Streamline flow and the equation of continuity

Smooth and rough flow

Water from a gently opened tap falls as a smooth, glassy stream. Open it fully and the stream turns rough and white. The first is steady or streamline flow; the second is turbulent.

In steady flow, the velocity at each fixed point does not change with time. Every particle that passes a point follows the same path, a streamline. Streamlines never cross, and where they crowd together the fluid moves fast.

The equation of continuity

A liquid hardly changes its volume, and it cannot pile up inside a pipe. So in each second the same volume must pass through every cross-section:

$Av$ is the volume flow rate, in $\text{m}^3/\text{s}$. Where the pipe is narrow, the liquid must move faster.

You should find that halving the diameter makes the water four times as fast, while the flow rate stays the same.

This is why you put your thumb over a hose to make the water shoot farther, and why a river runs faster where it is narrow and shallow.

Watch out. The area goes as the diameter squared. Halving the diameter quarters the area.

A worked example

A garden hose $1.8\ \text{cm}$ across fills a $15$ litre bucket in $30\ \text{s}$. Then a nozzle $0.60\ \text{cm}$ across is fitted.

1. Flow rate. $\dfrac{15 \times 10^{-3}\ \text{m}^3}{30\ \text{s}} = 5.0 \times 10^{-4}\ \text{m}^3/\text{s}$.
2. Speed in the hose. $A = \pi(0.0090\ \text{m})^2 = 2.54 \times 10^{-4}\ \text{m}^2$, so $v = 1.96\ \text{m/s}$.
3. At the nozzle. The diameter is a third, so the speed is $9$ times: $17.7\ \text{m/s}$.

Practise with new numbers

Try it in Python

The program works out the hose of the example.

Remember these

• Steady flow: fixed streamlines; fast flow turns turbulent.
• Continuity: $A_1v_1 = A_2v_2$; the flow rate $Av$ is the same everywhere.
• Narrow means fast.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A pipe's cross-section falls from $30\ \text{cm}^2$ to $6\ \text{cm}^2$. The water entered at $0.8\ \text{m/s}$. How fast does it leave?

2. Why does a stream of water from a tap get thinner as it falls?

3. Where are the streamlines of a flow closest together?

4. Water flows at $2.0\ \text{m/s}$ through a pipe of $5.0\ \text{cm}^2$. How many litres pass each second?