Continuity Equation (Mass Conservation)
The continuity equation expresses conservation of mass in fluid flow.
For incompressible flow, the product of cross-sectional area and velocity is constant.
The Formula
Compressible: ρ&sub1;A&sub1;v&sub1; = ρ&sub2;A&sub2;v&sub2;
General (differential form): ∂ρ/∂t + ∇·(ρv) = 0
The continuity equation is simply the statement that mass is conserved in a flowing fluid. For an incompressible fluid (liquids at moderate pressures), density is constant and drops out — leaving the simple result that the volumetric flow rate Q = Av is constant everywhere in the flow. If the pipe narrows, the fluid must speed up. If the pipe widens, the fluid slows down.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| A | Cross-sectional area of the pipe or channel | m² |
| v | Average fluid velocity at that cross-section | m/s |
| ρ | Fluid density (constant for incompressible flow) | kg/m³ |
| Q = Av | Volumetric flow rate (constant for incompressible flow) | m³/s |
| ∇·() | Divergence operator | — |
Example 1 — Narrowing Pipe
Water flows at 2 m/s in a 100 mm diameter pipe. The pipe narrows to 50 mm diameter. Find the velocity in the narrow section.
A&sub1; = π(0.05)² = 7.854 × 10−3 m²
A&sub2; = π(0.025)² = 1.963 × 10−3 m²
A&sub1;v&sub1; = A&sub2;v&sub2; → v&sub2; = v&sub1; × A&sub1;/A&sub2;
v&sub2; = 2 × (7.854/1.963) = 2 × 4 = 8 m/s
The velocity quadruples when the diameter halves (area reduces by factor of 4). Flow rate Q = 7.854 × 10−3 × 2 = 0.01571 m³/s throughout.
Example 2 — River Flooding
A river 20 m wide and 2 m deep flows at 0.5 m/s. It narrows to 10 m wide. Assuming the depth stays 2 m, find the flood velocity.
A&sub1; = 20 × 2 = 40 m², v&sub1; = 0.5 m/s
Q = A&sub1; × v&sub1; = 40 × 0.5 = 20 m³/s
A&sub2; = 10 × 2 = 20 m²
v&sub2; = Q/A&sub2; = 20/20 = 1.0 m/s — the river doubles in speed where it narrows. This is why narrow river channels flood so dangerously.
When to Use It
Use the continuity equation when:
- Analyzing flow through pipes, nozzles, diffusers, and channels
- Designing plumbing systems — pipe diameter determines flow speed and pressure
- Understanding blood flow — arteries branch and narrow near capillaries
- Aviation — airflow accelerates over the curved top of a wing
- Combining with Bernoulli's equation to solve pipe flow problems completely
For compressible flows (gases at high speed, above Mach 0.3), the density changes significantly and the compressible form ρAv = constant must be used instead.