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BlasiusBoundary Layer Solution With Slip FlowConditionsMichael J. Martin andlainD. Boyd

Department ofAerospaceEngineeringUniversity of MichiganAn nArbor,M I48109-2140

Abstract As the number of applications of m icro electro mechanical systems, or M EMS , increase, the varietyofflowgeom etries that must be analyzed at the micro-scale is also increasing. To date, most of the work onMEMS scalefluidmechanics hasfocusedon internalflowgeom etries, such as microchannels. As applicationssuch as micro-scale flyers are considered, it is becoming necessary to consider external flow geometries.Adding a slip-flow condition to the Blasius boundary layer allows these flowsto be studied without extensivecomputation.BOUND RYL YERWITHSLIP

TheBlasius boundary layer solutionforflowovera flatplateisamongthebest know solutionsin fluidmechanics [1]. The boundary layer equations assume the following: (1) steady, incompressible flow, (2)laminar flow, (3) no significant gradients of pressure in the x-direction, and (4) velocity gradients in the x-direction are small compared to velocity gradients in the y-direction. Only the last assumption isquestionable fo rMEMS scale flows.

No SlipBoundary Layer EquationsThe simplified Navier-Stokes Equations based on these assumptions, known as the boundary layer

equations, are given as:du dv + = 0 (1)dx dy

du du d 2uu v =v - (2)dx dy dywhere u and v are the x and y components of the velocity, and\)is the kinematic viscosity of thefluid.

In theBlasius solution,anon-dimensional positionT|combines boththe x and yposition:_ y

(**)1/2 (x/L)1/2 (where x* and y* are non-dimensional coordinates, u0is thefreestream velocity, and L is an arbitrary lengthscale that cancels itselfout.The non-dimensional velocities u* and v* are then functions of thenon-dimensional streamfunction/

CP585, Rarefied Gas Dynamics:22ndInternationalSymposium, edited by T. J. Bartel and M. A. G allis2001Am erican Ins titute of Physics 0-7354-0025-3/01/$18.00518

http://dx.doi.org/10.1063/1.1407604

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(4)A governingequationfor/canbefoundb ysu bstituting (3) and (4)intothe x-momentumequation(2):

1)= 0 5)

Fo r flow at non-rarefied length scales, the boundary conditions for the problem are no-slip, and nothroughflowat the w all, and u =u,,as yapproachesinfinity. Innon-dimensional variables,thesebecome:

M*(y=0)=0=>/ (n=0)=0

u * (y =lSlipBoundaryConditions

(6)(7)(8)

When the flow becomes rarefied, the no-slip condition (6) at the wall is replaced by a slip-flowcondition [2]. For an isothermal wall, the slip condition is given by

(2-Q). 3uUwall = A a dy (9)wallwhere X is themean free path,and a is thetangential mom entum accomm odation coefficient. Thiscan benon-dimensionalizedto obtain

/ O) K nx R ex1a =K1/ (O) (10)whereK nxand Rexare theK nudsena ndReynoldsnumbersbasedon x, and K I is anon-dimensionalparameterthatdescribesthebehaviorat thesurface:

KI = a U)

NUMERICAL SOLUTIONThese equations are solved using a shooting method,justas the no-slipboundary layerequations aresolved. Thereis oneunique valueof/ (0)and/'(0)foreach valueofKI . / (0) is showninfigure 1:

0.350.3

0.25

0.050 10 20 30K, 40 50

Figurel./ (0)versus

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Astheanalysisisexpandedtolarge valuesofKI,/ (0)w illasymptoticallyapproachzero.Figure2shows/'(0),or thenon-dimensionalslipvelocity,as afunctionofKI:

1090.80.70.6

^0.50.4)0.30.20.1

0

Figure2 .u *w anversusK IAs the Knudsen number approaches zero, KI also approaches zero, and the no-slip condition, and theclassical boundary layer solution, are recovered. As the Knudsen number becomes large, KI approachesinfinity,and thenon-dimensionalslip velocity app roaches 1,indicating 100 percent slip at the w all.The velocityprofile withinthe boundary layer willalso changeas a function of KI. Becausetheinitialvalueoff changesas wemovealong theplate, the self-similarity of theBlasius solutionislost. However,because conservation of mass and momentum are satisfied in the same approximate manner as in the

Blasius solution, theapproachremains valid. Figure 3 below shows the normalized velocityprofilein theboundarylayerf orvarious valuesof KI:

10.90.80.70.6

30.5

0.40.30.20.1

0

K =1.0K 2.0K = 3.0K : = 4 0

5.0

Figure3. u*versusT|f orvarious valuesof KIOne result thatcan be seen in figure 3 isthateven as thewallvelocity chang es drastically, the o verallboundary layerthicknessdoesnotchangeasrapidly.T hephysical thicknessof theboundary layeris

-TJ99 (12)

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Equation (12) can besu bstituted into(11) toobtaina K Ibasedon boundary layerthickness:K (2-a)Kn&a T ] 99 (13)

Fo r equilibrium flows, T|99 is aconstant with avalue of4.9. For anon-equilibrium boundary layer, T|99variesalong the plate. Figure4 show s the value ofr|99,w here u* is equal to .99, as afunctionof KI:

0 10 20 30 40 50Kl

Figure4.T |99versusK IThe non -equilibrium behavior at the wall, as measu red by K I, w ill be proportional to the boun dary layerthickness,w hich is am easureof the velocity gradient near the w all. How ever, sincer|99is afunctionof KI ,the originalform of K I ismore suitableforanalyzing real flows.Thefrictionat thesurfacewill change due tonon-equilibriumbehavior. The w allfrictionis given by:

T- = M V =ay) ay 1 2_(0) 14

The friction is proportional to the value o f f (0) given in figure (1), and the percent reduction infrictiondue tonon-equilibrium behaviorisgivenby(%Reduction)=100%(f'(0)|K = Q-/'(0)j=100%(.3321-/'(0)) (15)

T hepercent reduction in friction for aplate witha chord of 50microns, an dfreestreamconditionsof au0of 100 m /s, apressureof 0.1atmospheres,and a temperature of 298 K , is show asfigure 5:

xinnicransi

Figure5 . PercentReduction in FrictionThese resultsshow that thereductioninfriction w ill be verylargein theinitial portionof theplate,a ndstill m ay bemeasureddow nstream on theplate.

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There are tw o limitationsinusing thissimple model to studyflow around a MEM S scaleflatplate. Thefirst restriction is the singularity thatappears at x=0. Typically this problem is solved by combining theBlasius solution with a Stokes flow solution at the leading edge of the plate. Because the Stokes flowregion scales with plate thickness, it becomes less significant in boundary layer growth as the platethickness decreases to theorder of onemicron. Figure 6shows thecontours ofx-velocity for freestreamconditions of a velocity of 100m/s,a pressure of 0.1 atmospheres, and a temperature of 298 K.

201-10 15c5 10 99

0 5 10 15 20 25 30 35 40 45 50x microns)Figure6 . X-Velocity contoursfor u0= 100m/s,P= 0.1atm, T=298K

Thenext concern in theBlasiusmodel is theimportance of thevelocity gradients in the x-direction.Equation(2) can only beusedtodescribetheflow field w hen32u/3y2is much largerthan32u/3x2.Figure7shows the ratio ofthese derivatives, sugg estingthat the solution is valid for all but theextremeleading edgeof theplate.

20151050 10 15 20 25 30 35 40 45 50x microns)

9873543210.60.40.20

Figure 7. Ratio of32u/3y2versus32u/3x2for u0= 100 m/s, P= 0.1 atm, T =298 K

CONCLUSIONSThe resultsshow that the boundary layerequationscan beusedto studyflowat the MEM Sscale,and to

judgew hennon-equilibrium effects become important. While the self-similarity of theBlasiusbound rylayer is lost, theboundary layerequations continue toprovide useful information to study the effects ofrarefaction on the shear stress and structure of the flow. They also show the weakness ofusing a simplegeometricK nudsen number in describing the flow, an dprovidea new flow parameter,K I, for describingnon-equilibriumbehavior.

A Navier-Stokesflow solver [3], incorporatingtheeffects ofslip conditionsat theboundary,isbeingusedtostudyth eaccuracyo f these solutionsforflowover thinflatplates.

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These results are being used to evaluate test conditions for an experimental study of MEMS scaleairfoils. Theresultsofthismodel, and additional computational studies, suggestthat the reduction in dragdue totheseeffects shouldb emeasurableforflatplateswithchordso f10-40(im,a tpressuresrangingfrom0.1 to 1.0atmospheres.ACKNOWLEDGMENTS

The authors gratefully acknowledge support fo r this work from the Air Force Office of ScientificResearch throughM URIgrantF49620-98-1-0433.REFERENCES

1. Panton, Ronald L., Incompressible Fluid Flow, John Wiley and Sons, New York, 1996, pp. 581-591.2. Gad-el-Hak, Moham ed, Journal ofFluidsEng ineering 121,5-33 (1999).3. Fan, J.,Boyd, I.D.,Cai, C.P., Hennighausen, K. and Candler, G.V., Com putation of Rarefied gasflowsaround aNACA 0012Airfoil, AIAA, 99-3804, 1999.

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