Fluid Flow, Continuity and Bernoulli's Principle

1Put your thumb over the hose

When you water a garden and put your thumb over part of the end of the hose, the water shoots out much faster and further. You have not added any water: you have made the same flow pass through a smaller opening. And where water moves faster, something else happens that is far less obvious: its pressure drops. This trade between speed and pressure, described by Daniel Bernoulli in 1738, explains how a carburettor and a paint sprayer draw in liquid, how a Venturi meter measures the flow in a water main, why a strong wind can lift a roof, and part of how an aircraft wing gives lift.

2Describing fluid flow

In steady (streamline, or laminar) flow, the velocity of the fluid at each point does not change with time: every particle passing a given point follows the same path, called a streamline. Streamlines never cross. When the flow is too fast or meets a sharp obstacle, it becomes turbulent: irregular, with eddies and swirls, and the velocity at a point changes from moment to moment. To keep the mathematics simple we use an ideal fluid: incompressible (constant density), non-viscous (no internal friction), and in steady, non-rotational flow. Water in a pipe at moderate speed is close to ideal.

3The continuity equation

In time $\mathrm{\Delta }t\backslash Delta\; t$, the fluid entering section 1 moves a distance $v_1\mathrm{\Delta }tv\_1\backslash Delta\; t$, so a volume $A_1{v}_1\mathrm{\Delta }tA\_1v\_1\backslash Delta\; t$ flows in. An incompressible fluid cannot pile up between the sections, so the same volume must flow out at section 2:

$QQ$ is the volume flow rate in ${\mathrm{m}}^3{\mathrm{s}}^{-1}\backslash mathrm\{m^3\backslash ,s^\{-1\}\}$. Where the pipe narrows, the fluid speeds up in inverse proportion to the area; where streamlines crowd together, the flow is faster.

4Bernoulli's principle

Consider the fluid between two sections of a tube that may change both in width and in height. In a short time $\mathrm{\Delta }t\backslash Delta\; t$, a slice of volume $V=A_1\mathrm{\Delta }{x}_1=A_2\mathrm{\Delta }{x}_{2}V\; =\; A\_1\backslash Delta\; x\_1\; =\; A\_2\backslash Delta\; x\_2$ and mass $m=\rho Vm\; =\; \backslash rho\; V$ enters at 1, and an equal slice leaves at 2.

Apply the work–energy principle to the fluid between the sections. The fluid behind pushes it forward and does work $p_1{A}_1\mathrm{\Delta }{x}_1=p_{1}Vp\_1A\_1\backslash Delta\; x\_1\; =\; p\_1V$; the fluid ahead pushes back, doing work $-p_{2}V-p\_2V$. The net effect is as if the blue slice had been moved to the red position, so its changes in kinetic and gravitational potential energy are

Dividing by $VV$ and collecting terms gives Bernoulli's equation:

Each term is an energy per unit volume ($\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{P}\mathrm{a}\backslash mathrm\{J\backslash ,m^\{-3\}\}\; =\; \backslash mathrm\{Pa\}$). For a horizontal pipe the $\rho gh\backslash rho\; gh$ terms cancel, and the result is Bernoulli's principle: where the speed of a fluid is higher, its pressure is lower. If the fluid is at rest, the equation reduces to $p=p_0+\rho ghp\; =\; p\_0\; +\; \backslash rho\; gh$ from Lesson 1.

5The Venturi meter

A Venturi meter measures the flow rate in a pipe. The pipe narrows smoothly to a throat and widens again; a manometer measures the pressure difference between the wide section and the throat.

From continuity, $v_2=\frac{A_1}{A_2}{v}_{1}v\_2\; =\; \backslash frac\{A\_1\}\{A\_2\}v\_1$. From Bernoulli for horizontal flow, $p_1-p_2=\frac{1}{2}\rho (v_2^2-v_1^2)=\frac{1}{2}\rho v_1^2(\frac{A_1^2}{A_2^2}-1)p\_1\; -\; p\_2\; =\; \backslash frac\{1\}\{2\}\backslash rho\backslash left(v\_2^2\; -\; v\_1^2\backslash right)\; =\; \backslash frac\{1\}\{2\}\backslash rho\; v\_1^2\backslash left(\backslash frac\{A\_1^2\}\{A\_2^2\}\; -\; 1\backslash right)$. Solving:

If the manometer liquid has density ${\rho }_m\backslash rho\_m$ and the pipe carries a liquid of density $\rho \backslash rho$, the level difference $hh$ gives $p_1-p_2=({\rho }_m-\rho )ghp\_1\; -\; p\_2\; =\; (\backslash rho\_m\; -\; \backslash rho)gh$, because the pipe liquid also fills the manometer arms above the mercury.

6Bernoulli in everyday life

• Sprayers and carburettors. Air blown fast across the top of a narrow tube lowers the pressure there, so atmospheric pressure pushes liquid up the tube into the air stream, where it breaks into droplets.
• Bunsen burner. Gas leaving the narrow jet at high speed has low pressure, so air is drawn in through the air hole and mixes with it.
• Roofs in storms. Wind racing over a roof lowers the pressure above it while the air inside the house is still. A wind of $30\mathrm{m}{\mathrm{s}}^{-1}30\backslash ,\backslash mathrm\{m\backslash ,s^\{-1\}\}$ gives $\mathrm{\Delta }p=\frac{1}{2}\times 1.2\times {30}^2=540\mathrm{P}\mathrm{a}\backslash Delta\; p\; =\; \backslash frac\{1\}\{2\}\; \backslash times\; 1.2\; \backslash times\; 30^2\; =\; 540\backslash ,\backslash mathrm\{Pa\}$: on a 100 ${\mathrm{m}}^2\backslash mathrm\{m^2\}$ roof, an upward force of 54 kN, which is why roof sheets must be firmly nailed or strapped down.
• Aircraft wings. A wing's shape and tilt make air flow faster over the top surface than under it, so the pressure above is lower and there is a net upward force, the lift. Equivalently, the wing pushes air downwards, and the air pushes the wing up (Newton's third law); both descriptions are correct.

7Activity 4.3: two sheets of paper

Hold two sheets of paper vertically, about 3 cm apart, and blow hard between them. Before you try, predict: will they fly apart or come together? They come together: the moving air between them is at lower pressure than the still air outside. Repeat by blowing over the top of a single sheet held below your lips: the sheet rises into the air stream.