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

Oseen — where it appears

The linearisation of the inertial term about the uniform stream, used far from a body where the disturbance is small. It repairs the far field that creeping-flow theory misdescribes and produces a drag whose leading term is a logarithm of the Reynolds number rather than a power of it.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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

Named alongside it

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

Creeping flowMatched asymptoticsReynolds numberAsymptoticsBoundary conditionDimensionlessDragInertiaLogarithmModel limitParticle trackingSingular perturbation

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