Concept

Logarithm — where it appears

The function that appears in an expansion when a dimensional argument produces an exponent of exactly zero and something has to fill the gap. A quantity governed by one changes by a constant amount per decade, so it neither converges nor diverges at any useful rate.

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

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

Named alongside it

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

Creeping flow Matched asymptotics Model limit Asymptotics Boundary condition Convergence Dimensionless Drag Measurement Oseen Regime Reynolds number

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