Fluids in Motion and Bernoulli’s Principle — Why Airplanes Fly ✈️

Try this 😏: hold two sheets of paper parallel, close to your face, and blow between them. You’d expect them to fly apart — but they come together 🤯! And how does a 400-tonne airplane stay in the sky? Both are answered by a 280-year-old counterintuitive principle discovered by Daniel Bernoulli, and captured perfectly by a single elegant equation.

The core idea in one paragraph 📌

For a fluid that is steady, incompressible, and frictionless: along a streamline, the quantity \( P + \tfrac{1}{2}\rho v^2 + \rho g h \) is constant — Bernoulli’s equation. Direct consequence: wherever the flow is fast, the pressure is low. Combined with the continuity equation \( A_1 v_1 = A_2 v_2 \) (a narrower pipe forces higher speed), these two relations explain airplane flight, Venturi meters, spray bottles, and roofs ripping off in storms.

The simplified model — four assumptions 🎯

Real fluid flow is enormously complex (Navier–Stokes equations, turbulence, …). For grade 10 physics we adopt the ideal-flow model:

Assumption	Meaning
Steady	Velocity at any point is time-independent
Incompressible	Density \( \rho \) is constant (good for liquids, OK for slow gas flow)
Frictionless (inviscid)	No viscous dissipation
Laminar	Streamlines are smooth, no eddies

Continuity — conservation of mass 🚰

The water passing cross-section 1 per second must equal the water passing cross-section 2 per second (mass is conserved). That gives a constant volumetric flow rate:

\[ \boxed{A_1 v_1 = A_2 v_2} \]

• \( A \): cross-sectional area
• \( v \): mean velocity

Everyday example: pinch the end of a garden hose → water shoots out much faster 💦.

Numerical example: a pipe narrows from \( A_1 = 50 \) cm² to \( A_2 = 10 \) cm². Water in the wide section moves at \( v_1 = 2 \) m/s. What’s the speed in the narrow section?

\[ v_2 = v_1 \cdot \frac{A_1}{A_2} = 2 \times \frac{50}{10} = 10\ \text{m/s} \]

Five times faster!

Bernoulli’s equation — the heart of it 🫀

Along a streamline:

\[ \boxed{P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}} \]

Three terms:

• \( P \) — pressure energy per unit volume
• \( \tfrac{1}{2}\rho v^2 \) — kinetic energy per unit volume
• \( \rho g h \) — potential energy per unit volume

This is basically energy conservation applied to an ideal fluid (which we’ll formalize in Chapter 3).

The counterintuitive result: at constant height (\( h \) fixed), if \( v \) rises, \( P \) must fall. High speed ↔ low pressure.

The two-paper trick, solved 📄

When you blow between the papers → the air between them moves fast → \( P \) there drops. The air outside the papers is still → higher \( P \). The pressure difference squeezes the sheets together 🎯.

Why airplanes fly ✈️

An airfoil wing is shaped so that air on the top surface moves faster than the air below:

• Top surface: high \( v \) → low \( P \)
• Bottom surface: low \( v \) → high \( P \)

Pressure difference × wing area = the upward lift force:

\[ F_\text{lift} = (P_\text{bottom} – P_\text{top}) \times A_\text{wing} \]

Honest caveat: real flight involves multiple mechanisms (Bernoulli + turning the flow + Newton’s third law). But Bernoulli is the core quantitative one.

Why roofs tear off in storms 🏚️

Fast wind across a roof → low pressure on top. Still air inside the house → high pressure below. The difference pushes the roof upward — an accidental giant wing pointing the wrong way.

The Venturi tube — measuring speed via pressure 📏

A pipe with a narrower section in the middle. Measure the pressure difference between the two cross-sections, back out the speed:

\[ v_1 = \sqrt{\frac{2(P_1 – P_2)}{\rho \left[(A_1/A_2)^2 – 1\right]}} \]

Uses: carburetors, medical instruments, gas/water flow meters in industry.

Other applications 🌍

• Spray bottles 🧴: fast air across a nozzle drops the pressure, sucking liquid up
• Chimneys 🏭: wind over the top lowers pressure, pulling smoke upward
• A spinning ball’s curve (Magnus effect) ⚽: rotation makes air flow faster on one side → path curves (banana kicks!)
• Pitot tubes ✈️: the pointed sensor at an airplane’s nose measures speed from the difference between total and static pressure

Python analysis 🐍

1) Continuity — speed as pipe narrows

# A pipe with three different diameters
diameters_cm = [10, 5, 2]      # cm
A = [3.14159 * (d/2)**2 for d in diameters_cm]  # cm²
v1 = 2.0                        # m/s at the first diameter

for i, (d, a) in enumerate(zip(diameters_cm, A)):
    v = v1 * A[0] / a
    print(f"diameter {d:2d} cm → speed = {v:5.2f} m/s")
# diameter 10 → 2.00 m/s
# diameter  5 → 8.00 m/s
# diameter  2 → 50.00 m/s !!

2) Pressure difference on an airplane wing

rho_air = 1.225   # kg/m³ at sea level
v_top    = 250    # m/s airflow on top of the wing
v_bottom = 220    # m/s airflow below
A_wing   = 100    # m² wing area of a mid-size airliner

# From Bernoulli (constant height): P + 0.5·ρ·v² is constant
# So the pressure difference between top and bottom is:
dP = 0.5 * rho_air * (v_bottom**2 - v_top**2)
Lift = -dP * A_wing   # positive difference produces upward force

print(f"ΔP = {dP:.0f} Pa (top pressure is lower)")
print(f"Lift = {Lift/1000:.1f} kN")
print(f"Equivalent to ~{Lift/9.81/1000:.0f} tonnes of weight")
# Lift ≈ 864 kN ≈ 88 tonnes — enough to lift a small 737

3) Reverse Venturi — from pressure difference to speed

import numpy as np
rho = 1000     # water
A1, A2 = 50e-4, 10e-4   # m² (50 and 10 cm²)
P1_minus_P2 = 12_000    # Pa (measured pressure difference)

ratio = (A1/A2)**2 - 1
v1 = np.sqrt(2*P1_minus_P2 / (rho * ratio))
v2 = v1 * A1/A2

print(f"v1 (wide section) = {v1:.2f} m/s")
print(f"v2 (narrow section) = {v2:.2f} m/s")
# v1 ≈ 1.00 m/s, v2 ≈ 5.00 m/s

Take-home summary 🎁

Two equations capture the whole chapter:

• Continuity: \( A_1 v_1 = A_2 v_2 \) (mass conservation)
• Bernoulli: \( P + \tfrac{1}{2}\rho v^2 + \rho g h = \) constant (energy conservation)

Result: fast → low pressure. Explains flight, sprays, roofs blowing off, curveball trajectories. One equation, endless applications 😎.

Congratulations — Chapter 2 is complete 🎉

---

“Nice to know” box: the Bernoulli family 💡

The Bernoullis produced at least eight prominent mathematicians in the 17th–18th centuries — an extraordinary explosion of genetic or educational genius. Daniel Bernoulli (1700–1782) is our man. His father Johann was also a great mathematician, but he was so envious of Daniel’s success that in 1734 he actually backdated one of his own books by two years to claim priority over his son’s discoveries. History eventually got it right — the principle bears Daniel’s name. 🧠

---

Test yourself 📝

---

References and further exploration 📚

Articles and reference

• Wikipedia: Bernoulli’s principle, Venturi effect, Magnus effect, Lift (force)
• HyperPhysics — Bernoulli Equation
• Feynman Lectures — Vol. II, Ch. 40: The Flow of Dry Water, Ch. 41: The Flow of Wet Water
• NASA — How Wings Really Generate Lift (great correction to the “Bernoulli-only” myth!)

Videos (YouTube)

• Veritasium: How airplanes actually fly
• MIT OCW 8.01 Walter Lewin: Fluid dynamics — Bernoulli
• 3Blue1Brown: Bernoulli’s equation visualized
• SmarterEveryDay: Prince Rupert’s Drop & fluid physics
• Real Engineering: How Airplanes Work

External simulators

• PhET — Fluid Pressure and Flow
• NASA Foil Sim (interactive airfoil simulator)

On this site 🔗

• نسخه‌ی فارسی — اصل برنولی
• Previous section — Buoyancy
• Related — Pressure in Fluids

---

🎉 Congratulations — Chapter 2 (Physical Properties of Matter) is complete. You now know the states of matter, what intermolecular forces do, how pressure works in fluids, the buoyancy formula, and why Bernoulli’s principle makes flight possible. Ready for Chapter 3 — Work and Energy? 🚀