Fluids and Conservation Laws
Thumb over the hose — the water shoots out
- Put your thumb over the end of a garden hose and the water suddenly sprays far and fast.
- You narrowed the opening, and the same flow had to speed up to get through.
- Moving fluids obey conservation laws: mass and energy are both preserved.
- These give two powerful rules — continuity and Bernoulli's principle.
Continuity: what goes in must come out
- For a steady flow, the fluid passing each point per second is the same everywhere.
- Continuity: $A_1 v_1 = A_2 v_2$ — area times speed is constant along the pipe.
- So a narrow section forces a faster flow, and a wide one a slower flow.
- It is just conservation of mass: no fluid piles up or vanishes.
Follow the flow through a pipe
Step through a narrowing pipe and see how the flow speeds up where the pipe is narrow.
Water at $2\ \tfrac{\text{m}}{\text{s}}$ flows from a pipe of area $6\ \text{cm}^2$ into one of area $2\ \text{cm}^2$. What is the new speed, in $\tfrac{\text{m}}{\text{s}}$?
$A_1 v_1 = A_2 v_2 \Rightarrow 6 \times 2 = 2 v_2 \Rightarrow v_2 = 6\ \tfrac{\text{m}}{\text{s}}$.
The continuity equation $A_1 v_1 = A_2 v_2$ is a statement of the conservation of:
No fluid piles up or disappears — it is conservation of mass.
In a steady flow, fluid moves faster through a narrower section of pipe.
Continuity ($A_1 v_1 = A_2 v_2$) forces a higher speed where the area is smaller.
Bernoulli: faster flow, lower pressure
- Bernoulli's principle comes from conserving energy in a flowing fluid.
- Where a fluid flows faster, its pressure is lower; where it slows, pressure rises.
- It is the energy trade-off: speeding up the flow costs pressure energy.
- This is why a fast stream of air has a low pressure that things get pushed into.
By Bernoulli's principle, where a fluid flows faster, its pressure is:
Faster flow means lower pressure — the energy trade-off of Bernoulli's principle.
Bernoulli's principle comes from conserving ____ in a flowing fluid.
It is the energy conservation law for a flowing fluid.
Select all correct statements about flowing fluids.
Continuity (mass) speeds flow in narrow pipes; Bernoulli (energy) links faster flow to lower pressure.
Conservation laws explain everyday flow
- An aeroplane wing speeds air over the top, lowering the pressure there, producing lift.
- A spinning ball curves because it makes the air faster on one side than the other.
- A chimney draws better in wind, because fast air over the top lowers the pressure.
- Both continuity and Bernoulli are conservation laws wearing fluid clothing.
Bernoulli's principle is often stated backwards. It is faster flow ⇒ lower pressure, not higher. And continuity ($A_1 v_1 = A_2 v_2$) means fluid speeds up in a narrow pipe — the opposite of the intuition that "less room means slower".
Water flows at $2\ \tfrac{\text{m}}{\text{s}}$ through a pipe of area $6\ \text{cm}^2$, then into a narrow section of area $2\ \text{cm}^2$.
- Continuity: $A_1 v_1 = A_2 v_2 \Rightarrow 6 \times 2 = 2 \times v_2$.
- $v_2 = \dfrac{12}{2} = 6\ \tfrac{\text{m}}{\text{s}}$ — three times faster in the narrow part.
Flowing fluids conserve mass and energy. Continuity ($A_1 v_1 = A_2 v_2$) means a narrower pipe gives a faster flow. Bernoulli's principle says faster flow means lower pressure. Together they explain lift, curveballs and much of everyday fluid flow.