Boundary condition — where it appears
Named by 57 essays across 9 fields — each of them below, with the objects they name alongside it.
A wall made by reflection
Imposing a boundary condition on a plane is work. Putting a mirrored copy of everything on the far side of where the plane would be is not, and the plane then appears on its own — as a consequence of the symmetry rather than as a condition anybody enforced.
The wall that pushes back
A wing near the ground carries more lift than the same wing in free air, and the usual explanation — a cushion of compressed air underneath — is not what the equations say. The ground is a mirror, and what changes is not the pressure under the wing but the circulation the wing is forced to carry.
How many things a flow must be told
The equations of motion do not have one answer. They have as many as the conditions on the edge allow, and the number of those is decided by the order of the equation — which is why viscosity does not make the same problem harder, it makes a different problem.
Pressure has no speed
Take the divergence of the momentum equation for an incompressible flow and the time derivative disappears, the viscosity disappears, and what is left is Poisson's equation. Pressure is not carried anywhere: it is whatever satisfies an elliptic equation everywhere at once, and that is a statement about a fluid nobody has.
The other layer, and the one number that separates them
A wall in a stream carries two boundary conditions and grows two layers. Their thicknesses differ by a factor of twenty across ordinary fluids, and at exactly one Prandtl number the two profiles are not similar but identical.
Where vorticity comes from
Every scrap of vorticity in a flow past a body entered through its surface, and the rate at which it enters contains no viscosity at all — it is the pressure gradient along the wall. A flat plate makes none, and a closed body makes exactly as much of each sign.
The cushion that is not there
Ground effect is usually explained as air trapped under the wing and compressed into a cushion. At the heights an aeroplane actually flies at, the air under the wing is moving at four fifths of flight speed and most of what the ground gives is a drag saving the story never mentions.
The wing the equation is really solving
The Prandtl–Glauert rule is usually quoted as a factor on the answer. It is a change of shape — and in three dimensions the shape it changes is the aspect ratio, so a short wing is far less affected by compressibility than a long one.
Ask for the pressure, and see what shape that is
A designer knows what the pressure distribution has to do long before knowing what shape does it. Running the problem that way round is possible, it is exact, and it refuses more asks than it grants.
The air a wing does not carry
No slip says the air touching a surface moves with it, and the usual reading is that a wing drags a blanket of air along. The blanket is 1.7 millimetres thick per metre of chord, it is different air every instant, and the drag it costs falls as it gets thicker.
A wall that is not quite there
A porous surface has structure on every scale below the pore, and no calculation resolves it. The whole of it can be replaced by one length — the square root of the permeability — and the replacement is exact to first order, with what it leaves out identifiable as the flow the wall itself carries.
Nothing but the edge
Cut an aerofoil into panels, put a singularity on each, and require the surface to be a wall. The system that comes out has one more unknown than it has equations, and the row that is missing is not a bookkeeping slip — it is the fact that ideal flow round a closed body has no unique answer at all.
The hole that halves the flow
A jet leaving a sharp-edged hole is narrower than the hole, and by an amount that is not measured but computed. One geometry gives exactly one half from momentum alone; another gives exactly π/(π+2) from a conformal map in which the shape of the free surface is part of the answer.
Where a fluid stops being one
The Knudsen number is the mean free path over the size of the thing, and at one a molecule crosses the whole channel between collisions. The continuum equations with a no-slip wall are already one per cent wrong at one part in five hundred and ninety-four, which is a factor nothing about the definition would suggest.
Where the reaction to a wing's lift is
The force on a body can be computed on any contour drawn round it, and the answer is the same every time. How much of that answer is pressure and how much is momentum flux is not — it runs from three per cent to ninety-seven, and the difference is the shape of the contour.
A body with no lift, and a moment anyway
A fuselage in ideal flow carries no lift at any incidence and still tries to turn the aeroplane over. The couple is computable in one line, it is why tails are the size they are, and the line comes from applying the wall condition to a place where there is no wall.
The layer that stops at a depth
Every other boundary layer grows. This one does not — rotation supplies a frequency, the balance against diffusion supplies a length, and the transport that comes out contains the stress on the surface and not the viscosity underneath it.
The wall that heats itself
A surface told nothing about its temperature does not settle at the air's. It settles most of the way to the stagnation temperature, and the heat flux is driven from that invented temperature rather than from the free stream's — so a wall hotter than the air can be being heated by it.
How far before a duct forgets what was fed into it
A pipe is always drawn with its answer already in place. Getting there takes a distance proportional to the Reynolds number, which means a more viscous fluid is done sooner — and the entrance costs a fixed number of dynamic pressures however long the pipe is.
The sound that only leaves
A flow is a catastrophically bad radiator, and the reason is that it has no monopole and no dipole available to it. What is left is the eighth power of speed — and the equation is equally happy with sound converging on a jet, which is ruled out by a condition imposed at infinity.
Every flow is two flows
Any velocity field splits into a part carrying all of the divergence and a part carrying all of the vorticity. The theorem says so and does not say which split — the two halves can be moved between each other by anything harmonic, and on a bounded region that is an infinite family.
What a code says to a wall
A calculation that cannot afford to resolve the viscous sublayer has to tell the wall something else instead, and what it tells it is the law of the wall — an asymptotic result, applied at one grid point, on the assumption that the point lies in a region the calculation has not checked exists. Where it does, the answer is exact. Where it does not, the friction is out by tens of per cent, and refining the grid makes it worse.
The flow with the least energy in it
Draw a flow that conserves mass and does not go through the walls, and it will look exactly like a solution. There are infinitely many of them and one is the flow. What separates it from the others is not visible anywhere in the picture — it is a number, and the number is an energy.
The mirror that is a circle
A flat wall is made by reflecting everything in it. A round one is made the same way, except that the mirror is an inversion — the image of a point at distance d sits at a²/d, and a vortex acquires a second image at the centre that nothing about the wall requires.
Drag in the theory that forbids it
d'Alembert's paradox is a theorem about flows that close behind the body. Stop requiring that, let two streamlines leave the edges and never come back, and the same equations — no viscosity, no vorticity — produce a drag coefficient of 0.8798.
The cheapest shape the walls allow
The parabola in a pipe is usually presented as what the equations give. It is better understood the other way round — of every profile that could carry that flow between those walls, it is the one that destroys the least energy, and the equations give it for that reason.
Where the unknown boundary is the known one
A free surface is the hardest kind of boundary — its shape is part of the answer, so the region the problem is posed in is not known until the problem is solved. Draw the same flow in the plane of its own velocity and the shape becomes an arc of a circle, known in advance and exactly.
A force without the flow that makes it
A sphere carried along by a flow does not travel at the speed of the fluid at its centre. It travels at the average of the flow over its own surface — and getting that result needs no solution of the flow around the sphere at all, only the answer to a completely different problem that everybody already knows.
The one number that runs out at three dimensions
A stream function is one function where the velocity is two, and four essays here are built on it. It exists because the divergence vanishes, it is single-valued only if nothing inside is making fluid, and in three dimensions it is not one function at all.
The threshold the walls decide
A layer heated from below convects at a Rayleigh number of 1707.762, and nothing whatever happens at one. The free–free case has a closed form and the two that do not differ from it by a factor of 2.6 — produced by nothing but what the top and bottom surfaces are permitted to do.
The one rotational solution anybody can write down
A sphere of spinning fluid travelling steadily through fluid at rest, with no body anywhere in it — the boundary is a streamline and nothing else. It is exact, it is two lines long, and the reason it is the famous one turns out to be the reason it is the only one a real fluid can settle into.
A wake that closes on itself
Prandtl's best wing system is a rectangle, not a wing. Solving for the loading on a closed wake turns up a circulation that costs nothing and does nothing — a gauge freedom in the middle of an optimisation — and a drag that keeps falling with no floor under it.
The walls are in the answer
A wind tunnel measures a wing in a box the aeroplane will never fly in, and the box is worth a fifteenth of the induced drag. The correction is exactly an eighth for a closed circular section and exactly minus an eighth for an open jet, and the sign is the whole argument.
The surface that moves with the flow
A clean gas bubble feels two-thirds of the drag a rigid sphere of the same size would, and the formula has no density in it anywhere. What buys the third is that the bubble's surface is free to move — and real bubbles in ordinary water do not get it, for a reason that is a millionth of a per cent of the water by mass.
The drift a closed box will not allow
A wave in a wave tank carries mass forward, and the tank has nowhere to put it. So a return current appears carrying exactly the opposite transport — exactly, from mass conservation and nothing else. Which fixes a total and leaves the answer anybody wants entirely open.
The section that decides the river
A reach of open channel has a normal depth and a critical depth, and the water has neither. What it has is a profile obeying a first-order equation, which needs exactly one condition — and whether that condition belongs at the upstream end or the downstream end is not a choice, because the equation is stable in one direction and unstable in the other.
A swimmer that cannot go backwards
Taylor's waving sheet is the simplest self-propelled object in a viscous fluid, and its arithmetic contains a result that reads like a mistake — the work it does to travel a metre does not depend on how big its waves are. Doubling the amplitude quadruples both the speed and the power, and changes the bill for the journey by nothing at all.
The flow with no solution
Creeping flow past a sphere has a solution and everybody knows it. Creeping flow past a cylinder has none — not a difficult one, not one needing a clever method. The equations, the no-slip condition and the uniform stream at infinity are inconsistent, and the residual is exactly twice the free stream.
The corners that can be done with mirrors
The method of images works for a wall and for a circle, and for a corner it works only when the angle is pi over a whole number. At every other angle the reflections never come back, the image set is infinite and dense, and the flow exists anyway — which says the method is a statement about symmetry rather than about fluids.
The borrowed mass the boundary decides
A body accelerating near a solid wall has to squeeze out the fluid between them, and borrows more mass than it would in the open. The same body accelerating near a free surface, or inside an open-jet wind tunnel, borrows less. The fluid, the body and the speed are identical, and what reverses the answer is the one thing each boundary is allowed to tell the flow.
The skin that lags the flight
A wall can be told its temperature or told nothing, and both are solved problems. A real skin is told neither. It has heat capacity, so its temperature is a transient whose time constant is its own thickness divided by what the layer delivers — and the number the steady calculation returns is an upper bound a short exposure never collects.
How much more than the least
Of all the flows that conserve mass and stay inside the walls, the ideal one carries the least energy. That is a theorem, and the useful half of it is the part nobody quotes: it says by exactly how much every other flow misses, and the answer is the square of how wrong it looks.
The thermometer that heats itself
A total-temperature probe is a wall told nothing about its temperature, made small and put on a stalk, so it obeys the same recovery arithmetic an aircraft skin does. Its recovery factor is a calibration constant near 0.98 rather than a one — and the static temperature inferred from its reading amplifies that shortfall rather than inheriting it.
The vorticity a clean surface cannot refuse
A clean bubble's surface cannot hold a shear stress, and it is easy to conclude that it makes no vorticity. On a curved surface it must carry exactly 2κu — three times the speed over the radius at a sphere's equator, whatever the Reynolds number. That is so much weaker than a rigid wall's that the flow stays irrotational to leading order, and the bubble's drag is the dissipation of that irrotational flow: 48/Re, three to ten times below a rigid sphere's.
The wall the fluid is listening to
Water two millimetres above a moving wall is responding to what the wall did two thirds of a second ago — most likely. Half of its response is older than four and a half seconds, a tenth is older than two minutes, and the average age of what it is responding to does not exist at all.
Twice as slippery along as across
A surface of alternating gas and solid stripes lets a liquid slip, and a flow far above it sees one number in place of the pattern — but the number depends on which way the flow goes. Along the stripes it is Philip's logarithm; across them it is exactly half, for a reason that takes one substitution to show. And the logarithm means that the slip is bought by the pattern's period rather than by how much of it is gas.
The lowest pressure is on the body
In an ideal flow the minimum pressure is always on a surface — not usually, not for the shapes people draw, always. The proof is an identity about the velocity gradient, and the identity says exactly which flows are exempt.
A wall puts in exactly its own speed
A wall sliding in its own plane makes vorticity at a rate equal to its acceleration, with no viscosity in the rate. So however a wall is started, the vorticity it has put into the fluid is its speed, to the last digit; a wall that stops takes all of it back and leaves the fluid moving; and a spinning cylinder carries its whole circulation from the first instant, hidden behind an equal and opposite ring until viscosity carries the ring away.
The inside a flow does not decide
Every body on this site is built out of singularities that are not there. The exterior flow does not merely fail to determine them — it leaves an infinite family, whose members produce the identical field to the last bit outside and are nothing alike inside, and whose coefficients span five orders of magnitude for one unit free stream.
Slip is a memory of one mean free path
A molecule arriving at a wall last collided about a mean free path away and carries the velocity from there. Averaged over arrivals and departures, that leaves the gas at the wall moving — by two per cent of the centreline speed at a Knudsen number of a hundredth, and sixty per cent more flow through a microchannel at a tenth.
The drift a rotating planet takes back
In a wave tank the Stokes drift is cancelled by a return current because the tank has walls. The open ocean has none, and the drift is cancelled anyway: the Coriolis force acts on the water's real motion, drives an Eulerian current that answers it, and leaves the depth-integrated transport exactly zero at every viscosity. A float under steady swell with nothing to stop it goes round a circle instead of away.
A duct that forgets everything but one number
Whatever is fed into a pipe, what survives a little way down it is one shape. The disturbance's higher modes decay as the square of their mode number, so the sixth is gone in thirteen centimetres where the first survives four and a half metres — and the entrance length is that one mode's decay rate and an arbitrary threshold.
The drop a no-slip wall would never let spread
Liquid touching a solid moves with it, and nearly everywhere that is as close to exact as anything in fluid mechanics. At the edge of a spreading drop it cannot be: the stress in the corner rises as one over the distance from the edge, and the force needed to move the edge is infinite. Something slips over a nanometre, and because the answer depends on that length only through its logarithm, a drop spreads at almost the same rate whatever the something is.
One channel, one flux, two flows
Flow between two plane walls meeting at a line has an exact solution. Past a threshold that turns out to be a ratio of gamma functions, it has two — the same wedge carrying the same flux, once outward everywhere and once with the fluid running backwards along both walls, and nothing in the equations chooses.
Nothing in the present picks the flow
Six flows past one cylinder satisfy the same equation and let nothing through the wall, to the last bit of double precision. Their lifts run from zero to 37.7 and their peak suctions differ by a factor of twenty-one. The equations do not choose between them, and the thing that does is the history.
Why the list is this long
Every textbook list of exact solutions of the Navier–Stokes equations is about a dozen long, and the usual explanation is that the equations are hard. It is not the reason. A similarity reduction is a solution invariant under a subgroup of the equations' own symmetries, so the catalogue of possible reductions is the catalogue of subgroups — and that is a finite, countable object.
The floor that gives the drift back
In the open ocean the Coriolis force drives a current that cancels a swell's Stokes transport exactly. Over a continental shelf the sea floor holds a stress, and whatever it holds is transport the rotation does not take back. How much survives depends almost only on the depth in Ekman depths; which way it points depends on the wave.
Named alongside it
The objects these essays reach for when they reach for this one.
Potential flowMeasurementCirculationModel limitViscosityBoundary layerLaplace's equationModel validityThe no-slip conditionTransportVorticityConformal map