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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Derive a parabolic velocity profile between stationary plates.
- Justify the conclusion "No-slip gives v(y)=G y(h−y)/(2μ)" using the stated assumptions.
Before you start
Second derivatives and integration constants.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For steady unidirectional flow between y=0 and h with dp/dx=−G and viscosity μ, derive v(y).
Why this math matters
Derive a parabolic velocity profile between stationary plates. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The flow is fully developed and steady.
- G, μ, and h are positive constants; no additional body force is included.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Balance a pressure drop with viscous curvature
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For steady unidirectional flow between y=0 and h with dp/dx=−G and viscosity μ, derive v(y).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
0=−dp/dx+μv″=G+μv″
The pressure force is balanced by the viscous second derivative.
Work through the mathematics
v″=−G/μ; v=−Gy²/(2μ)+C₁y+C₂
Integrate twice before applying wall conditions.
Check the conclusion
No-slip gives v(y)=G y(h−y)/(2μ)
The profile is zero at both walls, symmetric, and maximal at the channel midpoint.
The result
No-slip gives v(y)=G y(h−y)/(2μ)
The profile is zero at both walls, symmetric, and maximal at the channel midpoint.
Common mistakes to catch
- The pressure gradient sign determines the flow direction.
- Flux per unit width has different units from velocity.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
With G=2, μ=1, h=2, find the maximum speed.
Show a hint
Evaluate at y=h/2.
Reveal answer and explanation
One
v(1)=1·1=1.
Practice 2
Find volume flux per unit width for these values.
Show a hint
Integrate y(2−y) from zero to two.
Reveal answer and explanation
4/3
The general flux is Gh³/(12μ), which gives 16/12.
Take the idea with you
Compare the strong cubic dependence of channel flux on plate separation with its linear dependence on pressure gradient.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Check viscous energy loss in an exact periodic flow
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