Playground
Two pipe sections with different widths; particles speed up in the narrow section. Sliders control inlet area and velocity.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Inlet Area Cross-sectional area at point 1 | m² | L² | 0.001 – 1 | |
| Inlet Velocity Fluid velocity at point 1 | m/s | L·T⁻¹ | 0 – 20 | |
| Outlet Area Cross-sectional area at point 2 | m² | L² | 0.0001 – 1 | |
| Outlet Velocityoutput Fluid velocity at point 2 | m/s | L·T⁻¹ | 0 – 200 |
Deep dive
Derivation
Consider a steady flow through a tube. In time Δt, mass entering at section 1 is ρ·A₁·v₁·Δt, and mass exiting at section 2 is ρ·A₂·v₂·Δt. Conservation of mass requires these be equal: ρA₁v₁ = ρA₂v₂. For an incompressible fluid (constant ρ), this reduces to A₁v₁ = A₂v₂.
Experimental verification
Verified by countless flow experiments with venturi meters, pitot tubes, and pipe networks. Modern PIV (particle image velocimetry) confirms continuity to high precision in liquids and low-Mach gases.
Common misconceptions
- Continuity does NOT say mass flow rate equals volume flow rate — only true for incompressible fluids
- It applies along a streamtube, not necessarily across an entire flow with branching
- Velocity here is the average over the cross-section, not the local maximum (which is higher in viscous flow)
Real-world applications
- Garden hose nozzles and fire hoses
- Carburetor and venturi meter design
- River flow narrowing at canyons
- Blood flow through narrowed arteries (stenosis)
Worked examples
Pinched garden hose
Given:
- A_1:
- 0.0004
- v_1:
- 1
- A_2:
- 0.0001
Find: v_2
Solution
v_2 = A_1·v_1 / A_2 = (0.0004 × 1) / 0.0001 = 4 m/s
River widening into a lake
Given:
- A_1:
- 50
- v_1:
- 2
- A_2:
- 500
Find: v_2
Solution
v_2 = (50 × 2) / 500 = 0.2 m/s
Scenarios
What if…
- scenario:
- What if you pinch the hose to 1/10 the area?
- answer:
- Speed multiplies by 10. The water exits 10× faster, which is why pinching makes it spray much further.
- scenario:
- What if the fluid is compressible (like air at high speed)?
- answer:
- You must use ρ₁A₁v₁ = ρ₂A₂v₂. Density changes (e.g., across shock waves in a rocket nozzle) become essential.
- scenario:
- What if the pipe branches into two?
- answer:
- The total mass flow splits: A₁v₁ = A₂v₂ + A₃v₃. Each branch gets a share proportional to the pressure-driven flow.
Limiting cases
- condition:
- A_2 → A_1
- result:
- v_2 → v_1
- explanation:
- Same pipe diameter throughout means uniform speed.
- condition:
- A_2 → 0
- result:
- v_2 → ∞
- explanation:
- Infinitely narrow opening would require infinite speed (unphysical — viscosity and compressibility kick in).
- condition:
- A_2 → ∞
- result:
- v_2 → 0
- explanation:
- Wide reservoir slows the flow to nearly zero.
Context
Leonhard Euler · 1757
Formulated as part of Euler's foundational equations of fluid dynamics, expressing mass conservation in differential form.
Hook
Why does water shoot out faster when you pinch the end of a hose?
A garden hose with cross-sectional area 4 cm² carries water at 1 m/s. You pinch the end to 1 cm². How fast does the water shoot out?
Dimensions: [A₁][v₁] = [A₂][v₂] → (L²)(L·T⁻¹) = (L²)(L·T⁻¹) → L³·T⁻¹ = L³·T⁻¹ ✓ (volume flow rate)
Validity: Steady, incompressible flow. For compressible flow use ρAv form; for unsteady flow use the differential form ∂ρ/∂t + ∇·(ρv) = 0.