How many things a flow must be told

The Navier–Stokes equations are usually introduced as though they were the whole of the subject, and then the difficulty is presented as computational: they are nonlinear, they are coupled, nobody can solve them. All of that is true and none of it is the first difficulty. The first difficulty is that a differential equation is not a question until somebody says what happens at the edge, and how many things may be said there is not a matter of taste.

It is a matter of the order of the equation. And the most famous result in this subject — that ideal flow gives every body zero drag — is a counting error rather than a physical one.

The count, and where it comes from #

Write the momentum equation and look only at the terms with a derivative across the wall. Euler’s equations have $u\,\partial u/\partial y$ and nothing else: first order. The viscous term is $\nu\,\partial^2 u/\partial y^2$: second order.

A first-order equation carries one constant of integration per direction and can therefore satisfy one condition; a second-order equation carries two. So an inviscid fluid may be told that nothing crosses the surface — the normal component, one number at each point — and that is the whole of what a body is allowed to say to it. A viscous fluid may be told that and that the tangential component matches the wall’s.

The word and is doing all the work. Viscosity does not make the same problem harder. It raises the order of the equation, and an equation of higher order admits another condition. At a Reynolds number of ten million the viscous term is negligible everywhere except in a layer a millimetre thick, and it is not negligible in the sense that matters: it is the term that lets the no-slip condition be imposed at all.

What the extra condition costs #

The cheapest place to watch this happen is one dimension, where the whole argument fits on a line. Take

$$\varepsilon u'' + u' = 0, \qquad u(0) = 0, \quad u(1) = 1,$$

with $\varepsilon$ small. It is the boundary layer with the fluid taken out of it: a small second derivative, a large first one, and a condition at each end.

The exact solution is $u = (1 - e^{-x/\varepsilon})/(1 - e^{-1/\varepsilon})$.

Set $\varepsilon$ to zero and the equation becomes $u' = 0$, whose solution is a constant. It can meet one condition. It meets the one at the far end and misses the one at the wall by the entire range of the variable.

Everything the boundary-layer chapter of this subject says in two dimensions is already here. The layer’s thickness goes to zero with $\varepsilon$; the gradient inside it goes to infinity like $1/\varepsilon$; and the product stays finite. Drag does not vanish as viscosity does — it converges. The limit of the viscous solution as $\nu \to 0$ is not the inviscid solution, and the difference between those two sentences is the whole of d’Alembert’s paradox.

The paradox is not that the ideal answer is inaccurate. It is that the ideal answer is exactly zero — not small, not a percentage of the real value, but identically nothing, at every Reynolds number the calculation could be run at, because no Reynolds number appears in it. A model that is wrong by a factor is a model that needs a correction. A model that is wrong by everything is a model that was asked a different question, and the different question was set at the wall.

Too many, and the answer stops meaning anything #

If too few conditions leave a solution unfixed, the obvious repair is to supply more. It does not work, and how it fails is worth seeing, because the failure has a name in this subject: well-posedness is not about whether an answer exists.

Hadamard’s example is Laplace’s equation with both the value and the normal derivative given on one line: $u = 0$ and $\partial u/\partial y = (\varepsilon/k)\sin kx$ at $y=0$. That is an elliptic equation handed the data a hyperbolic one would want. There is a solution and it is unique:

$$u = \frac{\varepsilon}{k^2}\sin(kx)\sinh(ky).$$

The data can be made as small as anybody likes by taking $k$ large; it is bounded by $\varepsilon/k$. The solution at $y=1$ is $\varepsilon\sinh(k)/k^2$, which is bounded by nothing at all.

The practical shadow of this is not obscure. It is the reason a pressure distribution measured on a surface cannot be extrapolated backwards into the flow that produced it, and the reason that inferring an interior field from boundary data is a regularisation problem rather than a calculation. An elliptic equation propagates information in every direction at once, which is why pressure has no speed; the same property makes the backwards question hopeless.

Impossible, and the arithmetic says so first #

The third way to get the count wrong is to impose the right number of conditions and make them inconsistent. That is not a hypothetical: it is what a body in a stream is.

Prescribe the normal velocity everywhere on a boundary and nothing else, and the interior problem is pure Neumann. Adding a constant to the potential changes nothing, so the constant is in the operator’s null space; the operator is symmetric; therefore a solution exists only if the right-hand side is orthogonal to that constant — which is to say that the prescribed flux must integrate to zero round the whole boundary.

That is the divergence theorem, arrived at without mentioning a fluid.

Mass conservation appears here as a property of a matrix. Not as an equation that is checked afterwards, not as a physical principle imposed on the answer, but as the reason a particular linear system has no solution. The physics is in the shape of the operator rather than in the modelling assumptions on top of it.

The same solve makes the other half of the point. The Neumann problem’s answer is unique only up to a constant, and adding 3.5 to every value of the potential changes the velocity by $7\times10^{-15}$ — which is why nothing that matters ever depends on the value of a velocity potential, and why the constant can be discarded without argument.

The condition that makes vorticity #

There is a fourth thing the count does, and it is the one that connects this essay to every figure on the rest of this site. The extra condition that viscosity permits is not merely a constraint. It is a source.

At a stationary impermeable wall both velocity components vanish, so the momentum equation collapses to

$$0 = -\frac{1}{\rho}\frac{\partial p}{\partial x} + \nu\frac{\partial^2 u}{\partial y^2},$$

and the second derivative there is exactly the diffusive flux of vorticity out of the surface. So the rate at which vorticity enters the fluid is fixed by the pressure gradient along the wall — and the viscosity that made the no-slip condition necessary has cancelled out of what the condition produces. That argument gets an essay of its own; what belongs here is only that it exists, and that it is the second boundary condition’s doing.

The edge that is not a wall #

Everything above counts at a solid surface, because that is where this subject’s arguments happen. But a boundary is wherever the domain stops, and three of the other kinds change the count in ways that decide whole chapters.

The far field is a boundary, and in two dimensions it does not say enough. An exterior problem has an edge at infinity, and the obvious thing to say there is that the velocity tends to the free stream. For a three-dimensional body that is the end of it. For a two-dimensional one it is not, and the shortfall is exactly one number. The region outside a closed curve in the plane is doubly connected — a circuit drawn round the body cannot be shrunk to a point without crossing it — and Laplace’s equation on such a region admits a solution whose potential is multivalued. That solution is a vortex at the body’s centre. Its velocity decays like $1/r$, so it satisfies the condition at infinity exactly; it is everywhere tangential to a circular body, so it satisfies the wall condition exactly too. Add any multiple of it to a solution and the result is another solution, meeting every condition that has been counted.

So the counting done at both boundaries leaves a one-parameter family, and no amount of care at either boundary removes it. The freedom is not an oversight in the statement of the problem; it is a property of the shape of the region. In the discrete problem it appears where the Neumann constant appeared above — as a null vector, one more direction the matrix cannot see — and the famous extra condition is literally an extra row supplied to a system that was short of one.

The sharp edge decides which member of the family is taken, and it does so in a way that is unlike every condition counted so far: it is imposed at a single point rather than over a surface, and what it asserts is not a velocity but that the velocity is finite. The lift of every aerofoil in this collection is fixed by that one sentence. In three dimensions the freedom is not there to be fixed — the region outside a closed body is simply connected, the multivalued potential does not exist, and the circulation round a wing is instead accounted for by the vorticity in its wake, which is what the far field remembers.

An interface has two of everything, and one condition that says where it is. Between two fluids neither side is a wall. The count applies to each side and the conditions couple them: both velocity components are continuous across the surface, and so is the traction — the shear stresses match, and the normal stresses differ by surface tension times curvature. That is four conditions for two second-order problems, which is the right number, and the arithmetic looks closed.

It is not, because the position of the surface is not given. It is an unknown of the problem on the same footing as the velocity and the pressure, and an unknown needs an equation: the kinematic condition, which says that a parcel on the surface stays on it. A problem whose boundary is part of its own answer is a different kind of problem, and it is nonlinear even when the equations are linear, because a condition is being applied at a location that depends on the solution. Linearising a water wave means agreeing to apply the surface conditions at the undisturbed level instead, and the small parameter that buys the agreement is the slope rather than the amplitude — which is why a surface that moves with the flow is harder than the flow underneath it.

And a computed domain has edges the fluid does not have. Truncating an exterior problem at a finite box puts a boundary where there is no physics, and the count there is a matter of arithmetic with nothing to appeal to. Say too little and the problem is not closed; say too much and the surplus is reflected back inward as a wave that the real flow never met, arriving later at the body as a disturbance indistinguishable from one it made itself. The repair is a condition that is transparent rather than correct — one built to let a disturbance leave, which means built out of which way the equation carries information rather than out of any statement about the flow. That is the counting argument used in reverse: not what an equation of a given order permits, but what a boundary must say in order to permit nothing.

What the picture cannot show #

The figures above are all of one-dimensional or two-dimensional model problems, and they are drawn that way deliberately, but the honest limits are worth stating.

The counting argument as given is local and linear. It says how many conditions an equation of a given order admits at a smooth wall. It does not say which combinations are well-posed for the full nonlinear system, and for compressible flow the answer depends on the local Mach number — a supersonic inflow takes a different number of conditions than a subsonic one, because the equation has changed type rather than order.

The layer’s structure is asserted here rather than derived. The one-dimensional problem has an exact solution and shows the mechanism; the two-dimensional one needs matched asymptotics, and its thickness $\sqrt{\nu x/U}$ is a result of that machinery rather than of this counting.

And nothing here says the extra condition is no slip. That the tangential velocity at a wall matches the wall’s is an empirical fact about ordinary fluids, arrived at after a century of argument about whether fluids slip; the equations merely leave room for a tangential condition. In a gas rarefied enough that the mean free path is comparable with the geometry, the room is filled by a slip condition instead, and the flow rate through a fine tube stops matching the no-slip prediction. That is a fact about the state of matter rather than about the flow, and this collection links it rather than deriving it.

The shape of the whole argument #

Three failures, three names, one cause.

Too few conditions and the solution is not fixed: the reduced problem has a family of answers and the missing condition is the one that picks the member with the layer in it. This is the inviscid limit, and it is why the drag of a body does not go to zero as its viscosity does.

Too many and the problem is unstable: it has a unique solution that depends on the data like $e^k$. This is the reason inference from surface measurements is hard.

Inconsistent and there is no solution at all: the operator’s null space refuses it, and it refuses in the arithmetic before anybody mentions physics. This is mass conservation.

The pattern is worth carrying into the rest of this collection, because it is what every essay in this field turns out to be about. A wing’s circulation is fixed by an extra condition at a sharp edge rather than by the equations. A designed pressure distribution is a body only if three integrals vanish, which are the same solvability conditions in another coordinate system. A wind tunnel measures something different from free air because its walls are conditions the aeroplane never meets. In each case the equation is the same equation, and what changed is what it was told at the edge.

Who counted, and when #

The counting itself is old — Cauchy and Kovalevskaya settled which data an equation of a given order admits in the nineteenth century — and Hadamard supplied the example above in 1902, in a paper about what makes a problem well-posed, a phrase he coined for the occasion. What took longer was recognising that fluid mechanics’ central embarrassment was an instance of it.

Prandtl’s 1904 paper is usually described as the introduction of the boundary layer, and it is; but what it does is restore a boundary condition that Euler’s equations had no room for, and then show that the fluid pays for it with a layer whose thickness vanishes and whose influence does not. The paper is eight pages long. It resolved a paradox that had stood for a hundred and fifty years, and it did it by counting.

Where the ladder goes next #

Two directions, and they leave this field in opposite ways.

Towards the equations: the same counting for the compressible case, where the type of the equation changes with the Mach number and the number of conditions changes with it. This collection has that argument in pieces — the domain of dependence and what a nozzle can be told — and not yet as one statement.

Towards the wall: what the second condition produces, rather than what it costs. That is the vorticity a surface makes, whose rate contains no viscosity at all and which integrates to nothing round any closed body — a result that arrives at Kelvin’s theorem from the wall rather than from a material loop.