Navier-Stokes Dynamic Balance in Cylindrical Coordinates
Real-time evaluation at probe radius r = 6.0 cm
As fluid elements orbit the drain with tangential velocity u_θ, their inertia creates an apparent outward centrifugal acceleration. For a steady vortex, this outward push must be precisely countered by the inward pressure gradient directed toward the low-pressure drain eye.
Radial Velocity & Surface Funnel Distribution Profiles
Cross-section across radius r from drain center (0 to 45cm)
Hydrodynamics of Bathtub & Drain Vortices
1. Conservation of Angular Momentum
As fluid parcels are sucked toward the center drain aperture, their radial distance r diminishes. Because angular momentum L = m · r · u_θ is conserved in inviscid flow, the swirl velocity u_θ increases sharply as 1/r.
2. Burgers-Rott Steady Navier-Stokes Solution
Real fluids have viscosity (μ > 0). Viscous shear diffuses momentum outward while radial sink flow advects it inward. The exact steady balance yields the Burgers-Rott vortex: a solid-body core (u_θ ∝ r) surrounded by potential decay (u_θ ∝ 1/r).
3. The Coriolis Myth vs Reality
In ordinary household sinks, Earth's Coriolis acceleration (2Ω sinφ ≈ 10⁻⁴ s⁻¹) is eclipsed by residual filling currents and basin geometry. Only in giant, meticulously settled tanks does a purely planetary Coriolis vortex emerge.