Fluid Mechanics
Arthur Wijaya
Joe Wu
Jayson Jochim
Physics 201: Summer 2004
Professor Richard Muirhead, Ph.D,M.D.
North Seattle Community College
Abstract:
The study of fluid mechanics is a dynamic
process. We have developed a systematic
algorithm which models the velocity
dependence of simple bodies in a viscous
incompressible fluid.
A three facet,
systematic process which enables users to
derive a working model of a systems
kinematical development position, velocity,
and acceleration as a function of time.
Conversely if kinematics are know a user may
them solve for fluid properties such as
density, viscosity.
Overall calculations
provide information regarding the drag
coefficient and order of velocity drag
dependence.
Contents:
I. Abstract
II. Introduction
III. Procedure
 Pre-Setup
 Setup[1]
 Setup[2]
 Data Analysis[1]
IV. Data
V. Analysis
 Drag Coefficient
 Experimentation
 Velocity Dependence
VI. Discussion
VII. Error
VIII. Conclusion
Introduction:
Fluid mechanics is a complex field. The motion
of spherical objects through a viscous fluid is a
highly dynamical process. There are many forces at
work; forces such as the easily identifiable force
of gravitational acceleration, forces easily
calculated such as the force due to buoyancy, and
mysterious forces which act on bodies as a function
of velocity such as drag forces.
The term drag force encompasses a large variety
of complex dynamic interactions between a body and
a viscous fluid. Such interactions create
different varieties of drag. The classic laminar
drag, which in viscous incompressible fluids occurs
at low velocity with dynamic objects, promotes a
smooth flow and relatively small drag. Turbulent
drag, which often occurs at higher velocities with
irregularly shape objects is complex and
mathematical analysis is chaotic in theory,
generally produces extreme drag forces.
Drag force is dependant on several parameter
including the cross-sectional area, the density of
the fluid in which the body is moving, the speed at
which the body is moving, and a drag coefficient
varying in magnitude for zero to one. This drag
coefficient is similar in function to that of the
frictional coefficient in classical mechanics.
Purpose:
In this lab experiment we intend to measure the
drag coefficient of various bodies in an
incompressible viscous fluid ( Water). We will
mainly study spherical bodies to reduce
unpredictable turbulent drag and easy the
calculation of cross sectional area.
Bodies will
be released from surface contact and allowed to
react freely in a descent though roughly a meter of
fluid. We will also perform some studies of bodies
entering a fluid with an initial velocity, having
been released from a predetermined height above the
surface of the water. This of course as we will
see adds addition consideration of forces as the
surface tension provides a considerable breaking
effect as the body enters the fluid.
In the end we hope to accurately model the
behavior of spherical bodies in freefall through
water. We will model such behavior using
Mathematica, Maple, and Venier Logger Pro software.
Procedure:
Tools:
 Ring stand
 2.00L graduate
cylinder
 1 meter ruler
 digital scale
 digital
micrometer
 digital video
camera
 logger proanalysis software
Materials
 backdrop (to
enhance exposure)
 various spherical
bodies
 water
Pre-Setup[1]:
There are substantial pre-lab procedures
involved in gathering the measurements required
for calculations. The mass of each body must be
made accurately.
Mass:
We used a Metler™ digital scale to measure
the mass in grams of each body we intended to
study.
Volume:
We used two separate methods to measure
the volume of the bodies. First by volumetric
displacement by submerging the bodies in a 100
ml graduate cylinder. Secondly by geometric
calculation using the diameter measured by
digital micrometry.
Cross-Sectional Area:
The cross-sectional area was
determined by geometric calculation using the
diameter measured by digital micrometry.
Setup[1]
We used a transparent 2.00L graduate cylinder along
side a meter ruler supported by a clapping ring
stand. The cylinder was filled with water to the
50 cm mark on the ruler. (see Fig.1) Alternating
backdrops were positioned behind the graduate
cylinder depending on the color of the body of
interest. A digital camera was positioned normal
to the plane of the backdrop. We them digitized
the progress of various bodies descent through the
viscous incompressible fluid.
QuickTime™ and a
Motion JPEG OpenDML decompressor
are needed to see this picture.
Setup[2]
For setup number two everything remained the
same except the bodies were released for an initial
position 20.0cm above the surface of the water.
(see Fig.2)
QuickTime™ and a
Motion JPE G OpenD ML decompressor
are needed to see this picture.
Figure 2 Ball 2: Trail 3 From Initial Height
Data Transfer[1]
The Venier product Logger Pro™ allows us to
digitize and analyze digital videos of AVI format.
We used a USB to Firewire cable to transmit the DV
video from the Digital Video Recorder to a Laptop
with Logger Pro installed. An intermediate program
Cannon Software Zoom™ was used to convert the DV
video to AVI format.
Data:
See table 1.
Analysis:
Pre-Analysis[1]:
Initial calculations included the
identification all the forces of nature acting on
our dynamical object of interest. Identification
and quantification of the gravitational force and
buoyant force were accessible from the drawing
board and readily calculated before the
experimentation. The drag force, however contained
several variables that would require experiment
results in order to narrow our estimations of their
magnitude.
The order of the velocity dependence,
and drag coefficient could only be determined by
fitting analytical solutions to the numerical data.
We spent a considerable amount of time running
simulations and calculations with both Maple and
Mathematica software with surprisingly accurate
results. The Algorithm is as follows:
I. Take some initial measurements:
It will be necessary collect some data about
the system intended for study before preceding with
the experiment.
 Volume and Cross- Sectional Area of the body
of interest
For spherical Objects :
o Volume =
4
3
r 3
Area = r 2
 Where r can be measure with a
micrometer or other device.

 Density of the Fluid:
Density :
o
mass

volume
 Mass:
o Measure by scale.
 Order of Velocity Dependence:

o Many fluids may be found in fluid index.
II. Calculating the Drag Coefficient:
After digitizing several trials with each
object we plotted the velocity with respect to
time. By inspection and comparison we found that
each object converged on a unique terminal
velocity.
Figure 3 Ball 1: Trial 1 –Velocity vs. Time
Note: limit of convergrence ~~ 1.6m/s
With a good estimate of the terminal velocity
we know at constant velocity the sum of the forces
on our dynamic object of interest is zero.
Consequently the acceleration is zero. We
developed a Maple™ worksheet that given the mass,
terminal velocity, radius, and density of the fluid
in which the body is moving we are able to
calculate the drag coefficient. (Assuming that the
order of the velocity dependence is of the order of
n=2)
Eq.1
r
 F  ma 
1
2
 DAvn  Vg  mg
Solving Eq.1 for D at a=0:
Eq.2
2g(V  m)
D
Av n
 is the density of the fluid
m is the mass of the object


v is the terminal velocity

where V is the volume of the object
A is the cross - sectional area of the object

g is gravitational acceleration
n is the order of velocity dependance

III. Modeling Motion through Incompressible
Viscous Fluid

Eq.3
d 2 x mg  12 D A dx
dt  Vg

dt
m
n
Plotting the differential Eq.3 we have a
working model which in comparison to actual data
proves an accurate approximation. (see overlays)
Side by sides:
Theoretical Position for Ball-1
Experimental Ball-1 Position:
Theoretical Velocity Ball-1:
Experimental Velocity Ball-1:
Theoretical Acceleration for Ball-1:
Experimental Acceleration for Ball-1:
As is apparent from the correlations between
experimental and theoretical the bodies act as our
model predicts.
Overlays:
Overlay 1 Position vs Time Actual Data Red
Overlay 2 Velocity vs Time Actual Data in Red
Discussion:
We have designed a system or algorithm in which
anyone may by measuring a few properties of a
spherical body my determine the coefficient of drag
of that body in any incompressible viscous fluid of
known density. By using Venier’s Logger Pro
technology were are able to measure the position
with respect to time of an object moving through a
viscous fluid an therefore identify the forces
acting on such an body. By determining the limit
or terminal velocity of the function by direct
analysis of the digital video of the bodies motion
we are able to determine the drag coefficient using
a Maple sub routine. By substituting the results
from the Maple sub routine into a Mathematica
differential equation algorithm we can plot a
position verse time and its subsequent derivatives
in order to analysis all possible characteristics
of our system of interest. By comparison of the
theoretical analysis of the objects kinematics we
are able to determine the most accurate order of
velocity dependence of the body’s motion though the
given fluid.
Error:
Several factors that were not taken into
consideration during the analysis of our data
include the interaction of the fluid with the
constraints of the container. Because the fluid is
incompressible the body as it moves through the
fluid it displaces an volume of fluid. This
displaced fluid interacts in a intricate manner
with the contours of the container and may produce
repercussive force that may effect the resulting
motion of the body. Similarly it has been proven
that the topography of a container base effect the
dynamics of the fluid which it contains. Therefore
the last few inches or more may be under the
influence of a force for which we did not
calculate.
Improvements:
This experiment would benefit vastly form video
equipment with a higher resolution rate and maximum
frames per second. On average we were able to
obtain 6 date points per trajectory. Doubling or
tripling this would have significant improving
effect on this experiment.
Conclusion:
In conclusion we have found the drag
coefficient of an unknown body by experimentation
using digital analysis of motion through fluid.
But more importantly we have developed systematic
algorithm in which anyone may determine the drag
coefficient and consequently the velocity
dependence of any object through any incompressible
viscous fluid of known density.
Maple, Mathematica programs as well as design
information and data available through:
JaysonJ@mac.com
At3rw@Yahoo.com
Joe5016@Juno.com