Fluid Dynamics: The Continuity Equation
The continuity equation expresses conservation of mass for a fluid in steady flow: for an incompressible fluid, A1v1 = A2v2, so the volume flow rate Q = Av (m3/s) is the same at every cross-section of a pipe, and the fluid speeds up where the pipe narrows. In brief: streamlines are closer together where the speed is higher. For a compressible fluid (a gas), the mass flow rate ρAv is constant instead. Branching pipes follow the rule that the total flow into a junction equals the total flow out, as in Kirchhoff’s current law.
Putting a thumb over the end of a garden hose makes the water spray faster, and rivers flow faster through narrow channels. These are consequences of continuity. This guide defines ideal fluid flow and streamlines, derives the continuity equation, discusses branching and blood flow, compares flow quantities in a table, works through numerical examples and corrects common mistakes.
What is ideal fluid flow?
Fluid dynamics describes fluids in motion. To start, an ideal fluid is assumed: incompressible (constant density), non-viscous (no internal friction), in steady flow (the velocity at each point does not change with time) and irrotational. Real fluids deviate from this, but the model works well for water in pipes and air at speeds well below the speed of sound. A streamline is the path of a fluid particle in steady flow; no fluid crosses the streamlines, and the velocity is tangent to them. A bundle of streamlines forms a flow tube. The streamlines are closer together where the flow is faster.
What does the continuity equation say?
In steady flow, the mass of fluid entering a section of a tube per second equals the mass leaving it per second. The mass flow rate is ρAv. For an incompressible fluid, ρ is constant and cancels:
A1v1 = A2v2 = Q,
where Q is the volume flow rate (m3/s). If the pipe narrows to half the cross-sectional area, the speed doubles. Since A = πd2 ÷ 4, halving the diameter increases the speed by a factor of four. For a compressible gas the equation is ρ1A1v1 = ρ2A2v2. The idea is also used in the design of nozzles, chimneys and rocket engines.
How does it apply to branching pipes and blood flow?
When a pipe branches, the flow rate in the main pipe equals the sum of the flow rates in the branches: Q = Q1 + Q2 + …, so A0v0 = ΣAivi. In the circulatory system, the aorta (area about 3 cm2) branches into many arteries, arterioles and about 1010 capillaries. The total cross-sectional area of all capillaries is about 1,000 times that of the aorta (roughly 3,000 cm2), so the blood speed falls from about 30 cm/s in the aorta to about 0.03 cm/s in the capillaries, giving time for the exchange of gases and nutrients. In a river, the water flows faster in narrow, deep sections and slower where the river widens, in accordance with the same equation (for the same depth).
Flow quantities compared
| Quantity | Symbol and formula | Unit |
|---|---|---|
| Volume flow rate | Q = Av | m3/s |
| Mass flow rate | ρAv | kg/s |
| Speed in a narrowing pipe | v2 = v1A1 ÷ A2 | m/s |
| Speed with diameter change | v2 = v1(d1 ÷ d2)2 | m/s |
Worked example: pipes, hoses and rivers
- Narrowing pipe. Water flows at 1.5 m/s in a pipe of diameter 4.0 cm, which narrows to 2.0 cm. v2 = 1.5 × (4.0 ÷ 2.0)2 = 1.5 × 4 = 6.0 m/s.
- Flow rate. Q = A1v1 = π(0.020)2 × 1.5 = 1.26 × 10−3 × 1.5 = 1.89 × 10−3 m3/s = 1.89 L/s. A 20 L bucket fills in 20 ÷ 1.89 = 10.6 s. In the narrow section, the same Q = π(0.010)2 × 6.0 = 1.89 × 10−3 m3/s ✓.
- Garden hose. Water flows through a hose of inner diameter 1.6 cm at 1.0 m/s. Covering the end so that the opening has a diameter of 0.50 cm gives v = 1.0 × (1.6 ÷ 0.50)2 = 10.2 m/s.
- Branching. A main pipe of diameter 10 cm carries water at 2.0 m/s and splits into three identical pipes of diameter 5.0 cm. The flow rate in the main pipe: Q = π(0.050)2 × 2.0 = 0.0157 m3/s. Each branch carries Q ÷ 3 = 5.24 × 10−3 m3/s, and the speed in each = 5.24 × 10−3 ÷ π(0.025)2 = 2.67 m/s.
- River. A river 20 m wide and 2.0 m deep flows at 0.50 m/s: Q = 20 × 2.0 × 0.50 = 20 m3/s. Where it narrows to 10 m width at the same depth, v = 20 ÷ (10 × 2.0) = 1.0 m/s.
Who formulated the equation?
Leonardo da Vinci noted around 1500 that “a river in each part of its length in an equal time gives passage to an equal quantity of water, whatever the width, the depth, or the slope”. The continuity equation in the general differential form was developed by Leonhard Euler in the 1750s as part of his equations for fluid motion. It is one of the basic equations of fluid dynamics, along with Bernoulli’s equation and the Navier–Stokes equations.
Which mistakes are common?
Mistake 1: Thinking that the flow rate is lower in the narrower section
The flow rate is the same; it is the speed that increases.
Mistake 2: Using the diameter ratio instead of the area ratio
Speeds are inversely proportional to area, so to the square of the diameter.
Mistake 3: Applying the equation to compressible gases without changing it
For gases with large density changes, use the mass flow rate ρAv.
Mistake 4: Forgetting that blood flows slower in the capillaries
The total area of the capillaries is much larger than that of the aorta, so the speed is lower.