Continuity and Bernoulli's Equation
Fluid Flow: Continuity and Bernoulli in a Narrowing Pipe
Tracked fluid parcel
Its length back at the inlet
Diameter drawn ×4 for visibility
Playback 0.40× real time
Inlet area A1
0 cm²
Throat area A2
0 cm²
Throat speed v2
0 m/s
Throat gauge p2
0 kPa
Flow rate Q
0 L/s
Cavitation. The predicted absolute pressure at station 2 is
—, below the vapour pressure of water
(2.3 kPa at 20 °C). Real water would boil into vapour pockets here, so the flow
stops being steady and continuous — Bernoulli's equation no longer describes it. The bars below are the
model's answer, not a physical one.
Continuity — the same volume passes every station each second
1A1 × v1
—
2A2 × v2
—
A1v1 = A2v2 = —
—
Bernoulli terms — energy per unit volume (kPa)
1Station 1
—
2Station 2
—
p (gauge)
½ρv²
ρgy
p + ½ρv² + ρgy = constant
the split changes — the total does not
12.0 cm
6.0 cm
1.50 m/s
30 kPa
0.40 m
1.20 m
A₁v₁ = A₂v₂ | p + ½ρv² + ρgy = constant
© The Science Cube
Why a narrower pipe runs faster at lower pressure
Water flows through a pipe that narrows and rises. You set the two diameters, the inlet speed, the inlet gauge pressure and the two heights. Everything else follows from two equations:
A₁v₁ = A₂v₂ and p + ½ρv² + ρgy = constant
Try this
- Halve the diameter — set d₁ = 12 cm, d₂ = 6 cm. The speed goes up ×4, not ×2, because area depends on d², not d. This is the most common slip on fluid questions.
- Watch the blue parcel. It stretches entering the throat but its volume never changes. The dashed outline at the inlet is its original length — that's A₁v₁ = A₂v₂ made physical.
- Watch the two bars. The split between p, ½ρv² and ρgy changes completely between stations; the totals stay identical. That is Bernoulli's equation — energy per unit volume is conserved, not any one term.
- Narrow the throat to ~4 cm. p₂ goes below atmospheric. That's real, not an error: it's how an atomiser and a carburettor pull liquid into a moving airstream.
- Keep narrowing. The sim eventually refuses the answer — absolute pressure would fall below water's vapour pressure and it would boil. That's cavitation, and it's the point where Bernoulli stops applying.
Pipe diameter is drawn ×4 so a 3 cm throat stays visible over a 6 m run. Heights and lengths are to scale; playback is 0.40× real time.
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