Fluid Dynamics

Migicovsky

As objects move through fluids, they are exposed to numerous forces that enhance or impede their progress. By analyzing and understanding these forces, one can predict the velocity of a moving object. Of the forces exerted on an object falling through a liquid, such as buoyant force or the force of gravity, the viscous or drag force appears to have the largest negative effect on the object.

The effect of aero and hydrodynamic drag forces and friction appears underrepresented in high school physics courses. Perhaps it is because concepts such as viscous and turbulent drag forces are difficult to predict and measure. My preliminary research indicated there are many factors affecting the forces on an object. These concepts fall in the field of fluid mechanics.

Initially, my study began with the idea of measuring the aerodynamic drag force exerted on a model rocket. My primary interest was in the factors that influenced the maximum height reached by a rocket with a set amount of propellant. I thought that launching a rocket on a particularly humid or hot day might result in a different maximum height than a launch on a colder day. It might be possible to theoretically identify the factors such as the pressure or density of the air, then relate them to the measured height. I soon realized that this experiment would not produce accurate data or a clear theoretical relationship because it involved a multitude of variables that were impossible to control without the use of a weather-controlling machine. Progressing fromthis first idea, a more controllable experiment evolved: measuring and comparing the terminal velocities of a ball falling through glycerine at various temperatures. Glycerine was selected because its high viscosity exhibits demonstrable results.

The amount of viscous drag within a given fluid appears to vary within a controlled container by temperature. This experiment was designed and conducted to test the following question: How is the terminal velocity of a ball falling through a column of glycerine affected by a change in the temperature of glycerine?

This study analysed the terminal velocities of an object falling under the force of gravity through a fluid at varying temperatures. My hypothesis states that as the temperature of the fluid decreases, the terminal velocity of the object will decrease by a proportion of. I base this prediction on the electrostatic bonding properties displayed in liquids. In order to prove this, an experiment will be conducted using a small metal ball and glycerine.

As an object falls through a fluid under the force of gravity, it is subject to three main types of forces: buoyant, drag and gravity (Figure 1).

Since vector addition can be performed with these forces, the net force is calculable using this equation.

(1)

The two easiest forces to understand are those of gravity and drag. The force of gravity on the ball is simply equal to its mass times the acceleration due to gravity, using Newton’s law of F=ma:

(2)

Mass can be substituted with its density,, times its volume,, to simplify later equations. As this is the largest force on an object with a smaller density than the fluid it is passing through, the two remaining forces will not be able to counter this or overcome it.

Buoyancy is derived from Archimedes’ principle, that the buoyant force, Fb, on an object will be equal to the weight of the displaced fluid. The equation:

(3)

describes this phenomena using the same substitution of density and volume for mass.

The third force exerted on the ball, drag, is always in the opposite direction of the movement. It is very similar to the friction force on a block sliding down a plane, which works in the opposite direction of the motion. The source of drag is derived from the viscosity of the fluid and the incidence of turbulence on the ball. At lower velocities, turbulence is small and laminar, whereas at higher velocities, turbulence becomes drastic and has an enormous effect on the drag force. One can use the Reynolds number, Re, a dimensionless value, to determine the nature of the flow (equation 4). It has been determined that Reynolds numbers of over 2000 indicate turbulent flow, while lower numbers signify laminar flow:

(4)

The viscosity of a fluid is the dominant factor in assessing the drag of a fluid. This value refers to the amount of internal friction, or resistance to flow, possessed by a fluid. Viscosity in fluids is due to a number of reasons. Certain physical properties of the fluid, for instance, strong intermolecular bonds, such as hydrogen bonding and other electrostatic bonds, will pull the molecules together and make it much harder for the fluid to “flow” and stretch around objects. Since the fluid is made up of particles, the size of these particles will also have an effect on the viscosity. For example, a liquid with significantly smaller particles will be more concentrated and therefore the particles will be more in contact with each other than if they were larger, causing more friction and therefore more viscosity.

One of the equations used to describe the effect of viscosity as a retardant force was derived by Stokes. His equation dealt solely with a laminar flow involving an extremely low Reynolds number and the fall of a sphere with a small radius through a viscous fluid. His equation was as follows:

(5)

where: Fv = viscous force

r = radius of the sphere

= coefficient of viscosity of the fluid.

v = velocity of sphere relative to fluid

The SI unit of the coefficient of viscosity is the: .

Now that all the forces exerted on the ball have been equated, it is possible to fill in Equation 1 with the appropriate forces.

(6)

This will form an equilibrium position. As the viscous force increases, the net force becomes closer to zero. The term of controls the force as it depends on the velocity of the ball.

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(5)

where: Fv = viscous force

r = radius of the sphere

= coefficient of viscosity of the fluid.

v = velocity of sphere relative to fluid

The SI unit of the coefficient of viscosity is the: .

Now that all the forces exerted on the ball have been equated, it is possible to fill in Equation 1 with the appropriate forces.

(6)

This will form an equilibrium position. As the viscous force increases, the net force becomes closer to zero. The term of controls the force as it depends on the velocity of the ball.

At the point where there is no net acceleration of the ball, a = 0, terminal velocity is reached. It can be isolated by rearranging Equation 6:

(7)

As I mentioned before, the viscosity of a fluid is also dependent on its temperature above all other factors. This is illustrated when syrup is heated; it becomes thinner and pours easily; and in cool climates motor oil thickens and can affect the performance of an automobile. This dependency on temperature is related as well to the intermolecular forces. As the temperature decreases in a fluid, the average velocity of the particles within it will decrease as well, due to the Kinetic Theory. Since the particles will now be moving slower, they will spend more time in close contact and therefore the intermolecular bonds will be stronger. If the temperature increases, the velocity will increase as well, causing the bond forces to weaken, as the particles are further apart from each other and less affected by their neighbours force.

The exact numerical correlation between temperature and viscosity is quite difficult to identify, since the relationship fluctuates greatly when the phase of the fluid changes, since the viscosity of a gas acts in the opposite. To be precise, in a gas, the opposite of this relationship is true: as the temperature increases, the viscosity increase because the particles are hitting each other more, and vice-versa. The only equation I have found that relates temperature to viscosity is:

(8)

where: = coefficient of viscosity of the fluid

A and B = constants

T = absolute temperature

Materials

1 litre pharmaceutical grade glycerine (aqueous solution of 99.5% to 100% by mass, USP)
1 m length clear plastic cylinder; diameter 0.051m
End cap for plastic tube
Level
Meter stick (accurate to ± 0.0005m)
Tripod
Digital-Video Camera (frame rate=29.97fps, accurate to ± 0.005s)
Ball from a bearing
Digital balance (accurate to ± 0.001g)
Graduated cylinder (accurate to ±1ml)
Digital Vernier Calliper (accurate to ± 0.0001m)
Bar magnet
Thermometer (accurate to ± 0.5ºC)
Freezer
Helpful and willing assistant
Right angle
Adobe Premiere and After Effects
Apple QuickTime
Microsoft Excel

Assembly of apparatus

The apparatus consisted of the plastic cylinder held in an upright position. The level was used to ensure that the tube was parallel to the earth’s gravity. The meter stick was attached to the cylinder in order to measure the descent of the ball. The video camera was placed on the tripod and set ≈ 1m away from the tube, so that it was possible to zoom in and include only the tube and the ruler on the side in the frame. The tripod also gave the ability to scroll up and down. This setup is illustrated in Figure 2. The tube was filled to within 3cm from the top with glycerine.

Figure 2

Experimental Procedure

Preliminary Measurements

In order to use many of the equations I mentioned in the theory, I needed to first discover or solve for constants such as the density of the ball and the glycerine, and the viscosity of the glycerine at room temperature.

I calculated the densities by first measuring the mass and volume of ball and fluid. I used an electronic balance to measure the masses. For the volume of glycerine, I used a graduated cylinder. To find the volume of the ball, I measured the radius using an electronic vernier calliper.

Base Temperature Measurements

I then measured the temperature of the glycerine at two points as illustrated in Figure 3:

Figure 3

I enlisted the aid of an assistant to drop the ball in the exact center of the tube while I controlled the video camera. I started the recording, the ball was dropped, and I followed the path of the ball by scrolling the camera down using the tripod. This recorded the descent of the ball by documenting the ruler measurement of the distance travelled by the ball corresponding with every frame of the video film.
After reaching the end of the scrolling distance possible on the tripod (around 30 cm) I stopped recording and used the magnet to retrieve the ball.
I repeated the test at the stable room temperature (fluid temperature 17ºC) 10 times in order to gain an accurate average terminal velocity.

Terminal Velocities at various temperatures

The glycerine was chilled in a freezer for 24 hours then poured carefully into the tube. The first measurement indicated a temperature of -12ºC at both 5cm and 15cm below the surface level of glycerine.
The same setup and procedure of recording as step 3 was repeated. Ten tests were conducted at intervals of several minutes, to allow measurements at various temperatures as the glycerine warmed up to towards room temperature.

Data Processing

For each test, the descent of the ball through the glycerine was captured on video film, which was then transferred to the computer for direct measurement on the monitor. To do this, I connected my camera via Firewire to Adobe Premiere and uploaded all the video. It was recorded at 29.97 frames per second or fps. From this I could measure the rate of change of distance of the ball: its velocity. I did this using the programs QuickTime and Excel. I opened the video in QuickTime, and using Adobe After Effects, added in a “time code” (elapsed frames) on the bottom right corner in order to make time measurements easier (Figure 4).

Figure 4

I then used a ruler with a right angle to measure the distance travelled by the ball by placing it up against my monitor and through the center of the ball. I recorded the absolute distance traveled for each 1/29.97th of a second for the first several seconds of each fall. When the change in distance began to be fairly constant, I started measuring at intervals of 2 seconds, 15 seconds and finally 30 seconds. As you can imagine, the spreadsheet that was created is huge and it is not possible to present the complete data in this paper.

Experimental Design Choices

During the pre-testing phase of the experiment, I became aware of several factors that I needed to change in order to limit potential sources of error.

When pouring the glycerine into the tube, I allowed it to stand for a while to let the bubbles created during the pouring process to rise to the top.
I found that sometimes the tripod scroll mechanism was rough and skipped ahead randomly so I oiled it and practiced scrolling to ensure that I could keep the ball centered in the frame.
In order to retrieve the ball from the bottom of the tube after a test, I originally planned to empty the whole tube, extract the ball, and pour the glycerine back into the tube. This would have created many bubbles which would have caused the path of the ball to become offset so I decided to use a magnet to carry the ball up the side.
Since the glycerine started heating up after I removed it from the freezer, I had to perform the tests quickly to collect sufficiently detailed data at varying temperatures.

Preliminary Measurements

The measurements of mass and density of both the glycerine and ball were essential to later relating the published data to the experimental data. They are presented in Table 1. All measurements were performed at 21ºC. The assumption was made that the fluid temperature variation would not have a noticeable effect on the volume of the glycerine and therefore would not affect the vt.

Table 1 – Measurements of ball and fluid characteristics

Base Temperature (17ºC) Vt Measurements

Since the raw data are numerous, due to the number of tests performed and the precision of the time measurements, they are not presented in full here, but in Appendix B.

For each test, distance travelled by the ball over time was quantified. Then the data were analysed to determine terminal velocity. Since the velocities were not completely regular, an average for each test was calculated. As the acceleration decreased to near zero, the values of velocity would be counted towards the test’s average. For the tests at 17ºC, terminal velocity seemed to be reached after 2 seconds. To aid in explanation, I offer this sample of data and calculation from test 2 (Table 2).

Table 2 – Test 2, Distance and Time measurements

Where: Real time = as marked on video frame

Measured d = the actual measurement of distance from the ruler

Actual d = the measure of d converted into metres and offset by how far

the fingers held the ball below the fluid level.

Velocity = , instantaneous velocity

Acceleration =

In this test, the averaging of the velocities went from frame 42;01, when the velocity became more constant. It did suffer two major acceleration changes, at 42;06 and 42;08, possibly due to various sources of error which will be described later. The average vt was calculated using equation 9 to be 0.0913 m/s2.

(9)

where: N = number of values averaged.

Ten tests were performed at the same temperature to determine a baseline average terminal velocity. This average should not be confused with the average from each separate test.

Table 3 – Average Terminal Velocity at 17ºC

Calculation of Standard Deviation

The standard deviation of the Table 3 of average vt at 17ºC was calculated using the Microsoft Excel command which in turn applied this formula:

(10)

The graph of the velocities with the error included (Figure 5):

Figure 5

The result of trial 5 appeared to be way out side of the standard deviation, so it was left out of the average vt. By dividing the standard deviation by the mean and multiplying it by 100, I calculated the percent of variation of my results, which was 7.36%.

Measurement of vt at varying temperatures

The same methodology was used to capture the data from the second part of the experiment, the variation of temperature. These values were much more numerous as the ball fell much slower and therefore there was more data. They are displayed in Appendix C.

Several problems arose during the variation of temperatures. As the glycerine was poured into the tube, a large number of air bubbles were formed. Since the fluid was so viscous, the bubbles floated up very slowly and I did not have the time to let them disperse completely, as the glycerine was heating up and I wanted to have some results at a very cold temperature. While I noticed in the video that the ball did not hit any extremely large bubbles, this may have introduced considerable error.

As the experiment proceeded, it appeared that the temperature of the glycerine varied between the two points of temperature measurement, at 5cm and 15cm. Luckily, there seemed to be consistency between the variations, which turned out to be a difference of 1ºC in each case. Since the temperature was constantly decreasing, the vt changed as the ball entered a new temperature range. When deciding how to average the velocities, I arbitrarily chose two points that seemed to display a fairly constant velocity, between 3.5cm and 13cm that were very close to the points at which the two temperatures were measured, and used the values between these points to average the velocities. The two points between which averages were calculated remained constant for the calculation of all tests. For example, this sample of test 14 shows the highlighted values that were used to calculate the average vt (Table 3),

Table 4 – Test 14 (fluid temperature: top -2ºC, bottom -3ºC)

Here, the highlighted values were used to calculate the average velocity. Fluid temperature was measured at -2ºC (5cm below), and -3ºC (15 cm below the fluid line). The average temperature was -2.5ºC for this test.

The standard deviation was also calculated in each test of the velocities used to create the average vt. These standards were then averaged to find the mean standard deviation of all of the tests. This was the value used to create the error bars in Figure 6.

These are the results of the variation in temperature (Table 5), including the average vt from the base temperature:

Table 5 – Results for Vt at varying temperatures

Figure 6

As the trend line indicates, the relationship between temperature and vt is clearly in exponential proportion, according to the equation of the trend line, Vt=0.0141e0.1297T. The line falls through the standard deviation for the lower temperature points but misses the last point by about two standards. This is probably due to the fact that there are not enough values between 5ºC and the test at 17ºC for the trend line to be accurate between these distances. As well, a reason for the deviation at 17ºC could be due to an increase in turbulence. At higher degrees of turbulence, the relation between velocity and viscous force begins take on the relationship of, therefore increasing the drag by the square of the velocity instead of a simple ratio. This is unlikely the reason, as the Reynolds number at 17ºC is:

This is clearly of laminar flow, as it is much less then 2000.

During my research, I came upon a table of values (displayed in Appendix D), which described the viscosity of pharmaceutical glycerine at various temperatures. The data for 100% concentration of glycerine was selected as I used between 99.5% and 100%. These values are the coefficients of viscosity. Therefore, using a rearrangement of equation 7, and the material constants measured in the first section of the data presentation, it is possible to translate these values into terminal velocity (Figure 7).

Figure 7

By superimposing the graph of the theoretical vt’s over the experimental results, Figure 8, it can be seen that both are in the same form, and exponential relationship. Also, the published data generally fell parallel and within the standard deviation of the experimental results, excluding values at the highest temperature of 17ºC. This discrepancy may have occured because the published data described the viscosity of a 100% concentration of glycerine, whereas the experimental glycerine may have ranged from 99.5% to 100%. As appendix D demonstrates, there is a major difference in viscosity between a 100% solution and a 99% solution of glycerine.

Figure 8

There were several main sources of error that most likely lead to the greatest divergence from the published values and several that might have had a smaller effect.

There is an underlying error relating to the variable glycerine temperatures in the experimental procedure. Since it was noted that the temperature varied along the height of the tube, the ball consequently fell through different layers of fluid. This may account for the lack of clear terminal velocity for these values. This problem could have been eliminated by conducting the experiment in closed system where it was possible to adjust the system temperature and therefore there would be no variation in the temperatures of the glycerine.
For the tests performed at variable temperatures, the temperature of the steel ball was not the same as the temperature of the glycerine, as the ball was not placed in the freezer. As the ball descended through the glycerine, it may have lost some of its energy as it was heating up the fluid surrounding it, thereby influencing the glycerine temperature. This could have been combated by either the methodology described above, or simply placing the ball in the freezer along with the glycerine.
There were sources of error present with respect to the tube containing the glycerine. The tube was of a relatively small diameter, and during several tests the ball descended fairly close to the side; the ball may have been subject to the “wall effect.” This happens as the ball forces the fluid closer to the wall, in turn causing the fluid to rebound and push back against the ball. This could have been be alleviated by using a tube of larger diameter. Human error could have contributed to this as well, as the ball was not always dropped perfectly in the center. A different method of dropping the ball, such as a fixed drop point, could have solved this.
The tube of glycerine, at lower temperatures, fogged up and obscured some measurements of distance. The tube was periodically rubbed off but fogged up immediately after. Bubbles, as mentioned, also concealed the ball at some points.
By measuring with a ruler precise to only ± 0.0005m, human error might have had an effect, as I was merely eyeballing the measurements of height. An improved method might have utilized a more precise ruler.

This experiment was designed to measure the terminal velocity of a solid ball falling through a column of glycerine at varying temperatures. The resulting experimental data and calculation of vt revealed an outcome that verified published data. The resulting relationship clearly supports my hypothesis. Based on my research and published data, the terminal velocity of an object falling through a viscous fluid will vary according to the temperature of the fluid, dependent on a proportionality of. I theorize that A and B are substance-specific constants, though I have not been able to support this through any research. This seems to be the reciprocal of the Equation 8, as. Since I researched many different sources in the field of fluid mechanics, it is interesting to note the relative lack of research completed to conclude on a direct numerical relationship between temperature and viscosity.

There are several possible applications for data and conclusions of this experiment. Since fluid temperatures usually vary, it would be helpful to know the precise terminal velocities of parachutists, so they would know when to deploy their parachutes, and for submarines, to calculate their dive angles. As well, relating back to my original experimental idea, the re-entry of rockets and the thrust needed for them to break free of earth’s atmosphere can be predicted knowing their terminal velocity and the viscosity of air. Precise, temperature-related terminal velocities can also be applied to other domains of science, such as the propagation of airborne diseases and the spread of pollen.

Giancoli 256

Giancoli 260

Appendix A, Section - Buoyancy

Ibid, Section - Drag

Ibid, Section - Viscosity

http://hypertextbook.com/physics/matter/viscosity/

Giancoli 353

applied fluid mechanics ROY 10

Adobe software

Microsoft Excel Help – STDEV function

Dow Glycerine = http://www.dow.com/glycerine/resources/table18.htm

MacDonald Pharmacy