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

Minimum-dissipation — where it appears

Helmholtz's theorem that a slow viscous flow is, of all flows meeting the same boundary conditions, the one that dissipates least energy. It turns every guessed velocity into a bound, and its stress-based twin supplies the bound from the other side.

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

The parabola is the cheapest shape the walls allow. The dissipation of the profile u = A(1 − |y/h|ⁿ) carrying a fixed flux between fixed walls, against the exponent n, divided by the parabola's. Every member of the family satisfies the same boundary conditions and carries the same fluid; they differ only in shape. The minimum is at n = 2 exactly, which is not a coincidence — it is Helmholtz's theorem, and the parabola is a solution of the equations because it is the least dissipative shape rather than the other way round.

The cheapest shape the walls allow

The parabola in a pipe is usually presented as what the equations give. It is better understood the other way round — of every profile that could carry that flow between those walls, it is the one that destroys the least energy, and the equations give it for that reason.

viscous · Minimum-dissipation Every velocity guesses low and every stress guesses high. Two bounds on the flow rate of a duct, in units of G a⁴/μ, against the number of mesh cells per half-side: from above, the best stress field in equilibrium with the pressure gradient (red); from below, the best velocity that vanishes on the walls (green). For the square the series value 0.562308 (grey) lies between them at every mesh. For the L-shaped duct, which has no closed form, they close on 0.21399 to 0.21415.

A flow pinned between two guesses

The minimum-dissipation principle says the true flow is the cheapest one the walls allow, so any guessed velocity carries too little. It has a twin that nobody teaches: any guessed stress in balance with the pressure carries too much, and it needs no wall condition at all. Between the two, the flow through a duct with no formula is pinned down to as many figures as anyone wants.

viscous · Minimum-dissipation

Named alongside it

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

Boundary conditionDissipationOptimisationPoiseuille flowStokes flowVariational principleViscosityIrreversibilityMeasurementModel limitTurbulence

All concepts