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Microfluidics and Nanofluidics

, Volume 13, Issue 6, pp 845–882 | Cite as

A review on slip models for gas microflows

  • Wen-Ming ZhangEmail author
  • Guang Meng
  • Xueyong Wei
Review Paper

Abstract

Accurate modeling of gas microflow is crucial for the microfluidic devices in MEMS. Gas microflows through these devices are often in the slip and transition flow regimes, characterized by the Knudsen number of the order of 10−2~100. An increasing number of researchers now dedicate great attention to the developments in the modeling of non-equilibrium boundary conditions in the gas microflows, concentrating on the slip model. In this review, we present various slip models obtained from different theoretical, computational and experimental studies for gas microflows. Correct descriptions of the Knudsen layer effect are of critical importance in modeling and designing of gas microflow systems and in predicting their performances. Theoretical descriptions of the gas-surface interaction and gas-surface molecular interaction models are introduced to describe the boundary conditions. Various methods and techniques for determination of the slip coefficients are reviewed. The review presents the considerable success in the implementation of various slip boundary conditions to extend the Navier–Stokes (N–S) equations into the slip and transition flow regimes. Comparisons of different values and formulations of the first- and second-order slip coefficients and models reveal the discrepancies arising from different definitions in the first-order slip coefficient and various approaches to determine the second-order slip coefficient. In addition, no consensus has been reached on the correct and generalized form of higher-order slip expression. The influences of specific effects, such as effective mean free path of the gas molecules and viscosity, surface roughness, gas composition and tangential momentum accommodation coefficient, on the hybrid slip models for gas microflows are analyzed and discussed. It shows that although the various hybrid slip models are proposed from different viewpoints, they can contribute to N–S equations for capturing the high Knudsen number effects in the slip and transition flow regimes. Future studies are also discussed for improving the understanding of gas microflows and enabling us to exactly predict and actively control gas slip.

Keywords

MEMS Microfluidic device Gas microflow Slip coefficient Slip model 

Abbreviations

AB

Augmented Burnett

MD

Molecular dynamics

BE

Boltzmann equation

MEMS

Microelectromechanical systems

CL

Cercignani–Lampis

MFP

Mean free path

HS

Hard sphere

N–S

Navier–Stokes

LBE

Linearized Boltzmann equation

N–S–F

Navier–Stokes–Fourier

LBM

Lattice Boltzmann method

QGD

Quasi-gas dynamic

BGK

Bhatnagar Gross Krook

TMAC

Tangential momentum accommodation coefficient

DSMC

Direct Simulation Monte Carlo

VHS

Variable hard sphere

IP

Information preservation

VSS

Variable soft sphere

KL

Knudsen layer

W–M

Weierstrass–Mandelbrot

List of symbols

aD, \( C_{\text{D}} \)

Constant with positive values

\( nn \)

Exponent constant

\( a_{{{\text{R}}1}} \), \( a_{{{\text{R}}2}} \)

Various coefficients

\( N_{\text{a}} \)

Total number of gas atoms

\( A_{\text{R}} \), \( D_{\text{R}} \), \( E_{\text{R}} \)

Curve-fitting coefficients

\( N_{\text{K}} \)

Index of the fluid lattices

\( b \)

Channel thickness

p

Pressure

\( b_{\text{BK}} \)

Generalized slip coefficient

\( P_{\text{m}} \)

Average pressure

\( c_{\text{m}} \)

Most probable speed

\( P_{\text{O}} \)

Outlet pressure

\( \bar{c} \)

Thermal speed of the gas

\( P_{\text{r}} \)

Prandtl number

\( C_{0} \)

Molar concentration

\( \vec{q} \)

Heat flux

\( C_{1} \), C2

First and second order slip coefficients

QN

Non-dimensional flow rate

CF

Correction factor

Qv

Volumetric flow rate

CL

Variable parameter

r

Traveling distance

Cp, rq

Constants

\( r_{\text{K}} \)

Fraction of gas particles

\( C_{{\tilde{y}}} \)

Variable parameter

\( R_{1} \), \( R_{2} \)

Inner and outer radius

\( C_{\text{Z}} \)

Variable \( \xi_{\text{s}} /\lambda \)(\( C_{\text{Z}} \in [0,1] \))

\( R_{\text{a}} \)

Average roughness

\( d \)

Mean molecular diameter

\( Re \)

Reynolds number

\( d_{\text{c}} \)

Collision molecular diameter

\( R_{\text{P}} \)

Specific gas constant

\( f \)

Roughness height function

\( S \)

Slip coefficient function

\( f_{\text{B}} \)

Distribution function

\( S_{\text{uy}} \), \( S_{\text{yy}} \)

Relative position and velocity parameters

\( h_{\text{B}} \)

Small perturbation;

\( T \)

Absolute temperature

\( H \)

Film thickness

Tr

Torque

I

Velocity defect function

u

Velocity

kB

Boltzmann constant

\( \tilde{u} \)

Velocity ratio \( \tilde{u} = u/\alpha_{\text{p}} \lambda \)

\( K_{\text{M}} \)

Variable parameter

\( u_{\text{n}} \)

Velocity normal to the wall

\( Kn \)

Knudsen number

\( u_{{{\text{N}}1}} \), \( u_{{{\text{N}}2}} \)

Velocity components

\( Kn_{\text{O}} \)

Knudsen number outlet

\( u_{\text{s}} \)

Slip velocity

ks1ks4

Constants

\( \vec{u}_{\text{s}} \)

Tangential slip velocity

\( k_{\text{u}} \)

Velocity gradient

\( u_{\text{w}} \)

Wall velocity

\( l_{\text{c}} \)

Knudsen layer thickness

\( u_{\lambda } \)

Tangential velocity component

\( L \)

Channel length

\( U_{\text{g}} \)

Gas flow velocity

\( L_{0} \)

Characteristic length

\( U_{\text{s}} \)

Non-dimensional slip velocity

\( L_{\text{c}} \)

Local characteristic length

\( U_{\text{w}} \)

Non-dimensional wall velocity

\( L_{\text{r}} \)

Inner cylinder length

\( v_{0} \)

Mixture velocity

\( L_{\text{s}} \)

Slip length

\( v_{\text{g}} \)

Kinematic viscosity

\( L_{\text{x}} \)

Width of the cell

\( V_{{{\text{g}}1}} \), \( V_{{{\text{g}}2}} \)

Fraction of components

\( m \)

Molecular mass

\( V_{\text{t}} \)

Particle information velocity

\( m_{1} \), \( m_{2} \)

Molecular mass of species

\( w \)

Channel width

\( Ma \)

Mach number

\( x_{t} \)

Coordinate tangential to the wall

\( n \)

Coordinate normal to the wall

\( y \)

Distance normal to the wall

\( n_{01} \), \( n_{02} \)

Equilibrium number densities

\( \tilde{y} \)

Relative variable \( \tilde{y} = y/\lambda \)

\( n_{\text{g}} \)

Number density of the gas

Greek symbols

αAC1, αAC2

Ratio coefficients

σv

Tangential momentum accommodation coefficient

αK

Adjustable coefficient

ωm, ωG

Constants

αM

Fraction parameter

ωM

Interaction parameter

αp

Applied parameter

ωr

Angle velocity

αs

Controversial coefficient

\( \vartheta_{\text{m}} \)

Variable coefficient

βM

Interaction parameter

μ

Gas viscosity (12)

βT

Difference constant

μf

First-order approximation

θP, \( \beta_{\text{P}} \)

Random variables

\( \chi_{\text{M}} \)

Parameter \( \chi_{\text{M}} = n_{02} m_{2} /(n_{01} m_{1} ) \)

δ

Mean molecular spacing

ξs

Distance

\( \delta_{{\tilde{y}}} \)

Variable parameter

\( \upsilon \)

Collision frequency

ρ

Gas density

τg

Relaxation time

λ

Mean free path

τN

Shear stress

λ1, λ2

MFP of the binary gas mixtures

τw

Wall shear stress

λs, λb

MFP from molecular and boundary scatterings

\( \vec{\tau } \)

Tangential shear stress

γ

Ratio of specific heats

\( \Upphi_{0} \)

Quantity (gas density, pressure or temperature)

γT

Molecular acceleration

\( \Upphi \)

Function of Knudsen number

σ

Standard deviation

\( \Uptheta \)

Probability density

σ22, σ25, σ55

TMAC Coefficients

ψ

Probability distribution function

σL0, σL1, σL2

Variable parameters

\( \Uppi \)

Pressure ratio

σn

Energy accommodation coefficient

ΔP

Pressure drop

σp

Slip coefficient

ΔU

Velocity drop

σT

Thermal accommodation coefficient

Superscripts

1st

First-order

M

Maxwell model

2nd

Second-order

S

Sharipov model

L

Loyalka model

Subscripts

c

DSMC cell

NS

Navier–Stokes

eff

Effective relations

r

Reflected gas molecule

i

Incident gas molecules

ref

Reference conditions

in, fin

Initial and final values

s

Slip boundary condition

IP

Information preservation

sm

Smooth surface

j

The order of the polynomial

S

Solid wall

l

Lower plate

u

Upper plate

Loc

Local value

Notes

Acknowledgments

This work was supported by the National Science Foundation of China under Grant No. 11072147 and the Specialized Research Fund for State Key Laboratory of Mechanical System and Vibration under Grant No. MSVMS201106, and sponsored by Shanghai Rising-Star Program under Grant No. 11QA1403400.

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.State Key Laboratory of Mechanical System and Vibration, School of Mechanical EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Department of EngineeringUniversity of CambridgeCambridgeUK

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