Explicit physics instruction
For steady incompressible flow, volume flow rate is conserved: A₁v₁=A₂v₂. Narrower cross-sectional area therefore requires greater average speed.
Pressure, speed, elevation, viscosity, and energy losses can all matter. Bernoulli-style reasoning is useful only when its assumptions are stated.
Real fluids are viscous; friction and turbulence can dissipate mechanical energy and weaken ideal predictions.
Core relation: A₁v₁=A₂v₂; ideal Bernoulli reasoning within stated assumptions.
Worked example
A flow moves through 4 cm² at 2 m/s and narrows to 2 cm². Continuity predicts v₂=(4×2)/2=4 m/s.
Guided question
A tube narrows from 6 cm² to 3 cm² while speed starts at 1.5 m/s. Find the ideal downstream speed.
Independent work
- Solve three continuity/flow calculations.
- State Bernoulli assumptions for one case.
- Identify one viscosity/turbulence limitation.
Misconception check
Faster flow does not universally mean lower pressure in every real situation; the simple Bernoulli relation has assumptions.
Formative check
If cross-sectional area halves for steady incompressible flow, what happens to average speed?
Reveal response
It doubles.
Mentor evidence conference
Ask Archaemenes to audit continuity assumptions, energy terms, and residual interpretation.
Teacher guidance
Foundation: Apply continuity.
Core: Combine continuity with bounded Bernoulli reasoning.
Extended: Analyze viscosity/turbulence residuals.