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ASCELEmTEB MOTION OF A SPHERE
by Marion Robert Carstens
A dissertation submitted in partial fulfillment of the require ments for the degree of Doctor of Philosophy, in the Department of Mechanics and Hydraulics, in the Graduate College of the State University of Iowa June, I960
ProQuest N um ber: 10991956
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uest P roQ uest 10991956 Published by ProQuest LLC(2018). C o p y rig h t of the Dissertation is held by the A uthor. All rights reserved. This work is p ro te cte d a g a in s t u n a u th o rize d co p yin g under Title 17, United States C o d e M icroform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6  1346
This dissertation is hereby approved as a credit able report on an engineering project or research carried out and presented in a manner which warrants its acceptance as a prerequisite for the degree for which it is submitted. It is to be understood* however* that neither the Department of Mechanics and Hydraulics nor the dissertation advisor is responsible for the statements made or for the opinions expressed.
ACKNOWLEDGMENTS
The writer wishes to express his appreciation to Dr, J, S. SfcNown with whose guidance the experimental program was conducted.
He
also wishes to express his appreciation to the members of the staff of the Iowa Institute of Hydraulic Research for their help,
Mr, Dale
Harris* shop superintendent* was helpful in designing the equipment, Mr. Philip 0, Hubbard* Research Engineer, very kindly assisted in the design and procurement of the electrical equipment. secretary of the Institute, typed the manuscript.
iii
Miss Leona kwelon,
TABLE CT CONTENTS
Chapter
Page ...................
Introduction 1 IX
Fundamental concepts
* ..........................
Sphere Oscillating in an Infinite Fluid
VJ
Sphere in an Oscillating Fluid Experimental Program
2
.......
A System with Free Oscillation and Viscous Camping ........ A System with Forced Oscillation and Viscous Da m p i n g.......................... III
X
7 •
14
..........
21
Suspended Sediment Diffusion Characteristics
.....
Comparison of the Amplitude Ratios Obtained by Using Stokes1 Solution and by Using the Experimental Solution for the Coefficients . . . . Comparison with the Data ofWagenschein ......... Application pf the Foregoing Concepts to the Problem of Suspended Sediment.................. VI
Conclusions
9
............
Dimensional Analysis ............ Design of Equipment . . Method of Operation . ....................... ........ Evaluation of Data Analysis of ExperimentalResults ............... V
7
........
23 24 26 31 34 63
64 66 66
64
Bibliography...................................
iv
66
TABUS OF FIGURES
Figure
Pag©
1.
Force Vectors at Resonance
. ............
H
2.
Sphere in an Oscillating F l u i d ...................... . .
20
3.
Schematic Drawing of Apparatus.
26
4.
Photograph of Apparatus
8*
Boundary Effect on Virtual Mass Coefficients.............
SO
6.
Mass at Resonance
36
7.
Mass at Resonance
8.
Mass at Resonance
9.
Mass at Resonance
.......................
. . . . . .....................
28
.......................... .
36
.....................
37
.......
38
......
10.
Viscous Damping Coefficients.........
39
11.
Viscous Damping Coefficients
40
12.
Viscous Damping Coefficients
13.
Virtual Mass Coefficients
14.
Viscous Damping Coefficients...............
16.
3Q
16.
Form of Curve
17.
Fluid Volume
18.
Diffusion Characteristics of Sediisent
•*?
vs.
7
......................... ............ ..........................
41 42 43
*..............................
48
.....................................
49
.................. * ...................
69
d ' uj
V
..................
62
TABLE m
TABLES
Table I 11
Page Summary of Physical Data
32
Couparison of the Relative Amplitude of Sphere ................................. to Fluid
vi
* 33
X INTRODUCTION In the study described herein# the forces acting on a sub merged particle have been determined experiinentally for simple harmonic particle motion.
Specifically# a method was sought to determine the
virtual mass and viscous damping coefficients associated with this motion.
Stokes performed an approximate solution of the UavierStokes
equations for this type of motion which several investigators have atteupted to verify.
However# the effeot of the amplitude of the motion
does not appear in this solution. In accelerated motion of a submerged body# the fluid being accelerated exerts a force on the body.
This force is represented by a
product of the body acceleration and the fictitious m s s called the virtual mass.
Usually the virtual mass is either neglected or its magni
tude Is determined from the irrotational solutions for rectilinear motion.
However# the type of motion and the viscous properties of the
fluid are certain to play a dominant role in the determination of the virtual mass. The determination of the magnitude of the forces on a submerged particle is important in a study of the diffusion of particles of foreign material in an accelerated fluid medium.
This type of diffusion process
is exemplified by suspended sediment diffusion in water# air transport by fluids# dust particle suspension in air# and mist droplet suspension in the atmosphere.
Lfethods were sought to approximate the magnitude of
the ratio of the diffusion coefficients of sediment and fluid momentum.
2
Chapter I F B m i E H m L CONCEPTS Prior to a detailed discussion of the methods of determining the magnitudes of the force components in the accelerated motion of a spherical particle* a brief discussion of three subjects is pertinent, the first is a brief review of the resistance forces on a sphere being moved with a constant velocity.
The second subject is a discussion of
the nature of the force components on a sphere which is moved with acceleration.
The third subject is the presentation of a mathematical
solution of the force components on a sphere being moved with simple harmonic motion.
When referring to motion of a spherical particle* the
writer always Implies that the motion is in a coordinate system with onedegreeoffreedom. The most common relation to express the drag force mi a sphere being moved with a constant velocity is F * C
V vn / in which 19
is defined implicitly in the relationship "ion

^co /vn
,
(3b)
(3/m  uj2*} The first term of the solution (Eq. (3a)) is a transient which decreases to a negligible magnitudeas
t
(time) increases. The phase angle
is the angle by which theforce P sincut precedes the displacement. The half amplitude after the motion is establishedis X
O

.
_
P/m
,_
/» v (3c)
•
This system can be tuned to maximum amplitude by the change in the value of m with the other quantities being maintained at a constant value.
The maximum amplitude occurs at the condition m

,
(3d)
to2' which is the resonant condition.
The phase angle
at this resonant
condition is ip

Tr/z,
radians .
Clearer insight into the meaning of the differential Eq. (3)
11 can be obtained by plotting the various force vectors at the resonant condition.
If =
in (cot.  II ^
%
t x
c o s a) t
then* k
=
X. oo c o s ^ o u t  II ^
r  X 0 CO Sin w t
;
and 
cousin (to i  I L ^
~  X Q co2*c o s cot
The quantities A , B * m , and P are scalar magnitudes. these relationships* (1) the force vector Ax
precedes
Prom
the force
vector Bx by an angle of "^/2 radians* and (2) the force vector mx precedes
the force vector Bx by an angle of tt radians. it
the force radians.
+ Ax
+
Bx
 psinu/i
Since
 O
(S)
P sinu*t is preceding the force Bx by an angle of TT fl This interrelationship of the force vectors is shown on Fig. 1 .
 P5»nout
Ax
mx
Figure 1. Force Vectors at Resonance.
12
This unique arrangement of the force vectors at resonance is the reason for the sinple relationships, m
=.
B coz
(3d)
and,
P
ft a
.
(5e)
x 0 co This type of mechanical system is satisfactory since the force components Ax and
hoc
can be separated*
Krishnaiyar [3] performed an experiment to determine the virtual mass coefficient of an oscillating sphere using this principle. His sphere was fastened to a fine wire midway between two knifeedge supports.
Krishnaiyar could adjust the tension in this wire.
The simple
harmonic Impulse P was supplied by means of fixed electromagnets in conjunction with the direct current in the wire supporting the sphere. The interaction of the alternating magnetic field surrounding the spheresupporting wire resulted in a simple harmonic force P sinu»t being applied to the oscillating system.
If the amplitude of the oscillation
is small, the elements of this mechanical system are identical with the differential Eq. (3). B
» 4 T/jl
T
* the tension in the wire.
S. * the length of the wire.
[8 ] H. C. Krishnaiyar, Philosophical Magazine, Series 6 , Vol. 46, pp. 10491053.
13 Krishnaiyar adjusted the tension in the wire to the point of resonance. At this resonant point
m
=■ 4 *T A tu*”
He performed experiments in air and in kerosene.
The difference in
m m , as determined from the resonant relation was the virtual mass* in kerosene* of the submerged sphere and suspension wire* since the virtual mass in air is negligible. The writer chose the mechanical system* for which the mathe matical model is Eq. (3)* as the medium to obtain experimental informa tion concerning the force components on a sphere oscillating with simple harmonic motion.
This type of mechanical system* if tuned to resonance
by mass changes* is suitable for an independent determination of the shear force Ax and inertial effective force nut (Figure 1). . j •f
14 Chapter 111 SFHEBE IN AN OSCIUATINQ FLUID Before discussing the details of the experimental program, the writer will discuss the motion of a sphere in a fluid oscillating with simple harmonic motion.
The motion of a sphere in an oscillating
fluid is of considerable interest to engineers because of its relation to foreign particle movement in turbulent flow.
Analysis of this motion
is mathematically more involved than any of the previously discussed systems, but is straightforward in a system with onedegreeoffreedom. The differential equation is developed term by terra for the understandof the individual forces involved. a *
a. position coordinate of the fluid,
a
a velocity coordinate of the fluid,
a
 acceleration coordinate of the fluid,
x
» position coordinate of the sphere,
X
“
velocity coordinate of the sphere,
X
»
acceleration coordinate of the sphere, mass of the sphere,
M
* mass of fluid displaced by the sphere, and
kH * virtual mass of the sphere. forces involved in this motion are as follows: l^x inertial reaction of sphere, U a
force exerted by the fluid on every fluid volume,
kM(a  x)
virtual mass effective force, and
A(a  x) dissipative force on the sphere.
15 On account o£ the pressure gradient in the fluid if the fluid is accel erated in the direction a • there is a force force kM(a  x)
Ma on the sphere,
the
is the virtual mass multiplied by the relative accel
eration between fluid and sphere.
This simple treatment is possible in
a system with onedegreeoffreedom, since the acceleration is a true vector in this coordinate system. all forces producing motion.
M
Q
The three forces mentioned last are
Therefore, this differential equation is
v =' M a + k
m
(o  x ') t A(qi)
The motion of the oscillating fluid is prescribed as a sq

is the half amplitude of the fluid motion.
ferential equation is
k
^
= C l + C w e~”t +
a.
i fM(i+K)
TTTZyU" \m u > /
..a
+( A Y
•»
>
r
+ / j f L >V  M ( l + k )  # I
\ (rncu IJ
im
The solution of the dif
\mtu / / Tf\
S in G u t+ v p ).
I I
J
(4a)
V kM.
/■ftVM. o + O
\W1 MJ / L IT) m ( u k 1
rn
+
( A.. Y Vimco / \i
 'I
J
(4b)
is
These expressions can be sinplified by introducing the equality m

M 0 + k M
= M +
k)
X x i U . , respectively.
IX
solution for the maximum established amplitude is & ^ ft, \Z
( tkf + f— v
____________ v H c o /
anoi
(^c)
+ k ') * +
0*kX^+kV(jfedz
+or» « = fffiLY1 ~ ^ 5/g ) 
xq is the half amplitude of the sphere motion. may be observed concerning the above solution.
(4d)
Some interesting points If
^ 1 *> the angle
tp is negative, and the motion of the sphere lags behind the fluid.
On
the other hand* if ^ s/ ^ < 1 , the phase angle is positive and the sphere precedes the fluid motion. zero.
If (Da/ ^ » 1 , the phase angle is
• m
As the density ratio is increased toward infinity, the phase
angle is
T
xfz .
On the other hand, as
^s/ ^
approaches zero, the
positive phase angle is increased to a limit less than TT/2, . The amplitude ratios are as follows:
O < x0/a0 O, x o /«© “ 1
^s/p > I j
j 'f
\< x0/aB
$l> 01
3
to CM
•
«?
i"?
P 0
of Physical Data
(4 g
d O
l{ •8
a E*3
eo £
d ua H•I ' 4*
0p •HI fll
d a •*1° 4& to
o * > *O * r I P » CD
to
to