14-7 Bernoulli’s Equation

14-7 Bernoulli’s Equation#

Prompts

  • State Bernoulli’s equation. What does each term represent (pressure, kinetic, gravitational)?

  • For horizontal flow (\(y\) constant), how are pressure and speed related? Where is pressure higher—in a narrow or wide section?

  • Why does a water stream from a faucet “neck down” as it falls?

  • A hole is punched in a water tank at depth \(h\) below the surface. What is the exit speed of the water? (Torricelli’s result)

  • Bernoulli’s equation is energy conservation. What are the three forms of energy per unit volume?

Lecture Notes#

Overview#

  • Bernoulli’s equation: Conservation of total energy density for ideal fluid flow.

  • Three energy densities (per unit volume): (1) Pressure \(p\)—work per volume, potential to expand; (2) Kinetic \(\frac{1}{2}\rho v^2\); (3) Gravitational \(\rho g y\).

  • Key result: Higher speed → lower pressure (for horizontal flow).

Bernoulli’s equation#

For steady, incompressible, nonviscous flow along a tube of flow:

(32)#\[ p_1 + \frac{1}{2}\rho v_1^2 + \rho g y_1 = p_2 + \frac{1}{2}\rho v_2^2 + \rho g y_2 = \text{constant} \]

Term

Physical name

Energy density

Intuitive meaning

\(p\)

Static pressure

Pressure energy density

Work per volume; energy stored from compression; fluid’s potential to expand and do work

\(\frac{1}{2}\rho v^2\)

Dynamic pressure

Kinetic energy (KE) density

KE per unit volume from macroscopic ordered motion

\(\rho g y\)

Hydrostatic pressure

Gravitational potential energy (PE) density

PE per unit volume from height in gravity field (\(y\) upward)

Physical meaning: The sum of pressure, kinetic, and gravitational energy per unit volume is conserved along a streamline. Bernoulli’s equation is energy conservation for an ideal fluid.

Note

Fluid at rest: Set \(v_1 = v_2 = 0\). Then \(p_2 = p_1 + \rho g(y_1 - y_2)\)—we recover hydrostatic pressure Eq. (16).

Horizontal flow: speed and pressure#

If \(y\) is constant (horizontal tube):

(33)#\[ p + \frac{1}{2}\rho v^2 = \text{constant} \]

Higher speed → lower pressure. Where streamlines are close together (fast flow), pressure is low; where they spread (slow flow), pressure is high.

  • Force perspective: A fluid element entering a narrow section must accelerate (\(F = ma\)). That requires higher pressure behind than in front—so pressure is lower in the narrow, fast region.

  • Energy perspective: Pressure energy is converted into kinetic energy.

Poll: Air tube with water columns

Air flows through a horizontal channel with a wide section and a narrow section. Two vertical tubes connect the channel to a basin of water (each tube has one end in the basin and the other connected to the channel). Which tube has the higher water level?

  • (A) Tube at the wide section

  • (B) Tube at the narrow section

  • (C) Same height

../images/14-7_air_tube.png

Fig. 11 Air channel (wide → narrow) with tubes to water basin; which tube has higher water level?#

Poll: Hurricane roof

High-speed wind blows over a flat roof. The roof can lift off. Compared to the still air inside, the fast air outside has:

  • (A) Higher pressure

  • (B) Lower pressure

  • (C) Same pressure

  • (D) Higher kinetic energy density but same pressure

Applications#

Faucet stream: Water falling from a tap speeds up (\(v\) increases). By continuity \(Av = \text{constant}\), so \(A\) must decrease—the stream “necks down.”

Leaky tank (Torricelli): A hole at depth \(h\) below the water surface. Take point 1 at the surface (\(p_1 = p_0\), \(v_1 \approx 0\), \(y_1 = h\)) and point 2 at the hole (\(p_2 = p_0\), \(y_2 = 0\)). Bernoulli gives

(34)#\[ v = \sqrt{2gh} \]

Same as the speed of free fall from height \(h\).

Example: Blowing over a penny

A penny (radius \(r \approx 1.0\;\text{cm}\), mass \(m \approx 2.5\;\text{g}\)) lies flat on a table. You blow air over the top. The fast-moving air has lower pressure than the still air below; the pressure difference \(\Delta p\) provides an upward force. For lift: \(\Delta p \cdot \pi r^2 \geq mg\). With \(\rho_{\text{air}} \approx 1.2\;\text{kg/m}^3\) and Bernoulli: \(\frac{1}{2}\rho v^2 \approx \Delta p\). So \(v \geq \sqrt{2mg/(\rho \pi r^2)} \approx 36\;\text{m/s}\)—you need to blow quite hard!

Example: Water emerging from pipe

Water exits a pipe at speed 4.0 m/s. The pipe is horizontal and open to the atmosphere. If the pipe narrows upstream, the pressure there is higher than \(p_0\). For a vertical pipe with exit at \(y = 0\) and a point at height \(h\) above it where \(v \approx 0\): Bernoulli gives \(p_0 + \rho g h = p_0 + \frac{1}{2}\rho v^2\), so \(h = v^2/(2g) \approx 0.82\;\text{m}\). The gauge pressure at the top equals the kinetic energy density at the exit.

Poll: Two buckets with same hole

Two buckets have the same hole size at the bottom and the same initial water volume and height. Bucket A is wide at the top and narrow at the bottom; Bucket B is narrow at the top and wide at the bottom. Which empties first?

  • (A) Bucket A empties first

  • (B) Bucket B empties first

  • (C) They empty simultaneously

  • (D) Cannot determine without more information

../images/14-7_buckets.png

Fig. 12 Two buckets with same initial water level and hole size: (A) wide top, narrow bottom; (B) narrow top, wide bottom.#

Summary#

  • Bernoulli’s equation: \(p + \frac{1}{2}\rho v^2 + \rho g y = \text{constant}\) (energy conservation)

  • Horizontal flow: \(p + \frac{1}{2}\rho v^2 = \text{constant}\); faster → lower pressure

  • Torricelli: Exit speed from hole at depth \(h\) is \(v = \sqrt{2gh}\)