At speed, resistance is not friction — it is the fluid you must shove aside, and shoving it twice as fast costs four times as much. F = ½ρv²CdA: the drag grows with the square of velocity, the wall every fast body eventually hits. Down the center, the numbers flow: fluid, speed and shape go in, the engine computes the force, the proven quadratic law comes out. The blue team builds and defends it; the red team tries to break it.
source Lord Rayleigh (J. W. Strutt), "On the Resistance of Fluids," Philosophical Magazine, Ser. 5, Vol. 2, pp. 430–441, 1876 — the dimensional argument behind the drag coefficient; no stable open DOI, see Scientific Papers, Vol. 1 (archive.org). Rendered, not quoted.
At high Reynolds number the drag is inertial: the body sweeps a column of fluid and hands it kinetic energy. Per unit time it accelerates a mass ∝ ρvA to a speed ∝ v, so force ∝ ρv²A. Collect the geometry and wake into one dimensionless number:
F = ½ · ρ (density) · v² (speed, squared) · Cd (shape) · A (frontal area).
The three exact dependences the engine holds to:
| vary | F multiplies by | law |
|---|
Resistance squares with speed — but only when speed is high. This sphere is the quadratic complement of the-stokes-drag, whose 6πμrv is linear in v and rules the creeping, low-the-reynolds-number world.
The crossover is Reynolds itself: Fhi/Flo = (Cd/12)·Re. Linear drag wins the small and slow; quadratic drag wins the fast — and it is the quadratic law that fixes a fast the-terminal-velocity at vt = √(2mg / ρCdA). Each sphere is the next one's premise.
The blue team's live check: re-run the pure drag functions and confirm the quadratic law, the linearities, the Cd definition and the Stokes crossover against known truth. If red tampers, this badge is where it shows.
Four measurable inputs, each with real units:
| symbol | is | unit |
|---|---|---|
| ρ | fluid density | kg·m⁻³ |
| v | relative speed | m·s⁻¹ |
| A | frontal area | m² |
| Cd | shape coefficient | — (dimensionless) |
Cd carries no units — it is exactly what is left when force is stripped of ½ρv²A. That is what you feed the panel below.
Cd is marked AMBER — a real sphere's Cd drifts with Re (the drag crisis near Re≈3×10⁵). Held fixed here for one regime.
Move any control — the force, the Reynolds number and the crossover are computed from ½ρv²CdA on the spot, never looked up.
What the machine proves, live: drag is quadratic in v (double the speed → ×4 the force, exact to 1e-12), linear in ρ and in A, its Cd is dimensionless, and it is the high-Re complement of linear Stokes drag — with the crossover verified in both regimes and an RK4 approach to terminal velocity matched to the tanh solution within 1e-9.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
"Frontal area A" is also a choice — reference area for a wing is planform, not frontal, so a lift/drag figure is meaningless without its stated A. The equation is a bookkeeping frame, not the aerodynamics.
"Drag is proportional to velocity." Cut. Only at low Re (Stokes). At high Re drag ∝ v² — both regimes live in the machine, and the crossover is Reynolds itself.
"Cd is a constant of the shape." Cut. A sphere's Cd falls from ~0.47 to ~0.1 at the drag crisis (Re≈3×10⁵). Held fixed per regime and marked AMBER.
"Newton derived F ∝ v²." Kept, corrected. Newton gave the inertial impact argument; the coefficient form and the ½ρv²CdA normalization trace to Rayleigh's dimensional analysis (1876).
The red team's move: linearize the drag — make F ∝ v instead of v² (secretly collapse it to the wrong low-Re form). The blue team's witness (window 7) is watching.
Drop one power of v and the quadratic F(2v)/F(v)=4 check collapses to 2 — the witness recomputes, disagrees with the known law, and turns red. Nothing is faked; the attack is real and it is caught.