THE ELECTROKINETIC DETERMINATION OF
THE STABILITY OF LAMINAR FLOWS
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
To my wife
The author wishes to express his sincerest appreciation to the
members of his supervisory committee for their cooperation and efforts,
and in particular to Drs. K. T. Millsaps and M. H. Clarkson. The
guidance of Dr. Millsaps avoided pedagogy of the sort usually bestowed
upon graduate students while introducing them to new fields of endeavor.
Anpreciation is also expressed to Dr. R. C. Anderson for his
many suggestions throughout the investigation and to Dr. J. E. Milton
for the help he has given.
Further appreciation is also expressed to the Air Force Office
of Scientific Research for grant AF-AFOSR--2-67 which has made this
TABLE OF CONTENTS
ACKNOWLEDGEMENTS . . . . . .. . . . . . . . iii
LIST OF FIGURES . . . . . . . . . . . . . v
LIST OF SYMBOLS .. . . . . . . .... .. ...... .vii
ABSTRACT . . . . . . . . . . . . . .. ix
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
1. Sketch of Flow Apparatus . . . . . . . ... 19
2. EDC Measurement . . . . . . . . . .. 22
3. ERMS Measurement . . . . . . . . . ... 22
4. Tracing of Typical Spikes (Reynolds Number is 2572,
Concentration No. 2) . . . . . . . .. 24
5. Pipe Calibration (Resistance Coefficient vs. Reynolds
Number) . . . . . . . . . . . .. . 26
6. Resistance Coefficient vs. Reynolds Number (Log-Log
Plot) . . . . . . . . . . . . .. 27
7. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 1 . . . . . . ... 29
8. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 2 . . . . . . .. 30
9. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 3 . . . .... . .. . 31
10. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 4 . . . . . . ... 32
11. R.M.S. Component of Streaming Potential vs. Reynolds
Number, Concentration No. 5 . . . . . . ... 33
12. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 1 . . . . . . . . ... 40
13. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 2 . . . . . . . . ... 41
14. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 3 . . . . . . . . .. 42
15. D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4 . . . . . . . .... 43
LIST O0 F1I.... (continued)
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
LIST OF SYMBOLS
radius of pipe bore
area of nipe cross section
diameter 0of pipe bore
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
lenghn of pipe
number density of it ionic species
number density of i ionic species at the wall
U velocity in the x-d1ircction
U mean velCcity
V velocity in the y-dircction
x cartesian coordinate
y cartesian coordinate
double layer thickness
S dielectric constant
K Boizzann's constant
2 specific conductivity
3 density (fluid)
T shear stress at wall
reciprocal of Debye length
S electric potenzial
noTential at wall
J electrical resistance
7 Lapacian operator
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
Addison Guy Hardee, Jr.
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
The investigation also revealed that the fluid flow alters the
eauicibrium state of the electric double layer. In addition, the
transition to turbulent flow can be detected by a change in slope of the
curve for streaming potential vs. pressure, but only if the concentra-
tion of the electrolyte is a very low value.
The present investiaion is te na rural outgrowth of previous
work by ArQnderson ',f wch it rprsents, at the same time, both
an extension an. c ariica io r.. U 1 .. the eectrokinetic phenomena
associated wi t he elecr ic .double ,.yer, Che transition from laminar
o turbuen- flow in a pipe is s-uuied by err..ning the growth of
erurbaions in e fow very close -co th pipe wall. The eleczrokinetic
pIenoenon o_ mes e s the followingg: when an electrolyte,
however oweak, flow-j a7rouh pi a potet difference can be
measure between a pair o eecods placed at the ends of the nine;
thi s n rern ,ont an.r is ac result of zhe electric double
layer 'nich exists a the sold-liquid interface at the wall of the pipe.
T.nhe electric double ayer is discussed m..ore fully on page 7.
The pupo o: Andersn's invest igaion was primarily that of
de .:.minin- p reisely ne .-nir critical R. ":.i number of Poiseuille
rfl ow in a p-ie, -u tr.e -l u on. which he discovered, of 71.e
strea&.ing potential were use o detect thne onset of turbulence in the
Te radia. -ocace.on of p5ions i the flow very close to
the wall is -ossil. for he ex-enson of the electric double layer
into the rflui is of the order of Angstroms and this extension (the
double layer thickness) is a function of the concentration of'the
electrolyte. The present investigation examines the nature of the
streaming potential fluctuations for double layer thicknesses ranr.ing
from 5S9 to about 5,000 A.
The Stability or Laminar 7Flows
The theory of the stability of laminar flows is usually traced to
Osborne Reynolds who, from experimental observation and theoretical
studies, postulated that the state of laminar flow is disrupted by the
amplification of small disturbances; Reynolds, however, gives credit to
Stokes for the concept.
Reynolds treated both inviscid and viscous fluid stability; many
workers since have spent a great deal of effort on the mathematics
underlying Reynolds' hypothesis--namely, to superimpose a small periodic
perturbation on the mean flow and examine the growth or decay of this
disturbance. Xost notable among the names of these workers are those
of Rayleigh(, Sommerfeld"4, Orr(, Tollmien and Prandtl(
Mention should also be made of Heisenberg(8) and Lorentz(9)
Just one example of Rayleigh's work will be mentioned here; he
showed that, for frictionless flow, a point of inflection in the velocity
profile is a necessary condition for instability, i.e., for amplifica-
tion of disturbances; later Tollmien was able to prove that this is also
a suffice ient condition.
The great majority of zteoretical work has been .directed toward
the sabity o :wo-dimensional flow, the application of the theory
being .hI superposition of u iwo-dimen&ional perturbation on the stream
function of the mean flow. N uglecting terms quadratic or higher in the
disturbance leads to the Sommerfcld-Orr equation, which is the fundamental
differential equation for the disturbance amplitude.
Use of a two-dimcnsional disturbance was questionable until
Scuire(10) showed that a tihree-dimensional disturbance is equivalent to
a two-dimensional one at a liwer Reynolds number--at least when applied
to a Two-dimensionI.. flow; t.his has become known as Squire's theorem.
Heisenberg is the first to deduce theoretically the instability
of plane Poiseuile flow for sufficiently large Reynolds numbers,
although he did not calculate a low,:r critical Reynolds number. The
theory of Heisenberg was c ended by Tollmien and Schlichting(
both calculated the neutral stability curves for the boundary layer over
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
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
flow has not been resolved with such clarity. Sexl was the first
to solve the viscous problem, although only for axisymmetric disturbances.
The results have been questioned because of his mtmacal simlifca-
tions. For a small regon near the wall, Pretsch(17) showed the problem
became the same as that of a disturbance applied to Dlane Couetne flow
and Pokeris obt.ai;d a solution for the region near the axis of the
pipe. Corcos and 'e jlars ) Sav a solution which accounts for the
work of both Pret.ch and Pckeris. The conclusion drawn from the work of
these investigators is that Hagen-Poiscuille flow is stable for small
disturbances and the work of Sexl and Spielberg(19) confirms this. (Sexl
and Spielberg also showed that Squire's theorem does not hold for axially
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
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
possibDle explanation has been given by Gill22, who reviewed the above
theoretical papers and indicated questionable steps in their procedures.
On the other hand, Tatsumi(23) has predicted theoretically that
the flow in the inlet of a pipe is unstable at a Reynolds number of
9.7 X 0. Exern.0taly, both :'. (unpublished) and Taylor
(unpublished) obtained laminar flow up to Reynolds numbers of 5 X 104
and 3.2 X respTectivrly; Ekmian made use of Reynolds original apparatus.
The Ex cri cnual Irv.st :atiu:.: o- Tlansition
(e24) firt notu the transitiLon from laminar to turbulent
flow hil deter :g law of resistance for pipe flow. He was
-low while d e t e ri:"' Ie 1wo-r lL ncc
aware that the "critical point" dcpendcd on the velocity, viscosity and
-he nine radius Te breakdown or what we today call laminar flow was
noted by the puling of the jet from the pipe and also by the addition
of sawdust to the flow, showing irregularities present above the critical
The fundamental investigation of the phenomenon of transition was
-.errorme by Reynolds, who showed conclusively that there exist two
nossible modes of fluid flow-laminar and turbulent. He was most
probably not aware of Hi 's work, which predated that of Poiseuille(25)
On the other hand, Reynolds was in possession of the Navier-Stokes ecua-
tions and, by dimensional reasoning, was able to determine the form of
the parameter governing the "critical point." The parameter, of course,
is the Reynolds number and, being the "similarity" parameter for viscous
flow, is more than just the parameter for transition. Though Reynolds'
paper is often quoted, one passage from his 1883 paper is worth noting,
especially with regard to the aforementioned "paradox." Concerning the
sudden disruption of the flow, he writes:
The :act that the steady motion breaks down suddenly
shows that the fluid is inr. a state of instability for
disturbances of the magnitude which cause it to break
down. the fact that in some conditions it will
break down for smaller disturbances shows that there is
Scertra:n r d.... l u tab'Ility so long as the cisturbances
co not exceed a given amount.
In the scon of ine jnen.dent exper mc"zs, Reynods ceItermined the
inimu:. critical l R umi'r of a :.._ straight pipe; the value he
found was aDpproximatly 2000.
Since that time, many workers Lave repeated Reynolds' experiments,
some with interesting variations. The most extensive repetition was
conducted by Stanton and Pannell(0. Barnc, and Coker(27) used a
thermal method c detection in which the walls of the pipe were heated
and the onset or urbuence was detected by a sharp rise in the tempera-
ture of the interior or the flow. RFiss and Hanratty(28) developed a
technique from which they could .infer the behavior within the so-called
viscous sublayer by measuring .he mass transfer to a small sink at the
wall. The sink was a polarized electrode, current limited by mass
transfer. One of the more interesting methods is that of Lindgren(29)
who made the disturbances visiloe by using polarized light and a bi-
refringent, weak solution of bentonite. A technique utilizing the
electric double layer was developed by Anderson(1) and used to determine
the lower critical Reynolds number for Poiseuille flow. That Reynolds
number was found to be 1907 3, indicating the sensitivity of the
The experiments above were mainly conducted to study phenomena
associated with "L-. e" disturbances, while it was not until Dryden's(30)
very low turbulence wind tunnel became available that experimentalists
were able to examine the small disturbance problem. This was undertaken
by Schubauer and Skramszad(37) who made an almost direct transfer of the
teoretcal mehod to, the h .cal situat: n--ocill.io:.. were induced
n a metal rbon aoe I rfat lace placed in the low turbulence tunnel
a:.d t.he amlifca'n of the flow disturbances : :.a:trd dwsream by
means of a hot wire. Thi. cx.eri:;en. s arc regarded as excellent
verification of stability teco,.
The experimen(.t of 2 r.c...cd e ,r was similar to that
of Schubauer and SXkramstad, but axi-symmeIric rbances were super-
.imposed to the flow of air in the inlet of a p,
The Electric Double Layer
The phenomenon known as the electric double layer has been
Studied extensively by chess, especially in connection with colloids
and with electrode processes.
..The electric doublee ay consists of an excess of charge present
at the interface between two phases, such as a solid and a liquid, and
an equivalent amount of ionic charge of opposite sign distributed in the
solution phase near the interface. Consider one phase to be a solid
such as the wall of a pipe and the other to be a weak electrolytic
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
flow anc h.e concept of the double layer developed by Quincke .
early workers in the field considered the double layer to be
Sr~ P- one fixe to the .a'l and one
ree to move w'th :hz flu-id. The more realistic model was proposed by
uy(25) (. 3 ) . ,( )
ouy ... p., w. inde rendentl.y ormua..... the theory of the
diffuse double i.yer, which is, in essence, the theory of ionic atmos-
iv -, - I- (3 ). err(38)
heres ivsn so:c -en years liter by liebye and Huckel ( Stern
.odified th th theory to account for the finite size of the ions at the
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
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
where T is the potential (having a value of Yo at the wall),
Sis the. cnhara density,
Sis the dielectric constant,
a- V is the Laplacian operator.
The number density of the i1h' ionic species is assumed to be given by
ne= Y0,-Z-t/KT) (2)
where Vtio is the number density of the ith species at the wall,
Z; is the valence,
_. is t-.e e ectronic charge,
Sin ~ Dl zmann's constant,
and T s t.: tzemerazture o the solution.
e. .'Ytv igven by
: --- i ',(3)
Combining equations (1), (2), ud (3), one obtains
7 = +-.-nflxQ-Z aY/KT) (4)
which is the differential equation for. the potential as a function of
the space coordinates.
Upon assuming an infinite plane wall, equation (4) is simplified
__ = Z C)(-Z2e.- TkT) (5)
with the boundary conditions
4)0 C'" = 0 00.
where M is the distance from the wall.
The first integration of (5) is carried out after multiplying
through by Z
2LY -..4-Z2Vje;?-C / I
Upon integration and applying the boundary conditions, the result
t 2 T k- Z i/x-2 IK T)
The equation can be simplified by considering a single binary
The" seo7 ri KT... -Z i
L 4 e ~ ...... p p-it/ Z 7Ts
Te Lecond inte-ration is performed after writing7 equation (7) as
Q ~ ~ [^(eZ'+j C t -, /"j,
X.ZMJ'~() =pze'o~r IAI------------ (a)
z 4 77" ;'L
For small oenzias, this :,ay be simplified by expanding the exponential
terms to yield
showing that the potenzial decreases exponentially to zero over a distance
of the order of magnitude of y ..,-. the thickness of the double
layer is of the order of V At room temperature % is approximately
3 X 10 Z V where C is the concentration in gram-moles/liter. As an
examp e, the double layer thickness is approximately 10-6 cm for a
0.001M KC solution.
S.:.r-:.i. -otcentll 's gnerated when >the- c crolytic fluid
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
where U is the hydrodynamic velocity and A is the cross-sectional
Substituting from Poisson's equation yields:
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
11 47 Tj (12
Successive inbegrazion by parts and application of the boundary condi-
tions 0 d C
^ S _[ (13)
1, rTj^V~ ^ ^ j ^ -
wh.ere S is the circumference o: the pipe and Y* is the value of
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,
where X is the coordinate along the axis of the pipe. Therefore,
equation (11) ben...
ii = ()~s-^^^
;77-j j)x j 4-
The second term in equation (14) will be neglected since it may be shown
that the ratio of the second t-erm to the first is the order of
where r is the radius of the pine. Therefore, if the pressure gradient
is constant over the cross-section, equation (14) may be written as
4T A T' dX A
The conduction currcnlt i Jiven by
where 9. is the specific conductance of the solution and E is the
;r 9, and *- are constant over the cross-section, equation
p ,; hence
Under the above .-: .tions, equation (18) shows that the streaming
potenzial is a linear function of the pressure drop between the ends of
.The first term on the right-hand side of equation (13) indicates
that the convection current, and hence the streaming potential, is a
function of the velocity radient at the wall.
THE L. i .'i :.T
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
Tne theory of Chapter I shows that the streaming potential is
directly proportional to the pressure gradient along the pipe. Also,
equation (13) shows that the streaming potential varies directly and
linearly with the velocity gradient at the wall.
if we now restrict the discussion to Poiseuille flow, the velocity
distribution across the pipe is given by
at a sufficient distance from. the inlet, where
.a : thre raclas of the pipe,
:x: ie ;. coordinate along the pipe,
is th. vi:cosiy o- the fluid,
r is the di nce from the center of the pipe,
and u is t veloct-y.
If this rlation is substituted into equation (13), the streaming
oteni is again found to be a linear function of the pressure gradient
along tLhe pipe; thisiZ-S bS.en o1 ;erved by ..,'.. -1 chemists() for many
years. Since cuation (13) indicates the dependence of the str- c. :...
otenial on the velocity gradient at the wall, Anderson reasoned that
streamrin potential ,easurements could be used to determine the transi-
tion from laminar to tubulent flow, for the velocity gradient near the
wall is different in :c.se to modes of flow.
n son looked for a change in the value of the quantity AE/AP
(i.e., a"brak" in ,e c1.- ve of E vs P) when transition occurred. His
ex-periment was conducted .....i ng an electrolytic solution of O.001N KCi
flow=n n a Pyrex capillary tube having a diameter of 0.0242 inches
and a length of 4.8 inches. The inlet to the pipe was artificially
Although he found a suggestion of such a break, Anderson's data
did not une.uivocaly show one; instead he discovered unique fluctua-
ins i the stre..m.n. potential which appeared at the transition
Reynolds number. Using these fluctuations as a guide, he was able to
determine the minimum criica Reynolds number very precisely. In
addition, he showed thait the streaming potential did not vary linearly
winpr<-.:.:* -.r........ ir ;. stream~ potential was measured across
tne full length of -?`h piDe. This s the effect "of the entrance length
(".t -.*.th. f :. ... p : ... cesay for est-biish:ing Foiseui e flow)
which the theory does not include. Most of the measurements of streaming
potential by physical chemists have not taken this into account.
The fluctuations appeared as "spikes," which always represented
an increase in (positive) voltage near the transition; above transition
the spikes appeared to be positive and negative.
The present investigation-was undertaken to investigate the effect
of the electric double layer thickness on streaming potential measure-
ments, the thickness being a function of the electrolytic concentration.
The electric double layer, for most liquids (including the electrolyte
used by Anderson), is well inside the so-called "viscous" sublayer;
thus the mechanism for producing the streaming potential fluctuations
was not apparent.
Little is understood of the actual conditions at a solid surface
in fluid flows; in a liquid, there is the electric double layer to be
considered. The electric forces in the double layer have been neglected
in solving the Navier-Stokes equations for the velocity in the pipe
(unless they are contained in the boundary conditions). In addition,
the theory given for the electric double layer, including its thickness,
is based on a stationary fluid--the analysis of the interaction of a
flow and the double layer is complicated. The double layer extends out
from the wall a distance on the order of Angstrom units. An aqueous
KC1 solution of 0.001N gives rise to a double layer thickness of 95.9A;
this, of course, is the distance from the wall to the "center of gravity"
of the double layer--the double layer can extend out much further. Such
small distances from the wall show that the double layer is within the
viscous sublayer. The measurements of Reichardt(44) and Laufer(45)
sem to point conclusively o tc existence of such a region near the wall
in which viscous stress ss gratly outweigh inertial stresses. In this
li.Sh, t xe ex:lanation of tze streamlng potential fluctuations becomes
more dlfficul4. Reichard(4 o predicted the fluctuations, but implied
that they were caused by perturbations in the mean pressure gradient
along re pipe--exactly what this means at a specific point very near
tme wall is unclear. eichard searched for the fluctuations, but was
unable to detect them using a quadrant electrometer. Also he found that
the value of AE/AP was essentially the same for both laminar and turbulent
It was decided to extend Anderson's investigation using a lr-.r.,er
-)e n a hi"a.g.er gain electrometer-amplifier and placing greater
on ise reduction. In addition, five different double layer
thicknesses (ranging from 589A to about 5,OOOA) would be used, so that
he doule layer would extend over varying distances into the viscous
sub-layer. The longer pipe (543 diameters) was used to reduce the effect
of the entrance length,
h.e aborazory was especially designed and constructed by Anderson
or streaming sng poentia! experiments. This was necessary since the work
is particularly sensitive to temperature and humidity. The thermal
dependency of the viscosity of the fluid necessitated the fine control
ofr th em-:-r'a-.r, ..l t]e humid 'ty of the room atmos-.ere was held
vtry ow ue to .e na-re of the electrical measurements. The currents
_nvc.v ; -/.\;r c. .e er i0- ,o 0 amperes with a source
resistance of the order of 109 to 10I ohms; a high humidity level could
cause surface leakage due to adsorbed water on the'exterior surfaces of
the glass apparatus.
The laboratory is a 10 by 16 foot room inside an air-conditioned
building and the entire room is vapor sealed with 10 mil polyethylene.
The floors and walls are insulated with 3 inches and the ceiling with 4
inches of Styrofoam. Temperature control is achieved by a special air-
conditioning system which holds the room temperature changes to 0.75 F.
In addition, the experimental flow apparatus was enclosed within a small
Styrofoam box. With this addition, temperature drifts are less than
0.02 C per hour.
The air drying is accomplished by use of a compression-cooling-
expansion cycle. This enables the room air to be held below 20% R.H.
The apparatus used is similar to that used by Jones and Wood(43)
Kruyt Anderson and others and is shown schematically in Fig. 1.
The reservoirs R1 and R2 are 5 liter boiling flasks fitted with ground
glass taper joints, J1 and J2. Tubes LI and L2 are 6mm Pyrex with
taper joints at the top for inserting electrodes. A1 and A2 are 1/4
inch tubes. The tube T is a precision bore capillary tube 0.0631
.0001 inches in diameter and 34.3 inches long. The capillary tube is
connected to A1 and A2 by flanges incorporating Teflon gaskets. In
operation, the liquid was forced up L1 and through the capillary by
applying gas pressure through VI.
The electrodes E1 and E2 are Ag-AgCl formed on spirals of 24
< \ itl
* platinum. -.... Ag-ACI coating was made with a mixture of 90% silver
oxide an- 10% silver chlorater formed into a paste, applied to the wire
and heuad at 500C for fi -een minutes.
The gas pressure was supplied by commercially bottled nitrogen.
The pressure was measured by a simple mercury manometer used in combina-
tion with a micro-telescope mounted on a vernier height -age. The
pressure could easily be controlled to within 0.005 inches Hg. The two
nhree-way stoecccks in the supply line provided a means of applying
pressure to the flasks or venting them to the atmosphere.
CleaninL of the Apparatus
The glassware in the apparatus was soaked in chromic acid for
several hours, cleaned with hot chromic acid and rinsed in hot conductiv-
ity water. The glassware was then leached in conductivity water over-
night. The electrodes were lcched in conductivity water for several
The conductivity water used as the solution was twice-distilled,
having -itrc .: .ubbled through it after each distillation.
Elecrrolyvic Solutions Used in the Experiment
The experiments were conducted with 5 concentrations of KCI
solutions, beginning with conductivity water as the first solution. Salt
was added to this to fo.rm the subsequent concentrations. Table 1 gives
th< vates cf rh five concentrations and the corresponding thickness of
THICKINESS OF THE ELE -..IC DOUBLE LAYE.
Concentration Nor:..ality 6 (cm)* 6e(A)*
S0 cc 00
2 0.5364XI0-6 4.16X10-5 4,160
3 0.8040X10-6 3.40X10-5 3,400
4 1.877X106 2.23X105 2,230
5 26.SOOXlO-6 0.589X10-5 589
= double layer thickness.
The Electrical measurementss
Three types of electrical measurements were made for each concen-
tration of electrolyte. The magnitude of the streaming potential and
-the root-mean-square value of the streaming potential fluctuations were
measured as functions of pressure. In addition, the resistance of the
eectrolyte-pipe-electrode "source" was measured before and after each
The streaming potential magnitude was determined by the system
shown in Fig. 2. Using a General Radio Model 1230-A Electrometer (input
impedance 10!2 ) as a null indicator, the streaming potential was
balanced by t.-he output of a 0-90V two-stage voltage divider. This was
then reduced by a factor of 1000 by the Leeds and Northrup Volt Box and
'c o a Leeds and Nortnru millivolt potentiometer. Streaming poten-
ti!s for th. l.77I0-N and 26.80X10N solutions were read directly
fro::. 'e G--neral Rd electrom7.ter.
Figure 2. E_- Measurement
Figure 3. ER Measurement
The R..S. values of the stre .. potential fluctuations were
determined usi.g .e system shown in Fig. 3.
Th.e string Ioential fluctuations were amplified by a
iffe....ia eecrometer-Cmplifier specially constructed for electro-
chemical. measurement; the input stages are Philbrick Model SP2A
.erti"...l amp1fiers. The inDut impedance of this amplifier is 10
ohms and -che gain was 8.80 .... .hout the experiment (gain can be set
at S.Z0, 88.C, or 830). The output of the volt-.ge divider was used to
cancel the D.C. comonen.-c of the streaming potential (this ranged up to
SOV). The output of the amplifier was fed into a strip-chart recorder
for a visual record of the fluctuations and into the Flcw Corp. Random
Signal Voltmeer :for their R.M.S. value.
The elctrical system represented by the electric double layer
is very sensiive to stray capacitance-it was found that an excessive
amount of "grassy" noise (which almost obliterated the smaller signals)
was generated by vibrations in the screen panels of the screen-room
(Fara"day c ). This was corrected by glueing aluminum foil shielding
onto the Styrofoam thermostat in which the flow system was placed. This
shielin, alo. with the usual care in reducing vibrations in all
electric components, gave a great reduction in noise level. Typical
1i 1 levels were:
sike magnitude (peak) 0.1V
sIike magnitude (R.M.S.) 0-50mv
.... .. .'n.. po-en. a! > (D.C.) 0-85V.
A typical spik e trace is s:own in Figure 4.
s prevics experimenters had noted, the values of the stre=.-in,
racr.n of Typica Spikes (Reynolds Number is 2572,
Co:.c.ratior No. 2)
L- 0.05 V
potentials could not be reproduced accurately from day to day. For
each separate electrolytic solution, the streaming potential showed a
long-time drift of a period ranging from a few days to weeks, reaching
a minimum, and then increasing slowly. The D.C. component of the
streaming potential, for a given electrolyte concentration, was taken
over a two-hour period on the day the potential reached a minimum.
Reproducibility was also improved if the fluid was pumped at high flow
rates before taking data; also, higher Reynolds number measurements were
alternated with those of lower Reynolds number. If this alternation
were not carried out, the D.C. values for the low flow rates (below
Reynolds numbers of about 900) would drift upward. This indicates that
the equilibrium state of the double layer is changed when the electrolyte
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.
SI I I I I I I
o vj -r 0. 0 oo C0) r C
C0!-i.-i I- I-- --I
60700 830 903 1000 1500 2000 30
c Coefficient vs. -enolds Number
600 700 800 300 1000 1500 2000 3000
-.-'-^, 6. Resisran~ce C "oefficient vs. Reynolds Number
RESULTS AND C.,h.LJiONS
R..S. C..n.. of Streamin;. Potential
The R.:.S. values of the fluctuations in the streaming potential
are shown in Figures 7 to 11, plotted against the Reynolds number; these
values have been divided by the gain of the electrometer-amplifier. The
points denoted by triangles represent laminar flow, i.e., no spikes
(fluczuations) on the streaming potential; those denoted by circles
reresen: the occurrence of spikes. Below transition (no spikes) and
above a Reynolds number of 2560, all of the data are shown on the graphs;
in between, only points selected to indicate the mean curve are given
o avoid crowding. The average deviation from the curve is about 7 in
Reynolds number. in Figure 11, all the available data are shown on the
> figure displays the same general trend; a peak, followed by
a dip, and a second, higher peak followed by a rapid decrease. Such
regular variatons are not detectable in mass flow rate measurements
tr.ough. the transition range of Reynolds numbers--only an irregular
increase (plot of resistance coefficient vs. R Fig. 6). At the
ies Reynolds number of the experiment, transition to fully-developed
urulence w:as no= complete (deduced from Fig. 6).
:.o frqu ncy 0of occurrence of the spikes at their first
n I .
I I I
,,,~~ ~ I
0 0 CD
0 0 :
oJ 0; 4
I <3 :r
C-C (N- o o
I :"i o
o <3 [-2
< 0 0
(D o -
0 0 0
0 o )
M CU )J
CO > (
appearance was taken as 0.002 per second (one spike in an e'-ght minute
period) anc at R 2755 was approximately 2.1 per second; this frequency
was defined by manually counting the peaks of the spikes over an
C! i...ute period of the chart from the recorder. Thus, altho'h the
frequency response of the electrometer-amplifier and electrodes is flat
-Or s range, the response of the system comprised of double layer-
eectrods-am.lifier may be responsible for the variations in the ERMS
vs. R_ curves. The period of an individual spike was about 2 seconds.
The existence of some fluctuation below the appearance of the
first sxikes is a measure of "free-stream" turbulence as well as fluctua-
tions fro:m the inlet of the pipe. The corresponding R.M.S. values at a
Reynolds number of 1000 was about half that at 2400.
The curves for each double layer thickness are shifted progressively
to lower Reynolds number as the double layer thickness decreases. The
first s.ikes appear at R = 2585 for the first concentration and at
R = 2470 for the last concentration. The peaks and other salient
features of the curves display corresponding, although smaller, shifts in
Reynolds number. The magnitudes of the shifts are too large to be
explained by changes in average viscosity of the fluid, but the same
cannot be said of the viscosity in the region of the double layer as
t-e double layer reacjusts to an overall concentration change. As will
be shown in the next section, the equilibrium state of the electric
doue layer is changed by the flow impressed upon it--whether this is
a cae in poCential distribution alone or with a simultaneous change
in noenial at the wall can not be determined. The part the electric
ou. layer it-f -ays in the stability of laminar flow is unknown,
although some insight into its role :,,t-h be gained by dimensional
Consider the two-dim-.nsional steady flow of a viscous fluid
aving an electric double layer present at the solid boundaries. The
Navier-Stokes equations are
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 .
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)
In ord-er that th.e equations of motion for the two flow fields may be
"dentic-l, the following must be true
ct -- C &
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
rL- L, ,
must also be true. This may be written as
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
where q is the electric charge contained in the element and Y is the
(electric) potential. For flow within the electric double layer
where is the charge density. Therefore, the parameter in equation
(10) may be written
Y, 'X( e-q'-ApL) L:
or, approximately (if the double layer thickness is taken as the
characteristic length) as
^^(- L1) L
Therefore, the parameter
^'e4'.L4 Y ) (13)
must be a constant for dynamically similar flows.
The charge density is given by
where n1 is the number density of the negative ions and n2 is the number
density of the positive ions. Again the number density is assumed to
be given by a Boltzmann distribution:
A; = Vt. 0e X?(-2i;e-Y/KT)15
where ni is the number density of the i th ionic species at the wall
(the wall is considered to have a negative charge).* Therefore (for a
single valence, binary electrolyte), the charge density is approximately
-similarity parameter now becomes
e M. L rxo. KT -1 P
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
The relation of the corresponding mean Reynolds number of the flow
to a local Reynolds number within the double layer depends on the rela-
tion of their respective velocities; whatever this exact relation may be,
the parameter given by the above analysis agrees qualitatively with the
xri ....T. results-the transition to turbulence, as indicated by the
double layer, occurs at higher Reynolds numbers for more dilute solutions.
- value of the D.C. com onent of the streaming potential for
the five solutions are shown as functions of pressure (non-dimensional)
in Figurs 12 to 17. Excepting a region at the -.ginning of transition,
he s :o for laminar flow is different than the slope for turbulent
flow. Thu., the change in -lope can be used to detect the transition
from laminar to turbulent flow. A comparison of the graph of EDC vs.
-ressure for the last concentration (Fig. 17) and resistance coefficient
(F) vs. Reynolds number (Fig. 6) shows that the streaming potential
indices a slightly higher critical Reynolds number than mass flow rate
measurements. The break in the F. vs. R curve begins when the eddies
grow enough in strength and number to increase appreciably the resistance
o the :flow. This should coincide very nearly with the establishment
of a different velocity profile near the wall--thus changing the value of
S as governed by equation (13), Chapter I. The difference in
critical Reynolds number given by the two methods, mass flow rate and
dreaming potential, may indicate that the turbulent velocity profile
is established "close" to the wall, i.e., on the order of 10- A, at a
higher Reynolds number than shown by an increase in flow resistance. A
cuestzion arises when one considers that the velocity gradient of equation
(13) must be that within the double layer, which means well within the
"viscous" sublayer; a.c. to the accepted model of this sublayer,
turbulent eddies (necessary to produce a change in velocity profile) so
near the wall must be very weak. Further, it is usually assumed that
the velocity profile within the viscous sublayer does not differ in
lainar nd turbunt flow. This says nothing about the magnitude of a
change in velocity profile which the double layer can Cdetect..n
S.;_- of a "viscous" sublayer, it should be kept in mind that the
I I I I I I I I
6 8 10 12
14 16 18 20
.C. C...n.r. of Strea~mi. n Potential vs. Pressure,
0 2 4
2 4 6 8 (v0 12 4 16 20
*P 13 (9I-4
Fi2u 3.D.C. Compconentll of Streaming Pote-ntiali vs. Pressure,
Concentration No. 2.
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.
II I I I I I I I
0 2 4 6 8
10 12 14 16 18 20
PA t -4
D. C. Component of Streaming Potential vs. Pressure,
Concentration No. 3
X I I I I I I I I I
0 2 4 6 8 10 12 14 16 18
D.C. Component of Streaming Potential vs. Pressure,
Concentration No. 4
rl io 0
1 1 1
0 2 4 6 8 10 12 14 16 18 20
F7Qu'c "C. P.C. Comnonent of Streaminj Pontial vs. Pressure,
C..r.:. trtion No. 4. Data TWkon from .'mcc integrating
Sl I I I I I I I I I
0 2 4 6 8 10 12
14 16 18
.C. ....t of Streaming Potential vs. Pressure,
Concentrcation No. 5
entire flow in the pipe is "viscous." Also the work which points to the
existence of the sublaycr is based largely on hot-wire measurements and
thes ave only been made over thie outer part of the sublayer.
A noter fact bearing on the above question is that in deriving
t-e eouarions for the streaming potential, it was assumed that the flow
condiiins" (including the velocity profile) were the same all alon. the
length of the pipe; in reality, the theoretical conditions are attained
cnlv =iter the entrance length of the pipe has been traversed. Further,
he entrance length necessary for establishment of such conditions is
much shorter for turbulent than for laminar flow. This means that the
entrance length is changing in the transition region of Reynolds numbers
(it is also a function of Reynolds number for laminar flow and, to a
lesser degree, for turbulent flow).
S. velocity profile for laminar flow in the entrance ien-th is
similar to that for established turbulent flow; thus, for laminar flow
a-: low Reynolds numbers, the value of AE/A? should be intermediate to
Those of established laminar flow and established turbulent flow. This
Sseen to be the case for all the data in Figures 12 to 17--the slope
of the laminar portion of the curve decreases with increasing Reynolds
Measuring the in potential and pressure across one segment
of the ie, excluding the entrance length, would better fulfill the
condiions of the theory and show whether or not the "non-developed"
f3w in .e ch_-ng.ng entrance length is responsible for the difference
in slops. Tis was considered in preliminary planning of the investiga-
ion; :.;ver, it was evident that any method of isolating one length
of the ie would introduce disturbances to the flow and would require
such a large system that purity of: the solution could not be maintained
The magnitudes of streaming potential given here include the
average value of the streaming 3otenzial fluctuations. These average
values should be --: the same order of magnitude as the R.M.S. values,
i.e., less than 50 millivolts. This is a sufficiently small part of the
7_ values to be ignored... data for the last, and most concentrated,
soutioc. (Fig. 17) shows an unexplained peak at the beginning of transi-
tion. The peak occurs roughly at the same Reynolds number as the maximum
..M.S. value (Fig. 11), but is about twice the expected magnitude.
6gure 16 ows EDtaken from a Dymec (Hewlett-Packard) Integrating
::u~ _. 6 ho s DC
Diital Voltmeter, :odel 2401A, set up as shown in the inset to this
f -.-._. Integration does not alter the basic curve, but does slightly
c.-.nge the ratio of the turbulent slope to the laminar slope. These
data were taken with concentration 4 on the same day as the data shown
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
3 -------- --- -------- 1 --I1--I--I-''I
S2 3 4 5 6 7 8 9 10
: (xSi'c. tS
* .: *'r ;..-:;:.o of T .:.-Inar Sic, to Turbulent Slope vs.
-o^ : L* ...k :%N ..';
The critical Recynolds number of pipe flow, as indicated by
streaming potcntiai fluctuations, depends on the thickness of the electric
double layer present at the wall of the pipe, i.e., on the concentration
of the electrolytric fluid.
The equilibrium state of the electric double layer in a stationary
fluid is different than that for a flowing fluid, at least for laminar
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.
Addison Guy !ardee, Jr., was born April 7, 1938, at Mulberry,
Florida. In June, 1955, he was graduated from Hillsboro High School
in Ta&mpa, Florida. From 1956 to 1960, he served as an electronics
technician in the United States Coast Guard and was stationed for a
tiM.e in Iceland. Following his discharge from the Coast Guard, he
enrolled in -the University of Florida and in December, 1964, he received
the degree of Bachelor of Aerospace Engineering. He received the
degree of Mastcr or Science in Engineering from the same school in
December, 1965. He received an appointment to the position of Research
A~sociate in the Aerospace E.-.,ineering Department in January, 1966,
wich: position h.. Ias held to the present time while : _i-l., his work
toward the degree of Doctor of Philosophy.
o- Dc-Lor of Philosophy.
Addison Guy Hardee, Jr., is married to the former Mildred Fe
Collar and is the father of two children. He is a member of Tau Beta
Pi, Phi Phi and Phi Eta Sigma.
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...-. .. .., -: iu over S-ke el:ectrolvten en
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This dissertation was prepared under the direction of the chairman
01o the candid 's supervisory committee and has been approved by all
meers of that committee. It was submitted to the Dean of the College
of Engineering and to the Graduate Council, and was approved as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
Dean, olleGraduate Shoolf E-eerin
Dean, Graduate School