Micro-Scale Gas Transport Modeling Continuum & Slip Flow Regimes: Navier-Stokes Equations Slip Boundary Conditions U g U w = ( σ ) σ U Kn n Slip, Transitional & Free Molecular: Direct Simulation Monte Carlo - DSMC f = f + akn + bkn + Chapman-Enskog expansion 0 (1...)
C.-M. Ho & Y.-C. Tai
Compressible Navier-Stokes Equations ρu1 ρu ρ ρu1 + p+ σ11 ρuu 1 + σ1 ρu1 + + t x 1 ρuu 1 σ 1 x ρu p σ ρu + + + ( E p σ11) u1 σ1u q 1 ( E p σ) u σ1u1 q + + + + + + + + Valid for continuum and rarefied Newtonian fluid Thermal stresses (derived from Boltzmann) not included T x x 1 T ij 3 x δ i j k Also the following term is not included in the energy equation: u x i j
Compressible Microflows: First-Order Models Stress Tensor: u u i j um um σ ij = µ + + µ δij ς δ ij xj x i 3 xm xm Boundary Conditions: σ v s 3 γ 1 Re U Kn T Us Uw = Kn + σ n π γ Ec s T s T = Non-dimensional Numbers: w v σ T γ Kn T σ γ + 1Pr n T 0 Re = ρ uh u T M Ec ( γ 1) M Kn λ πγ µ = C T = T = h = Re p M u = Mach number γ RT Reynolds Eckert Knudsen 0 Independent parameters: Re, Pr and Kn
Accommodation Coefficients (Breuer et al.) Argon Nitrogen σ v =0 corresponds to specular reflection σ v =1 corresponds to diffuse reflection σ v τ i τ r = τ τ i w References: Seidl & Steible (1974); Lord (1976)
Compressible Microflows: High-Order Models Stress Tensor: u µ u u D u u u T σ ij = µ + ω + ω + ω x p x x Dt x x x x x i k i i i k 1 3R j k j j k j i j Burnett Stress + + + u u µ 1 p T R T T i k ω 4 ω 5 ω 6 p ρt xi xj T xi xj xk xj Where bar defines a non-divergent symmetric tensor: f = ( f + f )/ δ f /3 ij ij ji ij mm Boundary Conditions: The Burnett equations are a second-order Chapman- Enskog expansion for Kn and they require second-order slip conditions. (sections.3, 4.4 and 5.1 in K & B (00).
Why Slip? Ο (δ) Boundary Layer O(λ) Knudsen Layer Uslip
Slip Boundary Conditions U g U w = ( σ ) U 3( γ 1) Kn + Kn σ n π Re T s Maxwell 1879 U 1 [ U + (1 σ U σ ] g = λ ) λ + U w Uλ λ S U g ( σ ) U Kn U U w = Kn + σ n n + Uslip=Ugas on S U g U w = ( σ ) σ Kn b Kn 1 U n b = U U '' o ' o s
Slip Flow Regime (Kn<0.1) Compressibility Rarefaction Viscous Heating Thermal Creep
Compressibility versus Rarefaction Increase in the mass flowrate compared to the No-Slip Flow M M slip no slip σ Knout = 1+ 1, σ Π + 1 P Π = P in out Decrease in the pressure gradient compared to the No-Slip flow
Viscous Heating T = 1 K T n 10 6 ( K / m)
Thermally Induced Flows: Thermal Creep U c 3µ R = 4 P T s Uc
Comparisons of Molecular versus Continuum Solutions (Velocity Contours)
Comparisons of Molecular versus Continuum Solutions Temperature Contours
Transitional & Freemolecular Flow Regimes (Kn>0.1) Analysis of the Burnett Equations for Isothermal Flow (ε=h/l<<1) πγ Px 1 Kn 3 πγ Py 1 Kn 3 out out M M out out Pout U y = P P out U y = P L 4 U yy + O( ε) πγ / Kn 3 out M out P P out h U y U yy + O( ε ) If Kn out 1 & M << 1 P P x y = = U yy 4 πγ / 3 Kn out M out out P P out U y U yy Parabolic Velocity Profile?
Transition and Free Molecular Flow Regimes and Knudsen s Minimum In 1909, Knudsen discovered that there is a minimum, when Q P i P o is plotted against the average pressure! Knudsen s Minimum
Transitional & Freemolecular Flow Regimes via Direct Simulation Monte Carlo (DSMC) U ( Y, Kn) = y h y Kn + + h 1 b Kn 1 Kn + 6 1 b Kn
Universal Velocity Scaling Centerline Velocity Slip Velocity U ( Y, Kn) = y h y Kn + + h 1 b Kn 1 Kn + 6 1 b Kn Maxwell s New Model
Modeling Flowrate dp = G, µ, h, λ dx Q Volumetric Flowrate Using Navier-Stokes w/ Slip Q W 3 h = 1µ dp dx 6Kn 1+ 1 bkn Correct for Rarefaction Q W 3 h = 1µ dp dx 6Kn 1+ C 1 bkn r ( Kn) Rarefaction Coefficient
Rarefaction Coefficient C r ( Kn) α 0 α α o = 1+ α Kn as Kn 0 as Kn α = α o tan π α & β 1 1 ( α Kn 1 are empirical parameters β )
Flowrate Scaling in Arbitrary Aspect-Ratio Rectangular Ducts (Based on the Freemolecular Limit) W h AR=W/h M M M FM FM = = C( AR) 6Kn (1 + αkn) 1 + 6 Kn 1 b Kn h W RT o P L Knudsen s Minimum
Flowrate Scaling in Arbitrary Aspect-Ratio Rectangular Ducts (Based on the Continuum Limit) First-Order Theory M M = C( AR)(1 + α Kn) 1 + FM 1 6Kn b Kn
Channel Flow, Nonlinear Pressure Distribution P/Po PCOMP PIC X/L 0 1
A Unified Model for Plane Couette Flows U Knudsen layer U U y d Uc( y) = α = a+ btanh( ckn ) 1+ α Kn D gθk+ 1 βku π P1 = { } k = Kn θk+ 1 1+ γk Macromodel Validation against: DSMC (argon, hard spheres) and linearied Boltzmann solutions P. Bahukudumbi, TAMU, MSME, August 00
A Unified Model for Plane Couette Flows 1. 1.1 1 0.9 0.8 Our model As ymptotic s olution Lin Boltzmann - Sone As ymptote -P 1 /u 0.7 0.6 0.5 10-10 -1 10 0 10 1 10 Analytical models of velocity profile & shear stress for 0 < Kn < 0.4 0.3 0. 0.1 k P. Bahukudumbi, TAMU, MSME, August 00
Gas Damping/Lubrication: Reynolds Equation General equation: 3 ρh ( ρh) i p = 1 + 6 ( ρhu ) µ t H 1 U L H 0 Inertial-free flow if: H0 Re << 1 L Then, leading-order solution: dp dx u = µ x where p=p(x) Constant flowrate: Slip-Flow: 3 dp 6µ UL HP =Λ ( PH) Λ= X dx X p H σ dp + Kn H P =Λ X σ v dx X 0 0 Bearing number PH v 3 [1 6 ] ( )
Slider Bearing Pressure Distribution 1.3 1.5 Fukui and Kane ko Our solution H 1 U H 0 P 1. 1.15 Λ = L 6µ UL p o H o 1.1 1.05 Kn=1.5, Λ=61.6, H 1 /H 0 = 1 0 0.5 0.5 0.75 1 X An Analytical model for generalized Reynolds equation 0 < Kn < P = p p o, X = x L Accurate Predictions of Pressure, Velocity, Shear Stress & Flowrate
Numerical Simulation for Gas Micro-Flows DSMC Method: Slow Convergence: ε 1 n Large Statistical Error: (10 8 samples) Extensive Number of Particles: 3 cells per λ and 0 particles per cell Multi-Domain Simulation: DSMC/Continuum Coupling Navier-Stokes-Slip Model Spectral Elements DSMC Unstructured Mesh Overlapping I. Boyd, AIAA 001-0876 (14:30)
Hash & Hassan (1995) Garcia et al. (1999) Hadjiconstantinou (1999) Liu (1999) Aluru (001) Flow Micro-Macro Interface Hand-Shake Region Oran et al. (1998) Zanolli iterative patching
Coupled Domain Simulations Titling rectangular accelerometer Gap of microns Generalized Reynolds equation with electrostatic actuation. Dynamic response of a micro accelerometer with holes. spring-like Courtesy of T. Veijola
Modeling Roughness in Micro-Geometries Regularized roughness Equivalent effect Random walls
Apparent Diffusion: Roughness
Roughness Effect on Pressure Drop
Slip Compressible Flow σ v =1.0 Re = 0.36; with enhanced viscosity Re = 0.76 Simulation - 0.640 10-5 kg/s, formula - 0.641 10-5 kg/s Simulation - 0.640 10-5 kg/s, formula - 0.643 10-5 kg/s(enhanced) Enhanced viscosity factor: 1.31
Slip Compressible Flow σ v =1.0 (continued) Slip walls make the flow less compressible than noslip walls Slip flow needs more extra viscosity than noslip flow
In-Phase, Out-of-Phase & Hybrid Channels These channels are hydrodynamically equivalent Artificial roughness patterns induce similar apparent diffusion
In-Phase, Out-of-Phase & Hybrid Channels (continued)
Slip Compressible Flow σ v =0.8 and 0.6 (continued) Increased surface smoothness condition enlarges the enhanced viscosity factor to 1.37 lessens the overall pressure drop needs more artificial viscosity makes the flow less compressible
The Effect of Surface Roughness Condition Improved surface roughness condition makes the flow less compressible decreases the overall pressure drop balances more extra artificial viscosity added into the flow
The Effect of Extra Artificial Viscosity Enhanced viscosity factor for σ v = 1.0 was found as 1.069, to match the case σ v = 0.8, enhanced viscosity factor 1.37 Enhanced viscosity increases pressure drop and compressibility Enhanced viscosity competes improvement of surface condition
Slip Velocity at In-Phase Curvilinear Wall Slip velocity fluctuates around artificial roughness Slip velocity increases as the main flow develops
Verifications M = 3 h p p 1 6σ i 1 + Π + 4µ RT L σ i i v v + α Kn i + σ σ v v ( b + α) 1 1 Π Kn i 1 bkn i ln Π bkn i p - pressure drop: p i -p o Π - pressure ratio: p o / p i L - total length of microscaled channels α - rarefaction factor, α=0 for Kn<0.5 h - channel width b slip parameter, b= 1 for fully developed channel flow Case Π µ i (kg/m/s) Kn i σ v M simu (kg/s) M formula (kg/s) Error 1 0.75638 1.7600 10-5 0.07 1.0 6.40 10-6 6.41 10-6 0.% 0.69803.3056 10-5 0.09 1.0 6.40 10-6 6.43 10-6 0.5% 3 0.7914 1.7600 10-5 0.07 0.8 6.40 10-6 6.4 10-6 0.3% 4 0.7447.411 10-5 0.10 0.8 6.40 10-6 6.45 10-6 0.7% 5 0.68743.411 10-5 0.10 1.0 6.41 10-6 6.45 10-6 0.6% 6 0.7473 1.8814 10-5 0.08 1.0 6.40 10-6 6.4 10-6 0.3% 7 0.83100 1.7600 10-5 0.07 0.6 6.40 10-6 6.4 10-6 0.3%
Channel flow with Random Boundary Conditions u = ξ y x f = ν Exact solution (uniform BCs): 1 y 1+ y u(y) = (1 y ) + σ1ξ1 + σξ u = ξ 1 Two-dimensional PC expansion Gaussian inputs : σ1 = %, σ = 1% Solution profile across the channel
Non-Uniform General Roughness Uncertainty -- Mean Solutionat Wall
Non-uniform Gaussian Random BC Exponential correlation C(x 1,x ) = σ Stochastic input: e x x σ = 0.1 1 / b D K-L expansion 4 th -order Hermite-Chaos expansion 15-term expansion U mean along centerline V mean along centerline
Mode 1