Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Generate divergence-free planar motion from a streamfunction.
- Justify the conclusion "u·∇ψ=0, so streamlines lie on xy=constant" using the stated assumptions.
Before you start
Mixed partial derivatives and gradients.
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 ψ(x,y)=xy and u=(∂yψ,−∂xψ), find the flow and its streamlines.
Why this math matters
Generate divergence-free planar motion from a streamfunction. 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:
- ψ is twice continuously differentiable.
- The velocity convention is u=(ψy,−ψx).
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.
Build incompressibility into a two-dimensional velocity field
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For ψ(x,y)=xy and u=(∂yψ,−∂xψ), find the flow and its streamlines.
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=(x,−y)
Differentiating the streamfunction gives the two velocity components.
Work through the mathematics
∇·u=∂x∂yψ−∂y∂xψ=0
Equal mixed partials enforce incompressibility automatically.
Check the conclusion
u·∇ψ=0, so streamlines lie on xy=constant
The velocity is tangent to streamfunction level sets wherever the field is nonzero.
The result
u·∇ψ=0, so streamlines lie on xy=constant
The velocity is tangent to streamfunction level sets wherever the field is nonzero.
Common mistakes to catch
- Some texts use the opposite sign convention.
- A two-dimensional streamfunction formula is not a general representation of every 3D field.
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 velocity comes from ψ=(x²+y²)/2?
Show a hint
Differentiate with the stated sign convention.
Reveal answer and explanation
(y,−x)
Its circular streamlines have clockwise orientation.
Practice 2
What is special at a stagnation point?
Show a hint
The velocity is zero there.
Reveal answer and explanation
A unique tangent direction is not supplied by the zero vector
Level-set and streamline descriptions need local care at critical points.
Take the idea with you
Design a planar incompressible test field by choosing a streamfunction first.
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.
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