
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00025843/00001
Material Information
 Title:
 The electrokinetic determination of the stability of laminar flows
 Creator:
 Hardee, Addison Guy, 1938
 Publication Date:
 1968
 Language:
 English
 Physical Description:
 ix, 53 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Charge density ( jstor )
Electrolytes ( jstor ) Ion density ( jstor ) Laminar flow ( jstor ) Reynolds number ( jstor ) Streaming ( jstor ) Turbulent flow ( jstor ) Velocity ( jstor ) Velocity distribution ( jstor ) Viscosity ( jstor ) Aerospace Engineering thesis Ph. D Boundary layer ( lcsh ) Dissertations, Academic  Aerospace Engineering  UF Fluid dynamics ( lcsh ) Laminar flow ( lcsh )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1968.
 Bibliography:
 Bibliography: leaves 5153.
 General Note:
 Manuscript copy.
 General Note:
 Vita.
 Statement of Responsibility:
 by Addison Guy Hardee, Jr.
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 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 030832991 ( ALEPH )
828922219 ( OCLC )

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Full Text 
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 AFAFOSR267 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 (LogLog
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 xd1ircction
U mean velCcity
V velocity in the ydircction
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 suuied 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, flowj 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 soldliquid 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 pie, 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 exenson 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' hypothesisnamely, 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 :wodimensional flow, the application of the theory
being .hI superposition of u iwodimen&ional perturbation on the stream
function of the mean flow. N uglecting terms quadratic or higher in the
disturbance leads to the SommerfcldOrr equation, which is the fundamental
differential equation for the disturbance amplitude.
Use of a twodimcnsional disturbance was questionable until
Scuire(10) showed that a tihreedimensional disturbance is equivalent to
a twodimensional one at a liwer Reynolds numberat least when applied
to a TwodimensionI.. 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 HagenPoiscuille 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 1wor 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 flowlaminar 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 NavierStokes 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 socalled
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: nocill.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 axisymmeIric 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 fluid. 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,Zt/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 = +.nflxQZ 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 ..4Z2Vje;?C / I
Upon integration and applying the boundary conditions, the result
is obtained
t 2 T k Z i/x2 IK T)
The equation can be simplified by considering a single binary
electrolyte, therefore
The" seo7 ri KT... Z i
L 4 e ~ ...... p pit/ Z 7Ts
N( E
Te Lecond interation is performed after writing7 equation (7) as
E(Ze~t/aCT)ex(ZL'^/cT) 
zeSo^
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 grammoles/liter. As an
examp e, the double layer thickness is approximately 106 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 crosssectional
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^^^
;77j j)x j 4
The second term in equation (14) will be neglected since it may be shown
that the ratio of the second term to the first is the order of
where r is the radius of the pine. Therefore, if the pressure gradient
is constant over the crosssection, 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 crosssection, 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 righthand 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 velocty.
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; thisiZS 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
experiment 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 estbiish: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 investigationwas 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 socalled "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 NavierStokes 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 fluidthe 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 layerthe 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 pipeexactly 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 electrometeramplifier 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
sublayer. 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 nare 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 airconditioned
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 compressioncooling
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 AgAgCl formed on spirals of 24
19
0
< \ itl
'* i>
0
U)
w
i
* platinum. .... AgACI 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 microtelescope mounted on a vernier height age. The
pressure could easily be controlled to within 0.005 inches Hg. The two
nhreeway 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 twicedistilled,
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 0.5364XI06 4.16X105 4,160
3 0.8040X106 3.40X105 3,400
6 5
4 1.877X106 2.23X105 2,230
5 26.SOOXlO6 0.589X105 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 rootmeansquare value of the streaming potential fluctuations were
measured as functions of pressure. In addition, the resistance of the
eectrolytepipeelectrode "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 1230A Electrometer (input
impedance 10!2 ) as a null indicator, the streaming potential was
balanced by t.he output of a 090V twostage 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.77I0N and 26.80X10N solutions were read directly
fro::. 'e Gneral 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
O80V
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 eecrometerCmplifier 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 stripchart 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 capacitanceit 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 screenroom
(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.) 050mv
.... .. .'n.. poen. a! > (D.C.) 085V.
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
longtime 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 twohour 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
(LogLog ?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 electrometeramplifier. 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 numbersonly an irregular
increase (plot of resistance coefficient vs. R Fig. 6). At the
ies Reynolds number of the experiment, transition to fullydeveloped
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
rI
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
4I
C2
0
0 ,
0' CC
I <3 :r
0 0
C)
C' C,)
CC (N o o
I :"i o
o <3 [2
32
0
0
r3
0 0
o Z
CN
0
4J
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 electrometeramplifier and electrodes is flat
Or s range, the response of the system comprised of double layer
eectrodsam.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 "freestream" 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
te 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 itwhether 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 itf ays in the stability of laminar flow is unknown,
although some insight into its role :,,th be gained by dimensional
analyst is.
Consider the twodim.nsional steady flow of a viscous fluid
aving an electric double layer present at the solid boundaries. The
NavierStokes 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 order that th.e equations of motion for the two flow fields may be
"denticl, 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( eq'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;eY/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. resultsthe 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 (nondimensional)
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 wallthus 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 (9I4
0
2" VL
Fi2u 3.D.C. Compconentll of Streaming Potentiali 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 TL
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 cylo4)
4AV 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# &o45
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 hotwire 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
te 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 ienth 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 17the 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 "nondeveloped"
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 (HewlettPackard) 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 I1II''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 : _il., his work
toward the degree of Doctor of Philosophy.
o DcLor 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 Ce 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.
15. Thom, L. H., Phyv. iv., (2), 91 (1953), 780.
16. Sexl, T., Arn. Phys., 83 (1927), 835; also 84, 807.
17. Pretsch, J., ZA7 , 21 (1941), 204.
1S. C S~, ". an rs,: J. R., J. Fluid. Mech., 5 (i1 59), 97.
Ce:l, ., ad C.:.:berg, K., Acta Phwy. Austr ac, 12 (1959), 9.
20. L
21. .k D and Stuart, J. T., Proc. Roy. Soc. A, 208 (1951), 517.
22. G111, A. E, J. F ulu MVch., 21 (1965), 145.
23. Tatsumi, T., Proc. Phys. Soc. Japan, 7 (1952), 489 and 495.
24. Hagcn, G., PoP7. Ann., 46 (1839), 423; also Abhandl. Akad. Wiss.,
(1854), 17 and (1869).
25. Poiscullie, J. Competes Rendus, 11 (1840), 961 and 1041; also 12
(1841); also : : : .. _ 9 (1846).
26. Sar.nton, T. E. and Pannell, J. R., Phil. Trans., A214 (1914), 199.
27. Barnes, H. T. and Coker, E. G., Proc. Roy. Soc., A74 (1905), 341.
28. Reiss, L. R. and Hanratty, T. J., Jour. A. I. Ch. E., (8), 2 (1962),
24S;
2. Lin.dgren, E. K., Arl:iv. for Fysik, 15 (1959), 97; 15 (1959), 503;
15 (1959), 103.
30. Dryden, H. 7. and Abbott, J. H., NACA TN 1755 (1948).
31. Schuauer, G. 3. and Skramstad, H. K., J. Aero. Sci., 14 (1947), 69.
. .. 7<> .r, A"i'*.. .Wi, 2 (lg7 ), l u,
33. E .:.holtz, H., rnn.  ., VII, 7 (1879), 22.
34. Quincke, G., Pog. Anr.., 7 (1879), 337.
35. CGuo G., J. ... (L4) 9 (1910), 457; also Ann. ..., (9), 7
( S1 7), 129.
36. Chr.an, D. L., Phil. ::ag., (6), 25 (1913), 475.
37. Debyc, P. and Huckel, E., Phy ik. Z., 24 (1923), 185.
33. Stern, 0., Z. Zlektrocherr.. 30 (1924), 508.
39. Smoluchcwski, v. von, .. ...
., .. 7 eez, Leipzig (1914), 366.
43 ,<...... ,. .. .... ?:.,s., 2 (19'34), 767.
.... .. .., : iu over Ske el:ectrolvten en
........ ...... * " ... .* :' .. Utrecht: ( ) 1.
53
42. Kruyt, 1. R. Colloid Science, I, Elsevier; Amsterdam (1952), 126.
43. Jones, G. and Wood, L. A., J. Cf.,. Phys., 13 (1945), 106.
4. R:ichard, ., ZAM':, 20 (1940), 297; trans. in NACA TM 1047, 1943.
45. L.ufr, NACA, ... 1174 (1954).
46. ,Reichardt, H., Z. PX.sik. Chem., A174 (1935), 15.
47. Kruyt, H. R., KolioidZ, 22 (1918), 81.
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 Eeerin
Dean, Graduate School
d/&6A4~
af~~
I

Full Text 
3
being "the superposition of a twodimensional perturbation on the stream
function of the mean flow. Neglecting terms quadratic or higher in the
disturbance leads to the SommerfeldOrr equation, which is the fundamental
differential equation for the disturbance amplitude.
Use of a twodimensional disturbance was questionable until
Squire^10^ showed that a threedimensional disturbance is equivalent to
a twodimensional one at a lower Reynolds numberat least when applied
to a twodimensional flow; this has become known as Squire's theorem.
Heisenberg was the first to deduce theoretically the instability
of plane Poiseuille flow for sufficiently large Reynolds numbers,
although he did not calculate a lower critical Reynolds number. The
theory of Heisenberg was extended by Tollmien^^ and Schlichting^"^;
both calculated the neutral stability curves for the boundary layer over
(13)
a flat plate. Much later, Lin performed the calculations again
obtaining more clarity, and succeeded in calculating the curve of neutral
disturbances for plane Poiseuille flow. Contrary to the conclusion of
(14)
Lin, Pekeris applied a different technique and concluded that
Poiseuille flow is stable for all Reynolds numbers. To resolve this
disagreement, Thomascalculated the critical Reynolds number by
direct numerical methods and found it to be 5780.
The more difficult analytical problem of the stability of pipe
flow has not been resolved with such clarity. Sexlv was the first
to solve the viscous problem, although only for axisymmetric disturbances.
The results have been questioned because of his mathematical simplifica
(17)
tions. For a small region near the wall, Pretsch showed the problem
became the same as that of a disturbance applied to plane Couette flow
23
The R.M.S. values of the streaming potential fluctuations were
determined using the system shown in Fig. 3.
The streaming potential fluctuations were amplified by a
differential electrometeramplifier specially constructed for electro
chemical measurements; the input stages are Philbrick Model SP2A
. 14
operational amplifiers. The input impedance of this amplifier is 10
ohms and the gain was 8.80 throughout the experiment (gain can be set
at 8.80, 88.0, or 880). The output of the voltage divider was used to
cancel the D.C. component of the streaming potential (this ranged up to
80V). The output of the amplifier was fed into a stripchart recorder
for a visual record of the fluctuations and into the Flew Corp. Random
Signal Voltmeter for their R.M.S. value.
The electrical system represented by the electric double layer
is very sensitive to stray capacitanceit 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 screenroom
(Faraday cage). This was corrected by glueing aluminum foil shielding
onto the Styrofoam thermostat in which the flow system was placed. This
shielding, along with the usual care in reducing vibrations in all
electrical components, gave a great reduction in noise level. Typical
signal levels were:
spike magnitude (peak) 0.1V
spike magnitude (R.M.S.) 050mv
streaming potential (D.C.) 085V.
A typical spike trace is shown in Figure 4.
As previous experimenters had noted, the values of the streaming
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
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1968
To my wife
ACKNOWLEDGEMENTS
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.
Appreciation 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 AFAFOSR84267 which has made this
research possible.
in
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF FIGURES v
LIST OF SYMBOLS vii
ABSTRACT ix
Chapters
IINTRODUCTION 1
Scope 1
The Stability of Laminar Flows 2
The Experimental Investigations of Transition 5
The Electric Double Layer 7
The Streaming Potential 10
IITHE 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
IIIRESULTS AND CONCLUSION 28
R.M.S. Component of Streaming Potential 28
Streaming Potential 38
Conclusion 47
BIOGRAPHICAL SKETCH 50
LIST OF REFERENCES 51
iv
LIST OF FIGURES
Figure Page
1. Sketch of Flow Apparatus 19
2. 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 (LogLog
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
v
LIST OF FIGURES (continued)
Figure Page
16. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4, Data Taken from Dymec
Integrating Voltmeter 44
17. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 5 45
18. Ratio of Laminar Slope to Turbulent Slope vs. Double
Layer Thickness 48
vi
LIST OF SYMBOLS
a
A
d
e
r
~DC
ERMS
Ei
T
*P
q
r
R.
S
T
radius of pipe bore
area of pipe cross section
constant
diameter of pipe bore
electronic charge
streaming potential
D.C. component of streaming 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
length of pipe
number density of iL^ ionic species
number density of i^*1 ionic species at the wall
pressure
electric charge
radial coordinate
Reynolds number
arc length
absolute temperature
Vll
u
velocity in the xdirection
Um
mean velocity
V
velocity in the ydirection
X
cartesian coordinate
y
cartesian coordinate
t
charge density
Â£
double layer thickness
dielectric constant
K
Boltzmann's constant
X
specific conductivity
s**
viscosity
nr
kinematic viscosity
density (fluid)
Tw
shear stress at wall
7i
reciprocal of Debye length
electric potential
potential at wall
n
electrical resistance
v"
Laplacian operator
viii
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
THE ELECTROKINETIC DETERMINATION OF THE
STABILITY OF LAMINAR FLOWS
By
Addison Guy Hardee, Jr.
August, 1968
Chairman: Knox T. Millsaps, Ph.D.
Major 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 examining the electrokinetic phenomena generated by the interaction
of the fluid flow with the electric double layer. It was found that
the 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
equilibrium 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.
IX
CHAPTER I
INTRODUCTION
Scope
The present investigation is the natural outgrowth of previous
/ i \
work by Anderson''', of which if represents, at the same time, both
an extension and clarification. Using the electrokinetic phenomena
associated with the electric double layer, the transition from laminar
to turbulent flow in a pipe is studied by examining the growth of
perturbations in the flow very close to the pipe wall. The electrokinetic
phenomenon of interest here is the following: when an electrolyte,
however weak, flows through a pipe, a potential difference can be
measured between a pair of electrodes placed at the ends of the pipe;
this is the streaming potential and is a result of the electric double
layer which exists at the solidliquid interface at the wall of the pipe.
The electric double layer is discussed more fully on page 7.
The purpose of Andersons investigation was primarily that of
determining precisely the minimum critical Reynolds number of Poiseuille
flow in a pipe; thus the fluctuations, which he discovered, of the
streaming potential were used to detect the onset of turbulence in the
flow.
The radial location of perturbations in the flow very close to
the wall is possible, for the extension of the electric double layer
1
2
into the fluid 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 ranging
from 589 to about 5,000 A.
The Stability of Laminar Flows
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' hypothesisnamely, to superimpose a small periodic
perturbation on the mean flow and examine the growth or decay of this
disturbance. Most notable among the names of these workers are those
of Rayleigh*'0), Sommerfeld'4^, Orr^^, Tollmien^^ and Prandtl^\
(8 ) ( 9)
Mention should also be made of Heisenberg' and Lorentz
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 sufficient condition.
The great majority of theoretical work has been directed toward
t.he stability of twodimensional flow, the application of the theory
3
being "the superposition of a twodimensional perturbation on the stream
function of the mean flow. Neglecting terms quadratic or higher in the
disturbance leads to the SommerfeldOrr equation, which is the fundamental
differential equation for the disturbance amplitude.
Use of a twodimensional disturbance was questionable until
Squire^10^ showed that a threedimensional disturbance is equivalent to
a twodimensional one at a lower Reynolds numberat least when applied
to a twodimensional flow; this has become known as Squire's theorem.
Heisenberg was the first to deduce theoretically the instability
of plane Poiseuille flow for sufficiently large Reynolds numbers,
although he did not calculate a lower critical Reynolds number. The
theory of Heisenberg was extended by Tollmien^^ and Schlichting^"^;
both calculated the neutral stability curves for the boundary layer over
(13)
a flat plate. Much later, Lin performed the calculations again
obtaining more clarity, and succeeded in calculating the curve of neutral
disturbances for plane Poiseuille flow. Contrary to the conclusion of
(14)
Lin, Pekeris applied a different technique and concluded that
Poiseuille flow is stable for all Reynolds numbers. To resolve this
disagreement, Thomascalculated the critical Reynolds number by
direct numerical methods and found it to be 5780.
The more difficult analytical problem of the stability of pipe
flow has not been resolved with such clarity. Sexlv was the first
to solve the viscous problem, although only for axisymmetric disturbances.
The results have been questioned because of his mathematical simplifica
(17)
tions. For a small region near the wall, Pretsch showed the problem
became the same as that of a disturbance applied to plane Couette flow
4
and Pekeris^14^ obtained a solution for the region near the axis of the
pipe. Coreos and Sellars^ gave a solution which accounts for the
work of both Pretsch and Pekeris. The conclusion drawn from the work of
these investigators is that HagenPoiseuille flow is stable for small
disturbances and the work of Sexl and Spielberg^2 ^ confirms this. (Sexl
and Spielberg also showed that Squire's theorem does not hold for axially
symmetric flows.) Experimentally, Leite^20^ failed to observe any
amplification of small axisymmetrical 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 pipesa seeming paradox when one regards the
mathematical results. The resolution of the "paradox" may lie in the
contrast between "small" and finite disturbances occurring in the flow
or in the symmetry of the disturbance. In this connection, Meksyn and
(21)
Stuart showed that, in a channel the lower critical Reynolds number
decreased as the amplitude of the superimposed oscillations increased,
which is in accord with the qualitative observations of Reynolds. A
(22)
possible explanation has been given by Gill who reviewed the above
theoretical papers and indicated questionable steps in their procedures.
(23)
On the other hand, Tatsumi has predicted theoretically that
the flow in the inlet of a pipe is unstable at a Reynolds number of
3
9.7 X 10 Experimentally, both Ekman (unpublished) and Taylor
(unpublished) obtained laminar flow up to Reynolds numbers of 5 X 104
4
and 3.2 X 10 respectively; Ekman made use of Reynolds original apparatus.
5
The Experimental Investigations of Transition
Hagen^24'* first noted the transition from laminar to turbulent
flow while determining the law of resistance for pipe flow. He was
aware that the "critical point" depended on the velocity, viscosity and
the pipe radius. The breakdown of what we today call laminar flow was
noted by the pulsing 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
performed by Reynolds, who showed conclusively that there exist two
possible modes of fluid flowlaminar and turbulent. He was most
probably not aware of Hagen's work, which predated that of Poiseuillev .
On the other hand, Reynolds was in possession of the NavierStokes equa
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 fact that the steady motion breaks down suddenly
shows that the fluid is in a state of instability for
disturbances of the magnitude which cause it to break
down. But the fact that in some conditions it will
break down for smaller disturbances shows that there is
a certain residual stability so long as the disturbances
do not exceed a given amount.
In the second of two independent experiments, Reynolds determined the
6
minimum critical Reynolds number of a long straight pipe; the value he
found was approximately 2000.
Since that time, many workers have repeated Reynolds' experiments,
some with interesting variations. The most extensive repetition was
conducted by Stanton and Pannell^^. Barnes and Coker^^) used a
thermal method of detection in which the walls of the pipe were heated
and the onset of turbulence was detected by a sharp rise in the tempera
ture of the interior of the flow. Reiss and Hanratty(28) developed a
technique from which they could infer the behavior within the socalled
viscous sublayer by measuring the 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 Lindgrenv ,
who made the disturbances visible by using polarized light and a bi
refringent, weak solution of bentonite. A technique utilizing the
electric double layer was developed by Anderson^^ 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
technique.
The experiments above were mainly conducted to study phenomena
associated with "large" disturbances, while it was not until Dryden's^^
very low turbulence wind tunnel became available that experimentalists
were able to examine the small disturbance problem. This was undertaken
by Schubauer and Skramstadv who made an almost direct transfer of the
theoretical method to the physical situationoscillations were induced
in a metal ribbon above a flat place placed in the low turbulence tunnel
and the amplification of the flow disturbances measured downstream by
7
means of a hot wire. Their experiments are v regarded as excellent
verification of stability theory.
(2 G
The experiment of Leite'' 7 mentioned e. er was similar to that
of Schubauer and Skramstad, but axisymmetric urbances were super
imposed to the flow of air in the inlet of a pip
The Electric Double Layer
The phenomenon known as the electric double layer has been
studied extensively by chemists, especially in connection with colloids
and with electrode processes.
The electric double layer 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.
If the solution is caused to flow past the wall, such as in
Poiseuille flow in a pipe, 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 Helmholtz gave an explanation based on Poiseuille
flow and the concept of the double layer developed by Quincke^34^.
The early workers in the field considered the double layer to be
composed of two distinct layers of charge, one fixed to the wail and one
free to move with the fluid. The more realistic model was proposed by
Gouy^'1'^ and Chapman^
who independently formulated the theory of the
8
diffuse double layer, which is, in essence, the theory of ionic atmos
(37) (38)
pheres given some ten years later by Debye and Huckel Stern' '
modified this theory to account for the finite size of the ions at the
wall.
An excellent summary of the classical physics of the effect has
( 30 )
been given by Smoluchowski' and extensive analyses of the approxima
tions used in the various theories are given by Kirkwood
(40)
and
Casimir^41 \
.(42)
The analysis given follows Kruyt The charge at the interface
is considered to be adsorbed on the solid surface and uniformly, distri
buted, while the solvent is assumed to be a continuous media, influencing
the double layer only through its dielectric constant. Coulomb inter
actions in the system are described by Poisson's equation
where T is the potential (having a value of at the wall),
(1)
^ is the charge density,
is the dielectric constant, .
rJ
and v is the Laplacian operator.
The number density of the i"^1 ionic species is assumed to be given by
= n,0exp(Z,eH'/KT) (2)
where ^Â¡o is the number density of the i^*1 species at the wall,
Z.\ is the valence,
is the electronic charge,
K is Boltzmann's constant,
and T is the temperature of the solution.
The charge density is given by
9
(3)
Combining equations (1), (2), and (3), one obtains
v> = 
4IT
2.zÂ¡n,.expCzÂ¡e.H'/fcT)
e (4)
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
= 25.Z,enue*P(ZÂ¡e. T/AT)
(5)
as
oo .
with the boundary conditions
AH'
^ = O c.n
where ^ is the distance from the wall.
The first integration of (5) is carried out after multiplying
At
through by 2 !
= Y/kt)
Upon integration and applying the boundary conditions, the result
is obtained
(6)
The equation can be simplified by considering a single binary
electrolyte, therefore
(7)
The second integration is performed after writing equation (7) as
10
Se.S'/fcA
"  \\]
STTHKT 4
t 1
J Â£xP(Zet/2kT)e>^p(Ze4/2ki)
'Zc'VoIkT
Therefore
[e*p(Zct/2KT)+ lj[e>cf (zcViKTV l]
[eypCZet/?^ J ] [exp (z %/sKt) +  "
(8)
where
ar
For small potentials, this may be simplified by expanding the exponential
terms to yield
o *
9t>j 
or
^ Hi (9)
showing that the potential decreases exponentially to zero over a distance
of the order of magnitude of Vx Thus the thickness of the double
layer is of the order of x/% At room temperature is approximately
rj
3 X 10 Z trc* where C is the concentration in grammoles/liter. As an
example, the double layer thickness is approximately 10D cm for a
0.001M KC1 solution.
The Streaming Potential
The streaming potential is generated when the electrolytic 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.
Assuming the charge density on the wall is independent of the
flow, the convection current is
A.
where U is the hydrodynamic velocity and A is the crosssectional
area.
Substituting from Poisson's equation yields:
(11)
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:
47T
A.
(12)
Successive integration by parts and
tions
=. o wd j
application of the boundary condi
12
and
yields
= o
. ^ Â£>
(13)
where S is the circumference of the pipe Â§nd H* is the value of 4*
at y=0.
If the flow is laminar,
/ JuA 
where is the shearing stress at the wall and is the viscosity
of the fluid. Also, if the flow is in a constant area pipe,
= ^=17 Vs
where X is the coordinate along the axis of the pipe. Therefore,
equation (11) becomes
>1 ~
J 477
U;
(14)
The second term in equation (14) will be neglected since it may be shown
that the ratio of the second term to the first is the order of
where r is the radius of the pipe. Therefore, if the pressure gradient
is constant over the crosssection, equation (14) may be written as
ili. A
1 4Ti>u. dX
(15)
13
The conduction current is given by
(16)
where % is the specific conductance of the solution and E. is the
streaming potential.
Â£
If "X and are constant over the crosssection, equation
(16) becomes
, 1
A X (17)
At equilibrium, i. = i^; hence
AE _
P ~
(18)
Under the above assumptions, equation (18) shows that the streaming
potential is a linear function of the pressure drop between the ends of
the pipe.
The first term on the righthand side of equation (13) indicates
that the convection current, and hence the streaming potential, is a
function of the velocity gradient at the wall.
CHAPTER II
THE EXPERIMENT
General Description
The present investigation was undertaken to examine more closely
several aspects of streaming potential phenomena which the experiments
of Anderson*'^ 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 experiment and the reasoning behind it will be
given.
The 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 is the radius of the pipe,
P is the pressure,
x is the coordinate along the pipe,
14
15
^ is the viscosity of the fluid,
r is the distance from the center of the pipe,
and u is the velocity.
If this relation is substituted into equation (13), the streaming
potential is again found to be a linear function of the pressure gradient
(43)
along the pipe; this has been observed by physical chemists' for many
years. Since equation (13) indicates the dependence of the streaming
potential on the velocity gradient at the wall, Anderson reasoned that
streaming potential measurements could be used to determine the transi
tion from laminar to turbulent flow, for the velocity gradient near the
wall is different in these two modes of flow.
Anderson looked for a change in the value of the quantity ae/ap
(i.e., a "break" in the curve of E vs P) when transition occurred. His
experiment was conducted using an electrolytic solution of 0.001N KC1
flowing in 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 unequivocably show one; instead he discovered unique fluctua
tions in the streaming potential which appeared at the transition
Reynolds number. Using these fluctuations as a guide, he was able to
determine the minimum critical Reynolds number very precisely. In
addition, he showed that the streaming potential did not vary linearly
with pressure gradient if the streaming potential was measured across
the full length of the pipe. This is the effect of the entrance length
(that length of the pipe necessary for establishing Poiseuille flow)
16
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 investigationwas 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 socalled "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 NavierStokes 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 fluidthe 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 layerthe 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^
17
seem to point conclusively to the existence of such a region near the wall
in which viscous stresses greatly outweigh inertial stresses. In this
light, the explanation of the streaming potential fluctuations becomes
more difficult. Reichardt^ predicted the fluctuations, but implied
that they were caused by perturbations in the mean pressure gradient
along the pipeexactly what this means at a specific point very near
the wall is unclear. Reichardt 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 longer
pipe and a higher gain electrometeramplifier and placing greater
emphasis on noise reduction. In addition, five different double layer
thicknesses (ranging from 589A to about 5,000A) would be used,, so that
the double layer would extend over varying distances into the viscous
sublayer. The longer pipe (543 diameters) was used to reduce the effect
of the entrance length.
Laboratory
The laboratory was especially designed and constructed by Anderson
for streaming potential 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
of the temperature, while the humidity of the room atmosphere was held
very low due to the nature of the electrical measurements. The currents
O
involved were of the order of 10 to 10
amperes with a source
18
9 11
resistance of the order of 10 to 10 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 airconditioned
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 compressioncooling
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^4'"^,
(47)
Kruyt Anderson and others and is shown schematically in Fig. 1.
The reservoirs and R2 are 5 liter boiling flasks fitted with ground
glass taper joints, and Tubes and are 6mm Pyrex with
taper joints at the top for inserting electrodes. A^ 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 A^ and by flanges incorporating Teflon gaskets. In
operation, the liquid was forced up and through the capillary by
applying gas pressure through V^.
The electrodes E1 and E2 are AgAgCl formed on spirals of 24
Figure 1. Sketch of Flow Apparatus
20
AWG platinum. The AgAgCl coating was made with a mixture of 90% silver
oxide and 10% silver chlorate formed into a paste, applied to the wire
and heated at 500C for fifteen 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 microtelescope mounted on a vernier height gage. The
pressure could easily be controlled to within 0.005 inches Hg. The two
threeway stopcocks in the supply line provided a means of applying
pressure to the flasks or venting them to the atmosphere.
Cleaning 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 leached in conductivity water for several
days.
The conductivity water used as the solution was twicedistilled,
having nitrogen bubbled through it after each distillation.
Electrolytic Solutions Used in the Experiment
The experiments were conducted with 5 concentrations of KC1
solutions, beginning with conductivity water as the first solution. Salt
was added to this to form the subsequent concentrations. Table 1 gives
the values of the five concentrations and the corresponding thickness of
the double layer.
21
TABLE 1
THICKNESS OF THE ELECTRIC DOUBLE LAYER
Concentration
Normality
6(cm)*
6(A)*
1
0
OG
oo
2
0.5364X106
4.16X10"5
4,160
3
0.8040X106
3.40X10"5
3,400
4
1.877X106
2.23X105
2,230
5
26.800X106
0.589X10'5
589
*6 = double layer thickness.
The Electrical Measurements
Three types of electrical measurements were made for each concen
tration of electrolyte. The magnitude of the streaming potential and
the rootmeansquare value of the streaming potential fluctuations were
measured as functions of pressure. In addition, the resistance of the
electrolytepipeelectrode "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 1230A Electrometer (input
impedance 10z ) as a null indicator, the streaming potential was
balanced by the output of a 090V twostage voltage divider. This was
then reduced by a factor of 1000 by the Leeds and Northrup Volt Box and
read on a Leeds and Northrup millivolt potentiometer. Streaming poten
tials for the 1.877X10 ^N and 26.80X10 ^N solutions were read directly
from the General Radio electrometer.
22
Figure 2. E^ Measurement
RMS
Figure 3. E.
Measurement
23
The R.M.S. values of the streaming potential fluctuations were
determined using the system shown in Fig. 3.
The streaming potential fluctuations were amplified by a
differential electrometeramplifier specially constructed for electro
chemical measurements; the input stages are Philbrick Model SP2A
. 14
operational amplifiers. The input impedance of this amplifier is 10
ohms and the gain was 8.80 throughout the experiment (gain can be set
at 8.80, 88.0, or 880). The output of the voltage divider was used to
cancel the D.C. component of the streaming potential (this ranged up to
80V). The output of the amplifier was fed into a stripchart recorder
for a visual record of the fluctuations and into the Flew Corp. Random
Signal Voltmeter for their R.M.S. value.
The electrical system represented by the electric double layer
is very sensitive to stray capacitanceit 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 screenroom
(Faraday cage). This was corrected by glueing aluminum foil shielding
onto the Styrofoam thermostat in which the flow system was placed. This
shielding, along with the usual care in reducing vibrations in all
electrical components, gave a great reduction in noise level. Typical
signal levels were:
spike magnitude (peak) 0.1V
spike magnitude (R.M.S.) 050mv
streaming potential (D.C.) 085V.
A typical spike trace is shown in Figure 4.
As previous experimenters had noted, the values of the streaming
24
Figure 4. Tracing of Typical Spikes (Reynolds Number is 2572,
Concentration No. 2)
25
potentials could not be reproduced accurately from day to day. For
each separate electrolytic solution, the streaming potential showed a
longtime 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 twohour 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
A? .
= 4Lp
where U.v is the mean velocity,
Lp is the length of the pipe
and a is the fluid density.
20
18
16
14
12
10
8
6
4
2
0
G 6/28/68
b 6/29/68
G
n
%
e
%
0
a
0
p
o
eo
o
G
G
0
0
1
1
1
1
500
1000
1500
2000
2500
Reynolds Number
Figure 5. Pipe Calibration (Resistance Coefficient vs. Reynolds Number)
o
I
3000
27
Reynolds Number
rigure 6. Resistance Coefficient vs. Reynolds Number
(LogLog Plot)
CHAPTER III
RESULTS AND CONCLUSIONS
R.M.S. Component of Streaming Potential
The R.M.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 electrometeramplifier. The
points denoted by triangles represent laminar flow, i.e., no spikes
(fluctuations) on the streaming potential; those denoted by circles
represent 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
to 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
graph.
Each figure displays the same general trend; a peak, followed by
a dip, and a second, higher peak followed by a rapid decrease. Such
regular variations are not detectable in mass flow rate measurements
through the transition range of Reynolds numbersonly an irregular
increase (plot of resistance coefficient vs. R Fig. 6). At the
highest Reynolds number of the experiment, transition to fullydeveloped
turbulence was not complete (deduced from Fig. 6).
The frequency of occurrence of the spikes at their first
28
Figure 7. R.M.S. Component of Streaming Potential vs, Reynolds Number, Concentration No. 1
K>
ID
Reynolds Number
Figure 8. R.M.S. Component of Streaming Potential vs. Reynolds Number, Concentration No. 2
GO
O
Figure 9. R.M.S. Component of Streaming Potential vs. Reynolds Number, Concentration No. 3
CO
Figure 10. R.M.S. Component of Streaming Potential vs. Reynolds Number, Concentration No. 4.
CO
fO
50
O spikes
A no spikes
30
20
10
0
2350
O
GO
,GO
eePo o
o
G
Aj A
A
G
G G
G
G
G
X
2400
2500
2600
2700
Reynolds Number
figure 11. R.M.S. Component of Streaming Potential vs. Reynolds Number, Concentration No. 5
Cl)
W
34
appearance was taken as 0.002 per second (one spike in an eight minute
period) and at Rn = 2755 was approximately 2.1 per second; this frequency
was determined by manually counting the peaks of the spikes over an
eight minute period of the chart from the recorder. Thus, although the
frequency response of the electrometeramplifier and electrodes is flat
for this range, the response of the system comprised of double layer
electrodesamplifier may be responsible for the variations in the
vs. R_ curves. The period of an individual spike was about 2 seconds.
The existence of some fluctuation below the appearance of the
first spikes is a measure of "freestream" turbulence as well as fluctua
tions from 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 spikes appear at Rn = 2585 for the first concentration and at
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
the double layer readjusts to an overall concentration change. As will
be shown in the next section, the equilibrium state of the electric
double layer is changed by the flow impressed upon itwhether this is
a change in potential distribution alone or with a simultaneous change
in potential at the wall can not be determined. The part the electric
double layer itself plays in the stability of laminar flow is unknown,
35
although some insight into its role might be gained by dimensional
analysis.
Consider the twodimensional steady flow of a viscous fluid
having an electric double layer present at the solid boundaries. The
NavierStokes equations are
(1)
where F. is the electrostatic force on a fluid element. If one now
i
considers two flows with geometrically similar boundaries and let
be any length of the first flow field and the corresponding length in
the second, then L^ C^L^. Similarly,
U.2. = j ~ Cj j =
(2)
"v2=CSV .
The individual terms for the second flow field in relation to the
corresponding terms of the first flow field are
(3)
(4)
c3 <=>^1
C,
(5)
and
(6)
36
In order that the equations of motion for the two flow fields may be
identical, the following must be true
ct_ = Â£1 _
c, ck
The equality
may be written
(7)
(8)
which means the Reynolds numbers of the two flows must be equal in order
for the flows to be geometrically similar.
The relation ^r = means that
(9)
must also be true. This may be written as
FI?
R,
= constant
(10)
where is the Reynolds number and F is the electrostatic force per unit
volume on a fluid element.
The electrostatic force per unit volume on the fluid element is
given by
(11)
37
where q is the electric charge contained in the element and H* is the
(electric) potential. For flow within the electric double layer
f =
(12)
where X is the charge density. Therefore, the parameter in equation
(10) may be written
X 1?
V R^
or, approximately (if the double layer thickness is taken as the
characteristic length) as
{i:) L) L
V Ra
Therefore, the parameter
7
/ % L e.yp(Q
V Rvv
(13)
must be a constant for dynamically similar flows.
The charge density is given by
+ e.nx
(14)
where n^ is the number density of the negative ions and is the number
density of the positive ions. Again the number density is assumed to
be given by a Boltzmann distribution:
A;
Aj0ex?( z:e Vkt')
(15)
where n^Q is the number density of the ith ionic species at the wall
(the wall is considered to have a negative charge).' Therefore (for a
38
single valence, binary electrolyte), the charge density is approximately
given by
The similarity parameter now becomes
(16)
The quantities 4o L and the exponential term in the numerator all
increase as the bulk concentration of the electrolyte decreases; there
fore, the Reynolds number must increase in order that the parameter
remain constant. For more dilute solutions (i.e., thicker double layer),
the higher must be the Reynolds number of flow phenomena within the
electric double layer. The Reynolds number here, of course, is based
on the thickness of the electric double layer and the local velocity of
the flow.
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
experimental resultsthe transition to turbulence, as indicated by the
double layer, occurs at higher Reynolds numbers for more dilute solutions.
Streaming Potential
The values of the D.C. component of the streaming potential for
39
the five solutions are shown as functions of pressure (nondimensional)
in Figures 12 to 17. Excepting a region at the beginning of transition,
the slope for laminar flow is different than the slope for turbulent
flow. Thus, the change in slope can be used to detect the transition
from laminar to turbulent flow. A comparison of the graph of vs.
pressure for the last concentration (Fig. 17) and resistance coefficient
(Fp) vs. Reynolds number (Fig. 6) shows that the streaming potential
indicates a slightly higher critical Reynolds number than mass flow rate
measurements. The break in the FP vs. Rn curve begins when the eddies
grow enough in strength and number to increase appreciably the resistance
to the flow. This should coincide very nearly with the establishment
of a different velocity profile near the wallthus changing the value of
, as governed by equation (13), Chapter I. The difference in
critical Reynolds number given by the two methods, mass flow rate and
streaming 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
question 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; according 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
laminar and turbulent flow. This says nothing about the magnitude of a
change in velocity profile which the double layer can detect. When
speaking of a "viscous" sublayer, it should be kept in mind that the
40
^ Cx/o4)
D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 1
Figure 12.
41
_r cx/o^)
^<*vLÂ¡>
Figure 13. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 2.
70
P.
sUi/L,
C*
JO
10
v
12
14
16
18
20
are 14.
D. C. Component of Streaming Potential vs. Pressure,
Concentration No. 3
43
wV 1p
Figure 15. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4
44
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2 t
r
I 1
GENERAL
RADIO
ELECTROMETER
PIPE
J I
0
.8
INTEGRATING
VOLTMETER
8 10 12 14 16 18 20
PAl U,o4)
uv' Li
Figure 16. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4. Data Taken From Dymec Integrating
Voltmeter.
45
Figure 17. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 5
46
entire flow in the pipe is "viscous." Also the work which points to the
existence of the sublayer is based largely on hotwire measurements and
these have only been made over the outer part of the sublayer.
Another fact bearing on the above question is that in deriving
the equations for the streaming potential, it was assumed that the flow
conditions (including the velocity profile) were the same all along the
length of the pipe; in reality, the theoretical conditions are attained
only after the entrance length of the pipe has been traversed. Further,
the 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).
The velocity profile for laminar flow in the entrance length is
similar to that for established turbulent flow; thus, for laminar flow
at low Reynolds numbers, the value of ae/ap should be intermediate to
those of established laminar flow and established turbulent flow. This
is seen to be the case for all the data in Figures 12 to 17the slope
of the laminar portion of the curve decreases with increasing Reynolds
number.
Measuring the streaming potential and pressure across one segment
of the pipe, excluding the entrance length, would better fulfill the
conditions of the theory and show whether or not the "nondeveloped"
flow in the changing entrance length is responsible for the difference
in slopes. This was considered in preliminary planning of the investiga
tion; however, it was evident that any method of isolating one length
47
of the pipe 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 potential fluctuations. These average
values should be of 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
E^ values to be ignored. The data for the last, and most concentrated,
solution (Fig. 17) shows an unexplained peak at the beginning of transi
tion. The peak occurs roughly at the same Reynolds number as the maximum
R.M.S. value (Fig. 11), but is about twice the expected magnitude.
Figure 16 shows taken from a Dymec (HewlettPackard) Integrating
Digital Voltmeter, Model 2401A, set up as shown in the inset to this
figure. Integration does not alter the basic curve, but does slightly
change 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
in Figure 10.
Although all the curves are similar, they represent a wide range
of streaming potential magnitude. In Fig. 18 is shown how the ratio of
the 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 sufficiently small double layer thickness.
Conclusion
The conclusions drawn from the experimental investigation are
delineated as follows.
48
1.5
INTEGRATING VOLTMETER
S
X
o
o
o
G
0
1.0
Cae/a?)t
(AE/A?)u
0.5
0
I
! 1 1 1 1 1 1
i t
01234567 89 10
Â£ (x/>_"c.wv')
Figure 18.
Ratio of Laminar Slope to Turbulent Slope vs.
Double Layer Thickness
49
The critical Reynolds number of pipe flow, as indicated by
streaming potential 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 transition from laminar to turbulent flow can be detected by
noting a change in slope of the graph of E^q vs. pressure, but only if
the electrolytic solution is very dilute.
BIOGRAPHICAL SKETCH
Addison Guy Hardee, Jr., was born April 7, 1938, at Mulberry,
Florida. In June, 1955, he was graduated from Hillsboro High School
in Tampa, Florida. From 1956 to 1960, he served as an electronics
technician in the United States Coast Guard and was stationed for a
time 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 Master of Science in Engineering from the same school in
December, 1965. He received an appointment to the position of Research
Associate in the Aerospace Engineering Department in January, 1966,
which position he has held to the present time while pursuing his work
toward the degree of Doctor 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 Kappa Phi and Phi Eta Sigma.
50
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53
42. Kruyt, H. R., Colloid Science, I, Elsevier; Amsterdam (1952), 126.
43. Jones, G. and Wood, L. A., J, Chem. Phys., 13 (1945), 106.
44. Reichardt, H., ZAMM, 20 (1940), 297; trans. in NACA TM 1047, 1943.
45. Laufer, NACA, TN 1174 (1954).
46. Reichardt, H., Z. Phsik. Chem., A174 (1935), 15.
47. Kruyt, H. R., KolloidZ, 22 (1918), 81.
'his dissertation was prepared under the direction of the chairman
of the candidate's supervisory committee and has been approved by all
members 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, 1968
Dean, Graduate School
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF FIGURES v
LIST OF SYMBOLS vii
ABSTRACT ix
Chapters
IINTRODUCTION 1
Scope 1
The Stability of Laminar Flows 2
The Experimental Investigations of Transition 5
The Electric Double Layer 7
The Streaming Potential 10
IITHE 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
IIIRESULTS AND CONCLUSION 28
R.M.S. Component of Streaming Potential 28
Streaming Potential 38
Conclusion 47
BIOGRAPHICAL SKETCH 50
LIST OF REFERENCES 51
iv
u
velocity in the xdirection
Um
mean velocity
V
velocity in the ydirection
X
cartesian coordinate
y
cartesian coordinate
t
charge density
Â£
double layer thickness
dielectric constant
K
Boltzmann's constant
X
specific conductivity
s**
viscosity
nr
kinematic viscosity
density (fluid)
Tw
shear stress at wall
7i
reciprocal of Debye length
electric potential
potential at wall
n
electrical resistance
v"
Laplacian operator
viii
18
9 11
resistance of the order of 10 to 10 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 airconditioned
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 compressioncooling
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^4'"^,
(47)
Kruyt Anderson and others and is shown schematically in Fig. 1.
The reservoirs and R2 are 5 liter boiling flasks fitted with ground
glass taper joints, and Tubes and are 6mm Pyrex with
taper joints at the top for inserting electrodes. A^ 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 A^ and by flanges incorporating Teflon gaskets. In
operation, the liquid was forced up and through the capillary by
applying gas pressure through V^.
The electrodes E1 and E2 are AgAgCl formed on spirals of 24
20
18
16
14
12
10
8
6
4
2
0
G 6/28/68
b 6/29/68
G
n
%
e
%
0
a
0
p
o
eo
o
G
G
0
0
1
1
1
1
500
1000
1500
2000
2500
Reynolds Number
Figure 5. Pipe Calibration (Resistance Coefficient vs. Reynolds Number)
o
I
3000
Figure 9. R.M.S. Component of Streaming Potential vs. Reynolds Number, Concentration No. 3
CO
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
THE ELECTROKINETIC DETERMINATION OF THE
STABILITY OF LAMINAR FLOWS
By
Addison Guy Hardee, Jr.
August, 1968
Chairman: Knox T. Millsaps, Ph.D.
Major 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 examining the electrokinetic phenomena generated by the interaction
of the fluid flow with the electric double layer. It was found that
the 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
equilibrium 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.
IX
10
Se.S'/fcA
"  \\]
STTHKT 4
t 1
J Â£xP(Zet/2kT)e>^p(Ze4/2ki)
'Zc'VoIkT
Therefore
[e*p(Zct/2KT)+ lj[e>cf (zcViKTV l]
[eypCZet/?^ J ] [exp (z %/sKt) +  "
(8)
where
ar
For small potentials, this may be simplified by expanding the exponential
terms to yield
o *
9t>j 
or
^ Hi (9)
showing that the potential decreases exponentially to zero over a distance
of the order of magnitude of Vx Thus the thickness of the double
layer is of the order of x/% At room temperature is approximately
rj
3 X 10 Z trc* where C is the concentration in grammoles/liter. As an
example, the double layer thickness is approximately 10D cm for a
0.001M KC1 solution.
The Streaming Potential
The streaming potential is generated when the electrolytic fluid
41
_r cx/o^)
^<*vLÂ¡>
Figure 13. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 2.
CHAPTER III
RESULTS AND CONCLUSIONS
R.M.S. Component of Streaming Potential
The R.M.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 electrometeramplifier. The
points denoted by triangles represent laminar flow, i.e., no spikes
(fluctuations) on the streaming potential; those denoted by circles
represent 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
to 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
graph.
Each figure displays the same general trend; a peak, followed by
a dip, and a second, higher peak followed by a rapid decrease. Such
regular variations are not detectable in mass flow rate measurements
through the transition range of Reynolds numbersonly an irregular
increase (plot of resistance coefficient vs. R Fig. 6). At the
highest Reynolds number of the experiment, transition to fullydeveloped
turbulence was not complete (deduced from Fig. 6).
The frequency of occurrence of the spikes at their first
28
9
(3)
Combining equations (1), (2), and (3), one obtains
v> = 
4IT
2.zÂ¡n,.expCzÂ¡e.H'/fcT)
e (4)
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
= 25.Z,enue*P(ZÂ¡e. T/AT)
(5)
as
oo .
with the boundary conditions
AH'
^ = O c.n
where ^ is the distance from the wall.
The first integration of (5) is carried out after multiplying
At
through by 2 !
= Y/kt)
Upon integration and applying the boundary conditions, the result
is obtained
(6)
The equation can be simplified by considering a single binary
electrolyte, therefore
(7)
The second integration is performed after writing equation (7) as
CHAPTER I
INTRODUCTION
Scope
The present investigation is the natural outgrowth of previous
/ i \
work by Anderson''', of which if represents, at the same time, both
an extension and clarification. Using the electrokinetic phenomena
associated with the electric double layer, the transition from laminar
to turbulent flow in a pipe is studied by examining the growth of
perturbations in the flow very close to the pipe wall. The electrokinetic
phenomenon of interest here is the following: when an electrolyte,
however weak, flows through a pipe, a potential difference can be
measured between a pair of electrodes placed at the ends of the pipe;
this is the streaming potential and is a result of the electric double
layer which exists at the solidliquid interface at the wall of the pipe.
The electric double layer is discussed more fully on page 7.
The purpose of Andersons investigation was primarily that of
determining precisely the minimum critical Reynolds number of Poiseuille
flow in a pipe; thus the fluctuations, which he discovered, of the
streaming potential were used to detect the onset of turbulence in the
flow.
The radial location of perturbations in the flow very close to
the wall is possible, for the extension of the electric double layer
1
43
wV 1p
Figure 15. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4
Figure 7. R.M.S. Component of Streaming Potential vs, Reynolds Number, Concentration No. 1
K>
ID
LIST OF FIGURES (continued)
Figure Page
16. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4, Data Taken from Dymec
Integrating Voltmeter 44
17. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 5 45
18. Ratio of Laminar Slope to Turbulent Slope vs. Double
Layer Thickness 48
vi
46
entire flow in the pipe is "viscous." Also the work which points to the
existence of the sublayer is based largely on hotwire measurements and
these have only been made over the outer part of the sublayer.
Another fact bearing on the above question is that in deriving
the equations for the streaming potential, it was assumed that the flow
conditions (including the velocity profile) were the same all along the
length of the pipe; in reality, the theoretical conditions are attained
only after the entrance length of the pipe has been traversed. Further,
the 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).
The velocity profile for laminar flow in the entrance length is
similar to that for established turbulent flow; thus, for laminar flow
at low Reynolds numbers, the value of ae/ap should be intermediate to
those of established laminar flow and established turbulent flow. This
is seen to be the case for all the data in Figures 12 to 17the slope
of the laminar portion of the curve decreases with increasing Reynolds
number.
Measuring the streaming potential and pressure across one segment
of the pipe, excluding the entrance length, would better fulfill the
conditions of the theory and show whether or not the "nondeveloped"
flow in the changing entrance length is responsible for the difference
in slopes. This was considered in preliminary planning of the investiga
tion; however, it was evident that any method of isolating one length
LIST OF FIGURES
Figure Page
1. Sketch of Flow Apparatus 19
2. 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 (LogLog
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
v
15
^ is the viscosity of the fluid,
r is the distance from the center of the pipe,
and u is the velocity.
If this relation is substituted into equation (13), the streaming
potential is again found to be a linear function of the pressure gradient
(43)
along the pipe; this has been observed by physical chemists' for many
years. Since equation (13) indicates the dependence of the streaming
potential on the velocity gradient at the wall, Anderson reasoned that
streaming potential measurements could be used to determine the transi
tion from laminar to turbulent flow, for the velocity gradient near the
wall is different in these two modes of flow.
Anderson looked for a change in the value of the quantity ae/ap
(i.e., a "break" in the curve of E vs P) when transition occurred. His
experiment was conducted using an electrolytic solution of 0.001N KC1
flowing in 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 unequivocably show one; instead he discovered unique fluctua
tions in the streaming potential which appeared at the transition
Reynolds number. Using these fluctuations as a guide, he was able to
determine the minimum critical Reynolds number very precisely. In
addition, he showed that the streaming potential did not vary linearly
with pressure gradient if the streaming potential was measured across
the full length of the pipe. This is the effect of the entrance length
(that length of the pipe necessary for establishing Poiseuille flow)
44
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2 t
r
I 1
GENERAL
RADIO
ELECTROMETER
PIPE
J I
0
.8
INTEGRATING
VOLTMETER
8 10 12 14 16 18 20
PAl U,o4)
uv' Li
Figure 16. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4. Data Taken From Dymec Integrating
Voltmeter.
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
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1968
6
minimum critical Reynolds number of a long straight pipe; the value he
found was approximately 2000.
Since that time, many workers have repeated Reynolds' experiments,
some with interesting variations. The most extensive repetition was
conducted by Stanton and Pannell^^. Barnes and Coker^^) used a
thermal method of detection in which the walls of the pipe were heated
and the onset of turbulence was detected by a sharp rise in the tempera
ture of the interior of the flow. Reiss and Hanratty(28) developed a
technique from which they could infer the behavior within the socalled
viscous sublayer by measuring the 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 Lindgrenv ,
who made the disturbances visible by using polarized light and a bi
refringent, weak solution of bentonite. A technique utilizing the
electric double layer was developed by Anderson^^ 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
technique.
The experiments above were mainly conducted to study phenomena
associated with "large" disturbances, while it was not until Dryden's^^
very low turbulence wind tunnel became available that experimentalists
were able to examine the small disturbance problem. This was undertaken
by Schubauer and Skramstadv who made an almost direct transfer of the
theoretical method to the physical situationoscillations were induced
in a metal ribbon above a flat place placed in the low turbulence tunnel
and the amplification of the flow disturbances measured downstream by
34
appearance was taken as 0.002 per second (one spike in an eight minute
period) and at Rn = 2755 was approximately 2.1 per second; this frequency
was determined by manually counting the peaks of the spikes over an
eight minute period of the chart from the recorder. Thus, although the
frequency response of the electrometeramplifier and electrodes is flat
for this range, the response of the system comprised of double layer
electrodesamplifier may be responsible for the variations in the
vs. R_ curves. The period of an individual spike was about 2 seconds.
The existence of some fluctuation below the appearance of the
first spikes is a measure of "freestream" turbulence as well as fluctua
tions from 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 spikes appear at Rn = 2585 for the first concentration and at
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
the double layer readjusts to an overall concentration change. As will
be shown in the next section, the equilibrium state of the electric
double layer is changed by the flow impressed upon itwhether this is
a change in potential distribution alone or with a simultaneous change
in potential at the wall can not be determined. The part the electric
double layer itself plays in the stability of laminar flow is unknown,
39
the five solutions are shown as functions of pressure (nondimensional)
in Figures 12 to 17. Excepting a region at the beginning of transition,
the slope for laminar flow is different than the slope for turbulent
flow. Thus, the change in slope can be used to detect the transition
from laminar to turbulent flow. A comparison of the graph of vs.
pressure for the last concentration (Fig. 17) and resistance coefficient
(Fp) vs. Reynolds number (Fig. 6) shows that the streaming potential
indicates a slightly higher critical Reynolds number than mass flow rate
measurements. The break in the FP vs. Rn curve begins when the eddies
grow enough in strength and number to increase appreciably the resistance
to the flow. This should coincide very nearly with the establishment
of a different velocity profile near the wallthus changing the value of
, as governed by equation (13), Chapter I. The difference in
critical Reynolds number given by the two methods, mass flow rate and
streaming 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
question 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; according 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
laminar and turbulent flow. This says nothing about the magnitude of a
change in velocity profile which the double layer can detect. When
speaking of a "viscous" sublayer, it should be kept in mind that the
47
of the pipe 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 potential fluctuations. These average
values should be of 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
E^ values to be ignored. The data for the last, and most concentrated,
solution (Fig. 17) shows an unexplained peak at the beginning of transi
tion. The peak occurs roughly at the same Reynolds number as the maximum
R.M.S. value (Fig. 11), but is about twice the expected magnitude.
Figure 16 shows taken from a Dymec (HewlettPackard) Integrating
Digital Voltmeter, Model 2401A, set up as shown in the inset to this
figure. Integration does not alter the basic curve, but does slightly
change 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
in Figure 10.
Although all the curves are similar, they represent a wide range
of streaming potential magnitude. In Fig. 18 is shown how the ratio of
the 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 sufficiently small double layer thickness.
Conclusion
The conclusions drawn from the experimental investigation are
delineated as follows.
53
42. Kruyt, H. R., Colloid Science, I, Elsevier; Amsterdam (1952), 126.
43. Jones, G. and Wood, L. A., J, Chem. Phys., 13 (1945), 106.
44. Reichardt, H., ZAMM, 20 (1940), 297; trans. in NACA TM 1047, 1943.
45. Laufer, NACA, TN 1174 (1954).
46. Reichardt, H., Z. Phsik. Chem., A174 (1935), 15.
47. Kruyt, H. R., KolloidZ, 22 (1918), 81.
Figure 10. R.M.S. Component of Streaming Potential vs. Reynolds Number, Concentration No. 4.
CO
fO
20
AWG platinum. The AgAgCl coating was made with a mixture of 90% silver
oxide and 10% silver chlorate formed into a paste, applied to the wire
and heated at 500C for fifteen 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 microtelescope mounted on a vernier height gage. The
pressure could easily be controlled to within 0.005 inches Hg. The two
threeway stopcocks in the supply line provided a means of applying
pressure to the flasks or venting them to the atmosphere.
Cleaning 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 leached in conductivity water for several
days.
The conductivity water used as the solution was twicedistilled,
having nitrogen bubbled through it after each distillation.
Electrolytic Solutions Used in the Experiment
The experiments were conducted with 5 concentrations of KC1
solutions, beginning with conductivity water as the first solution. Salt
was added to this to form the subsequent concentrations. Table 1 gives
the values of the five concentrations and the corresponding thickness of
the double layer.
13
The conduction current is given by
(16)
where % is the specific conductance of the solution and E. is the
streaming potential.
Â£
If "X and are constant over the crosssection, equation
(16) becomes
, 1
A X (17)
At equilibrium, i. = i^; hence
AE _
P ~
(18)
Under the above assumptions, equation (18) shows that the streaming
potential is a linear function of the pressure drop between the ends of
the pipe.
The first term on the righthand side of equation (13) indicates
that the convection current, and hence the streaming potential, is a
function of the velocity gradient at the wall.
35
although some insight into its role might be gained by dimensional
analysis.
Consider the twodimensional steady flow of a viscous fluid
having an electric double layer present at the solid boundaries. The
NavierStokes equations are
(1)
where F. is the electrostatic force on a fluid element. If one now
i
considers two flows with geometrically similar boundaries and let
be any length of the first flow field and the corresponding length in
the second, then L^ C^L^. Similarly,
U.2. = j ~ Cj j =
(2)
"v2=CSV .
The individual terms for the second flow field in relation to the
corresponding terms of the first flow field are
(3)
(4)
c3 <=>^1
C,
(5)
and
(6)
CHAPTER II
THE EXPERIMENT
General Description
The present investigation was undertaken to examine more closely
several aspects of streaming potential phenomena which the experiments
of Anderson*'^ 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 experiment and the reasoning behind it will be
given.
The 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 is the radius of the pipe,
P is the pressure,
x is the coordinate along the pipe,
14
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INGEST IEID EGF1V1LPW_L12QZ2 INGEST_TIME 20141014T22:33:27Z PACKAGE AA00025843_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
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.
Assuming the charge density on the wall is independent of the
flow, the convection current is
A.
where U is the hydrodynamic velocity and A is the crosssectional
area.
Substituting from Poisson's equation yields:
(11)
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:
47T
A.
(12)
Successive integration by parts and
tions
=. o wd j
application of the boundary condi
ACKNOWLEDGEMENTS
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.
Appreciation 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 AFAFOSR84267 which has made this
research possible.
in
36
In order that the equations of motion for the two flow fields may be
identical, the following must be true
ct_ = Â£1 _
c, ck
The equality
may be written
(7)
(8)
which means the Reynolds numbers of the two flows must be equal in order
for the flows to be geometrically similar.
The relation ^r = means that
(9)
must also be true. This may be written as
FI?
R,
= constant
(10)
where is the Reynolds number and F is the electrostatic force per unit
volume on a fluid element.
The electrostatic force per unit volume on the fluid element is
given by
(11)
50
O spikes
A no spikes
30
20
10
0
2350
O
GO
,GO
eePo o
o
G
Aj A
A
G
G G
G
G
G
X
2400
2500
2600
2700
Reynolds Number
figure 11. R.M.S. Component of Streaming Potential vs. Reynolds Number, Concentration No. 5
Cl)
W
16
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 investigationwas 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 socalled "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 NavierStokes 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 fluidthe 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 layerthe 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^
2
into the fluid 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 ranging
from 589 to about 5,000 A.
The Stability of Laminar Flows
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' hypothesisnamely, to superimpose a small periodic
perturbation on the mean flow and examine the growth or decay of this
disturbance. Most notable among the names of these workers are those
of Rayleigh*'0), Sommerfeld'4^, Orr^^, Tollmien^^ and Prandtl^\
(8 ) ( 9)
Mention should also be made of Heisenberg' and Lorentz
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 sufficient condition.
The great majority of theoretical work has been directed toward
t.he stability of twodimensional flow, the application of the theory
Figure 1. Sketch of Flow Apparatus
27
Reynolds Number
rigure 6. Resistance Coefficient vs. Reynolds Number
(LogLog Plot)
7
means of a hot wire. Their experiments are v regarded as excellent
verification of stability theory.
(2 G
The experiment of Leite'' 7 mentioned e. er was similar to that
of Schubauer and Skramstad, but axisymmetric urbances were super
imposed to the flow of air in the inlet of a pip
The Electric Double Layer
The phenomenon known as the electric double layer has been
studied extensively by chemists, especially in connection with colloids
and with electrode processes.
The electric double layer 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.
If the solution is caused to flow past the wall, such as in
Poiseuille flow in a pipe, 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 Helmholtz gave an explanation based on Poiseuille
flow and the concept of the double layer developed by Quincke^34^.
The early workers in the field considered the double layer to be
composed of two distinct layers of charge, one fixed to the wail and one
free to move with the fluid. The more realistic model was proposed by
Gouy^'1'^ and Chapman^
who independently formulated the theory of the
37
where q is the electric charge contained in the element and H* is the
(electric) potential. For flow within the electric double layer
f =
(12)
where X is the charge density. Therefore, the parameter in equation
(10) may be written
X 1?
V R^
or, approximately (if the double layer thickness is taken as the
characteristic length) as
{i:) L) L
V Ra
Therefore, the parameter
7
/ % L e.yp(Q
V Rvv
(13)
must be a constant for dynamically similar flows.
The charge density is given by
+ e.nx
(14)
where n^ is the number density of the negative ions and is the number
density of the positive ions. Again the number density is assumed to
be given by a Boltzmann distribution:
A;
Aj0ex?( z:e Vkt')
(15)
where n^Q is the number density of the ith ionic species at the wall
(the wall is considered to have a negative charge).' Therefore (for a
40
^ Cx/o4)
D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 1
Figure 12.
'his dissertation was prepared under the direction of the chairman
of the candidate's supervisory committee and has been approved by all
members 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, 1968
Dean, Graduate School
25
potentials could not be reproduced accurately from day to day. For
each separate electrolytic solution, the streaming potential showed a
longtime 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 twohour 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
A? .
= 4Lp
where U.v is the mean velocity,
Lp is the length of the pipe
and a is the fluid density.
21
TABLE 1
THICKNESS OF THE ELECTRIC DOUBLE LAYER
Concentration
Normality
6(cm)*
6(A)*
1
0
OG
oo
2
0.5364X106
4.16X10"5
4,160
3
0.8040X106
3.40X10"5
3,400
4
1.877X106
2.23X105
2,230
5
26.800X106
0.589X10'5
589
*6 = double layer thickness.
The Electrical Measurements
Three types of electrical measurements were made for each concen
tration of electrolyte. The magnitude of the streaming potential and
the rootmeansquare value of the streaming potential fluctuations were
measured as functions of pressure. In addition, the resistance of the
electrolytepipeelectrode "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 1230A Electrometer (input
impedance 10z ) as a null indicator, the streaming potential was
balanced by the output of a 090V twostage voltage divider. This was
then reduced by a factor of 1000 by the Leeds and Northrup Volt Box and
read on a Leeds and Northrup millivolt potentiometer. Streaming poten
tials for the 1.877X10 ^N and 26.80X10 ^N solutions were read directly
from the General Radio electrometer.
48
1.5
INTEGRATING VOLTMETER
S
X
o
o
o
G
0
1.0
Cae/a?)t
(AE/A?)u
0.5
0
I
! 1 1 1 1 1 1
i t
01234567 89 10
Â£ (x/>_"c.wv')
Figure 18.
Ratio of Laminar Slope to Turbulent Slope vs.
Double Layer Thickness
70
P.
sUi/L,
C*
JO
10
v
12
14
16
18
20
are 14.
D. C. Component of Streaming Potential vs. Pressure,
Concentration No. 3
Reynolds Number
Figure 8. R.M.S. Component of Streaming Potential vs. Reynolds Number, Concentration No. 2
GO
O
12
and
yields
= o
. ^ Â£>
(13)
where S is the circumference of the pipe Â§nd H* is the value of 4*
at y=0.
If the flow is laminar,
/ JuA 
where is the shearing stress at the wall and is the viscosity
of the fluid. Also, if the flow is in a constant area pipe,
= ^=17 Vs
where X is the coordinate along the axis of the pipe. Therefore,
equation (11) becomes
>1 ~
J 477
U;
(14)
The second term in equation (14) will be neglected since it may be shown
that the ratio of the second term to the first is the order of
where r is the radius of the pipe. Therefore, if the pressure gradient
is constant over the crosssection, equation (14) may be written as
ili. A
1 4Ti>u. dX
(15)
8
diffuse double layer, which is, in essence, the theory of ionic atmos
(37) (38)
pheres given some ten years later by Debye and Huckel Stern' '
modified this theory to account for the finite size of the ions at the
wall.
An excellent summary of the classical physics of the effect has
( 30 )
been given by Smoluchowski' and extensive analyses of the approxima
tions used in the various theories are given by Kirkwood
(40)
and
Casimir^41 \
.(42)
The analysis given follows Kruyt The charge at the interface
is considered to be adsorbed on the solid surface and uniformly, distri
buted, while the solvent is assumed to be a continuous media, influencing
the double layer only through its dielectric constant. Coulomb inter
actions in the system are described by Poisson's equation
where T is the potential (having a value of at the wall),
(1)
^ is the charge density,
is the dielectric constant, .
rJ
and v is the Laplacian operator.
The number density of the i"^1 ionic species is assumed to be given by
= n,0exp(Z,eH'/KT) (2)
where ^Â¡o is the number density of the i^*1 species at the wall,
Z.\ is the valence,
is the electronic charge,
K is Boltzmann's constant,
and T is the temperature of the solution.
The charge density is given by
LIST OF SYMBOLS
a
A
d
e
r
~DC
ERMS
Ei
T
*P
q
r
R.
S
T
radius of pipe bore
area of pipe cross section
constant
diameter of pipe bore
electronic charge
streaming potential
D.C. component of streaming 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
length of pipe
number density of iL^ ionic species
number density of i^*1 ionic species at the wall
pressure
electric charge
radial coordinate
Reynolds number
arc length
absolute temperature
Vll
BIOGRAPHICAL SKETCH
Addison Guy Hardee, Jr., was born April 7, 1938, at Mulberry,
Florida. In June, 1955, he was graduated from Hillsboro High School
in Tampa, Florida. From 1956 to 1960, he served as an electronics
technician in the United States Coast Guard and was stationed for a
time 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 Master of Science in Engineering from the same school in
December, 1965. He received an appointment to the position of Research
Associate in the Aerospace Engineering Department in January, 1966,
which position he has held to the present time while pursuing his work
toward the degree of Doctor 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 Kappa Phi and Phi Eta Sigma.
50
24
Figure 4. Tracing of Typical Spikes (Reynolds Number is 2572,
Concentration No. 2)
4
and Pekeris^14^ obtained a solution for the region near the axis of the
pipe. Coreos and Sellars^ gave a solution which accounts for the
work of both Pretsch and Pekeris. The conclusion drawn from the work of
these investigators is that HagenPoiseuille flow is stable for small
disturbances and the work of Sexl and Spielberg^2 ^ confirms this. (Sexl
and Spielberg also showed that Squire's theorem does not hold for axially
symmetric flows.) Experimentally, Leite^20^ failed to observe any
amplification of small axisymmetrical 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 pipesa seeming paradox when one regards the
mathematical results. The resolution of the "paradox" may lie in the
contrast between "small" and finite disturbances occurring in the flow
or in the symmetry of the disturbance. In this connection, Meksyn and
(21)
Stuart showed that, in a channel the lower critical Reynolds number
decreased as the amplitude of the superimposed oscillations increased,
which is in accord with the qualitative observations of Reynolds. A
(22)
possible explanation has been given by Gill who reviewed the above
theoretical papers and indicated questionable steps in their procedures.
(23)
On the other hand, Tatsumi has predicted theoretically that
the flow in the inlet of a pipe is unstable at a Reynolds number of
3
9.7 X 10 Experimentally, both Ekman (unpublished) and Taylor
(unpublished) obtained laminar flow up to Reynolds numbers of 5 X 104
4
and 3.2 X 10 respectively; Ekman made use of Reynolds original apparatus.
22
Figure 2. E^ Measurement
RMS
Figure 3. E.
Measurement
LIST OF REFERENCES
1.
Anderson, R. C., Ph.
D. dissertation, Univ. of Florida, 1965.
2.
Reynolds, 0., Phil.
Trans., (1883), 935.
3.
Lord Rayleigh, Sci.
Papers I, (1880), 474.
4.
Sommerfeld, A., Atti
. del congr. internat. dei Mat., Rome (1908)
5.
Orr, W. M. F., Proc.
Royal Irish Academy A, 17, (1907), 124.
6.
Tollmien, W., Nachr.
Ges. Wiss. Gottingen Math. Phys. Klasse,
Fachgruppe I, 1 (1935), 79; English trans.in NACA TM No.
792 (1936).
7. Prandtl, L. ZAMM, 1 (1921), p. 431 and Phys. Z. 23_ (1922), 19.
8. Heisenberg, W., Ann, d. Physik, 24 (1924), 577.
9. Lorentz, H. A., Akad. v. Wet. Amsterdam 6, (1897), 28.
10. Squire, H. B., Proc. Roy. Soc. A, 142 (1933).
11. Tollmien, W., Nachr. Ges. Wiss. Gottingen, Math. Phys. Klasse 21
(1929), 44; English trans, in NACA TM No. 609 (1931).
12. Schlichting, H., Nachr. Ges. Wiss. Gottingen, Math, Phys. Klasse,
(1933), 182; also ZAMM, 13 (1933), 171.
13. Lin, C. C., Quarterly A.ppl. Math., 3_ (July 1945), 117; also 3_
(Oct. 1945), 218 and 3_ (Jan. 1946), 277.
14. Pekeris, C. L. Proc. Nat. Acad. Sci., 34 (1948), 285.
15. Thomas, L. H., Phys. Rev., (2), 91_ (1953), 780.
16. Sexl, T., Ann. Phys., 83 (1927), 835; also 84, 807.
17. Pretsch, J., ZAMM, 21 (1941), 204.
18. Coreos, G. M. and Sellars, J. R., J. Fluid Mech., 5_ (1959), 97.
19. Sexl, T., and Spielberg, K. Acta Phys. Austriaca, 12 (1959), 9.
51
38
single valence, binary electrolyte), the charge density is approximately
given by
The similarity parameter now becomes
(16)
The quantities 4o L and the exponential term in the numerator all
increase as the bulk concentration of the electrolyte decreases; there
fore, the Reynolds number must increase in order that the parameter
remain constant. For more dilute solutions (i.e., thicker double layer),
the higher must be the Reynolds number of flow phenomena within the
electric double layer. The Reynolds number here, of course, is based
on the thickness of the electric double layer and the local velocity of
the flow.
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
experimental resultsthe transition to turbulence, as indicated by the
double layer, occurs at higher Reynolds numbers for more dilute solutions.
Streaming Potential
The values of the D.C. component of the streaming potential for
5
The Experimental Investigations of Transition
Hagen^24'* first noted the transition from laminar to turbulent
flow while determining the law of resistance for pipe flow. He was
aware that the "critical point" depended on the velocity, viscosity and
the pipe radius. The breakdown of what we today call laminar flow was
noted by the pulsing 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
performed by Reynolds, who showed conclusively that there exist two
possible modes of fluid flowlaminar and turbulent. He was most
probably not aware of Hagen's work, which predated that of Poiseuillev .
On the other hand, Reynolds was in possession of the NavierStokes equa
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 fact that the steady motion breaks down suddenly
shows that the fluid is in a state of instability for
disturbances of the magnitude which cause it to break
down. But the fact that in some conditions it will
break down for smaller disturbances shows that there is
a certain residual stability so long as the disturbances
do not exceed a given amount.
In the second of two independent experiments, Reynolds determined the
49
The critical Reynolds number of pipe flow, as indicated by
streaming potential 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 transition from laminar to turbulent flow can be detected by
noting a change in slope of the graph of E^q vs. pressure, but only if
the electrolytic solution is very dilute.
45
Figure 17. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 5
17
seem to point conclusively to the existence of such a region near the wall
in which viscous stresses greatly outweigh inertial stresses. In this
light, the explanation of the streaming potential fluctuations becomes
more difficult. Reichardt^ predicted the fluctuations, but implied
that they were caused by perturbations in the mean pressure gradient
along the pipeexactly what this means at a specific point very near
the wall is unclear. Reichardt 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 longer
pipe and a higher gain electrometeramplifier and placing greater
emphasis on noise reduction. In addition, five different double layer
thicknesses (ranging from 589A to about 5,000A) would be used,, so that
the double layer would extend over varying distances into the viscous
sublayer. The longer pipe (543 diameters) was used to reduce the effect
of the entrance length.
Laboratory
The laboratory was especially designed and constructed by Anderson
for streaming potential 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
of the temperature, while the humidity of the room atmosphere was held
very low due to the nature of the electrical measurements. The currents
O
involved were of the order of 10 to 10
amperes with a source
To my wife
52
20. Leite, R. J., J. Fluid Mech. 5_ (1959), 81.
21. Meksyn, D. and Stuart, J. T., Proc. Roy. Soc. A, 208 (1951), 517.
22. Gill, A. E., J. Fluid Mech., 21 (1965), 145.
23. Tatsumi, T., Proc. Phys. Soc. Japan, 7_ (1952), 489 and 495.
24. Hagen, G., Pogg. Ann., 46 (1839), 423; also Abhandl. Akad. Wiss.,
(1854), 17 and (1869).
25. Poiseuille, J. Comptes Rendus, 11 (1840), 961 and 1041; also 12_
(1841); also Memoires des Savants Etrangers, 9_ (1846).
26. Stanton, T. E. and Pannell, J. R., Phil. Trans., A214 (1914), 199.
27. Barnes, H. T. and Coker, E. G., Proc. Roy. Soc., A74 (1905), 341.
28. Reiss, L. R. and Hanratty, T. J., Jour. A. I. Ch. E., (8), 2_ (1962)
245; also (9), 2 (1963), 154.
29. Lindgren, E. R., Arkiv. for Fysik, 15 (1959), 97; 15_ (1959), 503;
15_ (1959), 103.
30. Dryden, H. L. and Abbott, J. H., NACA TN 1755 (1948).
31. Schubauer, G. B. and Skramstad, H. K., J. Aero. Sci., 14 (1947), 69
32. Zoliner, F,, Ann. Phygik, 2 (1873), 148.
33. Helmholtz, H., Ann. Phys. VII, 7_ (1879), 22.
34. Quincke, G., Pogg. Ann. 7_ (1879), 337.
35. Guoy, G., J. Phys., (4) 9 (1910), 457; also Ann. Phys., (9), 7
(1917), 129.
36. Chapman, D. L., Phil. Mag., (6), 25 (1913), 475.
37. Debye, P. and Huckel, E., Physik. Z. 24 (1923), 185.
38. Stern, 0., Z. Elektrochem. 30 (1924), 508.
39. Smoluchowski, M. von, Handbuch der Elekrizitat und des Magnetismus
II, Ed. Graetz, Leipzig (1914), 366.
40. Kirkwood, J. G., J. Chem. Phys., 2 (1934), 767.
41. Casimir, H. B. G., Tweede Symposium over sterke electrolyten en
over de electrische dubbeilaag, Utrecht (1944), 1.

