The paper presents an analytical solution of velocity, mass flow rate, and pressure distribution for fully developed gaseous slip flow in nonsymmetric and symmetric parabolic microchannels. The flow is considered to be steady, laminar, and incompressible with constant fluid properties. Fully developed gaseous slip flow in microchannels of parabolic cross section is solved analytically for various aspect ratios using a parabolic cylindrical coordinate system on applying the method of separation of variables. Prior to apply separation of variables, Arfken transform [Arfken, 1970, Mathematical Methods for Physicists, Academic Press, Orlando, FL, Ch. 2] was used on momentum equations and first-order slip boundary conditions at each channel wall were imposed. A simple model is proposed to predict the friction factor and Reynolds number product fRe for slip flow in parabolic microchannels. Through the selection of a characteristic length scale, the square root of cross-sectional area and the effect of duct shape have been minimized. The results of a normalized Poiseuille number for symmetric parabolic microchannels (ɛ=1) shows good agreement with the previous results [Morini et al., 2004, “The Rarefaction Effect on the Friction Factor of Gas Flow in Micro/Nano-Channels,” Superlattices Microstruct., 35(3–6), pp. 587–599; Khan and Yovanovich, 2008, “Analytical Modeling of Fluid Flow and Heat Transfer in Microchannel/Nanochannel Heat Sinks,” J. Thermophys. Heat Transf., 22(3), pp. 352–359] for rectangular microchannels. The developed model can be used to predict mass flow rate and pressure distribution of slip flow in parabolic microchannels.

References

References
1.
Karniadakis
,
G. E.
,
Beskok
,
A.
, and
Aluru
,
N.
,
2005
,
Microflows and Nanoflows: Fundamentals and Simulation
, Vol.
29
,
Springer
,
New York.
2.
Gad-el-Hak
,
M.
,
1999
, “
The Fluid Mechanics of Microdevices—the Freeman Scholar Lecture
,”
ASME J. Fluids Eng.
,
121
(
1
), pp.
5
33
.10.1115/1.2822013
3.
Jang
,
J.
, and
Wereley
,
S. T.
,
2006
, “
Effective Heights and Tangential Momentum Accommodation Coefficients of Gaseous Slip Flows in Deep Reactive Ion Etching Rectangular Microchannels
,”
J. Micromech. Microeng.
,
16
(
3
), pp.
493
504
.10.1088/0960-1317/16/3/004
4.
Arkilic
,
E. B.
,
Schmidt
,
M. A.
, and
Breuer
,
K. S.
,
1997
, “
Gaseous Slip Flow in Long Microchannels
,”
J. Microelectromech. Syst.
,
6
(
2
), pp.
167
178
.10.1109/84.585795
5.
Liu
,
J.
,
Tai
,
Y. C.
, and
Ho
,
C. M.
,
1995
, “
MEMS for Pressure Distribution Studies of Gaseous Flows in Microchannels
,”
Proceedings of IEEE International Conference on Micro Electro Mech. Systems
, Amsterdam, Netherlands, pp.
209
215
.
6.
Pfahler
,
J.
,
Harley
,
J.
,
Bau
,
H.
, and
Zemel
,
J. N.
,
1990
, “
Gas and Liquid Transport in Small Channels
,”
Proceedings of ASME Microstructures, Sensors and Actuators
,
19
, pp.
149
157
.
7.
Harley
,
J.
,
Huang
,
Y.
,
Bau
,
H.
, and
Zemel
,
J. N.
,
1995
, “
Gas Flows in Microchannels
,”
J. Fluid Mech.
,
284
, pp.
257
274
.10.1017/S0022112095000358
8.
Wu
,
S.
,
Mai
,
J.
,
Zohar
,
Y.
,
Tai
,
Y. C.
, and
Ho
,
C. M.
,
1998
, “
A Suspended Microchannel With Integrated Temperature Sensors for High Pressure Studies
,”
Proceedings of IEEE Workshop on Micro Electro Mechanical Systems
, Heidelberg, Germany, pp.
87
92
.
9.
Araki
,
T.
,
Kim
,
M. S.
,
Hiroshi
, I
.
, and
Suzuki
,
K.
,
2000
, “
An Experimental Investigation of Gaseous Flow Characteristics in Microchannels
,”
Proceedings of International Conference on Heat Transfer and Transport Phenomena in Microscale
, Begell House, New York, pp.
155
161
.
10.
Ebert
,
W. A.
, and
Sparrow
,
E. M.
,
1965
, “
Slip Flow in Rectangular and Annular Ducts
,”
ASME J. Basic Eng.
,
87
(
4
), pp.
1018
1024
.10.1115/1.3650793
11.
Eckert
,
E. G. R.
, and
Drake
,
R. M.
,
1972
,
Analysis of Heat and Mass Transfer
,
McGraw-Hill
,
New York.
12.
Arkilic
,
E. B.
,
Breuer
,
K. S.
, and
Schmidt
,
M. A.
,
1994
, “
Gaseous Flow in Microchannels
,”
Proceedings of ASME Application of Microfabrication to Fluid Mechanics
, ASME Paper No. FED-197, pp.
57
66
.
13.
Rostami
,
A. A.
,
Saniei
,
N.
, and
Majumder
,
A. S.
,
2000
, “
Liquid Flow and Heat Transfer in Microchannels: A Review
,”
Int. J. Heat Technol.
,
18
(
2
), pp.
59
68
.
14.
Rostami
,
A. A.
,
Majumder
,
A. S.
, and
Saniei
,
N.
,
2002
, “
Flow and Heat Transfer for Gas Flowing in Microchannels: A Review
,”
Heat Mass Transf.
,
38
(
4–5
), pp.
359
367
.10.1007/s002310100247
15.
Morini
,
G. L.
,
Spiga
,
M.
, and
Tartarini
,
P.
,
2004
, “
The Rarefaction Effect on the Friction Factor of Gas Flow in Micro/Nano-Channels
,”
Superlattices Microstruct.
,
35
(
3–6
), pp.
587
599
.10.1016/j.spmi.2003.09.013
16.
Colin
,
S.
,
Lalonde
,
P.
, and
Caen
,
R.
,
2004
, “
Validation of a Second-Order Slip Flow Model in Rectangular Microchannels
,”
Heat Transf. Eng.
,
25
(
3
), pp.
23
30
.10.1080/01457630490280047
17.
Barron
,
R. F.
,
Wang
,
X.
,
Ameel
,
T. A.
, and
Warrington
,
R. O.
,
1997
, “
The Graetz Problem Extended to Slip-Flow
,”
Int. J. Heat Mass Transf.
,
40
(
8
), pp.
1817
1823
.10.1016/S0017-9310(96)00256-6
18.
Duan
,
Z.
, and
Muzychka
,
Y. S.
,
2007
, “
Slip Flow in Non-Circular Microchannels
,”
Microfluidics Nanofluidics
,
3
(
4
), pp.
473
484
.10.1007/s10404-006-0141-4
19.
Duan
,
Z.
, and
Muzychka
,
Y. S.
,
2007
, “
Slip Flow in Elliptic Microchannels
,”
Int. J. Therm. Sci.
,
46
(
11
), pp.
1104
1111
.10.1016/j.ijthermalsci.2007.01.026
20.
Duan
,
Z.
, and
Yovanovich
,
M. M.
,
2010
, “
Models for Gaseous Slip Flow in Circular and Noncircular Microchannels
,”
Proceedings of ASME 2010 8th International Conference on Nanochannels, Microchannels, and Minichannels/3rd Joint US-European Fluids Engineering Summer Meeting
, Montreal, Canada, August 1–5, ASME Paper No. FEDSM-ICNMM2010-30320, pp.
421
431
.
21.
Arfken
,
G. B.
,
1970
,
Mathematical Methods for Physicists
, 2nd ed.
Academic Press
,
Orlando, FL
, Chap. 2.
22.
Maxwell
,
J. C.
,
1879
, “
On Stresses in Rarefied Gases Arising From Inequalities of Temperature
,”
Philos. Trans. Royal Soc.
,
170
, pp.
231
256
.10.1098/rstl.1879.0067
23.
Rohsenow
,
W. M.
, and
Choi
,
H. Y.
,
1961
,
Heat, Mass and Momentum Transfer
,
Prentice Hall
,
New York
.
24.
Churchill
,
S. W.
,
1988
,
Viscous Flows: The Practical Use of Theory
,
Butterworth-Heinemann
,
Boston.
25.
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, Proceedings of the International Symposium on Compact Heat Exchangers, Grenoble, France
, August 24, pp.
123
130
.
26.
Muzychka
,
Y. S.
, and
Yovanovich
,
M. M.
,
2002
, “
Laminar Flow Friction and Heat Transfer in Non-Circular Ducts and Channels Part II: Thermal Problem
,”
Compact Heat Exchangers, Proceedings of the International Symposium on Compact Heat Exchangers, Grenoble, France
, August 24, pp.
131
139
.
27.
White
,
F. M.
,
1974
,
Viscous Fluid Flow
,
McGraw-Hill
,
New York.
28.
Yovanovich
,
M. M.
, and
Muzychka
,
Y. S.
,
1997
, “
Solutions of Poisson equation within singly and doubly prismatic connected domains
,”
National Heat Transfer Conference
,
AIAA
Paper No. 97-3880.10.2514/6.1997-3880
29.
Muzychka
,
Y. S.
,
1999
, “
Analytical and Experimental Study of Fluid Friction and Heat Transfer in Low Reynolds Number Flow Heat Exchangers
,” Ph.D. thesis, University of Waterloo, Waterloo, ON, Canada.
30.
Bahrami
,
M.
,
Yovanovich
,
M. M.
, and
Culham
,
J. R.
,
2006
, “
Pressure Drop of Fully Developed, Laminar Flow in Microchannel of Arbitrary Cross-Section
,”
ASME J. Fluids Eng.
,
128
(
5
), pp.
1036
1044
.10.1115/1.2234786
31.
Muzychka
,
Y. S.
, and
Edge
,
J.
,
2008
, “
Laminar Non-Newtonian Fluid Flow in Non-Circular Ducts and Microchannels
,”
ASME J. Fluids Eng.
,
130
(
11
), p.
111201
.10.1115/1.2979005
32.
Sreekanth
,
A. K.
,
1969
, “
Slip Flow Through Long Circular Tubes
,”
Proceedings of the Sixth International Symposium on Rarefied Gas Dynamics
, Academic Press, San Diego, CA, pp.
667
680
.
33.
Khan
,
W. A.
and
Yovanovich
,
M. M.
,
2008
, “
Analytical Modeling of Fluid Flow and Heat Transfer in Microchannel/Nanochannel Heat Sinks
,”
J. Thermophys. Heat Transf.
,
22
(
3
), pp.
352
359
.10.2514/1.35621
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