Concept

Multiply-connected — where it appears

A region in which some closed curve cannot be shrunk to a point without leaving it — around an aerofoil, say, or around a hole. Circulation on such a curve is a free constant the equations do not fix, which is why lift needs a condition beyond the flow equations.

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

Five flows past one cylinder, every one of them a solution. Surface pressure round a cylinder in a stream, at five circulations. Each satisfies Laplace's equation, the tangency condition on the body and the condition at infinity, and each has a different lift. Nothing in the problem chooses between them: the domain has a hole in it, so the potential is many-valued and the circulation is a free constant.

The constant a hole leaves behind

In a region without holes, Laplace's equation and the boundary values have exactly one solution. Cut a hole and they have a one-parameter family. Nothing in the mathematics chooses between its members, which is why the Kutta condition has to exist and why it cannot be derived.

inviscid · Multiply-connected

Six flows past one cylinder, all of them legal. The tangential speed on the surface for six values of the circulation. Every one of them solves the same equation and lets nothing through the wall; the fastest point on the surface runs from twice the free stream to eight times it.

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.

inviscid · Multiply-connected

Named alongside it

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

Circulation Kutta condition Lift Uniqueness Boundary condition d'Alembert's paradox Image system Initial condition Kelvin minimum energy Kelvin's circulation theorem Kinetic energy Memory kernel

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