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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Check velocity, boundary conditions, and every term of a simple steady Navier–Stokes solution.
- Justify the conclusion "The steady incompressible momentum equation is satisfied" using the stated assumptions.
Before you start
Partial derivatives and incompressible momentum balance.
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
Between plates y=0 and y=h, verify u=(Uy/h,0,0) with constant pressure and zero body force.
Why this math matters
Check velocity, boundary conditions, and every term of a simple steady Navier–Stokes solution. 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:
- Plates are idealized as infinite in their tangential directions.
- The fluid has constant viscosity and density.
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.
Verify an exact flow between moving plates
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Between plates y=0 and y=h, verify u=(Uy/h,0,0) with constant pressure and zero body force.
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
At y=0, u=0; at y=h, u=(U,0,0)
The linear profile matches no-slip velocities of the two plates.
Work through the mathematics
∇·u=0, (u·∇)u=0, and Δu=0
The field depends on y but advects only in x, and its second derivatives vanish.
Check the conclusion
The steady incompressible momentum equation is satisfied
Acceleration, pressure gradient, viscous Laplacian, and forcing all vanish in the interior, while boundary motion maintains shear.
The result
The steady incompressible momentum equation is satisfied
Acceleration, pressure gradient, viscous Laplacian, and forcing all vanish in the interior, while boundary motion maintains shear.
Common mistakes to catch
- Zero viscous Laplacian does not imply zero shear stress.
- An exact solution for one geometry is not a theorem about arbitrary initial data.
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
For U=2 and h=0.5, find speed at y=0.25.
Show a hint
Substitute into Uy/h.
Reveal answer and explanation
One
The midpoint speed is half the moving plate's speed.
Practice 2
Is viscous shear stress necessarily zero because Δu=0?
Show a hint
Stress uses first derivatives.
Reveal answer and explanation
No
The shear rate U/h is nonzero even though the second derivative vanishes.
Take the idea with you
Distinguish a local momentum balance from the boundary work needed to sustain a shear flow.
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: Balance a pressure drop with viscous curvature
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