Underwater Cave Narrows: Continuity & Bernoulli Flow (2D)

Passage Geometry

Constriction ratio (A₂/A₁) 35%

Inlet passage width (W₁) 4.0 m

Flow Conditions

Inlet flow speed (v₁) 0.30 m/s

Water density (ρ) 1000 kg/m³

Dynamic viscosity (μ) 1.00 mPa·s

Narrows Live Readout

Flow speed at narrows

0.00 m/s

Speed-up factor

1.00×

Pressure drop (ΔP)

0.0 Pa

Reynolds number

Regime

Laminar

Narrows width

0.0 m

How it works

Water is incompressible, so mass conservation (the continuity equation) forces the flow to speed up wherever the passage narrows:

A1·v1 = A2·v2   →   v2 = v1 · (W1 / W2)

Cross-sectional area is taken proportional to the local passage width W(x) (constant depth). As the walls close in at "the narrows", v(x) rises exactly in proportion — this is the real reason cave-diving narrows carry dangerous currents even when the inlet flow looks gentle.

Bernoulli's equation (steady, incompressible, horizontal passage — no elevation term) then links that speed-up to a pressure drop:

P1 + ½ρv1² = P(x) + ½ρv(x)²
ΔP(x) = ½ρ(v1² − v(x)²)

Pressure falls exactly where speed rises. The diver-tracer particles are advected at the true local flow speed v(x) computed from the same width profile drawn on screen — nothing here is decorative. The Reynolds number Re = ρ·v2·W2/μ tracks whether the flow at the narrows is smooth (laminar) or churns (turbulent), shown as jitter on the tracers.

Flow-speed field (brighter = faster) Diver tracer Narrowest cross-section