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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Derive a divergence-free weak formulation of incompressible flow.
- Justify the conclusion "∫uₜ·φ−∫(u⊗u):∇φ+ν∫∇u:∇φ=∫f·φ" using the stated assumptions.
Before you start
Integration by parts, Sobolev spaces, and divergence-free tests.
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 smooth periodic u solving uₜ+(u·∇)u=−∇p+νΔu+f, test against a smooth divergence-free φ.
Why this math matters
Derive a divergence-free weak formulation of incompressible flow. 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 displayed derivation begins with smooth periodic fields.
- Test fields are divergence-free; a full time-weak definition also states initial data and integrability.
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.
Move derivatives onto test functions
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For smooth periodic u solving uₜ+(u·∇)u=−∇p+νΔu+f, test against a smooth divergence-free φ.
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
∫∇p·φ=−∫p∇·φ=0
Periodicity removes boundary terms, and divergence-free tests eliminate pressure from this identity.
Work through the mathematics
∫((u·∇)u)·φ=−∫(u⊗u):∇φ
Integrating the transport derivative uses ∇·u=0.
Check the conclusion
∫uₜ·φ−∫(u⊗u):∇φ+ν∫∇u:∇φ=∫f·φ
This identity motivates a weak formulation that requires fewer pointwise derivatives; time derivatives can also be transferred to spacetime tests.
The result
∫uₜ·φ−∫(u⊗u):∇φ+ν∫∇u:∇φ=∫f·φ
This identity motivates a weak formulation that requires fewer pointwise derivatives; time derivatives can also be transferred to spacetime tests.
Common mistakes to catch
- Writing a weak identity is not a proof that a weak solution is smooth.
- Boundary terms cannot be dropped for arbitrary domains or boundary conditions.
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
Why can pressure still matter physically if it disappears here?
Show a hint
The test space was specially restricted.
Reveal answer and explanation
It remains part of the full momentum equation
Elimination in one variational identity does not set p to zero.
Practice 2
Which condition removes the extra transport term involving div u?
Show a hint
Expand a product divergence.
Reveal answer and explanation
Incompressibility
Without div u=0, another term appears after integration by parts.
Take the idea with you
Track which derivatives a finite-element weak formulation asks of its unknown and test functions.
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: Separate a smooth energy identity from a weak energy inequality
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