The electrokinetic determination of the stability of laminar flows

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Title:
The electrokinetic determination of the stability of laminar flows
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ix, 53 leaves : ill. ; 28 cm.
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English
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Hardee, Addison Guy, 1938-
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Subjects / Keywords:
Fluid dynamics   ( lcsh )
Laminar flow   ( lcsh )
Boundary layer   ( lcsh )
Aerospace Engineering thesis Ph. D
Dissertations, Academic -- Aerospace Engineering -- UF
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1968.
Bibliography:
Bibliography: leaves 51-53.
Statement of Responsibility:
by Addison Guy Hardee, Jr.
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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oclc - 828922219
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THE ELECTROKINETIC DETERMINATION OF
THE STABILITY OF LAMINAR FLOWS













By

ADDISON GUY HARDEE, JR.


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
LN PARTIAL FULFILLMENT OF THIE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
1968










































To my wife















ACKNOWLEDC.,-...:I:-


The author wishes to express his sincerest appreciation to the

members of his supervisory committee for their cooperation and efforts,

and in particular to Drs. K. T. Millsaps and M. H. Clarkson. The

guidance of Dr. Millsaps avoided pedagogy of the sort usually bestowed

upon graduate students while introducing them to new fields of endeavor.

Anpreciation is also expressed to Dr. R. C. Anderson for his

many suggestions throughout the investigation and to Dr. J. E. Milton

for the help he has given.

Further appreciation is also expressed to the Air Force Office

of Scientific Research for grant AF-AFOSR--2-67 which has made this

research possible.
















TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS . . . . . .. . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . v

LIST OF SYMBOLS .. . . . . . . .... .. ...... .vii

ABSTRACT . . . . . . . . . . . . . .. ix

Chapters

I INTRODUCTION . . . . . . . . . . . .. 1

Scope . ....... ................................ 1
The Stability of Laminar Flows . . . . . . 2
The Experimental Investigations of Transition . . .. 5
The Electric Double Layer . . . . . . . .. 7
The Streaming Potential . . . . . . . ... 10

II THE EXPERIMENT . . . . . . . . . . . 14

General Description . . . . . . . . .. 14
Laboratory . . . . . . . . . . . 17
Apparatus. ........ ......................... .18
Cleaning of the Apparatus. .. . . . . . . 20
Electrolytic Solutions Used in the Experiment ..... 20
The Electrical Measurements . . . . . . ... 21
Determination of Reynolds Number . . . . . .. 25

III RESULTS AND CONCLUSION . . . . . . . . ... 28

R.M.S. Component of Streaming Potential . . . .... 28
Streaming Potential . . . . . . . .. . 38
Conclusion . . . . . . . . . . . . 47

BIOGRAPHICAL SKETCH . . . . . . . . . . ... 50

LIST OF REFERENCES . . . . . . . . . . . 51
















LIST OF FIGURES


Figure Page

1. Sketch of Flow Apparatus . . . . . . . ... 19

2. EDC Measurement . . . . . . . . . .. 22

3. ERMS Measurement . . . . . . . . . ... 22

4. Tracing of Typical Spikes (Reynolds Number is 2572,
Concentration No. 2) . . . . . . . .. 24

5. Pipe Calibration (Resistance Coefficient vs. Reynolds
Number) . . . . . . . . . . . .. . 26

6. Resistance Coefficient vs. Reynolds Number (Log-Log
Plot) . . . . . . . . . . . . .. 27

7. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 1 . . . . . . ... 29

8. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 2 . . . . . . .. 30

9. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 3 . . . .... . .. . 31

10. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 4 . . . . . . ... 32

11. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 5 . . . . . . ... 33

12. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 1 . . . . . . . . ... 40

13. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 2 . . . . . . . . ... 41

14. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 3 . . . . . . . . .. 42

15. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4 . . . . . . . .... 43











LIST O0 F1I.... (continued)


Figure Page

16. D.C. Component of Strea:.min Potential vs. ?r. e,
Conccnzration No. 4. Data Taken from Dymec
Integrating Voltmeiter . . . . . . . ... 44

17. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 5 . . . . . . . . .. 45

18. Razio of Laminar Slope to Turbulent Slope vs. Double
Layer Thickness . . . . . . . . . . 48






















a


1










DC


XRMS



- p

i-

-2










nio



c
n.
r0


R


LIST OF SYMBOLS



radius of pipe bore

area of nipe cross section

constant

diameter 0of pipe bore

electronic charge

strc.-.": potential

D.C. component of steaming potential

R.M.S. component of streaming potential

force per unit volume on fluid element

flow resistance coefficient

convective electric current

conductive electric current

characteristic length

lenghn of pipe

number density of it ionic species

number density of i ionic species at the wall

pressure

electric charge

radial coordinate




arc I..gr











U velocity in the x-d1ircction

U mean velCcity

V velocity in the y-dircction

x cartesian coordinate

y cartesian coordinate

charge density

double layer thickness

S dielectric constant

K Boizzann's constant

2 specific conductivity

viscosity

kinematic viscoszity

3 density (fluid)

T shear stress at wall

reciprocal of Debye length

S electric potenzial

noTential at wall

J electrical resistance

a2.
7 Lapacian operator


viii









Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of t:e Requirements for the Degree of
Doctor of Philosophy


THE L :. ', K T C .. : -. ...'...ION OF THE
STABILITY OF LAMINAR FL,..'L

By

Addison Guy Hardee, Jr.

August, 1968



al..., K ox T. 2,-b p h
ajor Department: Aerospace Engineering

An experimental investigation of the transition from laminar to

turbulent flow in a pipe of circular cross section has been conducted

by ...-i.l: the electrokinetIc phenomena generated by the interaction

of the fluid flow with the electric double layer. It was found that

-he critical Reynolds number of the flow depends on the electrolyte

concentration of the fluid, that Reynolds number being higher for lower

concentrations.

The investigation also revealed that the fluid flow alters the

eauicibrium state of the electric double layer. In addition, the

transition to turbulent flow can be detected by a change in slope of the

curve for streaming potential vs. pressure, but only if the concentra-

tion of the electrolyte is a very low value.


























The present investiaion is te na rural outgrowth of previous

work by ArQnderson ',f wch it rprsents, at the same time, both

an extension an. c ariica io r.. U 1 .. the eectrokinetic phenomena

associated wi t he elecr ic .double ,.yer, Che transition from laminar

o turbuen- flow in a pipe is s-uuied by err..ning the growth of

erurbaions in e fow very close -co th pipe wall. The eleczrokinetic

pIenoenon o_ mes e s the followingg: when an electrolyte,

however oweak, flow-j a7rouh pi a potet difference can be

measure between a pair o eecods placed at the ends of the nine;

thi s n rern ,ont an.r is ac result of zhe electric double

layer 'nich exists a the sold-liquid interface at the wall of the pipe.

T.nhe electric double ayer is discussed m..ore fully on page 7.

The pupo o: Andersn's invest igaion was primarily that of

de .:.minin- p reisely ne .-nir critical R. ":.i number of Poiseuille

rfl ow in a p-ie, -u tr.e -l u on. which he discovered, of 71.e

strea&.ing potential were use o detect thne onset of turbulence in the



Te radia. -ocace.on of p5ions i the flow very close to

the wall is -ossil. for he ex-enson of the electric double layer











into the rflui is of the order of Angstroms and this extension (the

double layer thickness) is a function of the concentration of'the

electrolyte. The present investigation examines the nature of the

streaming potential fluctuations for double layer thicknesses ranr.ing

from 5S9 to about 5,000 A.


The Stability or Laminar 7Flows


The theory of the stability of laminar flows is usually traced to
(2)
Osborne Reynolds who, from experimental observation and theoretical

studies, postulated that the state of laminar flow is disrupted by the

amplification of small disturbances; Reynolds, however, gives credit to

Stokes for the concept.

Reynolds treated both inviscid and viscous fluid stability; many

workers since have spent a great deal of effort on the mathematics

underlying Reynolds' hypothesis--namely, to superimpose a small periodic

perturbation on the mean flow and examine the growth or decay of this

disturbance. Xost notable among the names of these workers are those

of Rayleigh(, Sommerfeld"4, Orr(, Tollmien and Prandtl(

Mention should also be made of Heisenberg(8) and Lorentz(9)

Just one example of Rayleigh's work will be mentioned here; he

showed that, for frictionless flow, a point of inflection in the velocity

profile is a necessary condition for instability, i.e., for amplifica-

tion of disturbances; later Tollmien was able to prove that this is also

a suffice ient condition.

The great majority of zteoretical work has been .directed toward

the sabity o :wo-dimensional flow, the application of the theory











being .hI superposition of u iwo-dimen&ional perturbation on the stream
function of the mean flow. N uglecting terms quadratic or higher in the


disturbance leads to the Sommerfcld-Orr equation, which is the fundamental

differential equation for the disturbance amplitude.

Use of a two-dimcnsional disturbance was questionable until

Scuire(10) showed that a tihree-dimensional disturbance is equivalent to

a two-dimensional one at a liwer Reynolds number--at least when applied

to a Two-dimensionI.. flow; t.his has become known as Squire's theorem.

Heisenberg is the first to deduce theoretically the instability

of plane Poiseuile flow for sufficiently large Reynolds numbers,

although he did not calculate a low,:r critical Reynolds number. The
(11) (12)
theory of Heisenberg was c ended by Tollmien and Schlichting(

both calculated the neutral stability curves for the boundary layer over
(13)
a flat late. Much later, Lin performed the calculations again

c "..mo.e clarity, and succeeded in calculating the curve of neutral

disturbances for plane Poiseuille flow. Contrary to the conclusion of
(14)
Lin, Pekeris ..i.c a different technique and concluded that

Poiseuille flow is stable for all Reynolds numbers. To resolve this

.disagreement, Thomas(5) calculated the critical Reynolds number by

direct numerical methods and found it to be 5780.

The more difficult analytical problem of the stability of pipe
(16)
flow has not been resolved with such clarity. Sexl was the first

to solve the viscous problem, although only for axisymmetric disturbances.

The results have been questioned because of his mtmacal simlifca-
tions. For a small regon near the wall, Pretsch(17) showed the problem


became the same as that of a disturbance applied to Dlane Couetne flow











and Pokeris obt.ai;d a solution for the region near the axis of the

pipe. Corcos and 'e jlars ) Sav a solution which accounts for the

work of both Pret.ch and Pckeris. The conclusion drawn from the work of

these investigators is that Hagen-Poiscuille flow is stable for small

disturbances and the work of Sexl and Spielberg(19) confirms this. (Sexl

and Spielberg also showed that Squire's theorem does not hold for axially
(20) :
symmetric flows.) Experimentally, Leite failed to observe any

amplificarion of small axisy.mmetrical disturbances (placed in the inlet,

close to the wall) downstream in a circular pipe at Reynolds numbers as

high as 13,000.

All of the investigations to date imply that Poiseuille flow is

stable for small disturbances. It is an experimental observation that

turbulent flow occurs in pipes--_ seaming paradox when one regards the

mathematical results. The re ..lution of the "paradox" may lie in the

contrast between "small" a-. finite disturbances occurring in the flow

or in :he symmetry of the disturbance. In this connection, Meksyn and
(21 )
Stuart(21) showed that, in a channel the lower critical Reynolds number

decreased as the amplitude of tne superimposed oscillations increased,

which is in accord with the qualitative observations of Reynolds. A

(22)
possibDle explanation has been given by Gill22, who reviewed the above

theoretical papers and indicated questionable steps in their procedures.

On the other hand, Tatsumi(23) has predicted theoretically that

the flow in the inlet of a pipe is unstable at a Reynolds number of

9.7 X 0. Exern.0taly, both :'. (unpublished) and Taylor

(unpublished) obtained laminar flow up to Reynolds numbers of 5 X 104

and 3.2 X respTectivrly; Ekmian made use of Reynolds original apparatus.











The Ex cri cnual Irv.st :atiu:.: o- Tlansition


(e24) firt notu the transitiLon from laminar to turbulent

flow hil deter :g law of resistance for pipe flow. He was
-low while d e t e ri:"' Ie 1wo-r lL ncc

aware that the "critical point" dcpendcd on the velocity, viscosity and

-he nine radius Te breakdown or what we today call laminar flow was
noted by the puling of the jet from the pipe and also by the addition


of sawdust to the flow, showing irregularities present above the critical

point.

The fundamental investigation of the phenomenon of transition was

-.errorme by Reynolds, who showed conclusively that there exist two

nossible modes of fluid flow-laminar and turbulent. He was most
ill(25)
probably not aware of Hi 's work, which predated that of Poiseuille(25)

On the other hand, Reynolds was in possession of the Navier-Stokes ecua-

tions and, by dimensional reasoning, was able to determine the form of

the parameter governing the "critical point." The parameter, of course,

is the Reynolds number and, being the "similarity" parameter for viscous

flow, is more than just the parameter for transition. Though Reynolds'

paper is often quoted, one passage from his 1883 paper is worth noting,

especially with regard to the aforementioned "paradox." Concerning the

sudden disruption of the flow, he writes:

The :act that the steady motion breaks down suddenly
shows that the fluid is inr. a state of instability for
disturbances of the magnitude which cause it to break
down. the fact that in some conditions it will
break down for smaller disturbances shows that there is
Scertra:n r d.... l u tab'Ility so long as the cisturbances
co not exceed a given amount.

In the scon of ine jnen.dent exper mc"zs, Reynods ceItermined the











inimu:. critical l R umi'r of a :.._ straight pipe; the value he

found was aDpproximatly 2000.

Since that time, many workers Lave repeated Reynolds' experiments,

some with interesting variations. The most extensive repetition was
conducted by Stanton and Pannell(0. Barnc, and Coker(27) used a


thermal method c detection in which the walls of the pipe were heated

and the onset or urbuence was detected by a sharp rise in the tempera-

ture of the interior or the flow. RFiss and Hanratty(28) developed a

technique from which they could .infer the behavior within the so-called

viscous sublayer by measuring .he mass transfer to a small sink at the

wall. The sink was a polarized electrode, current limited by mass

transfer. One of the more interesting methods is that of Lindgren(29)

who made the disturbances visiloe by using polarized light and a bi-

refringent, weak solution of bentonite. A technique utilizing the

electric double layer was developed by Anderson(1) and used to determine

the lower critical Reynolds number for Poiseuille flow. That Reynolds

number was found to be 1907 3, indicating the sensitivity of the

technlcue.

The experiments above were mainly conducted to study phenomena

associated with "L-. e" disturbances, while it was not until Dryden's(30)

very low turbulence wind tunnel became available that experimentalists

were able to examine the small disturbance problem. This was undertaken

by Schubauer and Skramszad(37) who made an almost direct transfer of the

teoretcal mehod to, the h .cal situat: n--ocill.io:.. were induced

n a metal rbon aoe I rfat lace placed in the low turbulence tunnel

a:.d t.he amlifca'n of the flow disturbances : :.a:trd dwsream by











means of a hot wire. Thi. cx.eri:;en. s arc regarded as excellent

verification of stability teco,.

The experimen(.t of 2 r.c...cd e ,r was similar to that

of Schubauer and SXkramstad, but axi-symmeIric rbances were super-

.imposed to the flow of air in the inlet of a p,


The Electric Double Layer


The phenomenon known as the electric double layer has been

Studied extensively by chess, especially in connection with colloids

and with electrode processes.

..The electric doublee ay consists of an excess of charge present

at the interface between two phases, such as a solid and a liquid, and

an equivalent amount of ionic charge of opposite sign distributed in the

solution phase near the interface. Consider one phase to be a solid

such as the wall of a pipe and the other to be a weak electrolytic

solution.

I- the solution is caused to flow past the wall, such as in

Poisui le ow in a p there develops a potential difference between

the ends of the pipe due to the motion of the distributed charges. This

phenomenon, known as the streaming potential, was discovered by Zollner(32)

and subsequently Helmhoiz (33) gave an explanation based on Poiseuille

(3~4)
flow anc h.e concept of the double layer developed by Quincke .

early workers in the field considered the double layer to be

Sr~ P- one fixe to the .a'l and one

ree to move w'th :hz flu-id. The more realistic model was proposed by
uy(25) (. 3 ) . ,( )
ouy ... p., w. inde rendentl.y ormua..... the theory of the











diffuse double i.yer, which is, in essence, the theory of ionic atmos-

iv -, - I- (3 ). err(38)
heres ivsn so:c -en years liter by liebye and Huckel ( Stern

.odified th th theory to account for the finite size of the ions at the

wall.

An excellent sunm.rry of the classical physics of the effect has

been given by Smoluchowski and extensive analyses of the approxima-

tions used in the various theories are given by Kirkwood(40) and


(4Casimir2)
The analysis given follows Kruyt The charge at the interface

is considered to be adsorbec on the solid surface and uniformly distri-

buted, while the solvent is assumed to be a continuous media, influencing

the double layer only -hrough its dielectric constant. Coulomb inter-

actions in the system are described by Poisson's equation

= -(1)

where T is the potential (having a value of Yo at the wall),

Sis the. cnhara density,

Sis the dielectric constant,

a- V is the Laplacian operator.

The number density of the i1h' ionic species is assumed to be given by

ne= Y0,-Z-t/KT) (2)

where Vtio is the number density of the ith species at the wall,

Z; is the valence,

_. is t-.e e ectronic charge,

Sin ~ Dl zmann's constant,

and T s t.: tzemerazture o the solution.

e. .'Ytv igven by










: --- i ',(3)


Combining equations (1), (2), ud (3), one obtains

7 = +-.-nflxQ-Z aY/KT) (4)

which is the differential equation for. the potential as a function of
the space coordinates.

Upon assuming an infinite plane wall, equation (4) is simplified

to
__ = Z C)(-Z2e.- TkT) (5)
A
with the boundary conditions

4)0 C'" = 0 00.
C1


where M is the distance from the wall.

The first integration of (5) is carried out after multiplying

through by Z
2LY -..4-Z2Vje;?-C / I


Upon integration and applying the boundary conditions, the result

is obtained


t 2 T k- Z i/x-2 IK T)

The equation can be simplified by considering a single binary

electrolyte, therefore


The" seo7 ri KT... -Z i
L 4 e ~ ...... p p-it/ Z 7Ts
N( E
Te Lecond inte-ration is performed after writing7 equation (7) as













E(Ze~t/aCT)-ex(-ZL'^/cT) -
ze-So^


There:ford



Q ~ ~ [^(eZ'+j C t -, /"j,
X.ZMJ'~() =pze'o~r IAI------------ (a)




w..ere

z 4 77" ;'L
2 -



For small oenzias, this :,ay be simplified by expanding the exponential

terms to yield



cr



showing that the potenzial decreases exponentially to zero over a distance
of the order of magnitude of y ..,-. the thickness of the double

layer is of the order of V At room temperature % is approximately

3 X 10 Z V where C is the concentration in gram-moles/liter. As an

examp e, the double layer thickness is approximately 10-6 cm for a

0.001M KC solution.


S.:.r-:.i. -otcentll 's gnerated when >the- c crolytic fluid






11



is caused to flow by applying a pressure difference between the ends of

the pipe; the flow displaces the charges in the movable portion of the

double layer and, so, constitutes an electric convection current. As a

consequence of this current, a potential difference, the streaming

potential, arises between the ends of the pipe. In the equilibrium state,

an equal and opposite "conduction" current counterbalances the convection

current; the conductance determining the conduction current is usually

assumed to be the bulk conductivity of the fluid.

A:-...._ the charge density on the wall is independent of the

flow, the convection current is

(10)
A
where U is the hydrodynamic velocity and A is the cross-sectional

area.

Substituting from Poisson's equation yields:


((1

A

Assuming that the double layer extends a distance out from the wall

which is small compared to the radius of the pipe, equation (11) may be

written:


11 47 Tj (12

A

Successive inbegrazion by parts and application of the boundary condi-

tions 0 d C
^






12



and



yields

^ S _[ (13)
1, rTj^V~ ^ ^ j ^ -


wh.ere S is the circumference o: the pipe and Y* is the value of

at y=O.

if the flow is d....





where ,j is the shearing stress at the wall and is the viscosity

of the fluid. Also, if the flow is in a constant area pipe,


ax


where X is the coordinate along the axis of the pipe. Therefore,

equation (11) ben...

ii = ()~s-^^^
;77-j j)x j 4-


The second term in equation (14) will be neglected since it may be shown

that the ratio of the second t-erm to the first is the order of

where r is the radius of the pine. Therefore, if the pressure gradient

is constant over the cross-section, equation (14) may be written as


4T A T' dX A











The conduction currcnlt i Jiven by


= -.


(16)


where 9. is the specific conductance of the solution and E is the

straandilm 4ohontla-..
E
;r 9, and *- are constant over the cross-section, equation

(15) becomes


CiEl.


p ,; hence

2 ?~


(17)


(18)


Under the above .-: .tions, equation (18) shows that the streaming

potenzial is a linear function of the pressure drop between the ends of

the pi~e.

.The first term on the right-hand side of equation (13) indicates

that the convection current, and hence the streaming potential, is a

function of the velocity radient at the wall.

















CHAPTER II


THE L. i .'i :.T


General 7ascriotion


The present investigation was undertaken to examine more closely

several aspects of streaming potential phenomena which the experiments

of Anderson(1) brought to light.

The experiment is an extension of Anderson's work in which he

developed a technique for the detection of turbulence; therefore, a

short account of his c:*:. :iment and the reaso...: behind it will be

given.

Tne theory of Chapter I shows that the streaming potential is

directly proportional to the pressure gradient along the pipe. Also,

equation (13) shows that the streaming potential varies directly and

linearly with the velocity gradient at the wall.

if we now restrict the discussion to Poiseuille flow, the velocity

distribution across the pipe is given by







at a sufficient distance from. the inlet, where

.a : thre raclas of the pipe,




:x: ie ;. coordinate along the pipe,











is th. vi:cosiy o- the fluid,

r is the di nce from the center of the pipe,

and u is t veloct-y.

If this rlation is substituted into equation (13), the streaming

oteni is again found to be a linear function of the pressure gradient

along tLhe pipe; thisiZ-S bS.en o1 ;erved by ..,'.. -1 chemists() for many

years. Since cuation (13) indicates the dependence of the str- c. :...

otenial on the velocity gradient at the wall, Anderson reasoned that

streamrin potential ,easurements could be used to determine the transi-

tion from laminar to tubulent flow, for the velocity gradient near the

wall is different in :c.se to modes of flow.

n son looked for a change in the value of the quantity AE/AP

(i.e., a"brak" in ,e c1.- ve of E vs P) when transition occurred. His

ex-periment was conducted .....i ng an electrolytic solution of O.001N KCi

flow=n n a Pyrex capillary tube having a diameter of 0.0242 inches

and a length of 4.8 inches. The inlet to the pipe was artificially

roughened.

Although he found a suggestion of such a break, Anderson's data

did not une.uivocaly show one; instead he discovered unique fluctua-

ins i the stre..m.n. potential which appeared at the transition

Reynolds number. Using these fluctuations as a guide, he was able to

determine the minimum criica Reynolds number very precisely. In

addition, he showed thait the streaming potential did not vary linearly

winpr<-.:.:* -.r........ ir ;. stream~ potential was measured across

tne full length of -?`h piDe. This s the effect "of the entrance length

(".t -.*.th. f :. ... p : ... cesay for est-biish:ing Foiseui e flow)











which the theory does not include. Most of the measurements of streaming

potential by physical chemists have not taken this into account.

The fluctuations appeared as "spikes," which always represented

an increase in (positive) voltage near the transition; above transition

the spikes appeared to be positive and negative.

The present investigation-was undertaken to investigate the effect

of the electric double layer thickness on streaming potential measure-

ments, the thickness being a function of the electrolytic concentration.

The electric double layer, for most liquids (including the electrolyte

used by Anderson), is well inside the so-called "viscous" sublayer;

thus the mechanism for producing the streaming potential fluctuations

was not apparent.

Little is understood of the actual conditions at a solid surface

in fluid flows; in a liquid, there is the electric double layer to be

considered. The electric forces in the double layer have been neglected

in solving the Navier-Stokes equations for the velocity in the pipe

(unless they are contained in the boundary conditions). In addition,

the theory given for the electric double layer, including its thickness,

is based on a stationary fluid--the analysis of the interaction of a

flow and the double layer is complicated. The double layer extends out

from the wall a distance on the order of Angstrom units. An aqueous

KC1 solution of 0.001N gives rise to a double layer thickness of 95.9A;

this, of course, is the distance from the wall to the "center of gravity"

of the double layer--the double layer can extend out much further. Such

small distances from the wall show that the double layer is within the

viscous sublayer. The measurements of Reichardt(44) and Laufer(45)











sem to point conclusively o tc existence of such a region near the wall

in which viscous stress ss gratly outweigh inertial stresses. In this

li.Sh, t xe ex:lanation of tze streamlng potential fluctuations becomes

more dlfficul4. Reichard(4 o predicted the fluctuations, but implied

that they were caused by perturbations in the mean pressure gradient

along re pipe--exactly what this means at a specific point very near

tme wall is unclear. eichard searched for the fluctuations, but was

unable to detect them using a quadrant electrometer. Also he found that

the value of AE/AP was essentially the same for both laminar and turbulent

flow.

It was decided to extend Anderson's investigation using a lr-.r.,er

-)e n a hi"a.g.er gain electrometer-amplifier and placing greater

on ise reduction. In addition, five different double layer

thicknesses (ranging from 589A to about 5,OOOA) would be used, so that

he doule layer would extend over varying distances into the viscous

sub-layer. The longer pipe (543 diameters) was used to reduce the effect

of the entrance length,


Laboratory


h.e aborazory was especially designed and constructed by Anderson

or streaming sng poentia! experiments. This was necessary since the work

is particularly sensitive to temperature and humidity. The thermal

dependency of the viscosity of the fluid necessitated the fine control

ofr th em-:-r'a-.r, ..l t]e humid 'ty of the room atmos-.ere was held

vtry ow ue to .e na-re of the electrical measurements. The currents

_nvc.v ; -/.\;r c. .e er i0- ,o 0 amperes with a source











resistance of the order of 109 to 10I ohms; a high humidity level could

cause surface leakage due to adsorbed water on the'exterior surfaces of

the glass apparatus.

The laboratory is a 10 by 16 foot room inside an air-conditioned

building and the entire room is vapor sealed with 10 mil polyethylene.

The floors and walls are insulated with 3 inches and the ceiling with 4

inches of Styrofoam. Temperature control is achieved by a special air-

conditioning system which holds the room temperature changes to 0.75 F.

In addition, the experimental flow apparatus was enclosed within a small

Styrofoam box. With this addition, temperature drifts are less than

0.02 C per hour.

The air drying is accomplished by use of a compression-cooling-

expansion cycle. This enables the room air to be held below 20% R.H.


Apparatus


The apparatus used is similar to that used by Jones and Wood(43)
(47)
Kruyt Anderson and others and is shown schematically in Fig. 1.

The reservoirs R1 and R2 are 5 liter boiling flasks fitted with ground

glass taper joints, J1 and J2. Tubes LI and L2 are 6mm Pyrex with

taper joints at the top for inserting electrodes. A1 and A2 are 1/4

inch tubes. The tube T is a precision bore capillary tube 0.0631

.0001 inches in diameter and 34.3 inches long. The capillary tube is

connected to A1 and A2 by flanges incorporating Teflon gaskets. In

operation, the liquid was forced up L1 and through the capillary by

applying gas pressure through VI.

The electrodes E1 and E2 are Ag-AgCl formed on spirals of 24






19






























0



< \ itl


'* i>
0


U)







w


i-











* platinum. -.... Ag-ACI coating was made with a mixture of 90% silver

oxide an- 10% silver chlorater formed into a paste, applied to the wire

and heuad at 500C for fi -een minutes.

The gas pressure was supplied by commercially bottled nitrogen.

The pressure was measured by a simple mercury manometer used in combina-

tion with a micro-telescope mounted on a vernier height -age. The

pressure could easily be controlled to within 0.005 inches Hg. The two

nhree-way stoecccks in the supply line provided a means of applying

pressure to the flasks or venting them to the atmosphere.


CleaninL of the Apparatus


The glassware in the apparatus was soaked in chromic acid for

several hours, cleaned with hot chromic acid and rinsed in hot conductiv-

ity water. The glassware was then leached in conductivity water over-

night. The electrodes were lcched in conductivity water for several

days.

The conductivity water used as the solution was twice-distilled,

having -itrc .: .ubbled through it after each distillation.


Elecrrolyvic Solutions Used in the Experiment


The experiments were conducted with 5 concentrations of KCI

solutions, beginning with conductivity water as the first solution. Salt

was added to this to fo.rm the subsequent concentrations. Table 1 gives

th< vates cf rh five concentrations and the corresponding thickness of











TABLE 1

THICKINESS OF THE ELE -..IC DOUBLE LAYE.


Concentration Nor:..ality 6 (cm)* 6e(A)*

S0 cc 00

2 0.5364XI0-6 4.16X10-5 4,160

3 0.8040X10-6 3.40X10-5 3,400
6 5
4 1.877X106 2.23X105 2,230

5 26.SOOXlO-6 0.589X10-5 589


= double layer thickness.


The Electrical measurementss


Three types of electrical measurements were made for each concen-

tration of electrolyte. The magnitude of the streaming potential and

-the root-mean-square value of the streaming potential fluctuations were

measured as functions of pressure. In addition, the resistance of the

eectrolyte-pipe-electrode "source" was measured before and after each

day's run.

The streaming potential magnitude was determined by the system

shown in Fig. 2. Using a General Radio Model 1230-A Electrometer (input

impedance 10!2 ) as a null indicator, the streaming potential was

balanced by t.-he output of a 0-90V two-stage voltage divider. This was

then reduced by a factor of 1000 by the Leeds and Northrup Volt Box and

'c o a Leeds and Nortnru millivolt potentiometer. Streaming poten-

ti!s for th. l.77I0-N and 26.80X10N solutions were read directly

fro::. 'e G--neral Rd electrom7.ter.











GENERAL
RADIO
ELELCTO-
METER


0,0ioI
i lOIO





I


Figure 2. E_- Measurement


Z.IF :ERENTIAL
ELECTROMETER-
AMPLIFIER


I-

10~~I





--K


Figure 3. ER Measurement
RMS


O-80V
VOLTAGE
DIVIDER











The R..S. values of the stre .. potential fluctuations were

determined usi.g .e system shown in Fig. 3.

Th.e string Ioential fluctuations were amplified by a

iffe....ia eecrometer-Cmplifier specially constructed for electro-

chemical. measurement; the input stages are Philbrick Model SP2A
14
.erti"...l amp1fiers. The inDut impedance of this amplifier is 10

ohms and -che gain was 8.80 .... .hout the experiment (gain can be set

at S.Z0, 88.C, or 830). The output of the volt-.ge divider was used to

cancel the D.C. comonen.-c of the streaming potential (this ranged up to

SOV). The output of the amplifier was fed into a strip-chart recorder

for a visual record of the fluctuations and into the Flcw Corp. Random

Signal Voltmeer :for their R.M.S. value.

The elctrical system represented by the electric double layer

is very sensiive to stray capacitance-it was found that an excessive

amount of "grassy" noise (which almost obliterated the smaller signals)

was generated by vibrations in the screen panels of the screen-room

(Fara"day c ). This was corrected by glueing aluminum foil shielding

onto the Styrofoam thermostat in which the flow system was placed. This

shielin, alo. with the usual care in reducing vibrations in all

electric components, gave a great reduction in noise level. Typical

1i 1 levels were:

sike magnitude (peak) 0.1V

sIike magnitude (R.M.S.) 0-50mv

.... .. .'n.. po-en. a! > (D.C.) 0-85V.

A typical spik e trace is s:own in Figure 4.

s prevics experimenters had noted, the values of the stre=.-in,
































































racr.n of Typica Spikes (Reynolds Number is 2572,
Co:.c.ratior No. 2)


15 SEC.


L- 0.05 V


iru::e 4.










potentials could not be reproduced accurately from day to day. For

each separate electrolytic solution, the streaming potential showed a

long-time drift of a period ranging from a few days to weeks, reaching

a minimum, and then increasing slowly. The D.C. component of the

streaming potential, for a given electrolyte concentration, was taken

over a two-hour period on the day the potential reached a minimum.

Reproducibility was also improved if the fluid was pumped at high flow

rates before taking data; also, higher Reynolds number measurements were

alternated with those of lower Reynolds number. If this alternation

were not carried out, the D.C. values for the low flow rates (below

Reynolds numbers of about 900) would drift upward. This indicates that

the equilibrium state of the double layer is changed when the electrolyte

is moving.


Determination of Reynolds Number


All of the data was taken as a function of the pressure difference

across the pipe. The Reynolds number was determined by mass flow rate

measurements using the last (highest concentration) solution. Reynolds

numbers were taken from this data, shown in Figures 5 and 6. The

resistance coefficient is defined by


Yzp 4-- L 4- Lp

where kwm is the mean velocity,

Lp is the length of the pipe

and P is the fluid density.
























a) coQ





G 0
o%




G
'^

a




c cO



C CM
00


SI I I I I I I
o vj -r 0. 0 oo C0) r C
C0!-i.-i I- I-- --I













0.05 -








0.03 -






o




0.02





0
0.02
0

So
00



00
















60700 830 903 1000 1500 2000 30


Reynolds Number'
c Coefficient vs. -enolds Number



Qlot)

600 700 800 300 1000 1500 2000 3000

Reynolds Number

-.-'-^, 6. Resisran~ce C "oefficient vs. Reynolds Number
(Log-Log ?iotc)
















CHAPTER III


RESULTS AND C.,h.LJiONS


R..S. C..n.. of Streamin;. Potential


The R.:.S. values of the fluctuations in the streaming potential

are shown in Figures 7 to 11, plotted against the Reynolds number; these

values have been divided by the gain of the electrometer-amplifier. The

points denoted by triangles represent laminar flow, i.e., no spikes

(fluczuations) on the streaming potential; those denoted by circles

reresen: the occurrence of spikes. Below transition (no spikes) and

above a Reynolds number of 2560, all of the data are shown on the graphs;

in between, only points selected to indicate the mean curve are given

o avoid crowding. The average deviation from the curve is about 7 in

Reynolds number. in Figure 11, all the available data are shown on the



> figure displays the same general trend; a peak, followed by

a dip, and a second, higher peak followed by a rapid decrease. Such

regular variatons are not detectable in mass flow rate measurements

tr.ough. the transition range of Reynolds numbers--only an irregular

increase (plot of resistance coefficient vs. R Fig. 6). At the

ies Reynolds number of the experiment, transition to fully-developed

urulence w:as no= complete (deduced from Fig. 6).

:.o frqu ncy 0of occurrence of the spikes at their first












I0






p
0 0>
0 OG^'


0
o 9

00

0


00


01
00 co
n I .


U) U)

oK~


I I I


0 -~


U)


0 0


0
0
rN















to
0
0


C)


(U
r-I



0
Z



0
C)(


0 CN


I


,,,~~ ~ I


0


, !













0


Z,0


0
0

0


0


4^


C




0<


0 0 CD


D
3 C'.













0



00
0 co)

o 0
~o
0 0 :
0


C)
4_J




0 U
0
0 C)


o
E
z
0

0 *

0 o
0 -V


0 >
'0
-'-1 0

0
0 *H





0
oJ 0; 4





4-I-
C2

0


0 ,
0' CC


I <3 :r




0 0





C)
C-' C,)

C-C (N- o o
I :"i o

o <3 [-2







32




0


0



r-3
0 0
o Z
CN
0
4-J


0 0)
C)




Q) c
ri-
o 5
o







U) )
oE
0
0 4


0
oo




U-),
0 oC







CD)
00
z
0

< 0 0
0




0





(D o -
0 0 0



0 o )


044
0





C4




M CU )J
0
0 (1)e










00

(0
pQ)









CO > (
o











0

0


0



0

0


0

0


9
0


0
0


00

0 0


oK


0
0
c
I-













0
0
0c
(N

C)



0


V)
a
C;











appearance was taken as 0.002 per second (one spike in an e'-ght minute

period) anc at R 2755 was approximately 2.1 per second; this frequency

was defined by manually counting the peaks of the spikes over an

C! i...ute period of the chart from the recorder. Thus, altho'h the


frequency response of the electrometer-amplifier and electrodes is flat

-Or s range, the response of the system comprised of double layer-

eectrods-am.lifier may be responsible for the variations in the ERMS

vs. R_ curves. The period of an individual spike was about 2 seconds.
The existence of some fluctuation below the appearance of the


first sxikes is a measure of "free-stream" turbulence as well as fluctua-

tions fro:m the inlet of the pipe. The corresponding R.M.S. values at a

Reynolds number of 1000 was about half that at 2400.

The curves for each double layer thickness are shifted progressively

to lower Reynolds number as the double layer thickness decreases. The

first s.ikes appear at R = 2585 for the first concentration and at
- n

R = 2470 for the last concentration. The peaks and other salient

features of the curves display corresponding, although smaller, shifts in

Reynolds number. The magnitudes of the shifts are too large to be

explained by changes in average viscosity of the fluid, but the same

cannot be said of the viscosity in the region of the double layer as

t-e double layer reacjusts to an overall concentration change. As will

be shown in the next section, the equilibrium state of the electric

doue layer is changed by the flow impressed upon it--whether this is

a cae in poCential distribution alone or with a simultaneous change

in noenial at the wall can not be determined. The part the electric

ou. layer it-f -ays in the stability of laminar flow is unknown,










although some insight into its role :,,t-h be gained by dimensional
analyst is.

Consider the two-dim-.nsional steady flow of a viscous fluid

aving an electric double layer present at the solid boundaries. The

Navier-Stokes equations are


(1)




where F. is :he electrostatic force on a fluid element. If one now

considers two flows with geometrically similar boundaries and let L1

be any length of the first flow field and L2 the corresponding length in

'he second, then L2 = CiLI. Similarly,


q2_= C2- Lk% = c3 P% ? p = 4 .
C lb
(2)




The individual terms for the second flow field in relation to the

.orreonding .terms of the first flow field are


Lk^ ^ (3)



I F C (4)




iC Lx.- '- b-, (5)



and..(
-''i )











In ord-er that th.e equations of motion for the two flow fields may be

"dentic-l, the following must be true


ct -- C &
72.
C) I C C4 C, C1


-- Cuac y -Cz
- A. ------~.Y C1


may be written


which means the Reynolds nu..bers of the two flows must be equal in order

for the flows to be geometrically similar.

The relation C_- *= 1 means that
C9 C,


rL- L, ,


must also be true. This may be written as


-3
FL
_-U -I",Rn


= constant


(10)


where R. is the Reynolds number and F is the electrostatic force per u;

volume on a fluid element.

-. electrostaric force per unit volume on the fluid element is

given by



=-%-Z (l:


nit


L)










where q is the electric charge contained in the element and Y is the

(electric) potential. For flow within the electric double layer


F= o(12)


where is the charge density. Therefore, the parameter in equation

(10) may be written

Y, 'X( e-q'-ApL) L:
og..&% LZl.

or, approximately (if the double layer thickness is taken as the

characteristic length) as

^^(- L1) L


Therefore, the parameter

^'e4'.L4 Y ) (13)
,-44V R~

must be a constant for dynamically similar flows.

The charge density is given by

+ (14)


where n1 is the number density of the negative ions and n2 is the number

density of the positive ions. Again the number density is assumed to

be given by a Boltzmann distribution:


A; = Vt. 0e X?(-2i;e-Y/KT)15


where ni is the number density of the i th ionic species at the wall

(the wall is considered to have a negative charge).* Therefore (for a










single valence, binary electrolyte), the charge density is approximately

ivnby






-similarity parameter now becomes




e M. L rxo. KT -1 P
Tv (16)




The quaiies 4' L and the exponential term in the numerator all

increase as the bulk concentration of the electrolyte decreases; there-

fore, te Reynolds number must increase in order that the parameter

remain consant. For more dilute solutions (i.e., thicker double la'er),

the higher must be the Reynolds number of flow phenomena within the

electric double layer. The Reynolds number here, of course, is based

cn the thickness of the electric double layer and the local velocity of

toe fOW.

The relation of the corresponding mean Reynolds number of the flow

to a local Reynolds number within the double layer depends on the rela-

tion of their respective velocities; whatever this exact relation may be,

the parameter given by the above analysis agrees qualitatively with the

xri ....T. results-the transition to turbulence, as indicated by the

double layer, occurs at higher Reynolds numbers for more dilute solutions.


- value of the D.C. com onent of the streaming potential for











the five solutions are shown as functions of pressure (non-dimensional)

in Figurs 12 to 17. Excepting a region at the -.ginning of transition,

he s :o for laminar flow is different than the slope for turbulent

flow. Thu., the change in -lope can be used to detect the transition

from laminar to turbulent flow. A comparison of the graph of EDC vs.

-ressure for the last concentration (Fig. 17) and resistance coefficient

(F) vs. Reynolds number (Fig. 6) shows that the streaming potential

indices a slightly higher critical Reynolds number than mass flow rate

measurements. The break in the F. vs. R curve begins when the eddies

grow enough in strength and number to increase appreciably the resistance

o the :flow. This should coincide very nearly with the establishment

of a different velocity profile near the wall--thus changing the value of

S as governed by equation (13), Chapter I. The difference in

critical Reynolds number given by the two methods, mass flow rate and

dreaming potential, may indicate that the turbulent velocity profile

is established "close" to the wall, i.e., on the order of 10- A, at a

higher Reynolds number than shown by an increase in flow resistance. A

cuestzion arises when one considers that the velocity gradient of equation

(13) must be that within the double layer, which means well within the

"viscous" sublayer; a.c. to the accepted model of this sublayer,

turbulent eddies (necessary to produce a change in velocity profile) so

near the wall must be very weak. Further, it is usually assumed that

the velocity profile within the viscous sublayer does not differ in

lainar nd turbunt flow. This says nothing about the magnitude of a

change in velocity profile which the double layer can Cdetect..n

S.;_- of a "viscous" sublayer, it should be kept in mind that the











































Q
a


a


0

0


0
0


I I I I I I I I


6 8 10 12

Pat (y


14 16 18 20


.C. C...n.r. of Strea~mi. n Potential vs. Pressure,


-70 -


-50 1-


:: (v)


0 2 4


..... I


-20 -







41

-70



o
0

-60 0









0

- 0



0




-30






-20


2 4 6 8 (v0 12 4 16 20

*P 13 (9I-4



0














-2" -VL-














Fi2u 3.D.C. Compconentll of Streaming Pote-ntiali vs. Pressure,
Concentration No. 2.
00




-10-








0 2 4 6 8 10 12 1~4 16 18 20








Fiue1. DY'. Comp>onent of Streaming Pot ntial vs. Pressure,
Concentration No. 2.



































S (v)


-10 I-


II I I I I I I I


0 2 4 6 8


10 12 14 16 18 20


-5 PA t -4


D. C. Component of Streaming Potential vs. Pressure,
Concentration No. 3


-40






























0


X I I I I I I I I I
0 2 4 6 8 10 12 14 16 18

-P2 ^4)
,"-V VLy


Figure 15.


D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4


-10


2,. (v)


e (
cp
















rl io 0




-- PIPE
I T-L
L_ 7_7j


-1.4




-1.20



-0.8





-u 4


-0.2




o0B


0



0

0




1 1 1


0 2 4 6 8 10 12 14 16 18 20

P2 cylo-4)
-4A-V L.?


F7Qu'c "C. P.C. Comnonent of Streaminj Pontial vs. Pressure,
C..r.:. trtion No. 4. Data TWkon from .'mcc integrating
VujT::.tor.






























0
0 0
0
0

0

0



0


Sl I I I I I I I I I


0 2 4 6 8 10 12

L# &o-45


14 16 18


.C. ....t of Streaming Potential vs. Pressure,
Concentrcation No. 5


-3.0 1


- (v)


-1.5 -


-i.0 i-











entire flow in the pipe is "viscous." Also the work which points to the

existence of the sublaycr is based largely on hot-wire measurements and

thes ave only been made over thie outer part of the sublayer.

A noter fact bearing on the above question is that in deriving

t-e eouarions for the streaming potential, it was assumed that the flow

condiiins" (including the velocity profile) were the same all alon. the

length of the pipe; in reality, the theoretical conditions are attained

cnlv =iter the entrance length of the pipe has been traversed. Further,

he entrance length necessary for establishment of such conditions is

much shorter for turbulent than for laminar flow. This means that the

entrance length is changing in the transition region of Reynolds numbers

(it is also a function of Reynolds number for laminar flow and, to a

lesser degree, for turbulent flow).

S. velocity profile for laminar flow in the entrance ien-th is

similar to that for established turbulent flow; thus, for laminar flow

a-: low Reynolds numbers, the value of AE/A? should be intermediate to

Those of established laminar flow and established turbulent flow. This

Sseen to be the case for all the data in Figures 12 to 17--the slope

of the laminar portion of the curve decreases with increasing Reynolds

nu7er.

Measuring the in potential and pressure across one segment

of the ie, excluding the entrance length, would better fulfill the

condiions of the theory and show whether or not the "non-developed"

f3w in .e ch_-ng.ng entrance length is responsible for the difference

in slops. Tis was considered in preliminary planning of the investiga-

ion; :.;ver, it was evident that any method of isolating one length











of the ie would introduce disturbances to the flow and would require

such a large system that purity of: the solution could not be maintained

with certainty.

The magnitudes of streaming potential given here include the

average value of the streaming 3otenzial fluctuations. These average

values should be --: the same order of magnitude as the R.M.S. values,

i.e., less than 50 millivolts. This is a sufficiently small part of the

7_ values to be ignored... data for the last, and most concentrated,

soutioc. (Fig. 17) shows an unexplained peak at the beginning of transi-

tion. The peak occurs roughly at the same Reynolds number as the maximum

..M.S. value (Fig. 11), but is about twice the expected magnitude.

6gure 16 ows EDtaken from a Dymec (Hewlett-Packard) Integrating
::u~ _. 6 ho s DC

Diital Voltmeter, :odel 2401A, set up as shown in the inset to this

f -.-._. Integration does not alter the basic curve, but does slightly

c.-.nge the ratio of the turbulent slope to the laminar slope. These

data were taken with concentration 4 on the same day as the data shown

13.

Although all the curves are similar, they represent a wide range

of streaming potential magnitude. In Fig. 18 is shown how the ratio of

he turbulent slope to the laminar slope varies with double layer thick-

ness; the slope for laminar flow was taken to be that of the portion of

the curve immediately preceding transition. This ratio approaches unity

for a suffici.ently small double layer thickness.


The cocu. on.s drawn frcm the experimental :Investigation are











INTEGRATING VOLTMETER


0 0


3 -------- --- -------- 1 --I1--I--I-''I
S2 3 4 5 6 7 8 9 10

: (xSi'c. tS

* .: *'r ;..-:;:.o of T .:.-Inar Sic, to Turbulent Slope vs.
-o^ : L* ...k :%N ..';











The critical Recynolds number of pipe flow, as indicated by

streaming potcntiai fluctuations, depends on the thickness of the electric

double layer present at the wall of the pipe, i.e., on the concentration

of the electrolytric fluid.

The equilibrium state of the electric double layer in a stationary

fluid is different than that for a flowing fluid, at least for laminar

flow.

The ransition from laminar to turbulent flow can be detected by

noting a change in slope of the graph of EDC vs. pressure, but only if

the electrolytic solution is very dilute.
















BIOGRAPHICAL SKETCH


Addison Guy !ardee, Jr., was born April 7, 1938, at Mulberry,

Florida. In June, 1955, he was graduated from Hillsboro High School

in Ta&mpa, Florida. From 1956 to 1960, he served as an electronics

technician in the United States Coast Guard and was stationed for a

tiM.e in Iceland. Following his discharge from the Coast Guard, he

enrolled in -the University of Florida and in December, 1964, he received

the degree of Bachelor of Aerospace Engineering. He received the

degree of Mastcr or Science in Engineering from the same school in

December, 1965. He received an appointment to the position of Research

A~sociate in the Aerospace E.-.,ineering Department in January, 1966,

wich: position h.. Ias held to the present time while : _i-l., his work

toward the degree of Doctor of Philosophy.
o- Dc-Lor of Philosophy.

Addison Guy Hardee, Jr., is married to the former Mildred Fe

Collar and is the father of two children. He is a member of Tau Beta

Pi, Phi Phi and Phi Eta Sigma.
















LIST OF ;.;'L ;;


1. Andocrson, R. C., Ph.D. dissertation, Univ. of Florida, 1965.

2. Reynolds, 0., Pil. Trans., (ISJK), 935.

3. Lord Rayleigh, Sci. PaIers I, (1880), 474.

4. So..erfeld, A., Atti. del congr. internal. dei Mat., Rome (1908).

5. Orr, W F., rc. Royal Irish Academy A, 17, (1907), 124.

6. Tol -c W., .- .. ... . "-. .. -
CFclch r u C-e -I L ), 7 _; -r11 ..jr. "-.-.-r . -" .
792 (1936).

7. Pr.an.cdtl, L., ZAMY, 1 (1921), p. 431 and Phys. Z., 23 (1922), 19.

8. Hisenberg, W., Ann. d. Physik, 24 (1924), 577.

9. Lcrentz, H. A., N.kad. v. Wet. Amsterdam 6, (1897), 28.

10. Squire, H. B., Proc. Roy. Soc. A, 142 (1933).

11. To:ir.ien, W., Nachr. Ges. Wiss. Gottin)en, Math. .:. Klasse 21
(1929), 44 ; 1-_-,",- -, . .. IA:,- r : ,!.C7; (!: ,i .

12. Schlichting, H., Nachr. Ces. Wiss. Gottingen, Math, Phys. Klasse,
(1933), 182; also Z 3MM, 13 (1933), 171.

13. Lin, C. C., Quarterlv Ail. Math., 3 (July 1945), 117; also 3
(Ocr. 1945), 218 and 3 (Jan. 1946), 277.

14. Pekeris, C. L., Proc. Nat. Acad. Sci., 34 (1948), 285.

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...-. .. .., -: iu over S-ke el:ectrolvten en
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53



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This dissertation was prepared under the direction of the chairman

01o the candid 's supervisory committee and has been approved by all

meers of that committee. It was submitted to the Dean of the College

of Engineering and to the Graduate Council, and was approved as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.


August, 1098





Dean, olleGraduate Shoolf E-eerin

Dean, Graduate School


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