function of time — and which is not counted, because it acts on nothing a similarity reduction can use.

The six transformations leave a residual of a few parts in $10^8$, which is the finite-differencing floor and is what the untransformed solution leaves too. The five that are not symmetries leave between 0.048 and 4.3 — six to nine orders of magnitude more. There is nothing in between, which is what makes this a test: a transformation is either a symmetry or it is nowhere near being one.

The five are worth naming, because each is something a reader might reasonably believe. Anisotropic scaling — stretching one direction and not another. The Euler scaling, $x \to Lx$ with $t \to Lt$. Multiplying every velocity by a constant. Rotating the frame at a steady rate without adding the centrifugal and Coriolis terms, which is the one most often assumed. And reversing time with the velocity reversed, which is a symmetry of the Euler equations and is not one here, because diffusion has a direction.

The eleven can be checked against what is already known about each. Translation invariance is why a layer with no length in it exists at all; the scaling is why a similarity variable can be written down; and the rotation generator is the one a swirling flow uses when it reduces to a function of radius.

Why a boost is free and a turn is not #

The pair worth separating is the Galilean boost and the rotating frame, because they sound like the same kind of thing — change to a frame that is moving — and only one of them is a symmetry.

Boosting to a frame moving at constant velocity works because the acceleration of a fluid parcel is unchanged by it. Add a constant to every velocity and shift the coordinates to follow, and the extra term the time derivative picks up is exactly the extra term the convective derivative picks up, with the opposite sign. Nothing is left over, and the residual test confirms it at $2.7\times10^{-8}$.

Rotating to a frame turning at a constant rate does not work, and the reason is that a turning frame accelerates even when nothing in it is moving. A parcel sitting still in the rotating frame is going round a circle in the original one, so its acceleration is not zero, and the equations in the rotating frame have to carry that difference — the centrifugal and Coriolis terms. Applying the rotation without them leaves a residual of 4.3, which is the largest of the five and is by a wide margin.

The rotating frame is still usable; it is simply not free. The equations written in it are different equations, with two extra terms, and every rotating-flow result in this collection — the Ekman spiral’s companion, the Rossby number, the geostrophic balance — is a statement about those equations rather than about these. What the symmetry group says is that the transformation costs something, and what it costs is exactly the two terms.

That distinction also decides what belongs in the catalogue. A reduction built on the steady rotation would be a reduction of a different system, so it is not among the 66; the rotating-disc solution is in the catalogue because it uses the rotation generator — an instantaneous turn of the whole field, which is free — and not a turning frame.

The sequence’s own membership makes the division visible. The wall that shakes and the layer that stops growing are parallel flows and have formulae; the disc and the wedge are similarity reductions and have ordinary differential equations. The claim of this essay is that the two kinds are two choices of subgroup and not two levels of difficulty.

The scaling is one and not two, and that is the whole of it #

The most consequential entry in the list is the one there is only one of.

The Euler equations admit a two-parameter scaling group: lengths and times can be stretched independently, because with no viscosity there is no relation between them and the velocity scale can absorb whatever is left over. The viscous equations admit one. Viscosity supplies a relation — a length $L$ and a time $L^2/\nu$ are tied — and a relation is a constraint that removes a parameter.

Measured, the difference is stark: stretching with $t \to L^2 t$ leaves the residual at $8\times10^{-9}$ whatever $L$ is; stretching with $t\to Lt$ leaves 0.048, which is nearly six million times larger.

So the viscous equations have a smaller symmetry group than the inviscid ones, and therefore fewer similarity reductions. That inverts the usual intuition, which is that viscosity is a complication added to a simpler system. In the currency that decides how many exact solutions there are, viscosity is a simplification: it ties two free scalings into one and shortens the catalogue.

It is also the same fact as the dimensional argument one level up. The number of independent dimensionless groups a problem can depend on is decided by the rank of a matrix before anything is solved, and the scaling symmetry is what that rank is, seen from the other side.

An instrument that passed a test it should have failed #

One of the five near-misses failed to fail, and the reason is worth a section because it is the general trap in this kind of verification.

Multiplying every velocity by a constant $k$ multiplies the unsteady and viscous terms by $k$ and the convective and pressure terms by $k^2$. It cannot be a symmetry unless $k = 1$. Applied to the decaying Taylor–Green vortex, it passed at the residual floor for every $k$ tried.

The vortex is degenerate as an instrument. Its unsteady and viscous terms cancel each other exactly, and its convective and pressure terms cancel each other exactly, so the residual is a sum of two groups that vanish separately. Multiplying the two groups by different factors still leaves nothing.

Carrying the same solution past the observer at a steady speed — a Galilean boost, which is itself one of the symmetries — breaks the degeneracy without changing what the field is. Each group becomes $\mathbf U\cdot \nabla\mathbf u$, equal and opposite, and a transformation that treats them differently is visible again. The near-miss then fails by nine orders of magnitude.

A test field that is too symmetric is a test that cannot reject, and the failure mode is invisible: the transformation passed, the number was small, and nothing in the output said the instrument was blind. The standing rule here is that a check which has never rejected anything proves nothing; the sharper version is that a check needs an input on which its own subject matters.

The table, and what closes #

Knowing the eleven is not enough to count the reductions. What matters is which of them can be used together, and that is decided by the commutator: a set of generators can be imposed simultaneously only if the bracket of any two of them stays inside the set.

The table is computed rather than transcribed. Each bracket is evaluated numerically from the vector fields and then projected onto the eleven by least squares, and the largest amount left outside their span anywhere in the table is $1.5\times10^{-11}$. That number is what “the eleven close” means, and it is worth having as a measurement: a generator omitted from the list would show up as a bracket that does not fit.

Reading the table is reading the structure. Translations commute with each other and with time translation. A boost and a translation produce a time translation. Rotations produce rotations, and rotate boosts and translations into each other. And the scaling does not commute with anything except the rotations — it rescales every other generator — which is what makes it the awkward one and, as the next section shows, the one the interesting reductions need.

Counting the subsets that close gives the catalogue. There are 11 of size one, 37 of size two, 66 of size three and 80 of size four. The three-member ones are the interesting number: the equations have four independent variables, so removing three of them leaves one, and a partial differential equation in one variable is an ordinary differential equation.

Sixty-six is not a small number of candidates, and it is not the number of useful solutions either — most of those reductions produce an equation nobody can solve, or a flow with no boundary it can satisfy, or a solution already on the list under a different name. What the count establishes is the shape of the situation: the search space is a list of sixty-six entries, not an open field. Anybody who wants a new exact solution of this kind has to find it in that list, and the list has been enumerated.

What the sixty-six are not #

Two cautions belong beside the number before it is carried anywhere.

The first is that a closed subset is a candidate, not a solution. Imposing three generators fixes the form a solution must take and leaves an ordinary differential equation to solve; that equation may have no solution satisfying any sensible boundary condition, or no solution at all. The wedge’s reduction produces an equation with two answers at the same flux; the disc’s produces one with no closed form. Both are in the catalogue, and neither is in it because it was known to work.

The second is that the count is a property of the basis rather than of the algebra. Rotating the basis — a linear combination of two translations is a translation too — produces subsets that close and are not on this list, and produces others on the list that are the same reduction in other coordinates. The proper object is the set of subalgebras up to the algebra’s own automorphisms, which is smaller, and computing it is a piece of Lie theory rather than a piece of arithmetic. Sixty-six is an honest count of subsets of this basis and an over-count of the distinct reductions.

What survives both cautions is the shape of the claim, and the shape is what this essay is about. Whatever the exact number, it is a number: finite, enumerable, and fixed by the equations before anybody looks.

What each of these solutions is a fixed point of #

The claim becomes concrete when these solutions are put against it.

Stokes’ oscillating wall is invariant under all three space translations — it depends on the wall-normal coordinate and on time, and on nothing else. The asymptotic suction layer adds time translation, which is what makes it steady, and is left with one variable. Von Kármán’s disc is invariant under time translation, one rotation and the scaling, and the scaling is why the radius divides out: the similarity variable $z\sqrt{\Omega/\nu}$ is the combination the scaling leaves alone. The wedge is invariant under time translation and the scaling, and its similarity variable is the angle, which the scaling does not touch.

Read down that column and the pattern is the whole argument. The solutions are not five separate feats of ingenuity. They are five choices of which subgroup to impose, and they differ in what each choice leaves behind rather than in how hard anyone worked.

It also explains the division into two kinds. A flow invariant under all three translations is a parallel flow, whose nonlinear term vanishes automatically — so the parallel-flow half of every textbook list is the one subalgebra that kills the nonlinearity, and everything else on the list has it. The dozen entries are not a dozen tricks; they are a handful of subgroups, most of them chosen twice.

The inviscid catalogue, which is longer #

The contrast with the Euler equations is worth drawing out, because it runs the opposite way from what “exact solution” usually suggests.

The Euler equations have every symmetry listed here and one more scaling, since with no viscosity there is no relation tying a length to a time. A twelve-generator algebra has more subalgebras than an eleven-generator one — a good deal more, since the extra generator can join most existing closed subsets and still close — so the inviscid catalogue of similarity reductions is longer.

And the inviscid catalogue is longer for a second reason on top of that: dropping the viscous term lowers the order of the equation, so a reduction that would have produced a third-order ordinary differential equation produces a second-order one, and second-order equations are far more often solvable in closed form. The ideal-flow solutions — the cylinder, the doublet, the vortex, the whole conformal apparatus — are an enormous family by comparison with the dozen here.

So the honest summary is that adding viscosity to the equations shortens the list of exact solutions twice over: once by removing a symmetry, and once by raising the order of what each remaining symmetry leaves behind. The usual framing — that the inviscid equations are a simplification made for tractability — is true about the second and misses the first, which is a fact about the symmetry group and not about anybody’s patience.

A word about what the group does NOT explain, since it explains a good deal. It says nothing about why a layer thickens as the square root of distance rather than in some other way — that is the content of the reduced equation, not of the reduction. The group decides which variables survive; what the surviving equation says is a separate question with its own answer.

What the group does not settle #

The group of the equations is not the group of a problem. A boundary breaks whichever symmetries do not preserve it: a wall at $y=0$ destroys translation in $y$ and two of the three rotations, and a finite disc destroys the scaling. So the catalogue of 66 is an upper bound on what an unbounded problem admits, and the number available for any particular geometry is smaller — often much smaller, and sometimes one. That is why so few of the 66 have ever produced a solution anybody quotes.

The count is of subsets of one basis, not of subalgebras up to equivalence. Two subsets related by a symmetry of the algebra itself give the same reduction in different coordinates, and classifying subalgebras properly means quotienting by that — the object a Lie-group analysis calls an optimal system, which is smaller than 66 and is not computed here. The 66 is an honest count of a well-defined thing and an over-count of the thing it stands in for.

Point symmetries are not all the symmetries. The equations may admit transformations that act on derivatives as well as coordinates, and reductions built on those are not in this catalogue. None is known for the incompressible Navier–Stokes equations, which is a statement about what has been looked for rather than about what exists.

And a reduction is not a solution. Every one of the 66 produces an ordinary differential equation; the wedge’s has two answers at the same flux and the disc’s has none in closed form. Reducing the problem is the part the group decides, and solving what is left is a separate difficulty the group says nothing about.

The checks are the ones above. The untransformed solution’s residual sets the floor. The worst symmetry sits on it and the least near-miss is six orders above. No bracket falls outside the span of the eleven by more than $1.5\times10^{-11}$. And the calculation refuses a test with its tolerance set to zero, a near-miss test that could not reject, and a catalogue declared empty.

It is also worth being clear that a reduction is not a claim about stability. Several of the solutions this catalogue admits are unstable, and an unstable exact solution is still an exact solution — a pipe’s parabolic profile is the standing example, exact at every Reynolds number and observed at almost none of them.

Still open: which of the sixty-six have never been tried #

The count is the beginning of a question rather than the end of one. Sixty-six three-generator subsets close, a dozen or so reductions are in the textbooks, and nobody has published a list of which is which.

The calculation that would answer it is bounded and mechanical. For each of the 66, work out the similarity variable and the form the velocity must take, substitute into the equations, and record what comes out: an ordinary differential equation, an inconsistency, or a reduction already on the list under another name. The output is a table with 66 rows, and the interesting rows are the ones that produce a consistent equation nobody has written down. Some will have no boundary condition that makes sense; some will duplicate; and if even one is left, it is an exact solution of the Navier–Stokes equations that exists and has not been computed.

Beside it is the same count with a boundary imposed. Fixing a plane wall, or a pair of walls, or an axis, cuts the eleven down to the subgroup that preserves it, and the catalogue shrinks accordingly. Doing that for the three geometries this subject keeps returning to — a wall, a pipe, a wedge — would say for each of them exactly how many exact solutions are available, which is a sharper statement than any list of what has been found.