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

Stokes' drag — where it appears

The exact drag on a sphere in creeping flow, six pi times the viscosity, the radius and the speed. It is linear in the velocity rather than quadratic, and its exactness makes it the reference against which every small-particle calculation is checked.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

Particles at St = 1, against the flow that carries them. Particle paths and the streamlines they were released on, in this site's exact cylinder solution. At small Stokes number the two are indistinguishable and the body catches nothing; as the particles get heavier their paths straighten, cross the streamlines, and begin to strike. The paths are integrated with Stokes drag and nothing else — no gravity, no lift, no effect of the particles on the flow.

Whether the droplet turns

The air goes round the wing. Whether what is carried in it goes round too is decided by one number — and below a critical value of that number the body collects nothing at all, however many droplets are thrown at it, because the flow turns every one of them in time.

regimes · Particle 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 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 here: nothing above resolves 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 exact solution and its three approximations, at ε = 0.02. The outer solution is excellent everywhere except in a layer of width ε at the left, where it is wrong by a whole unit. The inner solution is excellent inside that layer and wrong everywhere else. The composite is their sum less the part they agree about, and it is within order ε of the exact solution across the whole interval — which is the entire content of matched asymptotics, drawn.

One formula for both ends

Two limits, each with its own description, neither valid everywhere. The composite is the sum less the part they agree about, and it is uniformly good — but the overlap region that justifies the construction does not exist at ε = 0.01, and the composite is still accurate to two per cent there.

regimes · Crossover Posts slip as one over the root of their fraction. The slip length of a surface of posts, over the period of their square array, against the fraction of the surface that is solid, beside Philip's stripes along and across and the dilute limit in which each post is a lone disc dragged edgewise, (3/16)√(π/φ). At a hundredth solid the posts slip 2.87 periods against the stripes' 1.32; at a tenth 0.597 against 0.591, nearly equal; at a half 0.0663 against 0.11 along the stripes and 0.0552 across them.

A post holds liquid back by its radius, a stripe by its length

A surface of no-slip posts standing in a gas-filled texture lets liquid slip over it, and the slip length grows as one over the square root of the solid fraction, where a surface of stripes grows only as its logarithm. The reason is the oldest contrast in slow flow: a disc dragged through a viscous liquid has a drag proportional to its radius, which is Stokes' law, and a line has no finite drag at all, which is Stokes' paradox. But sparse posts beat stripes only while they are sparse.

kinematics · Boundary conditions Two spheres settle faster together than apart, faster in line than side by side, and still faster at thirty radii than either alone. The settling speed of a pair of equal spheres over a single sphere's, against the distance between their centres in radii: along the line of centres, exactly (the bispherical solution) and by Batchelor's far-field series; across it, by the series; dashed, each sphere's Stokeslet alone, 1 + 3/(2ρ) and 1 + 3/(4ρ). Touching in line the pair falls 1.55 times as fast; at 5 radii, 1.286 in line and 1.154 side by side; at 30, 1.05 and 1.025.

Two spheres settling at a slant keep their slant for ever, and three do not

Two equal spheres settling through a viscous liquid fall faster together than either alone — half as fast again if they touch in line, and still five per cent faster thirty radii apart. Slanted, the pair drifts sideways as it falls, by up to three and a half degrees, and keeps the slant, the separation and the drift for as long as it falls: the two spheres always move with exactly the same velocity. Release a third and the shape is lost at once, the middle sphere drawing ahead and the outer two closing in behind it. What a pair keeps and a trio does not is a statement about symmetry, and reciprocity is half of it.

viscous · Reciprocity

Named alongside it

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

Creeping flowThresholdAsymptoticsConvergenceDimensionlessInertiaMatched asymptoticsModel limitParticle-trackingScalingStokes flowAveraging

All concepts