Fluid Flow: Continuity Before Bernoulli

Mechanical Engineering · CHAPTER 5 / 5

Level: introductory undergraduate · Study session: 15–20 minutes including practice.

Learning outcome: Relate pipe area, velocity and volumetric flow.

Before you begin: Complete earlier chapters in this discipline; revise algebra and units as needed.

1. Understand the idea

For steady incompressible flow through a pipe without leakage, volumetric flow is conserved. A narrower section therefore has higher mean velocity. Bernoulli relates pressure, elevation and speed along a streamline under restrictive assumptions; real pipes need loss terms.

2. Write the model

Q = Av; A1v1 = A2v2

Define the symbols from the worked case before substituting values. Keep units consistent, record the assumptions and round only after the calculation.

3. Follow a worked example

A flow of 0.010 m³/s passes through a 0.005 m² section: v=2 m/s. At 0.0025 m², v=4 m/s. Continuity gives these speeds without needing a pressure estimate.

4. Explore the model

Figure: calculated teaching cases; straight segments connect sampled points, not experimental observations. Read exact values below.

Illustrative scenario sweep — synthetic educational data
Area (m²)	Flow (m³/s)	Velocity (m/s)
0.01	0.01	1
0.005	0.01	2
0.0025	0.01	4

Halve the area at fixed flow and observe velocity doubling. Do not assume the pressure remains unchanged.

5. Reproduce it yourself

Copy the input cases into Excel or a notebook and name each quantity and unit.
Recalculate each result from the stated equation or rule. For qualitative cases, explain the state or partition decision instead.
Change one input, predict the direction of change, then calculate again.
Write one sentence explaining where the model stops being valid.

6. Check your understanding

What velocity corresponds to Q=0.020 m³/s and A=0.005 m²?

Reveal the answer and check your work

4 m/s.

Compare your result, units and reasoning. If they differ, return to the equation and test the worked case first.

7. Common mistake

Use internal flow area, not pipe outside diameter.

Further study and academic note

Explore the discipline reference and further learning resources. This is an original IIMEEE introductory teaching note with synthetic examples; it is not a peer-reviewed research result or a professional design specification. Figures were constructed from the listed teaching cases. Resource links support further study and do not imply endorsement.