Continuity / Mass Conservation
The continuity equation expresses mass conservation in a flow: what enters at one location must exit at another.
General form (compressible)
For a flow channel with cross-section A and mean velocity v:
ρ₁ · A₁ · v₁ = ρ₂ · A₂ · v₂
with ρ = density. The product ρ·A·v = mass flow rate (kg/s) is constant along the channel.
Incompressible (PPL-relevant)
At low speeds (Mach < 0.3) ρ ≈ constant:
A₁ · v₁ = A₂ · v₂
or A · v = constant along the channel.
→ At constriction (small A) flow accelerates (large v). → At enlargement (large A) flow decelerates (small v).
Application — Venturi
In a Venturi tube (convergent → divergent) flow accelerates at the narrowest point — and per Bernoulli the static pressure p drops there. This pressure depression is used in:
- Carburettor fuel intake.
- Vacuum system for gyro instruments.
- Speed measurement (Venturi anemometer historically).
Application — wing
Above the curved upper surface, the streamlines initially barely diverge (curved profile constrains the flow into narrower paths) → acceleration via Bernoulli/continuity.
Continuity and Bernoulli work together:
- Continuity: flow accelerates because cross-section narrows.
- Bernoulli: faster flow = lower p.
- Lift arises from pressure difference.
Limits
- Incompressible assumption valid only at M < 0.3. At higher Mach the compressible form must be used (ρ varies).
- Steady assumed — unsteady flow requires extension.
Mass-conservation example
In a Venturi tube: inlet A₁ = 100 cm², flow velocity v₁ = 5 m/s. Constriction to A₂ = 25 cm².
v₂ = (A₁/A₂) × v₁ = (100/25) × 5 = 20 m/s.
Velocity is quadrupled. With Bernoulli, pressure drops:
p₂ = p₁ − ½ρ(v₂² − v₁²) = p₁ − ½×1.225×(400 − 25) ≈ p₁ − 230 Pa.