Poiseuille Flow of Two Immiscible Fluids Between Flat Plates with Applications to Microfluidics

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1 PoiseuilleFlowofTwoImmiscibleFluidsBetweenFlatPlateswithApplicationsto Microfluidics BruceA.Finlayson ProfessorEmeritusofChemicalEngineering UniversityofWashington Seattle,WA Abstract An analytic solution is derived for fully developed flow of two immiscible fluids betweentwoflatplateswhenthefluidshavearbitraryviscosityratiosandarbitrary flowrateratios.thisgeneralizesthesolutioninbird,etal. 1 whenthetwofluidstake up exactly the same space (i.e. the thickness of each fluid is one half the total thickness).thissolutionisuseful,forexample,incertainmicrofluidicdeviceswhen twofluidsflowinlaminarcontact;knowingthepositionofthedividingstreamlineis critically important to accurate modeling of diffusion across the interface. Since diffusion is often slow in comparison to the channel transit time, an approximate expression is provided to identify the limits of diffusion about the dividing streamline. Thus, it is possible to obtain a priori a good estimate of the concentrationbandasthefluidsmovedownthedevice. 1.AnalyticalSolution Fully developed flow of two immiscible fluids between two flat plates is solved in Bird,etal. 1 whenthetwofluidstakeupexactlythesamespace(i.e.thethicknessof eachfluidisone halfthetotalthickness).thatsolutionisgeneralizedheretoallow any fraction, which ultimately will depend upon the flow rate ratio. The notation follows that of Bird, et al. 1 and the domain is illustrated in Figure 1. Let the total distancebetweenthetwoflatplatesbeh,andthefractionfilledwiththelowerfluid (I)bef.Thedomainstretchesfromx= fhtox=(1 f)h,i.e.theinterfaceisatx=0. Theviscositiesofthefluidsareµ1andµ2,andtheflowratesareQ1andQ2. Figure1.Flowoftwoimmisciblefluidsbetweenparallelplates 1

2 Thestressprofilesare τ 1 = Δp L x + C 1, τ 2 = Δp L x + C 1 The constant C1 is the same in both fluids since the stresses are the same at the interface,x=0.fornewtonianfluidswehave τ = µ dv dx, or µ dv 1 1 dx = Δp L x + C, and µ dv dx = Δp L x + C 1 Theseequationscanbeintegratedtogive v 1 = Δp 2µ 1 L x 2 C 1 µ 1 x + C 2, v 2 = Δp 2µ 2 L x 2 C 1 µ 2 x + C 2 Theboundaryconditionsrequire v 1 ( fh) = 0, v 2 [(1 f )H] = 0 ApplyingtheseconditionstothevelocitiesandsolvingforC1andC2gives Theflowratesare C 1 = ΔpH 2L µ 2 f 2 µ 1 (1 f ) 2, C 2 = ΔpH 2 fµ 2 + (1 f )µ 1 2L 0 Q 1 = v 1 dx, Q 2 = v 2 dx fh (1 f )H Q 1 = Δp 6µ 1 L ( fh)3 + C 1 2µ 1 ( fh) 2 + C 2 ( fh) Q 2 = ΔpH 3 6µ 2 L (1 f )3 C 1H 2 0 f (1 f ) fµ 2 + (1 f )µ 1 2µ 2 (1 f ) 2 + C 2 H(1 f ) Onecanspecifythefraction,f,andcalculatetheratioofflowratesdirectly.Whenf= 0.5itgivesthesolutioninBird,etal.Alternatively,theflowrateratio,R=Q1/Q2,can be specified, and the value of f is determined to meet this ratio. When the two viscosities are equal, this should give the equation for a fully developed flow betweentwoparallelplates,whichitdoes.amatlabprogramsisusedtosolvefor fwhengivenr=q1/q2. While this solution is for immiscible fluids, it may be a good approximation for misciblefluidsinamicrofluidicchannelwhenthereynoldsnumberissmall,which itusuallyis,sincethennorecirculationorinstabilitieswillbepresent.onerelevant situationisthelaminardiffusioninterface,whichisestablishedwhentwofluidsof different chemical compositions are brought into laminar contact. Since they have 2

3 differentconcentrations,diffusiontakesplaceperpendiculartotheinterfaceasthe fluidsmovedownthechannel.whilethecompletesolutiontotheprobleminvolves solving the convective diffusion equation, and including the possible effect of concentration on viscosity, there are situations in which an approximation is feasible.ifonefluidcontainsasubstancethatincreasesitsviscosityrelativetothe other fluid, but does not diffuse appreciably, then the viscosities of both fluids remain essentially constant as they move down the channel. Other chemicals, though,maydiffusionmorerapidly.forexample, 2 ifonefluidisserumandtheother is water, one fluid (serum) contains proteins (albumin representing the most concentrated among them), that diffuses slowly, and metabolites(creatinine being oneofmany),whichdiffusesmorerapidly.theproteinsaffecttheviscosity,butthe creatinine hardly does. In order to make use of such a channel to separate metabolites from proteins, one would like to know how far the albumin diffuses sidewaysastheflowproceedsdownthechannel,giventheflowratesforserumand water. This modeling capability would then permit identification of flow rates for which the ratio of metabolite to protein is particularly high. This is still an approximation, since there is some axial diffusion, but it is a good approximation andcangiveguidanceinthedesignandoperationofdevices. z Figure2.Diffusionproblemarounddividingstreamline Thus,considerthefollowingproblemfordiffusionaboutthedividingstreamlineatx =f, as sketched in Figure 2. Now the coordinate x is taken from 0 to H and the dividingstreamlineisatx=fh. v( f ) c z = D 2 c x, c 2 x (±,z) = 0 c(x,0) =1 for x f H,= 0 for x > f H The velocity is the velocity at the dividing streamline; this assumption can be removedbyusinganumericalsolution.withthisassumption,thesolutioncentered onx=fhis 3

4 c = 0.5erfc(η), η x f H 4 Dz v( f ) The thickness of the diffusion layer is taken as the distance over which the concentration falls from 0.5 to 0.01 (2% criterion). This happens when the complementaryerrorfunctionis0.01/0.5=0.02,oratanargumentofη= δ = 10.8 Dz v( f ) Thissolutionobviouslycannotbeusedoncethediffusionthicknessreachesoneof thewalls.thisisthethicknessthatthefluiddiffusesfromonefluidtoanotherina distancezdownstream.anapproximatesolutiontothisproblemis 3 c = 0.5(1 η 2 ),η = x f H,δ = 12 Dz δ v( f ) Eitherofthesefunctionscanbeusedtoestimatethediffusionthicknessaboveand below the dividing streamline under the stated assumptions. Other numerical experiencehasshownthattheseassumptionsarequitegoodinrealisticcases.inthe caseofalbuminandcreatinine,inalengthof2cmwithavelocityof2mm/sec,the creatinine diffusion length (D = m 2 /s) was 315 µm, but the albumin diffusion length (D = m 2 /s) was 85 µm. The microfluidic device thus providedonestreamthathadverylittlealbumininitprovideditwasobtainedata distancefh+δabovethebottom. 2.NumericalResults Figure 3 shows the dimensionless velocity profiles when the flow rate ratios vary from0.2to5andtheviscosityratiosis5.thecoordinatexisdimensionless,scaled by the height, H, with the dividing streamline at x = 0. The velocity is scaled such thatδph 2 /2L=1.Asexpected,whentheviscositiesaredifferent,theslopesofthe velocity profiles are different at the dividing streamline. Of more interest is the difference between the flow rate ratios and the physical location of the actual streamline. Figure 4 shows the difference between the f predicted by the analysis and the value of f that would be predicted based upon the ratio of flow rates, Q1/(Q1+Q2). As can be seen, there is considerable difference in those values when theflowrateratioissmall,regardlessoftheviscosityratio,upto100%differencein some cases. Thus, the ratio of flow rates does not give a good estimate of the locationofthedividingstreamline. Consider next a numerical solution of the diffusion problem given above with v(f) replacedbytheactualvelocity,scaledsothatthestreamlinevelocityis2mm/sec. The finite difference method 4 is used to solve the convective diffusion equation whentheheightis380µm,thediffusivitiesareasgiven,theviscosityratiois5and 4

5 the flow ratio is 1.0. In this case the f = Figure 5 shows the numerical solutionattheexitusingxscaledbyhandwiththedividingstreamlineatx=f.the velocityprofilehasadramaticeffectontherapidlydiffusingspecies(creatinine)but almost no effect on the albumin because it diffuses so slowly. If one takes the diffusionlayerasthedistancebetweenthestreamlinevalueandthepointwherec= 0.01or0.99,thediffusionlayerforalbuminis85µmononesideand95µmonthe other side. These numbers are comparable to the estimates given above but show theeffectofthevelocityprofileoneachsideofthedividingstreamline. Simulations for two dimensional and three dimensional configurations, including entryandexitgeometries,areavailableelsewhere. 5 Figure3.Velocityprofilefortwoimmisciblefluidsflowwhenµ 1/µ 2=5;all curvesareforthesamepressuredropperlength 5

6 Figure4.Fractionoftheheightshowinglocationofdividingstreamline;solid lineistheratioofflowrates,q1/(q1+q2) 6

7 Figure5.Numericalsolutionattheexit;viscosityratio=5,flowrateratio=1; (a,b)creatininewithavelocityprofile(a)andconstantvelocity(b); (c,d)albuminwithavelocityprofileandconstantvelocity 3. Conclusions Ananalyticalsolutionisderivedfortheflowoftwoimmisciblefluidsbetweentwo flat plates. It is valid for any ratio of viscosities and any ratio of flow rates, thus generalizingtheresultsofbird,etal. 1 Whencombinedwithananalyticalsolutionof diffusion in the transverse direction, it can be used to estimate the thickness of diffusionlayerswhenthetwofluidshavedifferentchemicalcompositions. 4. References 1. Bird, R. B., W. E. Stewart, E. N. Lightfoot, Transport Phenomena, 2 nd ed., Wiley (2002),pp Shaw,A.,B.Schattka,M.Alexander,S.Low Ying,A.Man,Metabolicfingerprinting ofbiofluidsbyinfraredspectroscopy:modelingandoptimizationofflowratesfor laminarfluiddiffusioninterfacesamplepreconditioning,submitted. 7

8 3. Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, AcademicPress(1972),p Finlayson,B.A.,NonlinearAnalysisinChemicalEngineering,McGraw Hill(1980). 5. Finlayson, B. A., R. A. Shaw, Modeling Microfluidic Separations using Comsol Multiphysics,ComsolConference2010,Boston. 5. Acknowledgment The author would like to thank R. Anthony Shaw, National Research Council of Canada, Institute for Biodiagnostics, Winnipeg, Manitoba, Canada, for introducing himtothisproblem. 8

Mixing of Liquids in Microfluidic Devices

Presented at the COMSOL Conference 2008 Boston Mixing of Liquids in Microfluidic Devices Bruce A. Finlayson Professor Emeritus of Chemical Engineering University of Washington and Andy Aditya, Vann Brasher,