Fluid Flow

Why this chapter

Why this chapter exists

So far the water in this book has stood still, in tanks, tubes and lakes. Now it moves: through pipes and nozzles, out of tanks, past wings and balls. Two laws you already know do almost all the work. Mass is conserved, so a fluid runs faster where a pipe is narrower. Energy is conserved, so where the fluid runs faster its pressure is lower. With these two ideas you will find how fast a tank empties and where its jet lands. You will also see why a cyclone lifts roofs and a spinning ball curves.

You will learn

By the end of this chapter you will

Describe steady and turbulent flow, streamlines and tubes of flow, and the four assumptions of an ideal fluid.
Use the volume flow rate \(Q = Av\) and the equation of continuity \(A_1v_1 = A_2v_2\) for pipes, rivers and a falling stream.
Derive Bernoulli's theorem from the work-energy theorem, and read its terms as energies per volume and as heads.
Say where Bernoulli's theorem holds and where it fails: pumps, viscosity and the limit of zero pressure.
Find the speed of efflux from an open or a closed tank, the thrust of the jet, and the time to drain a tank.
Find the range of a jet, the twin holes, the greatest range and the path \(y = x^2/4h\).
Handle a jet from a tank on a platform, or from a tilted nozzle.
Explain and use the venturimeter, the Pitot tube, the lift on a wing and a roof, the sprayer and the spinning ball.

Before you begin

From Motion in Two Dimensions you should know how to solve a projectile launched from a height, horizontally or at an angle. From Work, Kinetic Energy and Power you should know the work-energy theorem, the work done by gravity, and power. From Pressure in Fluids you should know \(P = P_0 + \rho gh\) at a depth \(h\), and what gauge pressure is.

Already know this?

Lessons

Describing a Flow

How to describe a moving fluid. Steady (streamline) flow and turbulent flow, the streamline as the path of a fluid particle, the tube of flow, and the ideal fluid that the rest of the chapter works with.

Steady and turbulent flowStreamlines and tubes of flowThe ideal fluid

12 min0%

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The Equation of Continuity

The volume of fluid that passes a section each second, \(Q = Av\). Mass is conserved, so an incompressible fluid crosses every section of a tube of flow at the same rate, \(A_1v_1 = A_2v_2\). A river speeds up in a gorge, and the stream from a tap thins as it falls.

Volume flow rateEquation of continuityThe narrowing stream of a tap

13 min0%

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Bernoulli's Theorem

The work-energy theorem applied to a tube of flow gives \(P + \tfrac12\rho v^2 + \rho gh = \text{constant}\) along a streamline. In a level pipe the fastest water has the lowest pressure. Each term is an energy per unit volume, or a head in metres. Where the theorem holds, and where it fails.

Bernoulli from work and energyPressure energy per volumeWhere bernoulli holds

15 min0%

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Speed of Efflux

Water leaves a small hole at depth \(h\) in an open tank at \(v = \sqrt{2gh}\), as fast as a stone that has fallen through \(h\). A closed tank with gas above adds the gas pressure to the head. A tank drains in \(t = (A/a)\sqrt{2H/g}\), fast at first and then slowly.

Torricellis lawEfflux from a closed tankTime to drain a tank

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The Range of a Jet

A jet from a hole in the side of a tank is a projectile launched horizontally at \(\sqrt{2gh}\). On the ground it lands at \(R = 2\sqrt{h(H - h)}\). Holes at \(h\) and \(H - h\) land together, a hole halfway down throws farthest, and the path is \(y = x^2/4h\).

Range of a jetTwin holes and the greatest rangeThe jet as a projectile

14 min0%

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Bernoulli at Work

Bernoulli's theorem as a measuring tool and as a force. The venturimeter and the Pitot tube measure the speed of a flow from a difference of levels. Faster air above a wing or a roof gives a lower pressure and a lift. Sprayers draw liquid up with a fast air stream, and a spinning ball curves.

The venturimeterThe pitot tubeLift on a wing and a roofSprayers and the spinning ball

15 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

Open →

Glossary

The key words of this chapter, with the meaning this chapter gives them.

Bernoulli's theorem: In the steady flow of an ideal fluid, \(P + \tfrac12\rho v^2 + \rho gh\) is the same at every point of a streamline.
cavitation: The forming of bubbles of vapour where the pressure of a flowing liquid falls to nearly zero. The bubbles collapse violently and damage pumps and propellers.
critical speed: The speed above which a steady flow becomes turbulent. It is lower in a wider pipe and for a less viscous fluid.
dynamic lift: The upward force on a wing, a roof or a spinning ball, caused by faster flow and lower pressure on its upper side.
equation of continuity: For the steady flow of an incompressible fluid, \(A_1v_1 = A_2v_2\): the same volume crosses every section of a tube of flow each second.
head: A term of Bernoulli's equation divided by \(\rho g\), which makes it a length: the pressure head \(P/\rho g\), the velocity head \(v^2/2g\) and the height \(h\).
ideal fluid: A model fluid that is incompressible and non-viscous, with a steady and irrotational flow.
incompressible: Having a density that does not change with pressure. Liquids are nearly incompressible, and slow air behaves so too.
irrotational: Describes a flow in which small elements of the fluid do not spin about their own centres as they move.
Magnus effect: The sideways force on a spinning ball moving through a fluid, towards the side where the spin speeds up the passing fluid.
mass flow rate: The mass of fluid that crosses a section each second, \(\rho Av\), in kg/s.
Pitot tube: A tube bent to face into a flow. With a side tube it measures the speed of the flow from the difference of levels, \(v = \sqrt{2gh}\).
pressure energy: The work that pressure does in pushing a volume of fluid forward, P per unit volume. It is the P term of Bernoulli's equation.
speed of efflux: The speed with which a liquid flows out of a hole in a vessel. For a small hole at depth \(h\) in a wide open tank it is \(\sqrt{2gh}\).
stagnation point: A point in a flow where the fluid is brought to rest, such as the mouth of a tube facing the flow. Its pressure is higher by \(\tfrac12\rho v^2\).
steady flow: Flow in which the velocity of the fluid at every fixed point stays the same as time passes. Every particle passing a point follows the same path.
streamline: A curve whose tangent at every point gives the direction of the fluid's velocity there. In steady flow it is the path of a fluid particle. Two streamlines never cross.
streamline flow: Another name for steady flow, in which the fluid moves along fixed streamlines. Also called laminar flow.
throat: The narrowest part of a venturimeter or a nozzle, where the fluid is fastest and its pressure lowest.
thrust: The push of a jet on the vessel it leaves, equal to the momentum it carries off each second. For a jet from a hole of area \(a\) at depth \(h\) it is \(2\rho gha\).
Torricelli's law: A small hole at depth \(h\) below the free surface of a wide open tank lets the liquid out at \(v = \sqrt{2gh}\). This is the speed of a body falling freely through \(h\).
total head: The sum of the pressure head, the velocity head and the height, \(P/\rho g + v^2/2g + h\). It stays the same along a streamline of an ideal flow.
tube of flow: A bundle of streamlines through a small closed loop. No fluid crosses its sides, so it acts as a pipe with invisible walls.
turbulent flow: Irregular flow in which the velocity at a point changes from instant to instant and eddies form, mixing the fluid and wasting energy.
twin hole: In a tank on the ground with liquid of depth \(H\), the hole at depth \(H - h\). Its jet lands at the same point as the jet from depth \(h\).
venturimeter: A pipe with a narrow throat and pressure tubes on the inlet and the throat. It measures the flow rate from the difference of levels.
volume flow rate: The volume of fluid that crosses a section each second, \(Q = Av\), in m³/s. One litre per second is \(10^{-3}\,\text{m}^3/\text{s}\).