Date of Experiment: 28/02/2021
Date of Report: 29/02/2021
Liquid Viscosity
Experiment conducted by
Abstract
The aim of this experiment is to find the liquid viscosities of three liquids, glycerine, castor
oil, and olive oil. This experiment is mostly dependent on the theory of viscosity and laminar
flow, and the factors contributing to their values. For instance, the radius of the ball bearing
falling through the viscous liquid was changed within the range of 1-4mm to observe the effects
of changing the size of the object on the velocity through which the ball falls through the tube.
The value of viscosity was calculated for each liquid using three different methods, but none of
which proved to be accurate because of the high percentage difference when compared to the
corresponding literature values (45-90% range). Therefore, it was concluded that the methods of
the experiment were flawed.
1. Introduction
The viscosity of a fluid is its resistance to flow and the resistance of the movement of
an object through a fluid (Nave, n.d.). Viscosity is an important property for diffusioncontrolled chemical reactions (“Properties of liquids”, n.d.). The aim of this experiment is
to investigate the viscosities of three liquids: castor oil, olive oil and glycerine by
dropping steel spheres of varying diameters through a cylinder filled with these fluids at
room temperature. This experiment examines dynamic viscosity; it is a property of fluids
that undergoes laminar flow (“Liquid Viscosity”, n.d.).
Figure 1. Laminar flow and parabolic velocity within tube
(“Hydraulic Resistance”, n.d.).
1
Laminar flow is a type of fluid motion that is defined by the smooth flow of fluids in
layers that do not mix (“Viscosity and Laminar Flow”, n.d.). In figure 1, the lines shown
are called streamlines, these are smooth and continuous because of laminar flow (ibid).
The velocity of flow is zero closer to the walls and increases near the center of the tube,
therefore resulting in a parabolic velocity distribution as seen in figure 1 (Nave, n.d.).
Applications of laminar flow are for cases in where there are somewhat high-viscosity
fluids, small flow channels, and slow-moving fluids (“Laminar Flow”, n.d.). Thus,
laminar flow can be used as a valid assumption in the experiment since the experiment
meets the conditions where laminar flow could be present. Therefore, the motion of the
steel balls through the viscous liquids in the tube could be predicted due to this flow.
Additionally, to aid in predicting the motion of the sphere balls, the forces acting on the
sphere should be known.
Figure 2. A sphere falling in a long cylindrical tube with viscous liquid, with three forces acting
on it (Al Raeesi, 2021).
2
There are three forces acting on the sphere as seen in figure 2:
1- the force of gravity 𝐹𝑔 acting downwards (Equation 1),
4
𝐹𝑔 = 3 𝜋 𝑟 3 𝜌𝑏 𝑔
(1)
where the mass m is given by the volume multiplied by the density of the steel sphere
4
3
𝜋 𝑟 3 𝜌𝑏
(r is the radius of the sphere and 𝜌𝑏 is the density of the steel sphere).
2- the drag force 𝐹𝑑 acting upwards (Equation 2),
𝐹𝑑 = 6𝜋𝜇𝑣𝑟
(2)
where v is the terminal velocity and 𝜇 is the dynamical viscosity of the fluid.
This force is a result of the friction of the fluid against the surface of the sphere (“Liquid
Viscosity”, n.d.). This drag force is given by Stokes’ law, which is an equation that represents
the “settling velocities of small spherical particles in a fluid medium” (“Stokes’ law”, n.d.). It
applies in measuring viscosity of fluids where there is laminar flow (ibid).
3- the buoyancy force 𝐹𝑏 acting upwards (Equation 3),
𝐹𝑏 =
4
3
𝜋 𝑟 3 𝜌𝑓 𝑔
(3)
where 𝜌𝑓 is the density of the fluid.
3
Buoyant force is the upward force exerted on an object that is immersed in a liquid (“Buoyant
Force”, n.d.). Buoyancy is the phenomena as a result of the buoyant force; it is a concept applied
for submarines, hot air balloons, swimmers, etc. (ibid). In the experiment, inside the cylindrical
tube, the pressure at the bottom is greater than that of the top of an object immersed in the fluid;
this pressure difference is what causes the net upward force on the object (ibid). This is defined
as buoyancy. This upward force is equal to the weight of the displaced liquid (Figure 3). This is
known as Archimedes principle: “Any object, totally or partially immersed in a fluid, is buoyed
up by a force equal to the weight of the fluid displaced by the object” (“Properties of liquids”,
n.d.). For an object to float, the density of the object must be less than that of the fluid; and for an
object to sink, the density of it should be greater than that of the fluid (“Buoyant Force”, n.d.).
Figure 3. The buoyant force is equal to the weight of the fluid displaced by object (“Buoyant
Force”, n.d.).
To calculate the dynamic viscosity 𝜇 in the experiment, the equations 1,2, and 3 are rearranged
such that the three forces are at equilibrium when the sphere is at terminal velocity (“Liquid
Viscosity”, n.d.). This is given by Equation 4,
2𝑟 2
𝜇 =
𝑔( 𝜌𝑏 − 𝜌𝑓 )
9𝑣
(4)
4
This equation does not take into consideration the effect of the drag of the fluid caused against
the cylinder walls (ibid); therefore Equation 5 is the true viscosity, which takes into account the
small effect of the drag. It is important that this is taken into consideration in order to have
accurate results with all the elements accounted for.
𝑑
𝑑 3
𝑑 5
𝜇𝑐 = 𝜇 [1 − 2.104 ( ) + 2.09 ( ) − 0.95 ( ) ]
𝐷
𝐷
𝐷
(5)
Here, 𝜇𝑐 is the true viscosity, d is the diameter of the sphere and D is the diameter of the tube.
2. Method
The equipment in this virtual experiment included a long cylindrical tube attached to a stand,
two rubber bands around the cylinder separated by a distance to determine position A at the top
and position B at the bottom, a spherical steel ball of radius 1, 2, 3, and 4 mm, three liquids at
room temperature (castor oil, glycerine, and olive oil), and a timer. Firstly, the radius of the steel
ball and the fluid was chosen from the list of options. Then, the ball was dropped from the top of
the tube into the center. After the ball passed the top rubber band A, the timer was started and
stopped after reaching the bottom rubber band B. The same procedure was repeated five times
for each radius and liquid chosen.
5
Figure 4. Screenshot of the virtual experiment with the equipment and settings.
3. Results
For all data:
-
Error in radius: ± 0.1mm
-
Error on time (instrumental): ± 0.01s
-
Error on distance: √(∆𝑥)2 + (∆𝑦)2 = ± 0.7mm
Literature values of density (𝑘𝑔/𝑚3) of:
-
Glycerine: 1259
-
Castor oil: 952
-
Olive oil: 911
-
Steel: 7850
Source for all the above: (“The Engineering Toolbox”, n.d.)
6
Method 1
Graph of 𝑟 2 𝑣𝑠. 𝑣 for steel ball in glycerine:
r^2 vs. v
0.000005
0.0000045
y = 2E-05x + 1E-06
0.000004
0.0000035
r^2 (m^2)
-
0.000003
0.0000025
0.000002
0.0000015
0.000001
0.0000005
0
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
v (m/s)
Graph 1. A graph for the results of 𝑟 2 𝑣𝑠. 𝑣 is plotted with a best fit straight line. The
slope m =
𝑟2
𝑣
= 2 × 10−5.
7
-
Graph of 𝑟 2 𝑣𝑠. 𝑣 for steel ball in castor oil:
r^2 vs. v
0.000005
0.0000045
y = 1E-05x + 1E-06
0.000004
r^2 (m^2)
0.0000035
0.000003
0.0000025
0.000002
0.0000015
0.000001
0.0000005
0
0
0.05
0.1
0.15
0.2
0.25
0.3
v(m/s)
Graph 2. A graph for 𝑟 2 𝑣𝑠. 𝑣 plotted with best fit straight line. Slope is m =
-
𝑟2
𝑣
= 1 × 10−5.
Graph of 𝑟 2 𝑣𝑠. 𝑣 for steel ball in olive oil:
r^2 vs. v
0.000005
0.0000045
0.000004
y = 2E-06x + 8E-07
r^2 (m^2)
0.0000035
0.000003
0.0000025
0.000002
0.0000015
0.000001
0.0000005
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
v (m/s)
Graph 3. Graph of 𝑟 2 𝑣𝑠. 𝑣 plotted with best fit straight line, with slope m =
𝑟2
𝑣
= 2 × 10−6.
8
Table 1: A table with the estimated viscosities calculated using equation (4) of three
liquids compared to their literature values and the percentage differences between them.
Liquid
Estimated
Literature value 𝝁
Percentage
viscosities 𝝁
(𝑷𝒂 ∙ 𝒔)
difference (%)
(𝑷𝒂 ∙ 𝒔)
Glycerine
0.950
0.287
70
(“The Engineering
Toolbox”, n.d.)
Castor oil
0.650
0.150
77
(“The Engineering
Toolbox”, n.d.)
Olive oil
0.0302
0.0181
67
(Nakamura, 2014)
9
Method 2
Table 2: The average viscosities for glycerine for each radius of the ball along with the errors.
Average viscosity 𝝁
̅
Error on average viscosity ∆𝝁
̅
(𝑷𝒂 ∙ 𝒔)
(𝑷𝒂 ∙ 𝒔)
0.001
0.16
±0.01
0.002
0.15
±0.01
0.003
0.16
±0.01
0.004
0.17
±0.01
Radius (m)
Table 3: The average viscosities for castor oil for each radius of the ball along with the
errors.
Average viscosity 𝝁
̅
Error on average viscosity ∆𝝁
̅
(𝑷𝒂 ∙ 𝒔)
(𝑷𝒂 ∙ 𝒔)
0.001
0.096
±0.01
0.002
0.096
±0.01
0.003
0.11
±0.01
0.004
0.094
±0.01
Radius (m)
10
Table 4: The average viscosities for olive oil for each radius of the ball along with the
errors.
Average viscosity 𝝁
̅
Error on average viscosity ∆𝝁
̅
(𝑷𝒂 ∙ 𝒔)
(𝑷𝒂 ∙ 𝒔)
0.001
0.01
±0.001
0.002
0.01
±0.001
0.003
0.009
±0.001
0.004
0.008
±0.001
Radius (m)
11
Table 5: The average of the viscosities for the liquids are found along with their errors and
compared to the literature values of the viscosity at room temperature.
Liquid
Average
Error on
Literature value at
Percentage
viscosity 𝝁
̅
average
room temperature
difference (%)
(𝑷𝒂 ∙ 𝒔)
viscosity ∆𝝁
̅
(𝑷𝒂 ∙ 𝒔)
(𝑷𝒂 ∙ 𝒔)
Glycerine
0.16
±0.01
0.950
83
Castor oil
0.10
±0.01
0.650
85
Olive oil
0.01
±0.01
0.1
90
12
Method 3
Table 6: the true viscosity is found with its error and compared to the literature value for each
fluid along with their percentage difference.
Liquid
Glycerine
True viscosity 𝝁𝒄
Error on true
Literature value
(𝑷𝒂 ∙ 𝒔)
viscosity ∆𝝁𝒄
of viscosity
0.14
(𝑷𝒂 ∙ 𝒔)
(𝑷𝒂 ∙ 𝒔)
±0.01
0.950
Percentage
difference (%)
85
(“The Engineering
Toolbox”, n.d.)
Castor oil
0.1
±0.01
0.650
85
(“The Engineering
Toolbox”, n.d.)
Olive oil
0.01
±0.01
0.0181
45
(Nakamura, 2014)
4. Discussion
-
Method 1: Here, the relationship between the radius of the ball and the velocity was
plotted as the graph of 𝑟 2 𝑣𝑠. 𝑣 for each liquid (glycerine, castor oil, olive oil). As seen
from the graphs, it is a linear relationship, with the values of the velocity v increasing with
the increasing radius of the sphere. This trend is present in all graphs 1,2, and 3.
Furthermore, it was found that glycerine, which is the most viscous liquid out of the three
had the highest readings for the fall times; while olive oil, which is the least viscous had
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the lowest reading. In table 1, the estimated values for the viscosities of the three liquids
were calculated (using equation (4)) and compared to their literature values. The
percentage difference between the estimated and literature value for each liquid was found
to be considerably high, ranging from 67-77%. This high percentage difference may have
been due to errors such as choosing an insufficient distance between the starting and
endpoint over which the falling sphere was timed, or random errors such as the reaction
time for starting and stopping the timer at the chosen points A and B, or it may be due to
the lack of knowledge of the temperature of the liquids used in the experiment while
comparing it to the literature values of the liquids which may be values at temperatures
different than that of the experiment.
-
Method 2: For this method, the average viscosities were calculated for each radius of the
ball (1-4 mm) and each liquid. Then, they were combined and compared to the literature
values of the viscosities at room temperature. The percentage difference was found to be
higher than in method 1, ranging from 83-90%. Again, a reason for this high error may be
from the unknown temperatures of the liquids used in the experiment, therefore leading to
an inaccurate comparison when compared to the literature value of the viscosity. Although
averaging the measurements is likely to reduce the errors, in this case it has done the
opposite. It may be concluded that some human, systematic, and random errors were
involved in this large percentage difference.
-
Method 3: Equation (5) was used here to find the true viscosity 𝜇𝑐 and this calculated
value was compared against the literature value of the viscosities of the three liquids. The
percentage difference was seen to have decreased for the viscosity of olive oil (45%),
which is the lowest percentage difference value calculated for this liquid, suggesting that
this calculated value is the closest to its literature value. This may be due to the fact that
the true viscosity 𝜇𝑐 is a more accurate method to measure the viscosity since it takes into
account the effect caused by the drag of fluid against the walls of the tube, unlike 𝜇 which
ignores these effects. Including these effects may be linked to reducing the error in
calculations because it takes this effect into account therefore leading to a more accurate
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calculation. However, for the other two liquids, glycerine and castor oil, this does not
seem to apply.
5. Conclusion
This experiment investigated the theory of dynamic viscosity which is a property of fluids
that undergo laminar flow. Laminar flow is described as a smooth flow of liquid without any
turbulence effects. This flow was observed in the experiment as the steel ball fell straight
through the viscous fluid inside the cylindrical tube smoothly. The viscosity was calculated
for each fluid using equation (5), and the values were compared to the literature values of the
viscosity of each liquid. In conclusion, there was a large percentage differences between the
calculated viscosities of the liquids and the literature values for all the three methods
mentioned. This may be due to accumulated errors ranging from calculation errors,
systematic, random and human errors. Another main error may have been from the unknown
values of temperature of the liquids used in the experiment, if they were known, the values
could have been compared to their literature values of the same temperature, therefore
resulting in more accurate values. Generally, method 1 seems to have the least percentage
difference for glycerine and castor oil’s viscosities while method 3 showed a more accurate
result for olive oil’s viscosity. Hence, the “most accurate” method could not be determined as
there is no consistent trend between the three methods.
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