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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Distinguish dynamic viscosity, kinematic viscosity, and shear rate.
- Justify the conclusion "If density is 1000 kg/m³, ν=μ/ρ=0.0005 m²/s" using the stated assumptions.
Before you start
Derivatives and dimensional units.
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 planar shear uₓ=2y per second and dynamic viscosity μ=0.5 Pa·s, find shear stress τxy.
Why this math matters
Distinguish dynamic viscosity, kinematic viscosity, and shear rate. 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 fluid is Newtonian with constant viscosity.
- The only nonzero velocity component depends linearly on y.
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.
Connect a velocity gradient with Newtonian shear stress
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For planar shear uₓ=2y per second and dynamic viscosity μ=0.5 Pa·s, find shear stress τxy.
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
∂uₓ/∂y=2 s⁻¹
The velocity difference per separation gives the shear rate.
Work through the mathematics
τxy=μ∂uₓ/∂y=0.5·2=1 Pa
In this unidirectional Newtonian shear, stress is proportional to the gradient.
Check the conclusion
If density is 1000 kg/m³, ν=μ/ρ=0.0005 m²/s
Kinematic viscosity is a momentum-diffusion coefficient, not the same quantity or unit as dynamic viscosity.
The result
If density is 1000 kg/m³, ν=μ/ρ=0.0005 m²/s
Kinematic viscosity is a momentum-diffusion coefficient, not the same quantity or unit as dynamic viscosity.
Common mistakes to catch
- Do not substitute kinematic viscosity directly into a dynamic-stress formula.
- The full stress tensor involves the symmetric velocity gradient; this is a special shear case.
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 stress results if the shear rate doubles with μ fixed?
Show a hint
Use the linear constitutive law.
Reveal answer and explanation
Two pascals
Newtonian shear stress doubles with the rate.
Practice 2
Would the same formula describe every non-Newtonian material?
Show a hint
The stress-rate law may not be linear.
Reveal answer and explanation
No
The constitutive assumption must match the material.
Take the idea with you
Use units to distinguish a measured stress from a velocity gradient and a viscosity.
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: Compare inertial and viscous scales without declaring a regime automatically
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