Dalembertia

Dalembertia, a mathematician-wraith of the Paradox Parlour, wraps a perfect fluid around a sphere, and the sphere feels no drag at all.

The Physics

In 1752 Jean le Rond d'Alembert showed that for steady, incompressible, inviscid and irrotational (potential) flow, a body moving at constant speed feels zero drag. The streamlines and the pressure are symmetric front to back, so the pressure pushes on the rear as hard as on the front. Yet every real body in air or water feels drag.

φ = U (r + a³/2r²) cos θ ⇒ F_drag = 0

• φ: velocity potential of steady flow past a sphere (m²/s)
• U: speed of the uniform flow far away (m/s)
• a: sphere radius (m)
• r, θ: distance from the centre and angle from the flow direction
• F_drag: net force along the flow (N)

The paradox was resolved by Ludwig Prandtl's boundary-layer theory (1904). However small the viscosity, a thin layer at the surface must slow to zero speed; on the rear side it separates and leaves a low-pressure wake, which causes drag. Drag does not go to zero as viscosity goes to zero, so the inviscid model fails exactly where it seems safest.

Honest caveat: The zero applies only to steady motion. An accelerating sphere in an ideal fluid does feel a force, as if it had extra mass (the added mass, half the mass of the displaced fluid for a sphere). Ideal-flow theory also gives lift correctly once circulation is included; only the drag fails.

On the sticker

Dalembertia holds a cyan sphere with streamlines wrapped symmetrically round it. Printed text: DALEMBERTIA with the line DRAG = 0 ?. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.