Section 6.6: Continuity Equation
The continuity equation expresses the conservation of mass for an incompressible fluid in steady flow. It relates the cross-sectional area and velocity at different points along a streamline:
Continuity Equation
\[
A_1 v_1 = A_2 v_2
\]
where:
\( A \) = cross-sectional area (m²)
\( v \) = fluid velocity (m/s)
This equation implies that if the pipe narrows, the fluid velocity increases, and vice versa, assuming incompressibility.
Example 1: Flow in a Tapered Pipe
A horizontal pipe narrows from 0.5 m² to 0.2 m². If the velocity in the wider section is 3 m/s, find the velocity in the narrow section.
Using \( A_1 v_1 = A_2 v_2 \):
\( 0.5 \times 3 = 0.2 \times v_2 \Rightarrow v_2 = \frac{1.5}{0.2} = 7.5 \, \text{m/s} \)
The fluid velocity in the narrow section is 7.5 m/s.
Example 2: Velocity Change
A pipe carries water with velocity 4 m/s in a 0.1 m² section. It widens to 0.25 m². Find the new velocity.
\( A_1 v_1 = A_2 v_2 \Rightarrow 0.1 \times 4 = 0.25 \times v_2 \)
\( v_2 = 0.4/0.25 = 1.6 \, \text{m/s} \)
The velocity in the wider section is 1.6 m/s.
Practice Problems
- A river narrows from 10 m width to 5 m. If the speed at the wider section is 2 m/s, find speed at the narrow section.
- A garden hose of diameter 3 cm delivers water at 2 m/s. If the hose end is squeezed to 1.5 cm, find the exit velocity.
- Water flows through a pipe with varying diameter: 0.3 m² to 0.1 m². Initial velocity is 5 m/s. Find velocity at narrow section.
- A horizontal pipe of 0.4 m² narrows to 0.1 m². If velocity at wide section is 6 m/s, calculate velocity at narrow section.
- Explain how the continuity equation is consistent with incompressibility of fluids.