Microfluidics and Nanofluidics

, Volume 3, Issue 4, pp 473–484 | Cite as

Slip flow in non-circular microchannels

Research Paper

Abstract

Microscale fluid dynamics has received intensive interest due to the emergence of Micro-Electro-Mechanical Systems (MEMS) technology. When the mean free path of the gas is comparable to the channel’s characteristic dimension, the continuum assumption is no longer valid and a velocity slip may occur at the duct walls. Non-circular cross sections are common channel shapes that can be produced by microfabrication. The non-circular microchannels have extensive practical applications in MEMS. Slip flow in non-circular microchannels has been examined and a simple model is proposed to predict the friction factor and Reynolds product fRe for slip flow in most non-circular microchannels. Through the selection of a characteristic length scale, the square root of cross-sectional area, the effect of duct shape has been minimized. The developed model has an accuracy of 10% for most common duct shapes. The developed model may be used to predict mass flow rate and pressure distribution of slip flow in non-circular microchannels.

Keywords

Slip flow Microchannels Non-circular Pressure distribution 

List of symbols

A

flow area (m2)

a

major semi-axis of ellipse or rectangle (m)

a

base width of a trapezoidal or double-trapezoidal duct (m)

b

minor semi-axis of ellipse or rectangle (m)

b

height of a trapezoidal or double-trapezoidal duct (m)

c

half focal length of ellipse (m)

c

short side of a trapezoidal or double-trapezoidal duct (m)

Dh

hydraulic diameter = 4A/P

E(e)

complete elliptical integral of the second kind

e

eccentricity \(={{\sqrt{1 - {b^{2}} \mathord{\left/ {\vphantom {{b^{2}} {a^{2}}}} \right. \kern-\nulldelimiterspace} {a^{2}}}}}\)

f

Fanning friction factor \(={\tau /{\left({\tfrac{1}{2}\rho \bar{u}^{2}} \right)}}\)

Kn

Knudsen number \({=\lambda \mathord{\left/{\vphantom {\lambda {\ell}}} \right. \kern-\nulldelimiterspace} {\ell}}\)

Kn*

modified Knudsen number =Kn(2−σ)/σ

L

channel length (m)

L+

dimensionless channel length \({=L \mathord{\left/ {\vphantom {L {D_{h} {Re}}}} \right. \kern-\nulldelimiterspace} {D_{\rm h} {Re}}_{{D_{\rm h}}}}\)

arbitrary length scale

Ma

Mach number = u/V s

\({\dot{m}}\)

mass flow rate (kg/s)

P

perimeter (m)

Po

Poiseuille number, \({={\bar{\tau}{\kern 1pt} {\ell}} \mathord{\left/ {\vphantom {{\bar{\tau}{\kern 1pt} {\ell}} {\mu {\kern 1pt} \bar{u}}}} \right. \kern-\nulldelimiterspace} {\mu {\kern 1pt} \bar{u}}}\)

p

pressure \({N \mathord{\left/{\vphantom {{\rm N} {{\rm m}^{2}}}} \right. \kern-\nulldelimiterspace} {m^{2}}}\)

R

specific gas constant \({J \mathord{\left/ {\vphantom {{\rm J} {{\rm kg}{\kern 1pt} K}}} \right. \kern-\nulldelimiterspace} {kg{\kern 1pt} K}}\)

Re

Reynolds number = \({{\ell}\bar{u}/\nu}\)

r

dimensionless radius ratio = r i /r o

ri

inner radius of a concentric duct (m)

ro

outer radius of a concentric duct (m)

T

temperature (K)

U

velocity scale (m/s)

u

velocity (m/s)

\({\bar{u}}\)

average velocity (m/s)

Vs

speed of sound \({={\sqrt{\gamma RT}}}\)

Xn

function of x/a

x, y

Cartesian coordinates (m)

z

coordinate in flow direction (m)

Greek symbols

α

constants

γ

ratio of specific heats

δn

eigenvalues

ε

aspect ratio =b/a

η, ψ, z

elliptic cylinder coordinates

η0

parameter of elliptic cylinder coordinates

λ

molecular mean free path (m)

μ

dynamic viscosity \({{N{\kern 1pt} s} \mathord{\left/ {\vphantom {{{\rm N}{\kern 1pt} {\rm s}} {{\rm m}^{2}}}} \right. \kern-\nulldelimiterspace} {m^{2}}}\)

ν

kinematic viscosity (m2/s)

σ

tangential momentum accommodation coefficient

τ

wall shear stress \({{N{\kern 1pt}} \mathord{\left/ {\vphantom {{{\rm N}{\kern 1pt}} {{\rm m}^{2}}}} \right. \kern-\nulldelimiterspace} {m^{2}}}\)

Φ

half angle rad

Subscripts

\({{\sqrt{A}}}\)

based upon the square root of flow area

c

continuum

Dh

based upon the hydraulic diameter

i

inlet

based upon the arbitrary length ℓ

o

outlet

Superscripts

E

ellipse

R

rectangle

Notes

Acknowledgments

The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Faculty of Engineering and Applied ScienceMemorial University of NewfoundlandSt. John‘sCanada

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