The Continuity Equation
Put your thumb over the end of a garden hose and the water suddenly shoots out much faster, even though you did not touch the faucet. Where did the extra speed come from? By the end of this page you will be able to explain it with one short equation.
Volume flow rate
Picture water moving through a pipe. How much water passes a point each second? That depends on two things: how wide the pipe is and how fast the water moves.
In a time $\Delta t$, water moving at speed $v$ travels a distance $v\,\Delta t$. That water fills a cylinder with cross-sectional area $A$, so the volume that passed is $A\,v\,\Delta t$.
Dividing by $\Delta t$ gives the volume flow rate:
For a round pipe, $A = \pi r^2$, where $r$ is the radius. If you are given the diameter $d$, use $r = d/2$ first.
Water flows steadily through a horizontal pipe whose cross-sectional area is $0.020\ \text{m}^2$. The water moves at $1.5\ \text{m/s}$. Calculate the volume flow rate…
Water flows steadily through a horizontal pipe whose cross-sectional area is $0.020\ \text{m}^2$. The water moves at $1.5\ \text{m/s}$. Calculate the volume flow rate through the pipe in $\text{m}^3/\text{s}$.
The continuity equation
Now follow the water along a pipe that changes width. We use the ideal fluid model: the fluid is incompressible (its density stays constant) and the flow is steady (nothing changes with time at any point).
Think about a closed pipe with no leaks. If more water entered the wide part each second than left the narrow part, water would pile up in between. If less entered, water would have to appear from nowhere. Neither can happen, so the mass passing each cross section per second is the same. With constant density, that means the volume per second is the same too.
Since $A2 < A_1$ in the figure, $v_2$ must be larger than $v_1$. The speed is inversely proportional to the area: halve the area and the speed doubles.
State the continuity equation for an ideal, incompressible fluid flowing steadily through a pipe whose cross-sectional area changes from $A_1$ to $A_2$. Say what…
State the continuity equation for an ideal, incompressible fluid flowing steadily through a pipe whose cross-sectional area changes from $A_1$ to $A_2$. Say what physical quantity is conserved, and define each symbol you use.
A garden hose has a cross-sectional area of $2.0\times10^{-4}\ \text{m}^2$, and the water inside it moves at $0.50\ \text{m/s}$. Emma partly covers the end with her…
A garden hose has a cross-sectional area of $2.0\times10^{-4}\ \text{m}^2$, and the water inside it moves at $0.50\ \text{m/s}$. Emma partly covers the end with her thumb so that the opening has an area of $5.0\times10^{-5}\ \text{m}^2$. Find the speed of the water as it leaves the opening.
Round pipes and flow rate problems
For circular pipes, $A=\pi r^2$ and the continuity equation becomes
$$\pi r_1^2 v_1 = \pi r_2^2 v_2 \quad\Rightarrow\quad r_1^2 v_1 = r_2^2 v_2$$The $\pi$ cancels. The speed depends on the square of the radius.
The flow rate also tells you how fast a container fills. Since $Q = \Delta V/\Delta t$, the time to add a volume $V$ is $\Delta t = V/Q$. For a tank with base area $A_{\text{tank}}$, a rise in level $\Delta h$ means $V = A_{\text{tank}}\Delta h$.
A pipe carries water with a radius of $0.050\ \text{m}$ in a wide section. It narrows to a radius of $0.020\ \text{m}$. The water speed in the wide section is…
A pipe carries water with a radius of $0.050\ \text{m}$ in a wide section. It narrows to a radius of $0.020\ \text{m}$. The water speed in the wide section is $1.2\ \text{m/s}$.
(a) Find the water speed in the narrow section.
(b) Find the volume flow rate in each section and compare them.
A town's water main delivers water at a volume flow rate of $0.080\ \text{m}^3/\text{s}$. A school's cylindrical tank has a base area of $12\ \text{m}^2$ and is filled…
A town's water main delivers water at a volume flow rate of $0.080\ \text{m}^3/\text{s}$. A school's cylindrical tank has a base area of $12\ \text{m}^2$ and is filled from this main with all of that flow. How long does it take the water level in the tank to rise by $1.5\ \text{m}$? Give your answer in seconds and in minutes.
Everyday consequences and branching pipes
A falling stream. Marcus's faucet stream gets thinner as it falls. The water speeds up under gravity. For the flow rate $Q = Av$ to stay the same at every height, a larger $v$ requires a smaller $A$. So the stream narrows.
Branches. When a pipe splits, the water that enters per second must equal the total that leaves per second. If the main pipe is 1 and the branches are 2, 3, and so on:
$$A_1 v_1 = A_2 v_2 + A_3 v_3 + \cdots$$The flow rates of the branches add up to the flow rate of the main pipe. If the branches are identical, each carries an equal share.
Marcus watches a smooth stream of water fall from a kitchen faucet. He notices that the stream gets thinner as it falls. Using the continuity equation, explain why the…
Marcus watches a smooth stream of water fall from a kitchen faucet. He notices that the stream gets thinner as it falls. Using the continuity equation, explain why the stream narrows. Assume the flow is steady and the water speeds up as it falls.
A main pipe of cross-sectional area $0.040\ \text{m}^2$ carries water at $2.0\ \text{m/s}$. It splits into three identical branch pipes, each with a cross-sectional area…
A main pipe of cross-sectional area $0.040\ \text{m}^2$ carries water at $2.0\ \text{m/s}$. It splits into three identical branch pipes, each with a cross-sectional area of $0.010\ \text{m}^2$.
(a) Find the speed of the water in each branch.
(b) One branch is then shut off, and the main pipe keeps the same volume flow rate. Find the new speed in each of the two remaining branches.
(c) Explain, using conservation of mass, why the speed in the remaining branches changes.
A horizontal pipe has three sections in a row. Section A has diameter $0.30\ \text{m}$, section B has diameter $0.15\ \text{m}$, and section C has diameter…
A horizontal pipe has three sections in a row. Section A has diameter $0.30\ \text{m}$, section B has diameter $0.15\ \text{m}$, and section C has diameter $0.10\ \text{m}$. Water enters section A at $0.40\ \text{m/s}$.
(a) Calculate the water speed in sections B and C.
(b) Calculate the volume flow rate in each section.
(c) A student claims that because the water moves faster in C, more water passes through C each second than through A. Evaluate this claim.
Back to the start: when you cover the hose end with your thumb, the opening area $A$ shrinks. The flow rate $Q = Av$ stays about the same, so the speed $v$ of the water leaving must go up.