A-level Physics Coursework - Measuring Viscosity

A2 Physics Coursework Matt R****

Page 1 of 25

Can I determine the viscosity of a fluid through experiment?

The aim of my investigation was to explore the relationships between a number of variables which are

described by Stokes’ Law and ultimately to use these in order to calculate the viscosity of a fluid from

experimental results; my experimental values could then be compared against known ‘textbook’ values to

assess the success of my measurements.

With the exception of superfluids; all fluids - both liquids and gasses – exhibit the property of viscosity. The

viscosity of a fluid, usually measured in Pascal-seconds (Pa·s), describes its resistance to deformation and

hence the ease with which it flows. It is caused by internal fluid friction due to intermolecular forces, thus it

is affected by both temperature and pressure in gasses, and just temperature in liquids. Newtonian fluids

have viscosities that remain constant under constant conditions, regardless of the forces acting upon them.

“Thick” liquids that flow slowly (such as glycerol) have higher viscosities, whereas “thin” liquids (like water)

that flow easily have lower viscosities. There are many ways to measure viscosity; however I chose to do so

using a falling-sphere viscometer as this was the most viable method to be performed in the lab using

standard apparatus.

Stokes’ Law is an expression (figure 1) derived by George Gabriel Stokes that describes the drag force ( )

exerted on a sphere in a Newtonian fluid. The drag force is a measure of the frictional force (or ‘resistance’)

in Newtons experienced by the sphere as a result of the fluid being made to flow around it.

Fig 1.

Where:

is the dynamic viscosity of the fluid in Pa·s

is the radius of the ball in m

is the velocity of the ball in m s-1

In the form shown in figure 1, this expression was of limited use to me; I planned to take measurements by

dropping ball bearings through a vertical tube containing the specimen fluid, however needs to be

known to calculate the viscosity using this equation and this cannot be directly measured in this method.

By looking at the forces acting on a ball when falling through the fluid at terminal velocity, another more

relevant equation could be derived. When a ball is falling due to gravity there are three forces acting upon

it; the force of gravity ( ) acts downwards, whilst the drag force of the fluid ( ) and the buoyancy of the

ball in the fluid ( ) act in the opposite direction.

At terminal velocity, the net force acting on the ball must be zero as there is no acceleration (from the

equation ). Therefore the following basic statement could be made about the forces acting upon

the ball when moving at terminal velocity:

Fig 2.


Page 2 of 25

The weight of the ball (i.e. gravitational force, ) can be expressed in terms of volume and density:

Fig 3.

Where:

is mass in kg

is the gravitational acceleration, taken to be 9.81 m s-2

and are density in kg m-3


The buoyancy ( ) can be considered to be the weight of the displaced fluid, therefore the same equation

can be used as figure 3 but with the density of the fluid ( ) substituted in place of the ball’s density ( ).

The volume of the fluid displaced will equal the volume of the sphere, and the weight of the displaced fluid

will be the product of the volume and density. Figure 4 shows how all these expressions can be combined.

Fig 4.

The expressions for the different forces are combined as shown in figure 2. This relationship describes a ball

travelling at terminal velocity (when the net force is zero), so the term for velocity in the original Stokes’ law

equation now has to be made specific to terminal velocity .

is made the subject and the similar expressions for and are combined:

( )

The whole expression is then divided by :

(( )

)

Finally it is divided by 6. Though this could be done as part of the previous step, I separated this out to make

it easier to identify where the fraction of

in the final equation comes from.

(( )

)


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This was now a much more useful form of Stokes’ law, as velocity is very easy to measure. If all the other

terms in the equation were known and kept constant, the terminal velocity of the ball (determined by

measuring the time taken to pass between two points) falling through the fluid could be used to calculate

the fluid’s viscosity. The tidied and rearrangement form of the equation to make viscosity ( ) the subject is

shown in figure 5.

Fig 5.

( )

Where:

is the terminal velocity of the ball in m s-1


is the gravitational acceleration, taken to be 9.81 m s-2

is the density of the ball in kg m-3

is the density of the fluid in kg m-3


Now that I understood the underlying theory behind calculating viscosity I could return to my aim and

planning how I would perform my measurements. I was given access to three different fluids: water and

two different types of bubble bath.

I assigned each fluid a number:

Fluid 1 was water

Fluid 2 was the blue Tesco branded bubble bath

Fluid 3 was the orange “Tesco Value” bubble bath

Unfortunately - but unsurprisingly - there were no known values of viscosity for the two bubble baths, so I

decided that I would first perform the experiment with water and compare my calculated value for viscosity

with known values (as originally planned) to verify the theory and method, then I would repeat the

experiment to calculate the viscosity of the other fluids and estimate the confidence I could have in these

results.

The three 1.8m tall tubes were filled with the fluid a few days in advance of my measurement-taking; this

was to ensure that any air bubbles trapped whilst filling them would have time to float to the surface and

that enough time was allowed for the fluids to reach room temperature. The room in which the tubes were

located was air-conditioned and usually maintained at an approximately constant temperature of 20.5°c.

I placed the three tubes against a white wall so they would have a plain backdrop – this was to make the

location of the ball easier to distinguish whilst it was falling. I later improved this further by carefully

backlighting the wall behind the tubes with several lamps to provide even greater contrast with the dark

balls. Because I had decided to use a camera to film the balls’ descents, it was also essential to ensure that

the light from the lamps did not reflect on the front of the tubes as these bright reflective glares could mask

the paths of the balls on the video.


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Next I chose four ball bearings with differing radii; I had to ensure they were all made from the same metal

so they would have the same density (see the density calculations later in this investigation). It was

important that they were magnetic so that I could recover them from the bottom of the tubes after each

drop by using a strong magnet. To make referencing each ball easier I gave them descriptive names based

on their relative sizes – the names and radii of the balls I used can be seen in figure 6 below. I mention how

I measured the balls later in this investigation.

Fig 6.

“Large” – 12.50mm (1.25x10-2

m)

“Medium” – 9.50mm (9.50x10-3

m)

“Small” – 7.55mm (7.55x10-3

m)

“Tiny” – 4.77mm (4.77x10-3

m)

Next I needed to devise an accurate method of measuring the velocity of the falling balls. I knew that I

would need to measure the time taken for the ball to travel between two points of known separation (as

suggested by the equation in figure 7), but I instantly knew that I would have issues achieving reasonably

accurate results with this if measuring the time myself by operating a stopwatch / timer.

Fig 7.

⁄

I had already marked a 0.5m vertical section at the bottom of each tube (two clearly marked lines drawn

50cm apart) with which to base the velocity measurements on. I deemed 0.5m to be an appropriate

distance for a number of reasons:

Though I later performed some preliminary tests to explore and confirm this, I had to be confident that the

ball would reach terminal velocity before its velocity was measured (because the expression for calculating

viscosity assumes terminal velocity, as shown in figures 2 and 4). Taking into account a 10cm space at the

bottom of the tube to allow for the balls to collect at the bottom between runs without interfering with the

measurement, 50cm was the furthest up the tube from this that I could mark and still remain wholly

confident that the ball would be travelling terminal velocity when it entered this section.

From a performing a few quick online tests and games, I knew that my average reaction time was roughly

0.2 seconds, and after briefly experimenting with my apparatus I estimated that the large ball took around

half a second or less to pass through this 0.5m section at the bottom of the tube. Alone this seemed bad

enough, but two reactions were to be required on my behalf for each measurement – one to start a

stopwatch when the ball crossed the first mark and one to stop it as it crossed the second.

Add to this that it was also difficult to judge when the fast-moving ball passed each mark, and this

accumulation of error condemned this method as being ineffective for the situation.

I did not have access to any light gates that would fit around the tube, and even if I had, it wasn’t very likely

that the balls (the ‘small’ and ‘tiny’ balls in particular) would consistently break the narrow beams of light

on each run and trigger the gates. Without access to more complex and expensive light gate equipment,

this was not a viable option.


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Eventually I decided that I would use a camera to film each ball drop and calculate the time between two

points by analysing the video on a frame by frame basis. Even a 25fps (frames per second) video camera

would improve on the limitations of my reaction times, allowing the time to be approximated to the

nearest 0.04 seconds (assuming the location of the ball was clear in each frame).

A higher frame rate camera could be used to further improve on this. I had access to a 120fps camera;

however the resolution was much lower, thus reducing the accuracy with which the location of a ball could

be distinguished. I decided this was not a very beneficial compromise and so settled on using my hard disk

camcorder which filmed with a widescreen resolution of 720x576 pixels at 25fps (interlaced).

After considering the details, I realised I could exploit the interlacing of the videos capture by the camera to

my advantage and effectively obtain 50fps video, increasing the accuracy of my measurement to the

nearest 0.02 seconds.

In an interlaced video, each frame is split into two ‘fields’ (usually named either “upper and lower”, or “odd

and even” fields) which only contain half the vertical resolution of the full frame. These fields carry

alternate horizontal lines of the frame, hence the naming ‘odd’ and ‘even’ (if the lines were to be

numbered). See figure 8 below for a basic diagram of a full frame being split into alternate fields.

Fig 8.

Figure 8 is a very simplified representation of interlacing in which there is no motion and so it can be said

that both fields are combined to form a full frame. However, when motion is involved, the content of each

field is captured at slightly different times and is independent from its other field; each field can no longer

be thought of as half of a full frame, but is effectively its own frame (just with half the vertical resolution).

This is essentially the purpose of interlacing – it has origins in CRT (cathode ray tube) line-based displays

and was intended as a way to increase the ‘refresh rate’ (analogous though not always directly

interchangeable with ‘frame rate’) of a video signal without increasing the bandwidth required to carry it.

Though each field does not contain the full vertical resolution, this is not perceived by the human eye and

instead is seen as smooth video when played back at full speed.

By filming the ball drop with the camera rotated through 90° so that the path of the falling ball would be

parallel to the interlacing lines, the reduced vertical resolution would not impede my ability to pinpoint the

location of the ball on each frame; this was assuming that the smallest ball would be wide enough to span a

width of at least several pixels when captured on the video, which it easily would do and so this didn’t

present a problem. Detail in horizontal motion of the ball (represented by the vertical resolution of the

camera) was not so important to my investigation, except possibly to indentify and examine drops where

the ball interacted with the edges of the tube and was subsequently slowed down.

Full frame Upper field Lower field


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Conversely, the full horizontal resolution of each field would be orientated such that detail in the vertical

motion of the ball remained the same as it would be if the video were not deinterlaced.

Fig 9.

So if I broke the interlaced video of my camcorder down into individual fields, I could in effect double the

frame rate to 50fps. A horizontal resolution of 720 pixels gave the potential of tracking the vertical motion

of the ball within a matter of millimetres, depending mainly on the framing of the camera shot.

Figure 9 shows sequential stills taken every 2 frames from a test video of the large ball dropping through

the marked 50cm section at the bottom of the tube containing fluid 2, demonstrating the level of accuracy

that could be achieved in following the motion of a ball using this method.

Fig 10.

At this point, I judged that my method posed no significant hazards or risks. There was a higher risk of the

tube containing water being knocked over as this had a less sturdy/stable base; however I ensured that the

tubes were located up against a wall where they were unlikely to be accidentally knocked. Unintentional

ingestion of the fluids (for example, if I had got them on my hands then eaten food without washing my

hands first) would have been unpleasant but not dangerous. The balls could have presented a trip hazard if

they dropped onto the floor, but I kept them in a plastic container when not in use.

I also had to ensure that the magnet I used to remove the balls from the tube did not get placed close to

my camcorder, which could have corrupted data on the hard disk drive that the videos are stored on.


Page 7 of 25

Before making any further adjustments to the method, I first needed to verify that the balls reached

terminal velocity for the section of tube that I had marked out. These preliminary runs of the measurement

method also allowed me to test out the processing of the videos on the computer.

To decompress, deinterlace and trim the videos I used free video editing software named VirtualDub. I used

filters to break the video into its individual fields, and then resized these to match the original aspect ratio.

I then imported the AVI files saved from VirtualDub into an open source Java program called Tracker (see

figure 9). Tracker allows you to set of scaled axes over a video and then track the path of an object moving

relative to these axes simply by plotting the location of the object on each frame. The data could then be

exported in a comma-separated format that can then be opened in Microsoft Excel for further analysis.

Fig 11.

Figure 11 shows the results from a test drop. On a graph of displacement plotted against time, the gradient

equals the velocity, so the linearity of the graph represents a constant velocity and thus it can be assumed

that the ball is travelling at terminal velocity.

Following this, the first modification I made was to check the tubes were vertical using a spirit level and to

adjust bases of the tubes to make corrections where necessary. Whilst it was impossible to prevent

altogether without having a much wider tube, this was one step I could take to try and reduce the

probability of the balls interacting with the edges of the tubes as they fell (which could affect their velocity).

When setting up the camera I realised that its position relative to the tube could have a significant effect on

the results due to perspective and parallax errors.

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4

Dis

pla

cem

en

t (m

)

Time (sec)

Test drop 1 (fluid 2, large ball)


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The first of this type of error I looked at is illustrated in figure 12 – whilst the two pairs of points along the

line are equally spaced, from the marked point-of-view; points B will appear much closer together than

points A. In the context of the experiment, if the tube were to be filmed from a high angle the ball may

erroneously appear to decelerate (as the perceived distance it travels over a set period of time appears to

decrease). Conversely, if filmed from a low angle, the ball may appear to accelerate when it is in fact

moving at constant velocity. This kind of error could seriously affect the accuracy of my measurements.

Fig 12.

I decided to perform a few more test drops to confirm this idea. The graphs in figures 13 and 15 show the

data obtained from four test drops; all were filmed the same distance from the tube, but for two the

camera was positioned at a very high angle relative to the tube, and the other two filmed from the floor.

Figure 13 which displays the data from the high angle drops clearly indicates a slight curve in the trend

which relates to a measured deceleration (the gradient of the graph equals the velocity, so a lessening in

the steepness of the gradient indicates a slower velocity). This deceleration did not in fact happen – if

anything the ball should accelerate - but is a result of the error introduced by filming from a high angle.

Fig 13.

0

20

40

60

80

100

120

140

160

180

200

0 0.2 0.4 0.6 0.8 1

Dis

tan

ce (

cm)

Time (sec)

High Angle Test Drops

Large Ball

Medium Ball

Two pairs of equally spaced points are

marked on a ruler. When viewed from an

oblique angle they appear farther apart.

viewpoint

Viewed almost head-on

Viewed at an angle


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To present this in a different way, I plotted a graph of velocity against time for the drop of the large ball.

The gradient of this graph represents the acceleration – a positive gradient would signify acceleration, no

gradient would mean that the velocity was constant, and a negative gradient would indicate deceleration.

As can be seen in the graph below (figure 14), there was a clear negative gradient which showed that the

ball appeared to decelerate over the course of the drop.

Fig 14.

As was expected, the opposite effect was observable in the drops filmed from a low angle (figure 15).

Fig 15.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.2 0.4 0.6 0.8 1.0

Ve

loci

ty (

ms-1

)

Time (sec)

High angle test drop, v/t graph

0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8

Dis

tan

ce (

cm)

Time (sec)

Low Angle Test Drops

Large Ball

Medium Ball


Page 10 of 25

Parallax error could also affect the moment when the ball is judged to be crossing a mark on the tube.

Figure 16 shows how a ball falling past a mark appears to be lined up with the mark at very different points

depending on whether the tube is viewed from A, B or C.

Refraction of light at the medium boundaries (fluid, tube, air) could also introduce similar errors if the

marks on the tube were viewed at an angle other than head-on.

Fig 16.

It crossed my mind that it was possible to accurately compensate for the first type of error using

trigonometry if the relevant distances and angles between the camera and tube were known, however this

would have introduced serious complication intro the processing and analysis of the videos. The Tracker

software is only intended for analysing videos with a constant scale applying across the whole frame.

I decided it was best to try and reduce these errors as much as possible through the positioning of the

camera before taking any measurements.

Each of these errors depended on the angle from which the tube or mark was viewed and ideally would be

viewed head-on. However the ball had to be observed crossing and travelling between two marks, and it

would be a physical impossibility to position and frame the camera shot such that both marks were filmed

head-on! The only solution was to increase the distance between the tube and the camera as much as

possible, thus reducing the angle that each mark was viewed from. Figure 17 illustrates this.

Fig 17.

At viewpoint A, the ball appears to be lined up

with mark after it actually crosses the mark

When viewed head on from viewpoint B, the

crossing of the mark can be correctly identified

At viewpoint C, the ball appears to be in line

with the mark before it crosses it

A single ball is

dropped through a

marked tube and

observed from three

different angles.

A greater distance between the camera

and the tube reduces the angle at which

both marks on the tube are viewed from.

Camera close to tube Camera further from tube

The camera is elevated and

positioned such that it

points horizontally to an

imaginary central point

between the two marks. 50 cm marked section

camera


Page 11 of 25

These two stills (fig 18) were taken from test videos and show the significant difference caused by the

distance between the camera and a marked section of tube – the first was filmed 3 metres away whilst in

the second the camera was placed just 85 cm away from the tube.

Fig 18.

The horizontal centreline of both images was marked as a reference point. The bottom tape marking is to

be ignored as this piece of tape was not applied to the tube straight to being with. In the first image, the

edges of the top tape marking appeared very straight and horizontal. In the second, the view of the tape

marking exaggerated the curvature of the tube due to the greater angle between the camera lenses’

midpoint and the tape marking. The perspective seen in the first image was much more desirable for taking

accurate measurements in my investigation.

Eventually I settled on filming the ball drops with a distance of 3m between the tube and the camera. The

camera would be elevated off the ground such that it would lie on an imaginary line drawn perpendicular

from the midpoint of the 50cm marked section of the tube (as shown in figure 17) – taking into account the

base/stand of the tube and the small segment below the marked section, this usually meant having the

camera at an elevation of roughly 35cm. I also checked that the camera was horizontal using a spirit level.

As discussed previously, the camera was rotated through 90 when filming the drops such that the

horizontal reference of the camera was parallel to the vertical tube.

Once the camera was set up to these ‘specifications’, it was usually already lined up with the marked

section and the shot could be framed simply by adjusting the zoom such that the marked area filled most of

the frame. I could tell when the camera was positioned properly because the top and bottom markings lay

3.00m 0.85m

Curvature of

tube is more

apparent

Greater angle between the

camera’s horizontally central

‘horizon’ and the marks.

Marks are

viewed from a

very shallow

angle.


Page 12 of 25

equal distances from the top and bottom of the frame respectively without any adjustment to the angle of

the camera needed.

Occasionally however, some adjustment was needed – this was due to the camera slipping in the clamp of

the stand I had made to hold it. The camera was held to a sturdy stand using a single clamp with some

protective polystyrene padding between the clamp and the camera in order to protect the camera.

The thick polystyrene allowed me to close the clamp more tightly around the camera as the pressure from

the small metal clamp ‘fingers’ (which would otherwise have damaged the camera) was spread across a

larger surface area and, within reason, the polystyrene would protect the camera from small over

tightening of the clamp by deforming and compressing under the extra pressure before the camera would.

To avoid knocking the camera out of position when starting and stopping the recording (and consequently

having to carefully reposition it), I used the infra-red remote control that came with the camera.

Lastly I considered the effect of air bubbles on my results – dropping the balls such that they were not

released whilst touching the surface of the fluid would result in a significant amount of air being trapped on

the underside of the ball for the duration of its descent. This would almost certainly have an effect on the

velocity of the ball for a number of reasons, including: the fluid resistance depends on the form (shape) of

the object passing through the fluid and the air would alter this; the air has a different density to the ball;

the air would cause turbulence in the fluid. Also, Stokes’ Law only applies to spherical objects.

Once a ball with air trapped underneath it reached the bottom of the tube, the air would rise back up the

tube as a bubble, spreading smaller air bubbles throughout the fluid as it descended; these would possibly

have an effect on any subsequent ball drops as they would affect the continuity of the fluid.

I filmed a large number of test drops to investigate the effect of air bubbles in the fluids on the falling balls,

but the interactions and behaviours I observed appeared to be very inconsistent and unpredictable. I

guessed that the theory and modelling of these interactions would be extremely complex and not worth

trying to pursue any further, especially considering that there would be no benefit for this investigation in

greater understanding of them, only in knowing that they occurred and that it was best to try and minimize

their incidence.

I concluded that I would try to release each ball from such a position that it was submerged in the fluid at

the top of the tube, and only do so once I had checked that there was no air bubbles trapped on the surface

of the ball.

As well as my velocity measurements, I also needed to know the values for some of the other terms in the

equation I derived (figure 5). These included the diameter of each different ball, the density of the balls,

and the density of the three fluids used. These can be seen in the tables below (figures 5 and 6).

I made measurements of the diameter (D) of each ball using digital veneer callipers that gave readings to

the nearest 0.001m. Spherical objects can be difficult to accurately measure with callipers, so to ensure

that I was always measuring the full diameter of each ball I would first roughly close the callipers on the ball

then check this was the largest part of the sphere by trying to pass it between the jaws – if it couldn’t pass

between the separated jaws then I knew that I hadn’t quite measured full the diameter. Usually, if this

were the case, then carefully pushing the ball through would widen the jaws to the correct width.


Page 13 of 25

Using the diameter measurements I then calculated the volume (V) of each ball using the equation for the

volume of a sphere shown as shown in figure 19 (where the radius in metres, R, is half of D).

Fig 19.

The mass measurements (M) were made using a set of digital balances with a precision of 0.05g. Three

measurements were made for each ball and then a mean value (m̅) calculated.

From all these measurements (shown in figure 20), the average density of the four balls could be

calculated. The equation for density can be seen in figure 21.

Fig 20.

Ball D(m) V (m³) m 1 (kg) m 2 (kg) m 3 (kg) m̅(kg) Density (kg m¯³)

Large 2.50x10-2 8.18x10

-6 6.38x10-2 6.39x10

-2 6.38x10-2 6.38x10

-2 7.81x103

Medium 1.90x10-2 3.59x10

-6 2.80x10-2 2.82x10

-2 2.81x10-2 2.81x10

-2 7.82x103

Small 1.51x10-2 1.79x10

-6 1.40x10-2 1.40x10

-2 1.40x10-2 1.40x10

-2 7.81x103

Tiny 9.53x10-3 4.53x10

-7 3.50x10-3 3.50x10

-3 3.60x10-3 3.53x10

-3 7.80x103

Average Density: 7.81x103

Fig 21.

From the lack of variation in the densities I could draw the assumption that all the balls were made from

the same metal. The very slight differences were likely to be caused by defects in the balls (such as

scratches in the surface) or small errors in the measurements. These were possibly emphasised when

rounding these measurements (and the subsequent calculated density results) to three significant figures.

For the purposes of the investigation I treated all the balls as sharing the same density of 7.81x103 kg m-3.

I also needed to know the densities of the fluids I was to be testing (values for ). I did this by weighing a

known volume of each fluid.

Fig 22.

Fluid tested Mass including container (kg) Mass of fluid (kg) Density (kg m¯³)

Fluid 1 0.1175 0.0998 998

Fluid 2 0.1196 0.1019 1019

Fluid 3 0.1198 0.1021 1021

These measurements were made under the following conditions:

Mass of measuring container (kg): 0.0177

Volume of fluid (m³): 0.0001

Temperature (°c):

20.6


Page 14 of 25

Now that I had developed and improved my method, I was ready to carry out my final set of

measurements. I chose to make a total of 16 velocity measurements for each fluid; 4 drops for each of the

4 balls. This large number of repetitions for each fluid would hopefully increase the accuracy with which I

could determine their respective viscosities.

It was a lot easier and more efficient to process the videos in large batches, so I chose to perform all my

measurements in one session and analyse them afterwards. This would also allow me to ensure that

affecting factors such as temperature remained close to constant throughout all the measurements.

I encountered very few difficulties when taking my measurements – I chose to discard some drops that

were visibly turbulent or appeared to be affected by air bubbles, but mostly the balls appeared to drop

smoothly and consistently. Completing my measurements took slightly longer than expected due to the

time I had to spend setting up my camera for each fluid, and in the end I utilised the entire remainder of my

allocated lab time.

Once I had analysed all my videos, I imported the distance/time data for each ball drop from Tracker into

Microsoft Excel, where I plotted a graph of this data for each fluid and ball size. I allowed Excel to plot its

own linear trend lines on the charts, and the gradient of each of these lines was equal to the terminal

velocity of the ball for that drop. The different runs had varying y-intercepts which were simply due to the

way in which the axes were setup up and the ball tracked in Tracker, and would not have any effect on the

results as it was only the gradient that was of importance.

The first set of results I analysed were those for fluid 1, water...

v1 = 1.00 m/s

v2 = 1.24 m/s

v3 = 1.28 m/s

v4 = 1.01 m/s

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4 0.5

Dis

tan

ce (

m)

Time (sec)

Fluid 1, tiny ball

Run 1 Run 2 Run 3 Run 4


Page 15 of 25

v1 = 1.43 m/s

v2 = 1.50 m/s

v3 = 1.31 m/s

v4 = 1.29 m/s

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4

Dis

tan

ce (

m)

Time (sec)

Fluid 1, small ball


v1 = 1.56 m/s

v2 = 1.62 m/s v3 = 1.57 m/s

v4 = 1.55 m/s

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4

Dis

tan

ce (

m)

Time (sec)

Fluid 1, medium ball



Page 16 of 25

At a glance I was relatively happy with how these results looked; the velocity measurements of the four

runs for each ball size all seemed to agree with each other (had similar gradients), more so was the case in

the larger balls – I observed that the range of the velocity measurements for each ball decreased as the size

of the ball increased, from a range of 0.28ms-1 for the tiny ball to just 0.04ms-1 for the large ball.

From these results I plotted a graph of Vt against R2. I chose to do this as it was a relationship implied by the

equation in figure 5; if the experimental data conformed to the theory then Vt should be proportional to R2

and result in a linear trend passing through the origin when plotted. The gradient (G) of this graph could

then be used to calculate a value for the viscosity of the fluid using the equation at the bottom of figure 23.

Fig 23.

( )

( )

This was a method of combining all the data from the 16 ball drops in order to calculate the viscosity.

Unfortunately, the plotted data for water did not fit the relationship that I expected. See graph “Vt against

R2, fluid 1“ at the end of the investigation.

v1 = 1.66 m/s

v2 = 1.62 m/s

v3 = 1.61 m/s v4 = 1.62 m/s

0.0

0.2

0.4

0.6

0.00 0.10 0.20 0.30 0.40

Dis

tan

ce (

m)

Time (sec)

Fluid 1, large ball



Page 17 of 25

There was some linearity between the values of the tiny, small and medium sized balls; however this did

not pass near the origin, nor could it be extended to include the large ball. I attempted to estimate an

approximate value for the viscosity based on the gradient of a line drawn through the first three data sets,

however the value I calculated of around 4.0 Pa·s, which differed from the textbook value for water at

room temperature (roughly 1.0x10-3 Pa·s) by such a large amount that I concluded this data set was of little

use except for determining the source of the error in my results.

By substituting a known value for the viscosity of water into the equation I could calculate the theoretical

linear gradient for my Vt against R2 graph – this revealed that the gradient should be several orders of

magnitude greater than the kind of relationship I was witnessing on my graph; this showed that the balls

were falling a lot slower than be expected, indicating that other forces were acting upon them.

I later hypothesised that a possible cause for this was associated to the relatively small diameter of the

tube – the equations for Stokes’ Law assume an ideal, though impossible to obtain, infinite body of fluid for

the object to fall through, yet I used cylindrical tubes I used with fairly narrow diameter of 5.5cm.

The compressibility of liquids is very small and generally negligible so they are usually treated as

incompressible. In an ideal volume of fluid, the fluid displaced by a ball as it falls can simply move around

sides of the of the ball to occupy the space above the ball, which in turn ‘pushes’ the fluid that was at the

sides of the ball further outwards – this is shown in figure 24. An infinite volume of fluid is theoretically

ideal for modelling the flow of an incompressible fluid as the effects of this horizontal displacement will

continue indefinitely throughout the entire body of the fluid.

Fig 24.

source: http://en.wikipedia.org/wiki/File:Stokes_sphere.svg

However, in the context of my experiment, this flow is constricted by the dimensions of the tube; the

bottom of the tube is sealed so the displaced fluid still has to move around the edges of the sphere to

occupy the space above it; however there is a limited cross-sectional area around the ball through which

this displaced fluid can flow. The velocity of the upwards flow around the edges of the ball has to increase.

This has the overall effect of slowing the ball and reducing the terminal velocity that the ball can reach.

This effect should become much greater as the radius of the ball increases (more correctly, as the ratio

between the radius of and tube to the radius of the ball decreases), as a relatively larger volume of fluid

would be displaced and have to flow through a smaller space.

Next I moved onto plotting the data for the second fluid.

http://upload.wikimedia.org/wikipedia/commons/a/ae/Stokes_sphere.svg


Page 18 of 25

v1 = 0.24 m/s

v2 = 0.24 m/s

v3 = 0.23 m/s

v4 = 0.26 m/s

0.0

0.2

0.4

0.6

0.0 0.5 1.0 1.5 2.0 2.5

Dis

tan

ce (

m)

Time (sec)

Fluid 2, tiny ball


v1 = 0.97 m/s

v2 = 0.98 m/s

v3 = 0.93 m/s

v4 = 0.99 m/s

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Dis

tan

ce (

m)

Time (sec)

Fluid 2, small ball



Page 19 of 25

v1 = 1.19 m/s

v2 = 1.24 m/s

v3 = 1.40 m/s

v4 = 1.17 m/s

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4 0.5

Dis

tan

ce (

m)

Time (sec)



v1 =1.28 m/s

v2 = 1.51 m/s v3 = 1.40 m/s

v4 = 1.41 m/s

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4

Dis

tan

ce (

m)

Time (sec)

Fluid 2, large ball



Page 20 of 25

Again the data did not closely fit the Vt R2 relationship as I had hoped, however it appeared to be more

useful than the data from fluid 1; I could draw a curve of best fit through all the points that looked as if it

would pass close to the origin should it be extrapolated.

The slope of this curve decreased as R2 increased. I postulated that the curve was due to the effect I

discussed on page 17 – the larger the radii of the ball (and thus the larger R2 is), the greater the strength of

this effect and so the more the terminal velocity of the ball was reduced from its hypothetical “true” value.

One could visualise where the theoretical linear trend might pass, and how the actual line curved further

away from this line as R2 increased.

I postulated that, as they would have been much less affected by this error, I could produce an estimate of

the viscosity by drawing a line, roughly a tangent to the curve, which passed through the points for the two

smallest balls. I did so, as can be seen on the chart “Vt against R2, fluid 2“, and calculated an estimated

viscosity of the fluid of 0.70 Pas based on the gradient of this tangent and the calculation in figure 25.

Fig 25.

(

)

Lastly was the data for fluid 3.

v1 = 0.31 m/s

v2 = 0.34 m/s

v3 = 0.33 m/s

v4 = 0.31 m/s

0.0

0.2

0.4

0.6

0.0 0.3 0.6 0.9 1.2 1.5 1.8

Dis

tan

ce (

m)

Time (sec)

Fluid 3, tiny ball



Page 21 of 25

v1 = 0.91 m/s

v2 = 0.84 m/s

v3 = 0.92 m/s

v4 = 0.92 m/s

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Dis

tan

ce (

m)

Time (sec)

Fluid 3, small ball


v1 = 1.16 m/s

v2 = 1.17 m/s

v3 = 1.11 m/s v4 = 1.20 m/s

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4 0.5

Dis

tan

ce (

m)

Time (sec)




Page 22 of 25

The chart “Vt against R2, fluid 3“ based on this data shared a lot of similarities with the chart for fluid 2,

though the range of velocities for each ball size appeared less than that for fluid 2 – there was slightly more

consistency in the measurement of the terminal velocities. The line of best fit also passed closer to the

origin. Again I calculated an estimate of the viscosity, as can be seen in figure 26 below.

Fig 26.

(

)

I had now estimated the viscosity of both fluids 2 and 3; however I could have little confidence in the

accuracy of these results. I had originally planned to compare my results for fluid 1 (water) with known

values in order to verify the method, but my results for fluid 1 were not even suitable to gain a reasonable

‘ball-park’ estimate of viscosity from, thus I could not have any certainty that my method was valid.

I identified a number of limitations in my investigation. Firstly there was the error related to the relatively

small diameter of the tube restricting the flow of fluid around the ball that I identified and briefly explored

on page 17 – I believe this had a significant effect on my results (reducing the terminal velocity, with

greater effect with the larger balls) and was one of my main sources of error.

I later discovered and researched another possible major source of error. The falling ball can experience

both viscous drag – the type described by Stokes’ Law – and inertial drag, which does not depend on

viscosity. The proportions of these forces being exerted on the ball are described by its Reynolds number.

v1 = 1.35 m/s

v2 = 1.43 m/s

v3 = 1.41 m/s v4 = 1.52 m/s

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4

Dis

tan

ce (

m)

Time (sec)

Fluid 3, large ball



Page 23 of 25

“In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the

ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two

types of forces for given flow conditions.”

Source: http://en.wikipedia.org/wiki/Reynolds_number

In the context of this investigation, the Reynolds number could be calculated using the equation outlined in

figure 27. As both the radius of the ball and the viscosity of the fluid are terms in the equation, the

Reynolds number would be different for each ball-size/fluid combination in my investigation.

Fig 27.

Where:

is the density of the fluid in kg m-3


is the diameter of the ball in m

is the velocity of the ball in m s-1

A condition of Stokes’ Law is that it is only applicable when the Reynolds number is less than 1, though this

would ideally be less than 0.1 as this is associated with completely laminar flow; when the fluid flows

smoothly without turbulence, as if in layers (as depicted by the streamlines on the diagram in figure 25).

When the Reynolds number is higher, inertial drag (which does not depend on viscosity) become more

dominant than viscous drag and so Stokes’ Law becomes much less relevant. According to this equation, a

ball falling with a greater velocity is more likely to experience turbulence and inertial drag, as is one falling

through a less viscous fluid (which would also allow it to fall faster).

I could not calculate the Reynolds numbers for fluids 2 and 3 as I was not confident in the validity of my

viscosity estimates; however I could calculate the hypothetical values for the four balls using the averages

of my measured velocities, my measured density and the textbook value for the viscosity of water at 20°c.

Fig 28.

All of these were a lot larger than the Re<1 condition of Stokes’ Law. This implied that the balls falling

through the water experienced a lot of turbulence and inertial drag and that their terminal velocities would

have been much less dependent on the viscosity of the fluid than I had assumed.


Page 24 of 25

As water was the least viscous fluid I used (I could determine this through basic observation), I could

assume that these were the largest values of Re for my investigation, and that the Re values for the other

more viscous fluids would be less and so the conditions closer (to some extent) to those required for the

correct application Stokes’ Law; this possibly explained why the data for fluids 2 and 3 fitted noticeably

closer to the Vt R2 relationship than for fluid 1.

Another possible limitation was the roughness of the surface of the balls. There are several components to

drag, including form drag and skin friction. According to the Wikipedia articles on “Drag (physics)” and

“Parastatic Drag”; for most situations involving spherical spherical objects, form drag contributes to

approximately 90% of the total drag. Form drag is related to the shape of the object and the way in which

the fluid is made to flow around it, and is the component of drag taken into account in Stokes’ Law.

Thus it could be assumed that not all the drag was accounted for in the equations I used. The remainder of

the total drag was likely to be skin friction – friction between the surface of the object and the fluid. A

rough surface will increase this friction and increase the drag force acting on the ball, thus reducing its

terminal velocity. This force is directly related to the surface area of the ball in contact with the fluid, and so

will increase in relation to the radii of the ball, much like the other error I identified.

I concluded that the terminal velocity of each ball was increasingly affected by the cumulative effect of

these identified factors as the size of the ball I used increased. My Vt against R2 graphs supported this idea,

as for all three fluids the slope decreased as R2 increased (where a linear relationship was expected). My

very linear distance against time graphs showed that I had been measuring the terminal velocity of the ball,

but this terminal velocity may have been reduced by other forces acting on the ball that I didn’t take into

consideration. It was for this reason that I deemed it acceptable to estimate the viscosity of fluids 2 and 3

based on the data from the two smaller balls as these should have been much less affected by the effects I

identified, however these estimates were probably still affected to some extend (I would reason they were

likely to be higher than the true value) by these effects.

In hindsight, I paid a lot of attention towards my approach for measurement taking and performed

numerous and quite extensive premilinary testing to improve it, however I didn’t do any testing or analysis

of my data to verify the accuracy of the method and theory as a whole prior to committing to taking my

final measurmements. In other words, I had checked that I was measuring the terminal velocity of the

falling balls and was able to measure it with considerable accuracy and precison, however I failed to

identify (until the analysis of my results) that there were considerable errors in the method; other forces

acting upon the balls and affecting their terminal velocity.

Overall I am happy with the measurement method I developed and would definitely continue to use a

camera if I were to develop this investigation any further. Taking the errors into consideration, an lage

improvement on this method would be to use tubes with a much larger diameter and balls with much

smaller radii (ideally as small as possible whilst still remaining discernible on the video footage). This would

lessen the problem of edge effects, reduce skin friction (due to a smaller surface area in contact with the

fluid), and would give the balls much lower Reynolds numbers along with the more desriable flow

properties associated with these lower values of Re.

In response to my original investigation title, I was unable to successfully determine the viscosity of a fluid;

however I learnt a lot about the falling sphere method, identified and researched several factors that might

have impeded my investigation, and deliberated improvements that would hopefully allow me to

successfully determine the viscosity of a fluid through experiment if I were to develop it further.


Page 25 of 25

Bibliography

Advanced Physics [Book] / auth. Adams Steve and Allday Jonathan. - [s.l.] : Oxford University Press, 2000. -

pp. 94-95. - ISBN 0-19-914680-2.

Buoyancy [Online] // Wikipedia. - 7 Nov 2009. - http://en.wikipedia.org/wiki/Buoyancy.

Determination of the viscosity of glycerol by the falling-sphere method [Online] / auth. Uselman Sonja //

Department of Chemistry, Concordia College. - 7 Dec 1998. - 05 Nov 2009. -

http://www.cord.edu/faculty/ulnessd/legacy/fall1998/sonja/stokeswrite-up.htm.

Drag Force [Online] // Wikipedia. - 22 Nov 2009. - http://en.wikipedia.org/wiki/Drag_force.

Experiment 8 - Falling Sphere Viscometer Experiment [Online] // Purdue School of Engineering and

Technology, IUPUI. - 14 Nov 2009. - http://www.engr.iupui.edu/me/courses/me310lab/experiment8.pdf.

Laminar flow [Online] // Wikipedia. - 24 Nov 2009. - http://en.wikipedia.org/wiki/Laminar_flow.

Laminar, Transitional or Turbulent Flow [Online] // The Engineering Toolbox. - 25 Nov 2009. -

http://www.engineeringtoolbox.com/laminar-transitional-turbulent-flow-d_577.html.

Parasitic Drag [Online] // Wikipedia. - 22 Nov 2009. - http://en.wikipedia.org/wiki/Parasitic_drag.

Reynolds number [Online] // Wikipedia. - 23 Nov 2009. - http://en.wikipedia.org/wiki/Reynolds_number.

Stokes' Law [Online] // Wikipedia. - 3 Nov 2009. - http://en.wikipedia.org/wiki/Stokes'_law.

Understanding Physics for Advanced Level [Book] / auth. Breithbaupt Jim. - [s.l.] : Stanley Thornes

Publishers, 1987. - 3rd Edition : pp. 142-143.

Viscosity [Online] // Wikipedia. - 21 Nov 2009. - http://en.wikipedia.org/wiki/Viscosity.

Viscosity [Online] // Transtronics. - Nov 17 2009. - http://wiki.xtronics.com/index.php/Viscosity.

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A-level Physics Coursework - Measuring Viscosity