Experiment #2
Fluid Properties: Viscosity
Jordan Hines
Performed on January 31, 2011
Report due February 7, 2011
Lab Group: Elizabeth Hildebrandt & Anthony Freeman
Table of Contents
Object ………………………………………..………………………….………….…. p. 1
Theory …………………………………………………………………………….....…p. 1
Procedure …………………………………………………………………………........p. 2
Results ………………………………………………………………...…………….pp. 3-4
Discussion and Conclusion …………………………………………………..……...pp.5-6
Appendix ……………………………………………………..…….………………pp. 7-8
Object
The purpose of this experiment was to determine the dynamic viscosity of a given fluid
using the falling sphere viscometer method.
Theory
A fluid’s dynamic viscosity is defined as its resistance to an externally applied shear
force. This definition implies that a fluid will resist any change in form. Due to this, if a solid
object is placed into a fluid of lesser density, the object will fall through the fluid, exerting a
shear force on the fluid surrounding it. The object will displace fluid as it falls which will cause
a buoyant reaction force from the fluid onto the object. During this process, there are three
forces present: the weight due to gravity, the buoyant force, and the drag force. The force
balance equation of these three forces is shown below:
Weight – Buoyant Force – Drag Force = 0 (Equation 1)
The force of weight in Equation 1 acts in the downward direction and is dependent upon
the acceleration due to gravity, the density of the sphere, and the size of the sphere. The density
and size of the sphere determine its mass, and this mass is multiplied by the gravitational
acceleration to determine the force of weight.
The buoyant force in Equation 1 is caused by the pressure gradient exerted by the fluid on
the object. The lateral forces of this pressure are equal and opposite and therefore negate one
another. The pressure on the submerged object acting in the vertical direction is lesser on the top
than on the bottom thus causing a net upward force on the object.
The drag force in Equation 1 is caused by the viscous effects of the fluid as it moves over
the surface of the submerged sphere. The force attempts to resist the sphere’s motion through
the fluid. As such, it acts in the vertical direction, but opposite to the motion of the sphere.
ρsg(4/3)πR3 – ρg(4/3)πR3 - 6πμVR = 0 (Equation 2)
Equation 2 represents the same force balance equation as Equation 1 but in less general
terms. In this equation, the variables ρs, ρ, g, R, μ, and V represent the density of the sphere
being placed into the fluid, the density of the fluid itself, the acceleration due to gravity, the
radius of the sphere, the dynamic viscosity of the fluid, and the terminal velocity of the sphere
respectively.
Equation 2 can be rearranged to solve for the dynamic viscosity of a fluid. This is shown
below in Equation 3:
μ = [ρg(4/3)πR3 - ρsg(4/3)πR3] / 6πVR (Equation 3)
1
Procedure
Equipment:
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
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Large, transparent cylinder with affixed scale
Suave Naturals Fresh Mountain Strawberry shampoo
Digital Calipers
Stainless Steel ball bearings
iPhone with default stopwatch application
Experiment:
1. Take note of the ambient air temperature. This will be used as the temperature of the
2.
3.
4.
5.
6.
fluid.
Measure the diameter of the stainless steel ball bearing using the digital calipers. Record
this measurement onto your data sheet. Also, take note of the density of the material out
of which the ball bearings are made.
Fill the large, transparent cylinder with shampoo. There should be an attached ruler so
that velocity measurements can be taken. The cylinder should be covered at the top by a
piece of material with a circular hole at its center having a diameter larger than that of the
largest ball bearing to be used.
Drop the ball bearing into the hole at the center of the cover of the cylinder. Observe the
bearing until it reaches terminal velocity and take note of its position with respect to the
ruler at that point; start the time measuring device at this point. Assign a second point a
fixed distance from the first point; this will be the stopping point for the time
measurement. Record the time taken to travel between the two points. Repeat this step
five more times, recording the time value each time.
Divide each time measurement by the length between the two points to determine the
velocity for each trial. These velocities will be used to determine the drag force.
Plug all known information into Equation 3 to solve for the dynamic viscosity of the fluid
for each trial. Average these values to determine the dynamic viscosity of the fluid.
2
Results
Table 2.1. Initial conditions for temperature, density, and vertical
distance traveled by sphere.
Stainless Steel Ball Bearing Density
ρs = 8000 kg/m3
Shampoo Density
Displacement on Scale
ρ = 1014 kg/m3
z = 0.3048 m
Ambient Lab Temperature
T = 23 oC
All initial conditions listed in Table 2.1 were measured except for the density of the ball
bearings. This value for density was obtained from Fundamentals of Material Science and
Engineering: An Integrated Approach by Callister and Rethwish. The density of the shampoo
was measured by another group in the laboratory using a hydrometer. The displacement distance
was found using the scale attached to the cylinder in which the experiment was performed, and
the ambient air temperature was found on the electronic thermostat in the laboratory.
After all six trials had been performed, the terminal velocity for each trial was obtained
using the following equation:
V = z/t (Equation 4)
Where z is the vertical distance (in meters) which the ball bearing travelled through the shampoo,
and t is the time (in seconds) measured during which the ball bearing travelled between the two
points. The calculated values can be found in Table 2.2 below.
Table 2.2. Measured results from falling sphere viscometer.
Sphere
Trial Time (sec.)
Diameter (m)
Velocity (m/s)
1
2
9.3
9.6
0.00950
0.00948
0.0328
0.0318
3
4
5
6
9.6
13.2
12.9
20.4
0.00948
0.00784
0.00789
0.00633
0.0318
0.0231
0.0236
0.0149
After obtaining the values for the terminal velocity for each trial, these values were used
(along with Equation 3) to calculate the value of the dynamic viscosity for each trial. These
values of viscosity can be found in Table 2.3.
3
Table 2.3. Calculated dynamic
viscosity for each trial
Trial
1
2
3
4
5
6
Viscosity (N∙s/m2)
10.484
10.777
10.777
10.135
10.031
10.211
Once the viscosity had been calculated for each trial, the Reynolds number of each trial
must be calculated in order to determine the validity of the calculations. Equation 1 above is
only valid for Reynolds numbers less than 1. The Reynolds number for each trial is shown in
Table 2.4 below:
Figure 2.4. Reynolds numbers
of individual time trials
Trial
Reynolds Number
1
0.03011
2
0.02832
3
0.02832
4
0.01811
5
0.01884
6
0.00939
As can be seen in the table above, the Reynolds number for the individual trials are all
below 1; thus, the equation and resulting calculations are valid. With this, the average of all the
viscosity values was taken to find the actual dynamic viscosity of the shampoo. The result is
shown below:
μAVG = 10.402 N∙s/m2
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Discussion and Conclusion
Terminal Velocity in meters per second
The terminal velocity of two spheres having different diameters would not be identical.
Assuming that both spheres are made of the same material, the difference in diameter would
cause a change in the geometry such that the weight of the spheres would be significantly
different. This change in size and weight would change all three forces listed in Equation 1
above. The weight would increase proportionately to the buoyancy force due to the fact that the
buoyancy force is directly related to the mass of fluid which is being displaced. The change in
size would also cause a change in drag force due to the increase in surface area of the sphere.
This increase in drag force would not be enough to counteract the relatively large increase in the
net downward force created by the weight minus the buoyancy force. Due to this, the terminal
velocity of the larger sphere would be significantly higher than that of the smaller sphere. This
can be seen in Figure 2.1 below. (1&2)
0.0350
Actual Data
0.0300
Poly. (Actual Data)
0.0250
0.0200
0.0150
0.0100
0.00600 0.00650 0.00700 0.00750 0.00800 0.00850 0.00900 0.00950 0.01000
Sphere Diameter in meters
Figure 2.1. Terminal velocity as a function of sphere.
The viscosity found for two different size spheres should, theoretically, be the same due
to the fact that viscosity is a property of the fluid through which the sphere is moving and not a
property of the sphere itself. (3)
The largest shortcoming for this experiment is human error. Because the starting and
stopping of the timing device is performed by a human, it is expected that some error will be
present. Also, the distance across which the sphere is travelling is measured by the human eye,
which accounts for more error. Also, in this experiment, the assumption is made that both the
fluid through which the sphere is travelling and the sphere itself are perfectly homogenous
materials with uniform densities. This is not likely to be the case due to the fact that all of the
bearings and all of the shampoo were not likely to have come from one batch of materials which
would allow for less variation in material properties. Further, the density of the shampoo was
obtained through experiment which allows for further human error. Finally, less error could
have been achieved by limiting variables within the experiment. By this, I mean to say that a
lower degree of error likely would have been achieved had we used the same ball bearing for
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each trial. This would have limited variables in the experiment and allowed us to average the
time for each trial and use that value of average time to calculate a terminal velocity and then a
single viscosity. (4)
The temperature of the ambient air should be recorded because the fluid is likely to be at
the same temperature given the amount of time that the fluid has been surrounded by air at the
same temperature. The temperature of the fluid is the significant measurement because the
density of the fluid is directly related to its temperature. If the temperature were greater, the
fluid would be less dense; conversely, if the temperature were lower, the opposite would be true.
(5)
This method could be used in gasses, however, there would likely be a much larger
degree of error due to the speed at which the sphere would fall and the ability of a human to react
quickly enough to measure such an occurrence. If some form of automation was used to
measure the time in which the sphere moved between two given points in a gas, this experiment
could likely be performed successfully in such an environment. (6)
This method could be successfully completed in opaque fluids, however, it would require
methods other that human observation due to a human’s inability to see in order to record the
time the sphere takes to pass from the starting point to the stopping point. A method such as
thermal imaging could be used to accomplish such an experiment. This method would not work,
however, with nonhomogeneous fluids such as those mentioned. (7 & 8)
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Appendix
Data Usage
The following is the calculation used to obtain the terminal velocity of the falling sphere:
V = z/t  V = 0.3048 m / 9.6 s  V = 3.175 x 10-2 m/s
The following calculation is used to obtain the dynamic viscosity of the shampoo:
μ = [ρg(4/3)πR3 - ρsg(4/3)πR3] / 6πVR
 μ = {[(1014 kg/m )(9.81m/s2)(4/3)(0.00475 m)3] – [(8000 kg/m3)(9.81m/s2)(4/3)
(0.00475 m)3]}/[6π(0.0328 m/s)(0.00475 m)]
 μ = 10.484 N∙s/m2
3
The following calculation is used to obtain the Reynolds number for each trial:
Re = ρVD/ μ Re = (1014 kg/m3)(0.0328 m/s)(0.00950 m)/(10.484 N∙s/m2)  Re = 0.03011
Bibliography
Introduction to Fluid Mechanics, 3rd Edition
William S. Janna (1993)
A Manual for the Mechanics of Fluids Laboratory
William S. Janna (2008)
Fundamentals of Material Science and Engineering: An Integrated Approach
W.D. Callister, Jr and D.G. Rethwish (2008)
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