# Fluid Dynamics: Continuity and Bernoulli's Equation

Physics I · Principles of Waves, Fluids and Thermodynamics · https://tryals.app/learn/physics-i/fluid-dynamics-continuity-and-bernoullis-equation

## Continuity and Ideal Fluids

Fluids obey a traffic law: what flows in must flow out. Squeeze the pipe and the fluid must speed up, and counterintuitively, its pressure **drops** where it moves fastest.

For an **ideal fluid** (incompressible, non-viscous, steady flow), the **continuity equation** enforces conservation of mass:

$$A_1 v_1 = A_2 v_2$$

Where pipe area $A$ shrinks, flow speed $v$ increases.

Energy conservation along a streamline is given by **Bernoulli's equation**:

$$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$

Here $P$ is pressure, $\rho$ is density, $v$ is speed, $g$ is gravity, and $h$ is height. Each term is energy per unit volume.

| Application | Key mechanism |
| --- | --- |
| **Torricelli's theorem** | $v = \sqrt{2gh}$ (tank drain speed) |
| **Venturi tube** | Pressure drops where speed increases |
| **Pitot tube** | Measures speed from pressure diff |
| **Airplane lift** | Fast air over wing drops pressure |

## Real Fluids and Viscosity

Real fluids introduce **viscosity** (friction). For laminar flow in a round pipe, **Poiseuille's law** governs volumetric flow rate $Q$:

$$Q = \frac{\pi r^4 \Delta P}{8\eta L}$$

Where $r$ is radius, $\Delta P$ is pressure drop, $\eta$ is dynamic viscosity, and $L$ is pipe length. Flow rate scales heavily with radius ($r^4$).

The **Reynolds number** ($\\text{Re}$) predicts whether flow remains smooth or breaks down:

$$\text{Re} = \frac{\rho v D}{\eta}$$

Where $D$ is pipe diameter. Regimes divide as follows:

| Regime | Reynolds Number | Behavior |
| --- | --- | --- |
| **Laminar** | $\text{Re} < 2000$ | Smooth flow |
| **Turbulent** | $\text{Re} > 4000$ | Chaotic flow |

> **Common pitfall:** Bernoulli's equation applies **only** along a single streamline of smooth, steady, frictionless flow. Applying it across turbulent or unconnected regions yields nonsense.

## Practice questions

4 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Sort each phenomenon as an example of Bernoulli effect or Continuity alone.

**Answer:**

- Bernoulli effect: Airplane wing generating lift, Venturi meter measuring flow speed, Spray atomiser uses fast airflow to reduce pressure
- Continuity (mass conservation): Water speeds up in a narrowing pipe, Fire hose outlet has higher speed than the intake, River narrows and current increases

**Why:** Bernoulli relates speed to pressure; continuity relates speed to cross-sectional area.

Page: https://tryals.app/practice/physics-i/fluid-dynamics-continuity-and-bernoullis-equation/sort-each-phenomenon-as-an-example-of-bernoulli-effect-or-continuity

### 2. You partly block a garden hose with your thumb and the water **jets out faster**. What made the water speed up?

A. Your thumb raises the pressure at the opening, and high pressure means high speed
B. The same flow must squeeze through a smaller area, continuity demands more speed
C. The water gains energy from friction with your thumb
D. It doesn’t speed up; it only looks that way because the stream is thinner

**Answer:** B. The same flow must squeeze through a smaller area, continuity demands more speed

**Why:** Continuity (mass conservation) forces $v = Q/A$: shrink the opening and the speed rises to carry the same flow. Rivers do it too, the same water that dawdles through wide pools races through narrow gorges.

Page: https://tryals.app/practice/physics-i/fluid-dynamics-continuity-and-bernoullis-equation/you-partly-block-a-garden-hose-with-your-thumb-and-the-water-jets-out

### 3. Order these milestones in fluid mechanics.

**Answer:**

1. Archimedes formulates buoyancy principle
2. Torricelli measures atmospheric pressure with mercury
3. Bernoulli publishes Hydrodynamica
4. Navier and Stokes derive viscous flow equations
5. Reynolds introduces dimensionless number for flow regime

**Why:** Archimedes (c. 250 BCE), Torricelli (1644), Bernoulli (1738), Navier–Stokes (1820s–1840s), Reynolds (1883).

Page: https://tryals.app/practice/physics-i/fluid-dynamics-continuity-and-bernoullis-equation/order-these-milestones-in-fluid-mechanics

### 4. Order the reasoning that explains why a roof can be **lifted off** in a storm (Bernoulli).

**Answer:**

1. Wind streams fast across the top of the roof
2. Along that streamline, high speed means low pressure
3. The still air inside the house keeps full atmospheric pressure
4. The pressure difference produces a net upward force on the roof

**Why:** Fast exterior flow lowers the outside pressure while indoor air still pushes up at full strength; the imbalance can exceed the roof’s weight. Opening windows on the lee side (equalizing pressure) is real storm advice for exactly this reason.

Page: https://tryals.app/practice/physics-i/fluid-dynamics-continuity-and-bernoullis-equation/order-the-reasoning-that-explains-why-a-roof-can-be-lifted-off-in-a