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Fluid Dynamics

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Introduction

Migicovsky

 Introduction

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.

Middle

1.001 7.50

Glycerine

208.45

166

1.26

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

 Time Code Real Time(s)(±0.005s) Measured d (cm) (±0.0005m) Real d (m)(±0.0005m Velocity(m/s) Acceleration(m/s2),a 41;23 0 1.1 0 0 0 41;24 0.033367 1.15 0.0005 0.014985 0.224546 41;25 0.066733 1.2 0.001 0.014985 1.35E-05 41;26 0.1001 1.3 0.002 0.02997 0.336813 41;27 0.133467 1.4 0.003 0.02997 -2.4E-15 41;28 0.166834 1.5 0.004 0.02997 1.95E-15 41;29 0.2002 1.7 0.006 0.059941 0.673692 42;00 0.233567 1.9 0.008 0.059939 -5.4E-05 42;01 0.266934 2.25 0.0115 0.104894 1.058575 42;02 0.3003 2.6 0.015 0.104897 9.42E-05 42;03 0.333667 2.95 0.0185 0.104894 -9.4E-05 42;04 0.367034 3.3 0.022 0.104894 -1.2E-14 42;05 0.4004 3.7 0.026 0.119883 0.421132 42;06 0.433767 4 0.029 0.089909 -1.04803 42;07 0.467134 4.4 0.033 0.11988 0.78595 42;08 0.500501 4.7 0.036 0.08991 -1.0479 42;27 1.134468 10 0.089 0.083601 -0.01033 43;12 1.634968 14 0.129 0.07992 -0.00752 43;24 2.035369 17.2 0.161 0.07992 -2.7E-17 44;05 2.435769 20 0.189 0.06993 -0.02673 44;11 2.635969 21.5 0.204 0.074925 0.024118 44;22 3.003003 24.3 0.232 0.076287 0.003678 45;01 3.33667 26.5 0.254 0.065934 -0.03346 45;13 3.73707 29.6 0.285 0.077423 0.026564 45;17 3.903904 30.7 0.296 0.065934 -0.07486

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

Conclusion

 Conclusion

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

 Microsoft Excel Help – STDEV function

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

 MacDonald Pharmacy

 http://www.rpi.edu/dept/chem-eng/Biotech-Environ/SEDIMENT/sedsettle.html

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