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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Relate a decaying Fourier shear mode to its kinetic-energy dissipation.
- Justify the conclusion "dE/dt=−νexp(−2νt)/2=−ν⟨|∇u|²⟩" using the stated assumptions.
Before you start
Periodic averages and exponential differentiation.
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
On a 2π-periodic box, take u=(exp(−νt) sin y,0,0), ν>0. Verify its mean kinetic-energy balance.
Why this math matters
Relate a decaying Fourier shear mode to its kinetic-energy dissipation. 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 domain is periodic, so there is no boundary energy flux.
- Energy is normalized as a spatial mean.
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.
Check viscous energy loss in an exact periodic flow
PausedQuestion: Start with the question. Paused.
Question
Start with the question
On a 2π-periodic box, take u=(exp(−νt) sin y,0,0), ν>0. Verify its mean kinetic-energy balance.
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
(u·∇)u=0 and Δu=−u, so uₜ=νΔu
This shear solves the unforced momentum equation with constant pressure.
Work through the mathematics
Mean energy E=⟨|u|²⟩/2=exp(−2νt)/4
The average of sin²y is one half.
Check the conclusion
dE/dt=−νexp(−2νt)/2=−ν⟨|∇u|²⟩
The average of cos²y supplies exactly the viscous dissipation rate.
The result
dE/dt=−νexp(−2νt)/2=−ν⟨|∇u|²⟩
The average of cos²y supplies exactly the viscous dissipation rate.
Common mistakes to catch
- Velocity amplitude and kinetic energy decay with different exponential rates.
- An exact special flow must not be generalized to all solutions.
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
What is the velocity amplitude at t=1/ν?
Show a hint
Evaluate its exponential factor.
Reveal answer and explanation
e⁻¹
Energy then scales by e⁻² relative to its initial value.
Practice 2
Does a decaying example prove every smooth 3D flow stays smooth forever?
Show a hint
One explicit solution covers only one data family.
Reveal answer and explanation
No
A universal regularity claim requires controlling arbitrary admissible data in its stated setting.
Take the idea with you
Verify an energy calculation both directly from a solution and from the governing balance.
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: Build incompressibility into a two-dimensional velocity field
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