Free-surface shape. In a steady axisymmetric swirl the radial Euler equation balances pressure gradient and centripetal acceleration; at the free surface p is atmospheric, so
dz/dr = vθ(r)² / (g r) z(r) = -(1/g) ∫r∞ vθ²/r' dr'
Rankine vortex: rigid core vθ = Γ r /(2π rc²) for r < rc, free vortex vθ = Γ/(2π r) outside. Integrating:
r > r_c : z = -Γ² /(8π² g r²)
r < r_c : z = ω² r²/(2g) - Γ²/(4π² g r_c²), ω = Γ/(2π r_c²)
depth z(0) = -Γ²/(4π² g r_c²) = -v_c²/g, v_c = Γ/(2π r_c)
The inside is the "rotating bucket" paraboloid; the two branches meet continuously at rc (z = -Γ²/(8π²g rc²), half the central depth), and the outer dip falls as 1/r².
Burgers vortex (radial inflow ur = -a r/2 with axial stretching uz = a z balancing viscous diffusion): vθ = Γ/(2π r)(1 - e-r²/r_c²), rc = 2√(ν/a). Its funnel is shallower than Rankine's because the core is smooth: z(0) = -(Γ² ln2)/(4π² g rc²), i.e. 0.693 of the Rankine value (checked numerically in Experiment 1). For the draining tank the strain is estimated from continuity as a = Q/(πR²h).
Drain. Torricelli: v = √(2gh), Q = Cd A √(2gh). The level obeys dh/dt = -(CdA/At)√(2gh), so t = (At/(CdA))√(2/g)(√h0 - √h) and the emptying time is (At/(CdA))√(2h0/g). Outside the core the converging flow conserves angular momentum, r vθ = Γ/2π = const, so the swirl speeds up as 1/r.
Kelvin's theorem. For an inviscid, barotropic flow the circulation Γ = ∮u·dl around a material loop is constant. The yellow loop of fluid points is advected through the converging flow; the graph shows Γ(t)/Γ(0), which stays at 1 while the loop remains outside the viscous core.
Coriolis myth. The Earth's rotation gives the planetary vorticity f = 2Ω sin(lat). The Rossby number Ro = U/(f L) compares inertia with rotation: for a bathtub (U ~ 0.1 m/s, L ~ 0.3 m) Ro is of order 103, so Coriolis is negligible against the residual stirring; the sense of a drain vortex is set by how the water was left moving. Only in carefully still, large, symmetric tanks (Ro < 1) has the hemisphere effect been demonstrated. The table in the readouts shows f, Ro and the Coriolis circulation fπR² against the actual Γ.
Ocean whirlpools. Tidal whirlpools (e.g. in Saltstraumen, Norway, or the Corryvreckan, Scotland; reported peak currents of order 10 m/s) are eddies shed where a strong tidal current is forced through a narrow channel over uneven topography. They are not drains; the model here reuses the same swirl profile with an illustrative converging flow, and the vertical exaggeration slider is needed because real surface dips are small compared with the eddy size (z(0) = vc²/g).
Simplifications. Steady, axisymmetric velocity field; the surface is a height field (no breaking, no air core geometry: the surface is clamped to the floor and flagged "air core"); water treated as incompressible with an effective viscosity; the refraction of the floor is approximated by absorption-based transparency; tracers and the boat are inertial particles (relaxation time τ) so they drift outward like heavy objects; the water surface is not a full Navier-Stokes solution. Burgers: J. M. Burgers (1948); Rankine vortex: W. J. M. Rankine (1858); Lamb-Oseen: Lamb (1932). Not a safety guide: never approach real whirlpools.