🌊 Fluid Dynamics & Continuity

Part 1 of 7 — Fluids in Motion

So far we've studied fluids at rest (hydrostatics). Now we analyze moving fluids — how they flow, what speeds them up, and why a garden hose squirts faster when you put your thumb over the end.

The Ideal Fluid Model

To make fluid dynamics manageable, AP Physics 2 uses the ideal fluid approximation:

What This Means in Practice

• Water flowing smoothly through pipes ✅
• Honey oozing slowly ❌ (viscous)
• Supersonic air ❌ (compressible)
• Whirlpools ❌ (rotational, turbulent)

The ideal fluid model works surprisingly well for water in pipes, blood in arteries, and many other everyday situations.

Streamlines and Flow

Streamlines

A streamline is the path a fluid particle follows during steady flow. Key properties:

• Streamlines never cross (if they did, a particle at the crossing would have two velocities!)
• Fluid velocity is tangent to the streamline at every point
• Closely spaced streamlines → fast flow; widely spaced → slow flow

Flow Tube

A flow tube (or stream tube) is a bundle of streamlines forming an imaginary tube. Think of it as a "pipe" made of flowing fluid.

In steady flow, fluid enters one end and exits the other — no fluid crosses the walls of the tube.

Volume Flow Rate

The volume flow rate $Q$ measures how much fluid passes a point per unit time:

$Q = A \cdot v$

Where:

• $Q$ = volume flow rate (m³/s)
• $A$ = cross-sectional area (m²)
• $v$ = fluid speed (m/s)

Units: 1 m³/s = 1000 liters/s. Typical garden hose: $Q \approx 3 \times 10^{-4}$ m³/s.

Concept Check

Flow Rate Drill (use $\pi \approx 3.14$)

Water flows through a circular pipe of diameter 4.0 cm at a speed of 3.0 m/s.

1. Cross-sectional area of the pipe (in m², scientific notation: e.g., "1.26e-3")
2. Volume flow rate (in m³/s, use same format)
3. How many seconds to fill a 50-liter (0.050 m³) bucket?

Round all answers to 3 significant figures.

Exit Quiz

📏 The Continuity Equation

Part 2 of 7 — What Goes In Must Come Out

The continuity equation is one of the most intuitive results in physics: fluid can't just appear or disappear in a pipe. Whatever enters one end must exit the other.

Deriving the Continuity Equation

Consider a pipe that narrows from area $A_1$ to area $A_2$:

In time $\Delta t$:

• Fluid entering at point 1: volume = $A_1 v_1 \Delta t$
• Fluid exiting at point 2: volume = $A_2 v_2 \Delta t$

Since the fluid is incompressible (constant density) and no fluid leaks:

$A_1 v_1 \Delta t = A_2 v_2 \Delta t$

$\boxed{A_1 v_1 = A_2 v_2}$

This is the Equation of Continuity.

What It Means

$Q = Av = \text{constant throughout the pipe}$

• Narrow section → fast flow (small $A$ → large $v$)
• Wide section → slow flow (large $A$ → small $v$)

This is why:

• A thumb over a garden hose makes water squirt faster
• River rapids occur where the channel narrows
• Wind accelerates between buildings (wind tunnel effect)

Continuity Concept Check

Continuity Equation Drill

Water flows through a circular pipe of radius 3.0 cm at 2.0 m/s. The pipe narrows to a radius of 1.0 cm.

1. Area of the wide section (in cm²)
2. Area of the narrow section (in cm²)
3. Speed in the narrow section (in m/s)

Round all answers to 3 significant figures.

Applications of Continuity

🩺 Blood Flow

The aorta (radius ≈ 1 cm) carries blood at ≈ 30 cm/s. Capillaries (radius ≈ 4 μm each) carry blood at ≈ 0.05 cm/s.

Total capillary area: $A_{\text{cap}} = A_{\text{aorta}} \times v_{\text{aorta}} / v_{\text{cap}} = \pi(0.01)^2 \times 30/0.05 \approx 0.19$ m²

That's about 6 billion capillaries! Continuity works even in biology.

🚿 Nozzle Design

Fire hose nozzles are narrow to increase exit speed: if the hose has $A = 50$ cm² and the nozzle has $A = 5$ cm², the exit speed is 10× the hose speed.

🏞️ River Flow

A river 20 m wide and 3 m deep flows at 0.5 m/s. It narrows to 10 m wide and 2 m deep:

$v_2 = \frac{A_1 v_1}{A_2} = \frac{(20)(3)(0.5)}{(10)(2)} = \frac{30}{20} = 1.5$ m/s

The current triples as the river narrows!

Exit Quiz

🔄 Mass Flow Rate & Conservation

Part 3 of 7 — Tracking Fluid Mass

Volume flow rate is great for incompressible fluids, but the deeper principle is mass conservation. Let's explore both forms and when to use which.

Mass Flow Rate

The mass flow rate $\dot{m}$ measures mass passing a point per second:

$\dot{m} = \rho A v$

Units: kg/s

Connection to Volume Flow Rate

$\dot{m} = \rho Q = \rho A v$

For incompressible fluids ($\rho$ = constant):

$\dot{m}_1 = \dot{m}_2 \implies \rho A_1 v_1 = \rho A_2 v_2 \implies A_1 v_1 = A_2 v_2$

The continuity equation is really mass conservation simplified for incompressible flow.

For Compressible Fluids (FYI)

$\rho_1 A_1 v_1 = \rho_2 A_2 v_2$

This form is needed for gases at high speeds. AP Physics 2 focuses on the incompressible version.

Flow Conservation Quiz

Special Flow Configurations

Pipe with a Leak

If fluid leaks out of a pipe, the flow rate downstream is reduced:

$Q_{\text{downstream}} = Q_{\text{upstream}} - Q_{\text{leak}}$

Pipe with an Inlet

If extra fluid is added:

$Q_{\text{downstream}} = Q_{\text{upstream}} + Q_{\text{inlet}}$

Multiple Branches (General Rule)

At any junction:

$\sum Q_{\text{in}} = \sum Q_{\text{out}}$

This is the fluid equivalent of Kirchhoff's Current Law in circuits!

Syringe

A syringe demonstrates continuity beautifully:

• Barrel area $A_1$ (large), plunger speed $v_1$ (slow)
• Needle area $A_2$ (tiny), fluid speed $v_2$ (fast!)
• $v_2 = (A_1/A_2) v_1$

If the barrel radius is 1 cm and the needle radius is 0.2 mm:

$v_2 = \frac{\pi(0.01)^2}{\pi(0.0002)^2} v_1 = \frac{10^{-4}}{4 \times 10^{-8}} v_1 = 2500 v_1$

The fluid exits 2,500× faster than the plunger moves!

Syringe Problem

A syringe barrel has diameter 2.0 cm. The needle has diameter 1.0 mm. The plunger is pushed at 1.0 cm/s.

1. Area of the barrel (in cm²)
2. Area ratio $A_{\text{barrel}}/A_{\text{needle}}$
3. Speed of fluid exiting the needle (in m/s)

Round all answers to 3 significant figures.

Exit Quiz

🔬 Types of Flow & Reynolds Number

Part 4 of 7 — When Ideal Flow Breaks Down

Not all fluid flow is smooth and orderly. Understanding when our ideal fluid model works — and when it fails — is key to applying it correctly.

Laminar vs. Turbulent Flow

Laminar Flow (Smooth)

• Fluid moves in parallel layers ("laminae")
• Each layer slides past adjacent layers without mixing
• Streamlines are smooth, parallel curves
• Low speeds, small pipes, viscous fluids
• Examples: honey flowing, slow river, blood in small vessels

Turbulent Flow (Chaotic)

• Fluid moves in irregular, swirling patterns
• Rapid mixing between layers
• Streamlines are chaotic, unpredictable
• High speeds, large pipes, low-viscosity fluids
• Examples: white water rapids, smoke rising (after initial laminar region), jet engine exhaust

The Transition

Flow transitions from laminar to turbulent as speed increases. The Reynolds number predicts when:

$Re = \frac{\rho v D}{\mu}$

Where $\mu$ is the fluid's viscosity and $D$ is the pipe diameter.

Note: AP Physics 2 rarely asks for Reynolds number calculations, but understanding the concept is valuable.

Flow Type Quiz

Viscosity: The "Thickness" of a Fluid

Viscosity ($\mu$) is a fluid's resistance to flow — its internal friction.

Effects of Viscosity

In a viscous fluid flowing through a pipe:

• Center of pipe: Fastest flow
• Near walls: Slowest flow (zero at the wall — "no-slip condition")
• The velocity profile is parabolic (Poiseuille flow)

Temperature Effects

• Liquids: Viscosity decreases with temperature (hot honey flows easily)
• Gases: Viscosity increases with temperature (opposite!)

AP Physics 2 Approach

AP treats fluids as non-viscous (ideal). But understanding viscosity helps you know when the ideal model fails and why real fluids behave differently.

Viscosity Quiz

Exit Quiz

🧮 Continuity Problem-Solving Workshop

Part 5 of 7 — AP-Level Practice

Time to tackle multi-step continuity problems — the kind that appear on AP exams with pipes splitting, merging, and changing size.

Problem-Solving Strategy

Step-by-Step Approach

1. Identify all inlets and outlets — draw the pipe system
2. Apply continuity at each junction: $\sum Q_{\text{in}} = \sum Q_{\text{out}}$
3. Use $Q = Av$ to relate area and speed
4. Convert units carefully (cm → m, L/s → m³/s)
5. For circular pipes: $A = \pi r^2 = \pi d^2/4$

Common Unit Conversions

Problem 1: Branching Pipe

A main water pipe (radius 5.0 cm, speed 2.0 m/s) splits into two branches. Branch A has radius 3.0 cm and Branch B has radius 4.0 cm.

1. Flow rate in the main pipe (in L/s)
2. If Branch A carries 60% of the flow, speed in Branch A (in m/s)
3. Speed in Branch B (in m/s)

Round all answers to 3 significant figures.

Problem 2: Filling a Tank

A circular pipe (diameter 4.0 cm) delivers water at 5.0 m/s into a cylindrical tank (diameter 2.0 m).

1. Volume flow rate (in m³/s, use "1.26e-3" format)
2. Rate at which the water level rises in the tank (in m/s, use same format)
3. Time to fill the tank to a depth of 1.0 m (in seconds)

Tricky Concept Questions

Exit Quiz

🫀 Biological & Engineering Applications

Part 6 of 7 — Continuity in the Real World

The continuity equation isn't just a textbook formula — it governs blood flow in your body, water distribution in cities, and aerodynamics of aircraft.

The Circulatory System

Your circulatory system is a masterpiece of fluid dynamics:

The Numbers (Moderate Activity)

Continuity in Action

$Q = A_{\text{aorta}} v_{\text{aorta}} = A_{\text{capillaries}} v_{\text{capillaries}}$

$(4.5)(40) = 180$ and $(4000)(0.05) = 200$ — approximately equal ✓

Blood slows down dramatically in capillaries because the total cross-sectional area is ~1000× larger than the aorta. This slow speed allows time for gas exchange!

Circulatory System Quiz

Engineering Applications

Water Distribution

A city water main (diameter 60 cm) supplies a neighborhood. Each house has a 2-cm diameter pipe.

If the main carries water at 1.5 m/s, how many houses can it supply at 0.5 m/s each?

$Q_{\text{main}} = \pi(0.30)^2(1.5) = 0.424$ m³/s

$Q_{\text{house}} = \pi(0.01)^2(0.5) = 1.57 \times 10^{-4}$ m³/s

$N = Q_{\text{main}}/Q_{\text{house}} = 0.424/(1.57 \times 10^{-4}) ≈ 2700$ houses

Wind Tunnels

Wind tunnels use continuity to accelerate air. A large fan pushes air through a converging section:

• Wide section: 4 m × 4 m, air at 5 m/s
• Test section: 1 m × 1 m

$v_{\text{test}} = (16/1)(5) = 80$ m/s — useful testing speed!

Aircraft Engines

Jet engines take in air through a wide intake and accelerate it through progressively narrower compressor stages, reaching extreme speeds before combustion.

Engineering Drill

A sprinkler system has one main pipe (radius 2.0 cm, speed 3.0 m/s) that feeds 8 identical sprinkler heads.

1. Total flow rate in the main pipe (in L/s)
2. Flow rate per sprinkler (in L/s)
3. If each sprinkler head has radius 0.30 cm, the exit speed at each head (in m/s)

Round all answers to 3 significant figures.

Exit Quiz

🎯 Fluid Dynamics Synthesis & AP Review

Part 7 of 7 — Complete Review

Let's consolidate everything from fluid dynamics and continuity before moving on to Bernoulli's equation.

Complete Concept Map

The Ideal Fluid Model

• Incompressible ($\rho =$ const)
• Non-viscous (no internal friction)
• Steady flow (pattern doesn't change)
• Irrotational (no swirling)

Key Equations

$Q = Av \quad \text{(Volume flow rate)}$

$A_1 v_1 = A_2 v_2 \quad \text{(Continuity equation)}$

$\dot{m} = \rho A v \quad \text{(Mass flow rate)}$

Decision Tree for Problems

1. Single pipe, two sections: Direct $A_1 v_1 = A_2 v_2$
2. Branching pipe: $Q_{\text{in}} = Q_1 + Q_2 + ...$
3. Merging pipes: $Q_1 + Q_2 + ... = Q_{\text{out}}$
4. Filling/draining: $Q = A_{\text{container}} \times (dh/dt)$

Top 5 Mistakes

Mixed Concept Quiz

Comprehensive Drill (use $\pi \approx 3.14$)

A water tower supplies water through a pipe of diameter 10 cm at 2.0 m/s. The pipe splits into two branches: Branch A (diameter 6.0 cm) and Branch B (diameter 8.0 cm), with equal flow rates.

1. Total flow rate from the tower (in L/s)
2. Speed in Branch A (in m/s)
3. Speed in Branch B (in m/s)

Round all answers to 3 significant figures.

AP-Style FRQ Preview

Typical Exam Setup

A horizontal pipe has a cross-sectional area that varies along its length. At point 1, the area is $A_1 = 40$ cm² and the water speed is $v_1 = 2.5$ m/s. At point 2, the area is $A_2 = 10$ cm².

(a) Find the water speed at point 2.

$v_2 = A_1 v_1 / A_2 = (40)(2.5)/(10) = 10$ m/s ✓

(b) Is the pressure at point 2 higher or lower than at point 1? Explain.

The pressure at point 2 is lower. By Bernoulli's equation (energy conservation), faster-moving fluid has lower pressure. This is the Venturi effect — we'll derive it in the next topic!

(c) If the pipe then rises vertically 5 m while maintaining area $A_2$, does the speed change?

No! By continuity, same area → same speed, regardless of height. The speed only depends on cross-sectional area.

This bridges perfectly into our next topic: Bernoulli's Equation!

Final Exit Quiz