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


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.


MEMS Microfluidic device Gas microflow Slip coefficient Slip model 



Augmented Burnett


Molecular dynamics


Boltzmann equation


Microelectromechanical systems




Mean free path


Hard sphere




Linearized Boltzmann equation




Lattice Boltzmann method


Quasi-gas dynamic


Bhatnagar Gross Krook


Tangential momentum accommodation coefficient


Direct Simulation Monte Carlo


Variable hard sphere


Information preservation


Variable soft sphere


Knudsen layer



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



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

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


Non-dimensional flow rate


Correction factor


Volumetric flow rate


Variable parameter


Traveling distance

Cp, rq


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

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




Velocity defect function




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



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

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


Tangential momentum accommodation coefficient


Adjustable coefficient

ωm, ωG



Fraction parameter


Interaction parameter


Applied parameter


Angle velocity


Controversial coefficient

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

Variable coefficient


Interaction parameter


Gas viscosity (12)


Difference constant


First-order approximation

θP, \( \beta_{\text{P}} \)

Random variables

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

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


Mean molecular spacing



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

Variable parameter

\( \upsilon \)

Collision frequency


Gas density


Relaxation time


Mean free path


Shear stress

λ1, λ2

MFP of the binary gas mixtures


Wall shear stress

λs, λb

MFP from molecular and boundary scatterings

\( \vec{\tau } \)

Tangential shear stress


Ratio of specific heats

\( \Upphi_{0} \)

Quantity (gas density, pressure or temperature)


Molecular acceleration

\( \Upphi \)

Function of Knudsen number


Standard deviation

\( \Uptheta \)

Probability density

σ22, σ25, σ55

TMAC Coefficients


Probability distribution function

σL0, σL1, σL2

Variable parameters

\( \Uppi \)

Pressure ratio


Energy accommodation coefficient


Pressure drop


Slip coefficient


Velocity drop


Thermal accommodation coefficient





Maxwell model




Sharipov model


Loyalka model



DSMC cell




Effective relations


Reflected gas molecule


Incident gas molecules


Reference conditions

in, fin

Initial and final values


Slip boundary condition


Information preservation


Smooth surface


The order of the polynomial


Solid wall


Lower plate


Upper plate


Local value



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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