B.3. Fluids and Fluid Dynamics (HL)
The bit of the course that a lot of teachers dread, fraught with tricky ideas and concepts.
However, there are plenty of good resources that should help you get your head around things.
Then you can look forward to the really juicy fluid mechanics when you get to university...
Fluids are anything that can flow, i.e. liquids and gases. Gases can be compressed, as discussed in Chapter 3. Liquids on the other hand are essentially incompressible. Hydraulic systems make use of this incompressibility and use the properties of liquid pressure to exert large forces.
Before we start, I wanted to introduce you all to one of the longest running science experiments int the world - the Pitch Drop experiment in Australia. Essentially it is a funnel of pitch (tar-like substance) over a beaker, and people watch it drip... very slowly. Once every 15 years slowly. Since it started in 1927 it has dripped 9 times, the most recent being in 2014. See it here:
There's even a livestream 24-7 waiting for the 10th drop.
Find out more through an episode of one of my favourite podcasts, Radiolab.
This topic has a few tricky concepts to get your head around, so I've broken it up into the following sections.
Static Fluids : Archimedes' and Pascal's Principles : Understanding the basics of fluids
Continuity Equation : Starting to look at fluids moving
Bernoulli Equation : The reason planes can fly.
Stoke's Law : Drag slows down movement through fluids.
Laminar and Turbulent Flow : What are the different ways that fluids can flow?
We start the fluids topic with the discussion of static fluids - not moving. This starts with a few of the basic GCSE ideas of pressure and density, and builds upon these (revision here if you are a little rusty). CrashCourse have a nice summary of static fluids, that builds upon these ideas.
First we introduce Archimedes' principle, which states:
What this means in practical terms :
A submerged object will displace it's volume of fluid, therefore the buoyant force will be equivalent to the weight of that volume of fluid
(e.g. a cannonball with a volume of 2 l submerged in water will have a buoyant force equivalent to the weight of 2 l of water, i.e. about 20 N)
A floating object has a buoyant force equal to its weight. This means the volume of fluid it displaces will have a weight equal to the weight of the object.
(e.g. a floating tree weighing 2000 N will displace 2000 N of water, i.e. 200kg or a volume of 200 l).
PHET has a nice simulation which allows you to play around with floating and sinking objects to understand Archimedes' principle
(this one requires Flash, so try opening in Edge, rather than Chrome).
The follow simulation by Geogebra is lovely, and illustrates clearly what is going on when an object is floating or sinking. Change the object's density and look at what happens to the Free Body Diagram (particularly as it moves).
We then move on to discuss Pascal's principle, which governs pressure within fluids. We remember from GCSE that the pressure under a fluid = density x graviational field strength x depth.
"The pressure applied at one point in an enclosed fluid under equilibrium is transmitted equally to all parts of the fluid."
PHET have another nice simulation here looking at fluid pressure. The key point is that the shape of container is irrelevant, only the depth of liquid in the container is important!
Geogebra have a couple more to see the maths in more detail:
1) Pressure in two beakers - change the dimensions and look at P1 and P2, what are the important factors?
2) Fluid in a reservoir - what is the difference between gauge pressure and absolute pressure?
3) U-Tube Manometer - A manometer is a device used to measure gas pressure. Take a look at how it works.
Pascal's principle is particularly important in the study of hydraulics, using fluid pressure to exert huge forces. In a hydraulic system, the pressure in the fluid acts the same in all directions. A bit about the theory of hydraulics can be found here.
In principle, if the pressure throughout the fluid is constant, then:
F1/A1 = F2/A2.
For the example to the right, a small force at F1 will generate a larger force at F2 (though the distance moved by piston 2 will be smaller to ensure the same work done/ energy is conserved).
This hydraulics simulation by Geogebra is very clear. Take a look what happens to the system as you change the diameters of the pipes. Remember, work done = force x distance, so if the force we can lift increases, it doesn't move as far.
For a bit of interest while we're on the topic, VSauce has a nice video talking about self starting syphons - something that makes use of Pascal's principles to cause fluid flow from a syphon
The continuity equation is us starting to look at fluids moving - CrashCourse have another nice video which introduces the various topics that will be explored for the remainder of the topic.
Essentially the equation is based off the basic principle of the conservation of mass - i.e. if a certain mass of fluid enters the pipe, then the same mass must leave. This applies for all fluids, but if we assume a fluid is incompressible (i.e. a liquid) - meaning constant density, this means that the volume of fluid entering a pipe must equal the volume leaving - in the below example the green volume equals the red volume. We can describe this mathematically using the following:
A1v1 = A2v2
A = cross sectional area of pipe (m²)
v = fluid velocity (m/s)
This illustration also shows the fluid streamlines, lines which represent the movement of particles. They will be moving fastest when the streamlines are closest together. Take some time to explore this using the below simulation.
Another nice one from PHET select the second tab titled 'flow'. Take a look at what happens to the fluid flow as you change the diameter of the pipe by opening the flux meter. Key thing here is that the flow rate (in m³/s or L/s) stays constant, so if area decreases, speed must increase.
This continuity equation simulation by Geogebra illustrates what happens if you change the area of each section of the pipe. Again, flow rate is constant, though you can see the streamlines getting close together and the fluid speed up as it enters a narrow section of pipe.
The Bernoulli equation looks a bit scary on the surface of it, but we can start by looking at what this actually shows. Essentially the fluid pressure depends on the speed of the fluid. This is the reason that a plane's wing generates lift and why a football kicked with spin will curve in the air.
Veritasium has a nice little clip which illustrates the Bernoulli effect.
Now, let's dive into the equation itself. It is given in your formula book as:
½ρv² + ρgz + p = constant
... which looks a bit unpleasant. However, it basically comes from conservation of energy. The first term is similar to the kinetic energy of fluid, and the second term is the potential energy of the fluid.
Khan academy has a nice walkthrough of the derivation of Bernoulli.
ρ = fluid density (constant for liquids) (kg/m³)
v = fluid velocity (m/s)
z = height of fluid (m)
p = pressure of fluid at height 'z' (Pa)
This equation only starts to make sense once we apply it to a particular context. Let's start by looking at the spinning ball example (sometimes called the Magnus effect- why a spinning ball curves).
Friction with the surface of the spinning ball causes changes in air speed. We can see on the right hand side the streamlines are closer together, meaning the air travels faster.
Using our equation qualitatively, this means if air speed (v) is higher, the pressure (p) decreases. The ball therefore has a higher pressure on it's left hand side than it's right, meaning a net force pushes the ball to the right.
The right of the pipe has a larger diameter than the left, and by the continuity equation, this means that the speed must be higher on the left, which will affect the 'kinetic energy bit' of the equation - i.e. ½ρv².
The right hand side is higher up than the left, which will affect the 'potential energy bit' of the equation - i.e. ρgz.
We can use these values to find the pressure difference of the fluid in the pipe.
Now, let's look at a general example to explore the full equation. The typical IB example is shown below:
If you flick the tab in the top right of this Geogebra simulation to 'Bernoulli's Principle', you can visualise what happens if you play around with the different variables.
If you lower the height of the left hand side, see what happens to the pressure on the right - clearly if it moves upwards, it gains GPE, therefore pressure must decrease (KE stays the same due to the continuity equation still being obeyed).
Applications of Bernoulli
This is only the very basics of Bernoulli explained. For a more in-depth dive into the Bernoulli equation I recommend looking at the relevant pages of Cambridge IB.
We need to look at a few different applications including Venturi tubes, Pitot static tubes and fluid flow out of a container. Geogebra have a nice Venturi tube simulation - take a look to see how the fluid level changes as the tube radius and fluid density are changed.
Have a go at a few questions now to make sure you are able to apply Bernoulli to various contexts.
Until now, we have looked at ideal fluids (no viscosity). Of course, real world fluids behave very differently depending on their viscosity. Viscous fluids (e.g. syrup, honey) flow much less easily than less viscous fluids (e.g. water, air). Viscosity is a fluid property, measured in Pa s.
Stokes' Law is used to describe the drag force acting on a sphere as it moves through a fluid. Fluid mechanics are complicated, so we assume a sphere to make things easier (see the famous Physics spherical cow).
The drag force is given by:
FD = 6πηrv
FD = drag force (N)
η = fluid viscosity (Pa s)
r = radius of sphere (m)
v = velocity of sphere (m/s)
If we apply our ideas about terminal velocity, we know that when the sphere is falling at constant speed it must be in equilibrium, as in the Free Body Diagram below.
This drag force is shown in red (with vt representing the terminal velocity). The buoyancy force is shown in yellow - found using Archimedes principle, i.e. the weight of fluid displaced (volume of sphere x density of fluid x g). As the sphere is in equilibrium, the sum of these will equal the weight force downwards in green.
School Physics have a simple simulation demonstrating this (requires Flash, try Edge rather than Chrome)
Laminar Flow, Turbulence & Reynolds Number
This image of smoke rising nicely illustrates the difference between laminar and turbulent flow. Immediately above the candle, the flow is laminar and the fluid streamlines would be parallel. As the smoke rises its flow becomes turbulent, as can be seen when the plume billows out.
Smarter Every Day put together a nice little video on laminar flow, to give a bit of context to this stuff.
Here we can see the difference in streamlines between laminar and turbulent flows.
With laminar flow (top) the fluid travels slowest closest to the walls, and the fluid 'layers' do not mix.
Turbulent flow causes mixing of these layers, and the formation of vortices/ eddy currents.
The Reynolds number is a simple equation which allows us to predict whether a fluid flow is likely to be turbulent or laminar.
R = vrρ
R = Reynolds number (dimensionless)
η = fluid viscosity (Pa s)
ρ = fluid density (kg/m³)
v = fluid velocity (m/s)
r = radius of pipe/ container (m)
Although these values are not exact, in general for Reynolds numbers below 1000, we can assume the flow is laminar. Above 2000 we can assume the flow is turbulent. For 1000 ≤ R ≤ 2000, the fluid is in a transition stage between laminar and turbulent flow.
Looking at the equation, we can see that increasing velocity, pipe diameter or fluid density are more likely to cause turbulence, while decreasing viscosity can increase the likelihood of turbulence.
IBPhysics.org have a few nice summary notes and resources on this fluids stuff.
Cambridge IB have some excellent IB-specific resources on this topic, including textbook pages IB style exam questions and self marking MCQs. It's all free to access, however, it's behind a sign-in wall, so you will have to create an account.
You will need to download the file called "Physics Student Material", which has a .zip file containing lots of textbook resources. Under the folder "Option B" there is a file containing the relevant textbook information - the following pages are useful.
Statics : Notes p35 - p37 ; Questions p46, Q38-45
Continuity Equation : Notes p38 ; Questions p47, Q46-47
Bernoulli Equation : Notes p41 - p44 ; Questions p47, Q48-54
Stokes' Law : Notes p4 ; Questions p47, Q57
Lainar and Turbulent Flow : Notes p46 ; Questions p47, Q55-56
A question by question breakdown of the IB papers by year is shown below to allow you to filter questions by topic. As this is the IB Option, you will need to use the 'Paper 3' tab at the bottom. Hopefully you have access to many of these papers through your school system. If available, there may be some links to online sources of questions, though please be patient if the links are broken! (DrR: If you do find some broken links, please contact me through the site)
Use this grid to practice past IB questions topic by topic. You can see from the colours how similar the question topic breakdown is year by year. The more you can familiarise yourself with the IB question style the better - eventually you will come to spot those tricks and types of questions that reappear each year.