1. Concept Overview
The continuity equation explains how a fluid maintains a constant flow rate as it moves through a pipe or channel. If the path becomes narrower, the fluid speeds up; if it widens, the fluid slows down. This happens because an incompressible fluid cannot pile up or disappear — whatever enters must leave at the same rate.
This idea helps explain why rivers flow faster at narrow points and why water shoots out more quickly from a squeezed hose pipe.
2. Definition
3. Understanding the Principle
3.1. Incompressible Fluids
Liquids are nearly incompressible, meaning their density stays the same as they flow. So the amount of fluid entering a section per second must be the same as the amount leaving it.
3.2. Flow Rate
The volume flow rate is the amount of fluid crossing a surface per second. It is given by:
\( Q = A v \)
where:
- \(A\): cross-sectional area
- \(v\): speed of the fluid
3.3. Conservation of Mass
The continuity equation comes from the fact that mass cannot be created or destroyed. For steady flow, the mass entering one section must equal the mass leaving the next.
4. Mathematical Form
4.1. Basic Equation
For an incompressible fluid:
\( A_1 v_1 = A_2 v_2 \)
This means a smaller area forces the fluid to move faster, and a larger area allows it to move slower.
4.2. General Mass Flow Form
The mass flow rate is:
\( \dot{m} = \rho A v \)
For incompressible fluids, \( \rho \) is constant. So:
\( A_1 v_1 = A_2 v_2 \)
5. Physical Meaning
5.1. Speed Adjusts to Maintain Flow
If the cross-section becomes narrower, the fluid speeds up so the same amount of fluid passes through per second.
5.2. No Fluid Piling
Fluid cannot accumulate in a section of a pipe during steady flow. The continuity equation ensures a smooth transfer without build-up.
6. Common Flow Patterns Explained
6.1. Flow in Pipes
Water flows faster in narrower parts of a pipe. This is why squeezed sections of a hose produce faster jets.
6.2. River Flow
A river flows faster when its width or depth decreases. A narrow stream speeds up naturally due to the continuity condition.
6.3. Blood Flow in Vessels
Blood speeds up in narrow vessels and slows in wider ones. The total flow rate remains the same through the circulatory pathway.
7. Examples to Build Intuition
7.1. Squeezing a Garden Hose
When the hose end is partially blocked, the opening becomes smaller. To maintain the same flow rate, the speed increases, producing a strong jet of water.
7.2. Flow Through a Funnel
As the funnel narrows toward the bottom, the fluid speeds up. This is a direct demonstration of the continuity equation.
7.3. Airflow Over a Wing
Air moves faster over curved surfaces where the effective flow path narrows. This difference in speed later connects to Bernoulli’s principle.