When a fluid flows through a pipe or channel, the volume flow rate must remain constant along the path if the fluid is incompressible (like water). This is the continuity equation, a direct consequence of mass conservation.
Continuity equation derivation
Consider a fluid flowing through a pipe of varying cross-sectional area. In time \( dt \), a volume of fluid \( dV \) passes through any cross-section. The volume flow rate is:
\[ Q = \frac{dV}{dt} = A v \]
where \( A \) is the cross-sectional area and \( v \) is the fluid's average velocity at that section. For an incompressible fluid with no sources or sinks of mass, the flow rate must be the same everywhere along the pipe:
\[ Q = A_1 v_1 = A_2 v_2 = \text{constant} \]
This is the continuity equation. If the pipe narrows (area decreases), the fluid must speed up to maintain the same volume flow. Conversely, if the pipe widens, the fluid slows.
Intuitive picture
Imagine water flowing through a flexible hose. Where the hose is pinched to half its radius, the cross-sectional area becomes \( (1/2)^2 = 1/4 \) the original. By continuity, the velocity must quadruple to compensate. The same mass of water that enters the narrow section must exit—it just moves faster.
Worked example: Flow through a constriction
Water flows steadily through a horizontal pipe. The pipe has two sections: a wide section with diameter \( d_1 = 10 \) cm and velocity \( v_1 = 2 \) m/s, and a narrow section with diameter \( d_2 = 5 \) cm. What is the velocity in the narrow section?
Step 1: Calculate areas \[ A_1 = \pi \left(\frac{d_1}{2}\right)^2 = \pi (0.05)^2 = 0.00785 \text{ m}^2 \] \[ A_2 = \pi \left(\frac{d_2}{2}\right)^2 = \pi (0.025)^2 = 0.00196 \text{ m}^2 \]
Step 2: Apply continuity \[ A_1 v_1 = A_2 v_2 \] \[ 0.00785 \times 2 = 0.00196 \times v_2 \] \[ v_2 = \frac{0.0157}{0.00196} = 8 \text{ m/s} \]
Check: The area ratio is \( A_1/A_2 = 0.00785/0.00196 = 4 \). The velocity ratio is \( v_2/v_1 = 8/2 = 4 \). Consistent.
The fluid quadruples in speed as the cross-section shrinks by a factor of 4. This accelerates the water, converting gravitational or pressure energy into kinetic energy.
Volume flow rate in common units
The volume flow rate \( Q \) is often expressed in liters per second (L/s) or gallons per minute (gal/min).
- A typical shower head: \( Q \approx 10 \) L/min \( \approx 0.17 \) L/s
- A fire hose: \( Q \approx 1000 \) L/min \( \approx 17 \) L/s
- A major river: \( Q \approx 10^8 \) L/s (Amazon discharges ~209,000 m³/s \( \approx 2 \times 10^8 \) L/s)
Mass flow rate vs volume flow rate
For a compressible fluid (like a gas), mass is conserved but volume may change as pressure and temperature vary. The mass flow rate is:
\[ \dot{m} = \rho Q = \rho A v \]
where \( \rho \) is the local fluid density. For incompressible fluids, \( \rho \) is constant, so mass and volume flow rates are proportional.
Continuity in 3D: the general form
For flow through a closed surface, the rate of mass flow out equals the rate of mass flow in:
\[ \oint \rho \vec{v} \cdot d\vec{A} = 0 \]
This is the continuity equation in integral form, the mathematical expression of mass conservation. For a fluid element with volume \( \delta V \):
\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0 \]
This is the differential form, used in detailed simulations of fluid flow.
Real-world applications
Garden hose: Placing your thumb over the opening reduces the area, increasing the jet velocity and allowing you to spray further. The same principle used in fire hose nozzles.
Cardiovascular system: Blood flow rate through vessels of different diameters must be continuous. Where arteries branch and get narrower, blood accelerates; in capillaries, the total cross-sectional area increases, and blood slows (critical for nutrient and waste exchange).
Wind tunnel: To accelerate air flow, engineers use a converging nozzle that narrows the flow channel, forcing air to speed up by continuity.
Limitations: compressibility and turbulence
Continuity in the simple form \( A_1 v_1 = A_2 v_2 \) assumes:
- Incompressible flow — density is constant (valid for liquids, high-speed air compresses slightly)
- Steady flow — no time-dependent changes (flow pattern doesn't fluctuate)
- No source/sink — mass isn't injected or removed along the path
At very high speeds, sound speed, or in gases, compressibility becomes significant.
What you should be able to do
- State and apply the continuity equation \( A v = \text{constant} \)
- Calculate velocity or area in different pipe sections
- Compute volume flow rates and convert between units
- Explain the physical meaning: faster flow in narrower sections
- Distinguish between volume and mass flow rates
- Recognize practical applications in plumbing, medicine, and engineering
Preview of §14.5
Bernoulli's equation: relating pressure, velocity, and height in flowing fluids, merging energy conservation with dynamics.
📚 See also: Halliday Vol 2, Ch 14, §14.4.
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