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Int. J. Mech. Eng. & Rob. Res. 2014 Vishnu S B and Ajith C Menon, 2014

MATHEMATICAL MODELLING OFELECTROOSMOTIC FLOW THROUGH A SLITMICROCHANNEL UNDER THE INFLUENCE OF

TIME VARYING ELECTRICFIELD

Vishnu S B1* and Ajith C Menon1

*Corresponding Author: Vishnu S B,vishnusb90@gmail.com

Electrokinetics involves the study of liquid or particle motion under the action of an electric field.The applications of electrokinetics in the development of microfluidic devices have been widelyattractive in the past decade. Electrokinetic devices generally require no external mechanicalmoving parts and can be made portable by replacing the power supply by small battery. Therefore,electrokinetic based microfluidic systems can serve as a viable tool in creating a Lab-On-a-Chip (LOC) for use in biological and chemical assays. Here analytical procedure is carried outto find out the solution of electroosmotic flow. The governing equations including the linearizedPoisson-Boltzmann equation, the Cauchy momentum equation are solved to seek velocitydistribution and flowrate. Then effect of electricfield frequency on velocity profile is studied. Theeffect of frequency,electric field and electric double layer thickness on flow rate is also studied.The problem is first treated by using analytical methods, but the quantitative estimates areobtained numerically with the help of the software MATHEMATICA. The study reveals that theflow is oscillatory and maximum positiveflow is obtained for electricfield frequency of 20 Hz andas electricfield strength increases, flow rate also increases.

Keywords: Electrical double layer, Electroosmotic flow, Zeta potential

INTRODUCTIONA lab on a chip device is a micro scalelaboratory with an array of micro channels,electrodes, sensors and electronic circuits.Electrodes are placed at different locations sothat by applying electric field we could control

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Int. J. Mech. Eng. & Rob. Res. 2014

1 Mechanical Department, Marian Engineering College, Kazhakuttam, Kerala, India.

the liquid flow and other operations on the chip.Almost every application which can do onroom sized laboratories can be duplicated byLOC devices. So a reduction in samples andreagents, very short reaction time, portability,etc., is possible with LOC devices. We can

Research Paper

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Int. J. Mech. Eng. & Rob. Res. 2014 Vishnu S B and Ajith C Menon, 2014

also perform several applications likepumping, mixing, thermal cycling, separatingetc. For example in a typical heart attacksituation, the patient would require suddendiagnostic such as electrocardiogram andblood test. If blood test is conducting on acommon laboratory, it will require a greatertime and it may affect their life. But a LOCdevice could provide rapid diagnosis of thepatient’s cardiac condition. The recentdevelopment in electro kinetics and LOCdevice brings a strong demand ofunderstanding electro kinetic phenomena andmicro fluidics.

When electric fields are applied acrosscapillarities or micro channels, bulk motion isobtained. The velocity of this motion is linearlyproportional to the applied electric field, andis depends on the material used to constructthe micro channel and also the solution incontact with the channel wall. This motion isreferred as electro osmosis and stems fromelectrical forces on ions in the electrical doublelayer, a thin layer of ions that is located near awall exposed to an aqueous solution. If the fluidvelocity is interrogated at micrometerresolution, the fluid flow in a channel of uniformcross-section appears to be uniform. If the fluidvelocity is interrogated at nanometreresolution, the fluid velocity would be uniformfar away from the wall, but it would decay tozero at the wall over a length scale rangingfrom approximately 0.5-200 nm.

This fluid flow can be immensely useful inmicro fluidic systems, because it is often muchmore straight forward experimentally toaddress voltage signals sent to electrodesrather than implement and control aminiaturized mechanical pressure pump. This

flow however comes with its owncomplications, as it s velocity distributions isdifferent from pressure driven flow , it issensitive chemical features at the interface ,and because the act of applying electric fieldscan also move particles relative to fluid orcause joule heating throughout the fluid. Thisphenomenon provides an effective way formixing applications. By applying an externaloscillatory electric field, a time dependantsecondary flow can be excited. It is found thatthe mixing is improved due to an enlargementand folding of the virtual interface between theliquids. Thus diffusion can act more efficientlyand virtual interface between the liquids canbe maximized in area by cooling an optimumexcitation frequency (Meisel and Ehrhard,2006). The analytical solutions for channel andpipe flows of visco elastic fluids under themixed influence of electro kinetic and pressureforces are found out (Afonso et al., 2009). Theanalyses are restricted to cases with smallEDL. The viscoelastic fluids used aredescribed by Phan Thien and Tanner (PTT)model with linear Kernel for the stresscoefficient function and zero second normalstress difference and FENE – P model. In theabsence of an imposed pressure gradient, theelectro osmotic flow exhibits a plug like velocityprofile. When visco elastic flow is induced bya combination of both electrical and pressurepotential, in addition to a single contributionsfrom these two mechanisms, there exist anextra term in velocity profile which is absentfor the Newtonian case where superpositionprinciple applies. This non linear term can becontribute significantly to the total flow rate ie,under conditions of favourable pressuregradient it act as a drag reducer and foradverse pressure gradients, it will act as a

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Int. J. Mech. Eng. & Rob. Res. 2014 Vishnu S B and Ajith C Menon, 2014

drag increaser. The exact solution of unsteadyflow due to an infinite plate oscillating in its ownplane is found out.

Oldroyd fluid A and B are non Newtonianfluid but they show constant viscosity. The mainfeatures of the general solution are the fluidinitially at rest, oscillates harmonically in the x– direction. The envelope of these oscillationsgrows with increasing time‘t’ and decreaseswithin the distance from the surface (Hayatet al., 2001). The solution obtained representsa transverse wave: its velocity is perpendicularto the direction of propagation. Also transversewaves can occur in an Oldroyd – B fluid, butthey are rapidly damped as moved away fromthe solid surface. The exact and approximatesolutions are presented for fully developed pipeand planar flows of multimode differentialviscoelastic equations (Cruz et al., 2005). Theexact solutions are semi analytic and pertainto models based on PTT, Giesekus andFENE-P equations, whereas the approximatesolutions are fully analytical and concern thePTT and FENE-P models. In all cases dueaccount is taken of the presence of Newtoniansolvent (Cruz and Pinho, 2007). The analyticalapproximate solutions was obtained with aperturbation technique, but calculations withthese formulae are limited to low values of flowDeborah number for the PTT model and alsofor FENE-P model. In spite of its length theanalytical solution is advantageous at lowDeborah numbers because it eliminates theneed for an iterative numerical procedure. Theelectro osmotic flow of a third grade fluidbetween micro parallel plates is considered(Akgül and Pakdemirli, 2008). Equations ofmotion are derived and made dimensionless.Temperature and velocity profiles are

calculated using approximate solutions and byperturbation technique. Effect of nonNewtonian parameter, joule heating, andviscosity index and electro kinetic parameterson the velocity and temperature profiles isdiscussed. Approximate solutions arecontrasted with numerical solution. Twodifferent cases ie constant viscosity andvariable viscosity cases are treated separately.Analytical solutions are contrasted with thenumerical solutions of the original equations.Within the validity range, solution matches witheach other.

Electro viscous effects in a power law non-Newtonian fluid taken to be a symmetric 1:1electrolyte solution are investigated for fullydeveloped pressure driven flow through acylindrical micro fluid pipe (Bharti et al., 2009).The Poisson Boltzmann equation for theelectrical potential, and the Navier stokesequation with electrical forcing and power lawfluid rheology, are solved numerically using afinite difference method. The induced axialelectrical field shows only a weak dependenceon the power law index with the dependencebeing greatest for shear thinning liquid. Theelectro viscous effect is stronger in shearthinning and weaker in shear thickening liquids,than it is when the liquid is Newtonian. Thefrequency dependent flow of electrolytebetween pairs of parallel platemicroelectrodes is analyzed (Talapatra andChakraborty, 2008). It is found that the impactof double layer overlap on ac electro osmosisflow turns out to be more predominant atfrequencies of the order relaxation frequencyof electrode electrolyte system. Closed formexpressions are derived to depict the influenceof EDL overlap on ac electro osmosis flow

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Int. J. Mech. Eng. & Rob. Res. 2014 Vishnu S B and Ajith C Menon, 2014

patterns. If the applied potential is below theionization potential, electrolysis doesn’t occurat electrode surfaces. The EDL formedbehaves in a capacitive increases resulting instrong potential gradients across the electrodesurface. A satisfactory explanation of ac electroosmotic fluid flow characteristic betweenparallel plate micro channels under overlappedEDL conditions could be obtained.

Non Newtonian fluids such as blood, colloidsand cell suspensions are often manipulated inmicro fluidic devices and exhibit extraordinaryflow behaviour, not existing in Newtonian fluids(Xiao-Xia Li and Ze Yin, 2012). An analyticalsolution of transient velocity for electro osmoticflow generalized Maxwell fluids through both amicro parallel channel and a micro tube usingmethod of Laplace transform is found out. Thesolution involves analytically solving thelinearized PB equations together with Cauchymomentum equation and the general Maxwellconstitutive equation. Using the method ofLaplace transform the transient EOF velocityprofiles are obtained. The computationalresults show that velocity profile depend on thenormalized relaxation time, the EOF velocityapproaches steady and flow needs longertime to attain steady status with the increaseof normalized relaxation time. The velocity andtemperature distributions of the thermally fullydeveloped electro osmotic flow through arectangular micro channel are obtained (JieSu and Yongjun Jian, 2012). Based on linearPB equation, Navier stokes equation andthermally fully developed energy equation,analytical solution of normalized velocity,temperature and nusselt number are derivedusing the method of variable separation. Theygreatly depend on the electro kinetic width,

width to height ratio and joule heating to heatflux ratio. By numerical computations, it is clearthat for prescribed electro kinetic width (k),increased heating to heat flux ratio (s) yieldsgreater variation of the temperature over therectangular micro channel cross section. Thedependence of temperature on‘s’ is moresignificant for small k, while at a larger k, thetemperature profiles are almost identical.

The analytical solution of the time periodicEOF of the general Maxwell fluids betweenmicro parallel plates under the Debye-Huckelapproximation is presented (Quan-Sheng Liuand Yong-Jun Jian, 2011). The solutioninvolves analytically solving the linearizedPoisson-Boltzmann equation, together withCauchy momentum equation and the generalMaxwell constitutive equation consideringelectro osmotic forces as the body forces.With the method of separation of variables,the velocity profile and volumetric flow ratesfor the time periodic 1-D EOF are obtainedanalytically.

GOVERNING EQUATIONSTheoretical Model and Analysis ofEDLThe relation between the electrical potential and local net charge density per unit volumee at any point in the solution is described byPoisson equation

0

2

...(1)

From Boltzman distribution, the numberconcentration of type i ion in symmetricelectrolyte solution is:

KbTZie

ii enn

..(2)

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Int. J. Mech. Eng. & Rob. Res. 2014 Vishnu S B and Ajith C Menon, 2014

where ni∞ = bulk ion concentration

Zi = valency

e = charge of a proton

e is propotional to the concentration differencebetween symmetric cations and anions.

nnzee ...(3)

KbTZezen sinh2 ...(4)

The net volume charge density,

ie zien ...(5)

KbTZie

ii enze

...(6)

So equation reduces to Poisson Boltzmanequation.

KbTZezen

sinh02

0

2 ...(7)

KbT

Ze

ii enZe

0

2 ...(8)

By defining Debye Huckel parameter,

KbTnzeK

0

2 22

...(9)

And non dimensional electrical potential

KbTze

...(10)

The final equation becomes

sinh22 k ...(11)

AC electroosmotic flows in a rectangularmicrochannel

Electroosmotic flow induced by unsteadyapplied electric fields or electroosmotic flowunder an alternating electrical (AC) field hasits unique features and applications

This section reviews a theoretical approachthat was taken to investigate the time periodicelectro osmotic flow in a rectangular microchannel . An analytical solution to the velocityfield is presented for a linearized Poisson-Boltzmann double layer model and an appliedsinusoidal electric field. As explainedpreviously, when a liquid comes into contactwith a solid, the formation of an interfacialcharge causes a rearrangement of the localfree ions in the liquid and produces a thinregion of non-zero net charge density near theinterface, referred to as the Electrical DoubleLayer (EDL). The application of an externalelectric field then results in a net body force onthe free ions within the EDL inducing a bulkfluid motion called electro osmotic flow. Forelectro osmotic flows of incompressibleliquids, the Navier-Stokes equations may takethe following form,

FPDtDV

...(12)

where is the density, V denotes the velocityvector, p is the pressure, represents thestress tensor, and F is the body force.

Assuming that the body force acts in the xdirection, this term is replaced by eEx, wheree represents the local net charge density, andEx is the external electric field intensity appliedalong the axis direction. Considering steady-state and hydrodynamically fully developedflow, the shear stress for power-law fluidbecomes

dydV

...(13)

Applying the above values, the momentumequation becomes,

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Int. J. Mech. Eng. & Rob. Res. 2014 Vishnu S B and Ajith C Menon, 2014

tie eE

yV

xP

tv 02

2

...(14)

tiedydE

yV

xP

tv

2

22

20

...(15)

Applying V = V0 Sint and tieEE 0

tiedydE

xP

ttV

ytV

2

sin02sin02

0

...(16)

xPt

yVt

cos202sin

0

coshcosh2

0 tiw e

KHKyKE

By solving the momentum equation, thevelocity V0 can be obtained.

The above equation can be solvedanalytically and velocity can be obtained as

ttk

V sincot

120

thk

cotcos2

...(17)

Discharge(Q)

cotcot

cotsinsin

cotsec2

2

2

k

hk

...(18)

RESULTS AND DISCUSSIONAC ElectroosmosisVariation of velocity with time is plotted withconstant frequencies.

It is found that upto a time of 0.4 seconds,the magnitude of velocity is negative. It meansthat upto a time of 0.4 seconds, there will be

Figure 1: Variation of DimensionlessVelocity at Constant Electric Field

Frequency

Figure 2: Variation of Normalized EOFVelocity of Newtonian Fluid in the

Rectangular Micro Channel (Electric FieldFrequency = 10)

Figure 3: Variation of Normalized EOFVelocity of Newtonian Fluid in the

Rectangular Micro Channel (Electric FieldFrequency = 15)

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Int. J. Mech. Eng. & Rob. Res. 2014 Vishnu S B and Ajith C Menon, 2014

reverse flow and after that positive flow occurs.Another peculiarities of the flow is, it isoscillating in nature.

Variation of Velocity with DifferentFrequencyVariation of velocity with different frequency isplotted against channel height.

From figures, it is clear that at very lowelectric field frequency, the flow is reverse innature. As electric field frequency increasesgradually, slightly forward flow occurs. Furtherincrease in frequency result in sinusoidal typeflow. Nearer to the channel wall surface reverseflow occurs and at centre line, positive flowoccurs. It is observed that the maximumpositive flow occurs when electric fieldfrequency is 20 Hz.

Variation of Discharge withFrequencyFrom Figure 8 it is found that at electric fieldfrequency of 20 Hz, maximum flow velocityoccurs and hence discharge. If the frequencyis further increased, the net dischargedecreases. It is due to sinusoidal nature ofvelocity profile.

Figure 4: Variation of Normalized EOFVelocity of Newtonian Fluid in the

Rectangular Micro Channel (Electric FieldFrequency = 20)

Figure 5: Variation of Normalized EOFVelocity of Newtonian Fluid in the

Rectangular Micro Channel (Electric FieldFrequency = 30)

Figure 6: Variation of Normalized EOFVelocity of Newtonian Fluid in the

Rectangular Micro Channel (Electric FieldFrequency = 50)

Figure 7: Variation of Normalized EOFVelocity of Newtonian Fluid in the

Rectangular Micro Channel (Electric FieldFrequency = 75)

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Int. J. Mech. Eng. & Rob. Res. 2014 Vishnu S B and Ajith C Menon, 2014

CONCLUSIONThe flow behaviour of the electro osmotic flowof the Newtonian fluid in a slit micro channel isstudied. An exact solution of the velocitydistribution is presented under the Debye-Huckel approximation. This solution isobtained by solving the linear zed Poisson-Boltzmann equation and the modified Cauchyequation. The approximate solutions of thevelocity distribution are also obtained by theapproximation of the hyperbolic sine functions.For AC electro osmosis, electricfield frequencyis the major parameter which determines thenature of the flow. At low frequencies, there isonly reverse flow. But at about 20 Hz, maximumpositive flows occur and hence discharge. It ismost suited for mixing applications.

REFERENCES1. Afonso A M, Alves M A and Pinho F T

(2009), “Analytical Solution of MixedElectroosmotic/ Pressure Driven Flows ofViscoelastic Fluids in Microchannels”,Journal of Non-Newtonian FluidMechanics, Vol. 159, pp. 50-63.

2. Akgül M B and Pakdemirli M (2008),“Analytical and Numerical Solutions of

Figure 8: Variation of NormalizedDischarge of Newtonian Fluid in the

Rectangular Micro Channel with Variationin Freequency

Figure 9: Variation of NormalizedDischarge of Newtonian Fluid in the

Rectangular Micro Channel with Variationin K Value

Variation of Discharge with K ValueFor AC electro osmosis, it is clear that thevariation of k value doesn’t affect the dischargesignificantly, i.e., electrical double layerthickness doesn’t have much impact on flowvelocity.

Figure 10: Variation of NormalizedDischarge of Newtonian Fluid in the

Rectangular Micro Channel with Variationin Electricfield

Variation of Discharge with ElectricFieldFrom Figure 10, it is clear that dischargeincreases proportionally to electric fieldstrength. The increase in electric field strengthwill result in greater body force, which will resultin increase in discharge.

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7. Jie Su and Yongjun Jian (2012), “ThermallyFully Developed ElectroosmoticflowThrough a Rectangular Micro Channel”,International Journal of Heat and MassTransfer, Vol. 55, pp. 6285-6290.

8. Meisel I and Ehrhard P (2006), “Electrically-Excited (Electroosmotic) Flows inMicrochannels for Mixing Applications”,European Journal of Mechanics – B/Fluids, Vol. 25, pp. 491-504.

9. Quan-Sheng Liu and Yong-Jun Jian(2011), “Time Periodic ElectroosmoticFlow of the Generalized Maxwell FluidsBetween Two Micro-Parallel Plates”, J.Non Newtonian Fluid Mech., Vol. 166, pp.478-486.

10. Talapatra S and Chakraborty S (2008),“Double Layer Overlap in ACElectroosmosis”, European Journal ofMechanics – B/Flu ids , Vol . 27,pp. 297-308.

11. Xiao-Xia Li and Ze Yin (2012), “TransientElectro-Osmotic Flow of GeneralizedMaxwell Fluids Through a Micro Channel”,Journal of Non Newtonian FluidMechanics, Vol. 187-188, pp. 43-47.

Electroosmotically Driven Flow of a ThirdGrade Fluid Between Micro-ParallelPlates”, International Journal of Non-Linear Mechanics, Vol. 43, pp. 985-992.

3. Bharti R P, Harvie D J E and Davidson MR (2009), “Electroviscous Effects inSteady Fully Developed Flow of a Power-Law Liquid Through a CylindricalMicrochannel”, International Journal ofHeat Fluid Flow, Vol. 30, pp. 804-811.

4. Cruz D O A and Pinho F T (2007), “FullyDeveloped Pipe and Planer Flows ofMultimode Viscoelastic Fluids”, Journalof Non-Newtonian Fluid Mechanics,Vol. 141, pp. 85-98.

5. Cruz D O A, Pinho F T and Oliveira P J(2005), “Analytical Solutions for FullyDeveloped Lamina Flow of SomeViscoelastic Liquids with a NewtonianSolvent Contribution”, J. Non-NewtonianFluid Mech., Vol. 132, pp. 28-35.

6. Hayat T, Siddiqui A M and Asghar A(2001), “Some Simple Flows of anOldroyd-B Fluid”, International Journalof Engineering Science , Vol. 39,pp. 135-147.

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Eo External electric field (N/C)

Fx Driving force due to the interaction of the applied electric Dield (N/m)

H Half distance between plates (m)

m Flow consistency index

n Viscosity index

u Velocity (m/s)

ni∞ Bulk ion concentration

e Charge of a proton

Zi Valency

Kb Boltzmannconstant

Greek Symbols

Electric field frequency (Hz)

Oscillating velocity frequency (Hz)

Dynamic viscosity (Pa s)

Density (kg m-3)

e Net charge density (C/m)

w Zeta potential (mV)

-1 Debye length (m)

Dielectric constant

Rate of strain tensor

T Shear stress (N/m2)

Subscripts

ws Wall

APPENDIX

Nomenclature