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When the mean free path of the gas is comparable to the channel’s characteristic dimension, the continuum assumption is no longer valid and a velocity slip may occur at the duct walls. Noncircular cross sections are common channel shapes that can be produced by microfabrication. The noncircular microchannels have extensive practical applications in MEMS. The paper deals with issues of hydrodynamic flow development. Slip flow in the entrance of circular and parallel plate microchannels is first considered by solving a linearized momentum equation. It is found that slip flow is less sensitive to analytical linearized approximations than continuum flow and the linearization method is an accurate approximation for slip flow. Also, it is found that the entrance friction factor Reynolds product is of finite value and dependent on the Kn and tangential momentum accommodation coefficient but independent of the cross-sectional geometry. Slip flow and continuum flow in the hydrodynamic entrance of noncircular microchannels has been examined and a model is proposed to predict the friction factor and Reynolds product fRe for developing slip flow and continuum flow in most noncircular microchannels. It is shown that the complete problem may be easily analyzed by combining the asymptotic results for short and long ducts. Through the selection of a characteristic length scale, the square root of cross-sectional area, the effect of duct shape has been minimized. The proposed model has an approximate accuracy of 10% for most common duct shapes.

1.
Ged-el-Hak
,
M.
, 2001,
MEMS Handbook
,
CRC
,
Boca Raton, FL
.
2.
Karniadakis
,
G. E.
,
Beskok
,
A.
, and
Aluru
,
N.
, 2005,
Microflows and Nanoflows
,
Springer
,
New York
.
3.
Nguyen
,
N. T.
, and
Wereley
,
S. T.
, 2003,
Fundamentals and Applications of Microfluidics
,
Artech House
,
London
.
4.
Eckert
,
E. G. R.
, and
Drake
,
R. M.
, 1972,
Analysis of Heat and Mass Transfer
,
McGraw-Hill
,
New York
.
5.
Rohsenow
,
W. M.
, and
Choi
,
H. Y.
, 1961,
Heat, Mass, and Momentum Transfer
,
Prentice-Hall
,
New Jersey
.
6.
Arkilic
,
E. B.
,
Breuer
,
K. S.
, and
Schmidt
,
M. A.
, 1994, “
Gaseous Flow in Microchannels
,”
Application of Microfabrication to Fluid Mechanics
, Vol.
FED-197
,
ASME
,
New York
, pp.
57
66
.
7.
Arkilic
,
E. B.
,
Breuer
,
K. S.
, and
Schmidt
,
M. A.
, 1997, “
Gaseous Slip Flow in Long Microchannels
,”
J. Microelectromech. Syst.
1057-7157,
6
(
2
), pp.
167
178
.
8.
Pong
,
K.
,
Ho
,
C.
,
Liu
,
J.
, and
Tai
,
Y.
, 1994, “
Nonlinear Pressure Distribution in Uniform Microchannels
,”
Application of Microfabrication to Fluid Mechanics
, Vol.
FED-197
,
ASME
,
New York
, pp.
51
56
.
9.
Liu
,
J.
,
Tai
,
Y. C.
, and
Ho
,
C. M.
, 1995, “
MEMS for Pressure Distribution Studies of Gaseous Flows in Microchannels
,”
Proceedings of the IEEE International Conference on Micro Electro Mechanical Systems
, Amsterdam, The Netherlands, pp.
209
215
.
10.
Araki
,
T.
,
Kim
,
M. S.
,
Hiroshi
,
I.
, and
Suzuki
,
K.
, 2000, “
An Experimental Investigation of Gaseous Flow Characteristics in Microchannels
,”
Proceedings of the International Conference on Heat Transfer and Transport Phenomena in Microscale
,
G. P.
Celata
, ed.,
Begell House
,
New York
, pp.
155
161
.
11.
Pfahler
,
J.
,
Harley
,
J.
,
Bau
,
H.
, and
Zemel
,
J. N.
, 1990, “
Gas and Liquid Transport in Small Channels
,”
Micromechanical Sensors, Actuators, and Systems
, Vol.
DSC-19
,
ASME
,
New York
, pp.
149
157
.
12.
Pfahler
,
J.
,
Harley
,
J.
,
Bau
,
H.
, and
Zemel
,
J. N.
, 1991, “
Gas and Liquid Flow in Small Channels
,”
Micromechanical Sensors, Actuators, and Systems
, Vol.
DSC-32
,
ASME
,
New York
, pp.
49
58
.
13.
Harley
,
J.
,
Huang
,
Y.
,
Bau
,
H.
, and
Zemel
,
J. N.
, 1995, “
Gas Flows in Micro-Channels
,”
J. Fluid Mech.
0022-1120,
284
, pp.
257
274
.
14.
Choi
,
S. B.
,
Barron
,
R. F.
, and
Warrington
,
R. O.
, 1991, “
Fluid Flow and Heat Transfer in Microtubes
,”
Micromechanical Sensors, Actuators, and Systems
, Vol.
DSC-32
,
ASME
,
New York
, pp.
123
134
.
15.
Wu
,
S.
,
Mai
,
J.
,
Zohar
,
Y.
,
Tai
,
Y. C.
, and
Ho
,
C. M.
, 1998, “
A Suspended Microchannel With Integrated Temperature Sensors for High Pressure Flow Studies
,”
Proceedings of the IEEE Workshop on Micro Electro Mechanical Systems
, Heidelberg, Germany, pp.
87
92
.
16.
Schaaf
,
S. A.
, and
Chambre
,
P. L.
, 1961,
Flow of Rarefied Gases
,
Princeton University Press
,
New Jersey
.
17.
Maurer
,
J.
,
Tabeling
,
P.
,
Joseph
,
P.
, and
Willaime
,
H.
, 2003, “
Second-Order Slip Laws in Microchannels for Helium and Nitrogen
,”
Phys. Fluids
1070-6631,
15
, pp.
2613
2621
.
18.
Aubert
,
C.
, and
Colin
,
S.
, 2001, “
High-Order Boundary Conditions for Gaseous Flows in Rectangular Microducts
,”
Microscale Thermophys. Eng.
1089-3954,
5
, pp.
41
54
.
19.
Deissler
,
R. G.
, 1964, “
An Analysis of Second-Order Slip Flow and Temperature-Jump Boundary Conditions for Rarefied Gases
,”
Int. J. Heat Mass Transfer
0017-9310,
7
, pp.
681
694
.
20.
Colin
,
S.
,
Lalonde
,
P.
, and
Caen
,
R.
, 2004, “
Validation of a Second-Order Slip Flow Model in Rectangular Microchannels
,”
Heat Transfer Eng.
0145-7632,
25
, pp.
23
30
.
21.
Barber
,
R. W.
, and
Emerson
,
D. R.
, 2006, “
Challenges in Modeling Gas-Phase Flow in Microchannels: From Slip to Transition
,”
Heat Transfer Eng.
0145-7632,
27
, pp.
3
12
.
22.
Jeong
,
N.
,
Lin
,
C. L.
, and
Choi
,
D. H.
, 2006, “
Lattice Boltzmann Study of Three-Dimensional Gas Microchannel Flows
,”
J. Micromech. Microeng.
0960-1317,
16
, pp.
1749
1759
.
23.
Barber
,
R. W.
, and
Emerson
,
D. R.
, 2001, “
A Numerical Investigation of Low Reynolds Number Gaseous Slip Flow at the Entrance of Circular and Parallel Plate Microchannels
,”
ECCOMAS Computational Fluid Dynamics Conference
, Swansea, Wales, UK.
24.
Shah
,
R. K.
, and
London
,
A. L.
, 1978,
Laminar Flow Forced Convection in Ducts
,
Academic
,
New York
.
25.
Sparrow
,
E. M.
,
Lundgren
,
T. S.
, and
Lin
,
S. H.
, 1962, “
Slip Flow in the Entrance Region of a Parallel Plate Channel
,”
Proceedings of the Heat Transfer and Fluid Mechanics Institute
,
Stanford University Press
,
Stanford
, pp.
223
238
.
26.
Sparrow
,
E. M.
,
Lin
,
S. H.
, and
Lundgren
,
T. S.
, 1964, “
Flow Development in the Hydrodynamic Entrance Region of Tubes and Ducts
,”
Phys. Fluids
1070-6631,
7
, pp.
338
347
.
27.
Kennard
,
E. H.
, 1938,
Kinetic Theory of Gases
,
McGraw-Hill
,
New York
.
28.
Atkinson
,
B.
,
Brocklebank
,
M. P.
,
Card
,
C. C. H.
, and
Smith
,
J. M.
, 1969, “
Low Reynolds Number Developing Flows
,”
AIChE J.
0001-1541,
15
, pp.
548
553
.
29.
Chen
,
R. Y.
, 1973, “
Flow in the Entrance Region at Low Reynolds Numbers
,”
ASME J. Fluids Eng.
0098-2202,
95
, pp.
153
158
.
30.
Friedmann
,
M.
,
Gillis
,
J.
, and
Liron
,
N.
, 1968, “
Laminar Flow in a Pipe at Low and Moderate Reynolds Numbers
,”
Appl. Sci. Res.
0003-6994,
19
, pp.
426
438
.
31.
Sreekanth
,
A.
, 1969, “
Slip Flow Through Long Circular Tubes
,”
Proceedings of the Sixth International Symposium on Rarefied Gas Dynamics
,
L.
Trilling
and
H. Y.
Wachman
, eds.,
Academic
,
New York
, pp.
667
680
.
32.
Shapiro
,
A. H.
,
Siegel
,
R.
, and
Kline
,
S. J.
, 1954, “
Friction Factor in the Laminar Entry Region of a Smooth Tube
,”
Proceedings of the US Second National Congress of Applied Mechanics
, New York, pp.
733
741
.
33.
Muzychka
,
Y. S.
, and
Yovanovich
,
M. M.
, 2002, “
Laminar Flow Friction and Heat Transfer in Non-circular Ducts and Channels: Part I-Hydrodynamic Problem
,”
Compact Heat Exchangers, A Festschrift on the 60th Birthday of Ramesh K. Shah
, Grenoble, France, pp.
123
130
.
34.
Yu
,
S. P.
, and
Ameel
,
T. A.
, 2001, “
A Universal Entrance Nusselt Number for Internal Slip Flow
,”
Int. Commun. Heat Mass Transf.
,
28
(
7
), pp.
905
910
.
35.
Renksizbulut
,
M.
,
Niazmand
,
H.
, and
Tercan
,
G.
, 2006, “
Slip-Flow and Heat Transfer in Rectangular Microchannels With Constant Wall Temperature
,”
Int. J. Therm. Sci.
1290-0729,
45
, pp.
870
881
.
36.
Duan
,
Z. P.
, and
Muzychka
,
Y. S.
, 2007, “
Slip Flow in Non-circular Microchannels
,”
Microfluid. Nanofluid.
1613-4982,
3
, pp.
473
484
.
37.
Duan
,
Z. P.
, and
Yovanovich
,
M. M.
, 2009, “
Pressure Drop for Laminar Flow in Microchannels of Arbitrary Cross-Sections
,”
Semi-Therm 25 Semiconductor Thermal Measurement and Management Symposium
, San Jose, CA.
38.
Churchill
,
S. W.
, and
Usagi
,
R.
, 1972, “
A General Expression for the Correlation of Rates of Transfer and Other Phenomena
,”
AIChE J.
0001-1541,
18
, pp.
1121
1128
.
39.
Hornbeck
,
R. W.
, 1964, “
Laminar Flow in the Entrance Region of a Pipe
,”
Appl. Sci. Res., Sect. A
0365-7132,
13
, pp.
224
232
.
40.
Liu
,
J.
, 1974, “
Flow of a Bingham Fluid in the Entrance Region of an Annular Tube
,” MS thesis, University of Wisconsin, Madison.
41.
Lawal
,
A.
, and
Mujumdar
,
A. S.
, 1984, “
Forced Convection Heat Transfer to a Power Law Fluid in Arbitrary Cross-Section Ducts
,”
Can. J. Chem. Eng.
0008-4034,
62
, pp.
326
333
.
42.
Curr
,
R. M.
,
Sharma
,
D.
, and
Tatchell
,
D. G.
, 1972, “
Numerical Predictions of Some Three Dimensional Boundary Layers in Ducts
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
1
, pp.
143
158
.
43.
Gangal
,
M. K.
, 1974, “
Some Problems in Channel Flow
,” Ph.D. thesis, University of Calgary, Calgary, Alberta, Canada.
44.
Niazmand
,
H.
,
Tercan
,
G.
, and
Renksizbulut
,
M.
, 2005, “
Entrance Region Flows in Rectangular Microchannels With Constant Wall Temperature
,”
Proceedings of the Third International Conference on Minichannels and Microchannels
, Toronto, Canada, Paper No. ICMM2005-75064.
45.
Quarmby
,
A.
, 1968, “
A Finite Difference Analysis of Developing Slip Flow
,”
Appl. Sci. Res.
0003-6994,
19
, pp.
18
33
.
This content is only available via PDF.
You do not currently have access to this content.