Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Creeping flow — where it appears

The limit in which inertia is entirely negligible, leaving a linear and exactly reversible equation. A flow in it undoes its own stirring when run backwards, and a body in it feels a drag proportional to its speed rather than to its square.

Named by 16 essays across 5 fields — each of them below, with the objects they name alongside it.

Reynolds number: one number, four different flows. Reynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.

One number decides which physics applies

A bacterium and a whale both swim, and they are not doing the same thing at different sizes. The ratio of inertia to viscosity separates them, and crossing it changes the rules rather than the magnitudes.

regimes · Reynolds Creeping flow, and the same body with inertia. The exact creeping-flow solution beside a solved field at a Reynolds number where inertia matters. The creeping flow is a mirror image of itself front to back — a photograph of it run backwards is a photograph of it — and the field with inertia has a wake, which is what a direction of time looks like.

The world with no inertia

Drop the viscosity and the equations become exactly solvable and wrong about drag. Drop the inertia instead and they become exactly solvable again — and for a cylinder in an unbounded fluid there is no solution at all, which took fifty years to notice and longer to fix.

regimes · Reynolds 20.45 MPa out of a film 25 µm thick. The pressure along a tapered pad, from the closed-form solution of Reynolds' equation, with the same equation's tridiagonal grid solve drawn over it as points. The peak is 20.45 MPa — enough to yield mild steel — and it sits at 69 per cent of the way along rather than in the middle, because the pressure gradient vanishes where the film equals the harmonic mean of its two ends and the harmonic mean is biased towards the thinner one. The pad's own shape is drawn along the top, to a vertical scale of its own. Nothing pumps this oil: the runner drags it into a narrowing gap and the gap does the rest.

Nothing but the shape of the gap

A machine that holds a steel shaft off its bearing with a film of oil twenty-five microns thick has no pump in it, and the pressure it generates would yield mild steel. The mechanism is not the oil and not the speed; it is that the gap narrows.

viscous · Lubrication 45° of corner, and an infinite number of eddies in it. The creeping flow in a corner of 45 degrees, drawn from Moffatt's similarity solution. Each eddy turns the opposite way to its neighbours and is 3.17 times smaller and 1.6e+3 times weaker than the one outside it. The contour levels are rescaled inside each eddy, because they differ in strength by three orders of magnitude per step and a single set of levels would show the first and nothing else — which is itself the reason nobody has seen the third. The dividing lines between eddies are drawn where the stream function changes sign, and the dots are the centres, both found from the solved field.

The eddies nobody stirs

A slow flow past the mouth of a sharp corner does not simply fail to get in. The corner fills with an infinite sequence of counter-rotating eddies, each about three times smaller and sixteen hundred times weaker than the last, and the whole structure is decided by one complex number.

viscous · Corner How small is small enough. The error in Stokes' law for the drag on a sphere, against the Reynolds number, both logarithmic. Oseen's correction is the first term the neglected inertia puts back, and it says the error is 3Re/16: one per cent at Re = 16/297 = 0.054, five per cent at 0.28, and already sixteen per cent at Re = 1 — which is the value at which the two terms the Reynolds number compares are equal, and is where every textbook draws the boundary of creeping flow.

How small is small enough

Creeping flow drops the inertia terms and gets an exact answer for the drag on a sphere. The Reynolds number at which those terms are equal is one — and at one, Stokes' law is already sixteen per cent low. The honest boundary is 0.054, and the reason it is so far down is the same reason the theory needed repairing in the first place.

regimes · Reynolds How far downstream the heat is still being made. The dissipation accumulated from a station ahead of a cylinder to a station behind it, as a fraction of the whole of what is made inside the frame, at five Reynolds numbers. At Reynolds number 1 the fluid has finished paying by about a diameter behind the body. At 100 it has not finished at five, and the curve is still climbing at the edge of the picture — the drag is a force on the body, and the heat it stands for is somewhere else.

Where the heat of a drag is made

The power it takes to tow a body through a fluid becomes heat, all of it, eventually. None of the interesting words in that sentence are the first four. It is the "eventually" that decides how a wake behaves, how far a disturbance reaches, and why no box drawn round a body contains its own bill.

viscous · Dissipation A sphere does not go at the speed of the flow it is in. How far a force-free sphere on the axis of a round tube lags the fluid at its own centre, against its size. The lag is two-thirds of the square of the size ratio and comes from the reciprocal theorem in one line: the sphere moves at the average of the ambient flow over its own surface, and a parabolic profile is slower everywhere on that surface than at the middle. At a fifth of the tube's radius the lag is 2.7 per cent, which is the difference between a tracer and a thing being measured.

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.

viscous · Reciprocity Two velocities, 2.50 apart, and only one of them is anybody's. The velocity in Darcy's law is the flow rate divided by the whole cross-section, solid included: a speed no fluid particle ever has, since the fluid occupies only the fraction ε of that area. The speed the fluid actually averages is larger by exactly 1/ε — 2.50 times here — and it is the one that belongs in a residence time, in a pore Reynolds number and in any statement about when a tracer arrives. The grains are drawn to say that the pore-scale flow is not computed anywhere: no figure on this site claims to resolve it.

A velocity nobody has

The velocity in Darcy's law is the flow rate divided by the whole cross-section, solid included — a speed no fluid particle in the bed ever has. Averaging buys a linear law and charges for it in exactly this coin, and the constant that comes with it is an area of about a square micron.

applied · Porous The same flow, from two different physics. Solid lines: the depth-averaged flow between two plates a small distance apart, with an obstacle standing between them. Dashed: the ideal-flow solution for the same obstacle. They are the same field to a part in ten billion, because averaging Stokes flow across a narrow gap gives a velocity that is the gradient of a harmonic potential. The cell has no inertia at all, which is the one hypothesis the ideal theory cannot do without.

The exact theory, drawn by viscosity

Two flat plates a millimetre apart, syrup between them, an obstacle in the gap. The Reynolds number is a hundredth, inertia is absent, and the streamline pattern is the potential flow past that obstacle — exactly, to a part in ten billion. The one hypothesis ideal flow cannot do without is the one this flow most conspicuously breaks.

inviscid · Hele shaw Where Einstein's line stops being the measurement. The viscosity of a suspension of rigid spheres relative to the liquid's, against the volume fraction. Einstein's 1 + 5φ/2 is exact for one sphere and holds while the spheres cannot feel one another, which is up to about five per cent by volume. Batchelor and Green's two-sphere term takes it a little further; beyond about a fifth nothing derived works and the curve drawn is a fit, which diverges at a maximum packing that is itself a measurement.

A viscosity made of particles

Stir rigid spheres into a liquid and the mixture is thicker, by five halves of the volume fraction. Einstein's coefficient is not an empirical constant — it is a dissipation calculation on one sphere — and doing it as an energy rather than as a stress shows that four-fifths of it comes from somewhere nobody mentions.

viscous · Suspension Two-thirds to one, and nothing outside it. The drag on a spherical drop as a fraction of Stokes' drag on a rigid sphere of the same size, against the ratio of the viscosities. It runs from exactly two-thirds for a clean gas bubble to exactly one for a rigid particle and takes no value outside that range, however light or heavy the drop is — the formula has no density in it anywhere. A drop of water in air and a drop of mercury in water sit at opposite ends of it.

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.

viscous · Mobile interface Drop deformation in simple shear, against the capillary number. The shape model's steady deformation for six viscosity ratios. Every curve is linear in the capillary number at small Ca — which is Taylor's result — and every one of them saturates, at 5/2(2λ+3), because the shear's own rotation turns the drop out of the stretching direction. A drop in simple shear cannot be deformed beyond that however hard it is sheared.

The number that cannot break a drop

The capillary number sets the stress that stretches a drop against the stress that holds it round, and it predicts the deformation beautifully. It cannot predict the breakup, because above a viscosity ratio of about four a drop in simple shear cannot be broken at any shear rate — and the theory that says the ratio hardly matters is the same theory that gets the deformation right.

regimes · Capillary A swimmer that dissipates the same everywhere. Taylor's waving sheet, with the wave drawn along the bottom and the dissipation drawn against height above it. The dissipation function works out at 4μb²c²k⁶y²e^{−2ky} — with no x in it at all, so the sheet is destroying energy at the same rate under every part of the wave and at every instant of the cycle. It peaks one radian of wavelength above the sheet and is gone within about three.

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.

viscous · Swimming The only candidate the far field allows, and the wall it slips past. The general Stokes solution has four constants; the condition at infinity kills two of them and fixes a third, leaving one to satisfy two conditions at the wall. Setting the stream function to zero there uses it up, and the tangential velocity that remains is exactly twice the free stream — for every radius, every speed, and every fluid.

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.

viscous · Stokes' paradox And they are one piece of arithmetic. All three corrections against one over the logarithm of their own large parameter. Each is a straight line through the origin, with its own slope: one third exactly for the enstrophy range, near one for the wall layer, and near ln 10 for Oseen's per-decade measure. Three disappointments in three different fields, written up separately, are the same function.

The three that never converge

Oseen's drag coefficient, the overlap layer's power-law exponent and the enstrophy range's slope are three separate disappointments in three different fields. They are one piece of arithmetic, and the arithmetic says why none of them will ever be reached.

regimes · Crossover The best efficiency runs from nine-eighths of φ² to one. The best efficiency a peristaltic pump can reach, against the fraction of the channel its wave closes, with its two limits. For a shallow wave it is 9φ²/8, which is small — a wave closing a fifth of the channel is at best 4.5 per cent efficient. As the wave closes the channel the best efficiency tends to one, and its shortfall shrinks in proportion to the remaining gap: about 1.9(1 − φ). Nothing in between is independent of the amplitude.

The pump that is better the more it squeezes

A waving sheet swims at a cost per metre with no amplitude in it. A waving wall pumping fluid is the same mechanism turned round, and its efficiency is nothing like that: it starts at nine-eighths of the amplitude ratio squared, is exactly 2 − √3 at half closure, and rises towards one as the wave closes the tube — where the pump stops being a wave and becomes a piston.

kinematics · Continuity

Named alongside it

The objects these essays reach for when they reach for this one.

Reynolds numberStokes flowDissipationDragModel limitBoundary conditionViscosityAsymptoticsDimensionlessMatched asymptoticsThe no-slip conditionRegime

All concepts