A Rankine combined vortex: solid-body rotation inside the core radius r₀, an irrotational 1/r free vortex outside it — the standard bathtub-drain model. A radial sink adds the inward drift toward the plughole:
- v_θ(r) = ω·r for r ≤ r₀, with ω = Γ / (2π r₀²)
- v_θ(r) = Γ / (2π r) for r > r₀
- v_r(r) = −Q / (2π r) for r > r₀ (continuity of the drain flux)
- free-surface depth: z(r) = z∞ − v_θ(r)² / (2g) (Bernoulli along the free surface)
Higher circulation Γ spins the surface faster and deepens the funnel; higher drain flow Q pulls particles inward faster without changing the funnel shape.