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Slip flow in non-circular microchannels

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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When the mean free path of the gas is comparable to the channel’s characteristic dimension, the continuum assumption is no longer valid and a velocity slip may occur at the duct walls.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.

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Abbreviations

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)

D h :

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

ℓ:

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

r i :

inner radius of a concentric duct (m)

r o :

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)

V s :

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

X n :

function of x/a

x, y :

Cartesian coordinates (m)

z :

coordinate in flow direction (m)

α:

constants

γ:

ratio of specific heats

δ n :

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

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

based upon the square root of flow area

c :

continuum

D h :

based upon the hydraulic diameter

i :

inlet

ℓ:

based upon the arbitrary length ℓ

o :

outlet

E :

ellipse

R :

rectangle

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Acknowledgments

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

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Authors and Affiliations

  1. Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John‘s, NF, A1B 3X5, Canada

    Zhipeng Duan & Y. S. Muzychka

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  1. Zhipeng Duan
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  2. Y. S. Muzychka
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Correspondence to Zhipeng Duan.

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Duan, Z., Muzychka, Y.S. Slip flow in non-circular microchannels. Microfluid Nanofluid 3, 473–484 (2007). https://doi.org/10.1007/s10404-006-0141-4

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