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A review on slip models for gas microflows

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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.

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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

a D, \( C_{\text{D}} \) :

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

\( C_{1} \), C 2 :

First and second order slip coefficients

Q N :

Non-dimensional flow rate

C F :

Correction factor

Q v :

Volumetric flow rate

C L :

Variable parameter

r :

Traveling distance

C p, r q :

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

T r :

Torque

I :

Velocity defect function

u :

Velocity

k B :

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

k s1–k s4 :

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

α AC1, α AC2 :

Ratio coefficients

σ v :

Tangential momentum accommodation coefficient

α K :

Adjustable coefficient

ωm, ωG :

Constants

α M :

Fraction parameter

ω M :

Interaction parameter

α p :

Applied parameter

ω r :

Angle velocity

α s :

Controversial coefficient

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

Variable coefficient

β M :

Interaction parameter

μ :

Gas viscosity (12)

β T :

Difference constant

μ f :

First-order approximation

θ P, \( \beta_{\text{P}} \) :

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

τ g :

Relaxation time

λ :

Mean free path

τ N :

Shear stress

λ 1, λ 2 :

MFP of the binary gas mixtures

τ w :

Wall shear stress

λ s, λ b :

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

σ 22, σ 25, σ 55 :

TMAC Coefficients

ψ:

Probability distribution function

σ L0, σ L1, σ L2 :

Variable parameters

\( \Uppi \) :

Pressure ratio

σ n :

Energy accommodation coefficient

ΔP :

Pressure drop

σ p :

Slip coefficient

ΔU :

Velocity drop

σ T :

Thermal accommodation coefficient

1st:

First-order

M:

Maxwell model

2nd:

Second-order

S:

Sharipov model

L:

Loyalka model

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

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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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  1. State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

    Wen-Ming Zhang & Guang Meng

  2. Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK

    Xueyong Wei

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Zhang, WM., Meng, G. & Wei, X. A review on slip models for gas microflows. Microfluid Nanofluid 13, 845–882 (2012). https://doi.org/10.1007/s10404-012-1012-9

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