# Slip flow in non-circular microchannels

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s10404-006-0141-4

- Cite this article as:
- Duan, Z. & Muzychka, Y.S. Microfluid Nanofluid (2007) 3: 473. doi:10.1007/s10404-006-0141-4

- 48 Citations
- 325 Views

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the mean free path of the gas is comparable to the channel’s characteristic dimension, the continuum assumption is no longer valid and a velocity slip may occur at the duct walls.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

*A*flow area (m

^{2})*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 = 4

*A*/*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)

### Greek symbols

- α
constants

- γ
ratio of specific heats

- δ
_{n} eigenvalues

- ε
aspect ratio =

*b*/*a*- η, ψ,
*z* elliptic cylinder coordinates

- η
_{0} 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 (m

^{2}/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

*D*_{h}based upon the hydraulic diameter

*i*inlet

- ℓ
based upon the arbitrary length ℓ

*o*outlet

### Superscripts

*E*ellipse

*R*rectangle