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THE ELECTROKINETIC DETERMINATION OF THE STABILITY OF LAMINAR FLOWS By ADDISON GUY HARDEE, JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA LN PARTIAL FULFILLMENT OF THIE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1968 To my wife ACKNOWLEDC.,...:I: The author wishes to express his sincerest appreciation to the members of his supervisory committee for their cooperation and efforts, and in particular to Drs. K. T. Millsaps and M. H. Clarkson. The guidance of Dr. Millsaps avoided pedagogy of the sort usually bestowed upon graduate students while introducing them to new fields of endeavor. Anpreciation is also expressed to Dr. R. C. Anderson for his many suggestions throughout the investigation and to Dr. J. E. Milton for the help he has given. Further appreciation is also expressed to the Air Force Office of Scientific Research for grant 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_,", , . .. 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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 