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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Distinguish vorticity transport, stretching, and diffusion.
- Justify the conclusion "In genuinely planar flow u=(u₁(x,y),u₂(x,y),0), this stretching term vanishes" using the stated assumptions.
Before you start
Curl and vector differential identities.
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 incompressible unforced flow, interpret ωₜ+(u·∇)ω=(ω·∇)u+νΔω.
Why this math matters
Distinguish vorticity transport, stretching, and diffusion. 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:
- Velocity is smooth enough for the curl identity.
- The planar reduction includes independence of the third coordinate.
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.
Identify the three-dimensional vorticity stretching term
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For incompressible unforced flow, interpret ωₜ+(u·∇)ω=(ω·∇)u+νΔω.
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
ω=∇×u; taking curl removes ∇p
Curl of a smooth pressure gradient is zero.
Work through the mathematics
The term (ω·∇)u differentiates velocity along the vorticity direction
It describes stretching or tilting of vortex direction in three dimensions.
Check the conclusion
In genuinely planar flow u=(u₁(x,y),u₂(x,y),0), this stretching term vanishes
Vorticity points in z while the velocity has no z dependence, yielding scalar vorticity transport-diffusion.
The result
In genuinely planar flow u=(u₁(x,y),u₂(x,y),0), this stretching term vanishes
Vorticity points in z while the velocity has no z dependence, yielding scalar vorticity transport-diffusion.
Common mistakes to catch
- Two-dimensional and three-dimensional vorticity mechanisms differ.
- Local stretching does not by itself prove finite-time singularity.
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 local axial strain ∂zu₃=a and ω aligned with z, what stretching contribution appears in the z component?
Show a hint
Apply ω₃∂z to u₃.
Reveal answer and explanation
aω₃
Positive a amplifies that aligned vorticity component locally.
Practice 2
Does νΔω necessarily decrease ω at every spatial point?
Show a hint
A Laplacian redistributes a field.
Reveal answer and explanation
No
Diffusion has global smoothing properties but its local sign depends on curvature.
Take the idea with you
Identify what extra mechanism is lost when a 3D flow is reduced to a planar model.
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: Find which velocity norm survives Navier–Stokes scaling
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