Blasius Boundary Layer Solution With Slip Flow Conditions

Michael J. Martin and lain D. Boyd

Department of Aerospace Engineering University of Michigan Ann Arbor, MI 48109-2140

Abstract. As the number of applications of micro electro mechanical systems, or MEMS, increase, the variety of flow geometries that mus analyzee b t micro-scale th t da alss ei o increasing dateo T . ,wor e mosth n kf o o t MEMS scale fluid mechanic focuses sha internan do l flow geometries, suc microchannelss ha applications A . s suc s micro-scalha e flyer e consideredar s s becomini t i , g necessar consideo yt r external flow geometries. Addin gslip-floa w conditio Blasiue th o nt s boundary layer allows these flowstudiee b o st d without extensive computation.

BOUNDARY LAYER WITH SLIP

The Blasius boundary layer solution for flow over a flat plate is among the best know solutions in fluid mechanics boundare [1]Th . y layer equations assum followinge eth ) steady(1 : , incompressible flow) (2 , lamina significanro flown ) (3 , t gradient pressurf so x-directione th velocit) n ei (4 d an ,y gradient- x e th n si directio e smalar n l compare o velocitt d y gradient e y-directionth n i s . Onl e lasth yt assumptios i n questionabl MEMr efo S scale flows.

No-Slip Boundary Layer Equations

The simplified Navier-Stokes Equations base thesn do e assumptions, knowboundare th s na y layer equations, are given as:

du dv ) (1 0 = — + — dx dy

du du d 2u ) (2 - — v = + u— v— dx dy dy where u and v are the x and y components of the velocity, and \) is the kinematic viscosity of the fluid.

Blasiue Inth s solution non-dimensionaa , l position T| combine positiony d an s botx :e hth

_ y (**)1/2 (x/L)( 1/2

non-dimensionae ar * y d wheran * ex l coordinates free th e s streai 0 u , arbitrarmn a velocitys i L yd lengtan , h scale that cancels itself non-dimensionaoute Th . thee ar ln velocitie * functionv d none an th * sf u -s o dimensional stream function/

CP585, Rarefied Dynamics:s Ga 22nd International Symposium, . GalliA . . Barte sJ M edite . d T lan y db © 2001 American Institute of Physics 0-7354-0025-3/01/$18.00 518 (4)

A governing equation for/can be found by substituting (3) and (4) into the x-momentum equation (2):

1) = 0 (5)

For flo t non-rarefiewa d length scales boundare th , y condition problee th r no-slipe fo so m n ar d an , through flow at the wall, and u = u,, as y approaches infinity. In non-dimensional variables, these become:

M*(y=0)=0=>/'( 0)=n= 0 (6)

(7)

u * (y = l (8)

Slip Boundary Conditions

Whe e flonth w becomes rarefied e no-slith , p e walconditio th s replace i lt a a slip-flo) y n(6 db w conditio isotherman a n r [2]Fo . l wall slie pth , conditio gives ni y nb

(2-Q . )3 u Uwall =——————— — A (9) a dy wall

where X is the mean free path, and a is the tangential momentum accommodation coefficient. This can be non-dimensionalize obtaio dt n

1 /'(O) = Knx Rex = K /"(O) (10) a 1

Knudsee th e Reynoldd ar x nan Re d whernon-dimensionasa an numbers x i I eKn K d san base , x n do l parameter that describes the behavior at the surface:

= KI a (U)

NUMERICAL SOLUTION

These equation solvee sar d usin gshootina g method no-slie , th jus s a tp boundary layer equatione sar solved. There is one unique value of/ " (0) and/' (0) for each value of KI. / " (0) is shown in figure 1:

0.35 0.3 0.25

0.05 0 10 20 30 40 50 K,

Figure l./"(0) versus

519 As the analysis is expanded to large values of KI,/ " (0) will asymptotically approach zero. Figure 2 shows/ ' (0), or the non-dimensional slip velocity, as a function of KI:

1

09

0.8

0.7

0.6

^0.5 3 0.4)

0.3

0.2

0.1

0

Figure 2. u*wan versus KI

As the Knudsen number approaches zero, KI also approaches zero, and the no-slip condition, and the classical boundary layer solution recoverede ar ,Knudsee th s A .n number becomes large approacheI K , s infinity, and the non-dimensional slip velocity approaches 1, indicating 100 percent slip at the wall. The velocity profile within the boundary layer will also change as a function of KI. Because the initial value off" change move w s esa alon platee gth self-similarite th , Blasiue th f yo s solutio losts ni . However, because conservation of mass and momentum are satisfied in the same approximate manner as in the Blasius solution approace th , h remains valid. Figur beloe3 w show normalizee sth d velocity profile th n ei boundary layer for various values of KI:

1

0.9

0.8

0.7

0.6

*30.5

0.4 K'=1.0 0.3 K,= 2.0 K = 3.0 0.2 0 4. K= : 5.0 0.1

0

Figure 3. u* versus T| for various values of KI

One resul seee tb figurthan n i tha s i ca t e3 twale eveth ls nvelocita y changes drastically overale th , l boundary layer thickness does not change as rapidly. The physical thickness of the boundary layer is

-TJ9-n 9 (12)

520 Equation (12) can be substituted into (11) to obtain a KI based on boundary layer thickness:

(2-a)Kn K = & (13) a T]99

r equilibriuFo m flows, constana T| s 9i 9 t wit valuha non-equilibriua r f 4.9eo Fo . m boundary layer, T|99

varies along the plate. Figure 4 shows the value of, r| where u* is equal to .99, as a function of KI: 99

0 10 20 30 40 50 Kl

Figur . T|e4 9 9 versuI sK

The non-equilibrium behavior at the wall, as measured by KI, will be proportional to the boundary layer thickness, which is a measure of the velocity gradient near the wall. However, since r|99 is a function of KI, the original form of KI is more suitable for analyzing real flows. frictioe surface Th th t na e will chang non-equilibriuo t e edu m behavior wale Th l. frictio gives ni : nby

„_ (14) ay Tay -) = M- V = 1/2 (0)

frictioe Th s proportionai n value th eo t l off) give(0 "figurn ni percene eth (1)d tan , reduction ni friction due to non-equilibrium behavior is given by

(%Reduction) = 100%(f'(0)|K =Q -/'(0)j=100%(.3321-/'(0)) (15)

The percent reduction in friction for a plate with a chord of 50 microns, and freestream conditions of a u0 of 100 m/s, a pressure of 0.1 atmospheres, and a temperature of 298 K, is show as figure 5:

nnicransxi i Figure 5. Percent Reduction in Friction

These results show tha reductioe tth friction i n wil vere lb y initiae largth n ei l portio platee th d f nan ,o measuree b y stillma d downstrea platee th mn .o

521 There are two limitations in using this simple model to study flow around a MEMS scale flat plate. The first restriction is the singularity that appears at x=0. Typically this problem is solved by combining the Blasius solution wit hStokea s flow solutio leadine th t na gplatee edgth f .eo Becaus Stokee eth s flow region scales with plate thickness t becomei , s less significan boundarn i t y layer growte platth es a h thickness decreases to the order of one micron. Figure 6 shows the contours of x-velocity for freestream conditions of a velocity of 100 m/s, a pressure of 0.1 atmospheres, and a temperature of 298 K.

201-

10 15 c -99 5 10

I 5

0 5 5 4 0 4 5 3 0 3 5 2 0 2 5 1 0 1 5 0 x (microns)

Figur . X-Velocite6 atm1 m/s0. 0 ,y = T=29 10 P ,contour = 0 8u K r sfo

The next concer Blasiue th n ni s modeimportance th s i lvelocite th f eo y gradient x-directione th n si . Equation (2) can only be used to describe the flow field when 32u/3y2 is much larger than 32u/3x2. Figure 7 show ratie sth thesf oo e derivatives, suggestin gextreme tha solutioth e t tth bu el valileadins nal i r dfo g edge platee ofth .

10 20 9 8 15 7 3 5 10 4 3 5 2 1 0.6 0 0.4 0 5 5 4 0 4 5 3 0 3 5 2 0 2 5 1 10 0.2 x (microns) 0

2 2 2 2 Figure 7. Ratio of 3 u/3y versus 3 u/3x for u0 = 100 m/s, P= 0.1 atm, T=298 K

CONCLUSIONS

The results show tha boundare tth y layer equation usee b studo dn t sca y MEMe floth t wa S o scalet d an , judge when non-equilibrium effects become important. Whil self-similarite eth Blasiue th f yo s boundary layer is lost, the boundary layer equations continue to provide useful information to study the effects of rarefactio sheae th n r no stres structurd sflowan e th f .e o They also sho weaknese wth usinf so gsimpla e geometric Knudsen number in describing the flow, and provide a new flow parameter, KI, for describing non-equilibrium behavior.

A Navier-Stokes flow solver [3], incorporating the effects of slip conditions at the boundary, is being used to study the accuracy of these solutions for flow over thin flat plates.

522 These results are being used to evaluate test conditions for an experimental study of MEMS scale airfoils resulte Th .thif so s model additionad an , l computational studies, suggest thareductioe th t dran i g due to these effects should be measurable for flat plates with chords of 10-40(im, at pressures ranging from 0.1 to 1.0 atmospheres.

ACKNOWLEDGMENTS

The authors gratefully acknowledge support for this work from the Air Force Office of Scientific Research through MURI grant F49620-98-1-0433.

REFERENCES

1. Panton, Ronal , IncompressibldL. e Fluid Flow, John Wile Sons d Yorkw yan ,Ne , 1996 . 581-591,pp . 2. Gad-el-Hak, Mohamed, Journal of Fluids Engineering 121, 5-33 (1999). 3. Fan, J., Boyd, I.D., Cai, C.P., Hennighausen, K. and Candler, G.V., "Computation of Rarefied gas flows around a NACA 0012 Airfoil," AIAA, 99-3804, 1999.

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