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.