The effects of hydrodynamic and thermal slip on heat transfer in a thermally developing, steady, laminar Couette flow are investigated. Fluid temperature at the inlet to a parallel plate channel is prescribed, as various combinations of isothermal and adiabatic boundary conditions are along its surfaces. Analytical expressions incorporating arbitrary slip are developed for temperature profiles, and developing and fully developed for Nusselt numbers. The results are relevant to liquid and gas flows in the presence of apparent and molecular slip, respectively.

References

1.
Quéré
,
D.
,
2005
, “
Non-Sticking Drops
,”
Rep. Prog. Phys.
,
68
, pp.
2495
2532
.10.1088/0034-4885/68/11/R01
2.
Wang
,
E. N.
,
Bucaro
,
M. A.
,
Taylor
,
J. A.
,
Kolodner
,
P.
,
Aizenberg
,
J.
, and
Krupenkin
,
T.
,
2009
, “
Droplet Mixing Using Electrically Tunable Superhydrophobic Nanostructured Surfaces
,”
Microfluid. Nanofluid.
,
7
(
1
), pp.
137
140
.10.1007/s10404-008-0364-7
3.
Lam
,
L. S.
,
Hodes
,
M.
, and
Enright
,
R.
,
2013
, “
Galinstan-Based Microgap Cooling Enhancement Using Structured Surfaces
,”
Proceedings of the ASME 2013 Summer Heat Transfer Conference
.
4.
Lifton
,
V.
,
Taylor
,
J.
,
Vyas
,
B.
,
Kolodner
,
P.
,
Cirelli
,
R.
,
Basavanhally
,
N.
,
Papazian
,
A.
,
Frahm
,
R.
,
Simon
,
S.
, and
Krupenkin
,
T.
,
2008
, “
Superhydrophobic Membranes With Electrically Controllable Permeability and Their Application to Smart Microbatteries
,”
Appl. Phys. Lett.
,
93
(
4
), p.
043112
.10.1063/1.2965615
5.
Cassie
,
A. B. D.
, and
Baxter
,
S.
,
1944
, “
Wettability of Porous Surfaces
,”
Trans. Faraday Soc.
,
40
, pp.
546
551
.10.1039/tf9444000546
6.
Vogelpohl
,
G.
,
1951
, “
Die Temperaturverteilung in Schmierschichten zwishen parallen warmendurchlassigen Wanden
,”
Z. Angew. Math. Mech.
,
31
, pp.
349
356
.10.1002/zamm.19510311105
7.
Hudson
,
J.
, and
Bankoff
,
S.
,
1965
, “
Heat Transfer to a Steady Couette Flow With Pressure Gradient
,”
Chem. Eng. Sci.
,
20
(
5
), pp.
415
423
.10.1016/0009-2509(65)80054-9
8.
Bruin
,
S.
,
1972
, “
Temperature Distributions in Couette Flow With and Without Additional Pressure
,”
Int. J. Heat Mass Transfer
,
15
, pp.
341
349
.10.1016/0017-9310(72)90079-8
9.
Davis
,
E. J.
,
1973
, “
Exact Solutions for a Class of Heat and Mass transfer problems
,”
Can. J. Chem. Eng.
,
51
, pp.
562
572
.10.1002/cjce.5450510506
10.
El-Ariny
,
A.
, and
Aziz
,
A.
,
1976
, “
A Numerical Solution of Entrance Region Heat Transfer in Plane Couette Flow
,”
ASME J. Heat Transfer
,
98
, pp.
427
431
.10.1115/1.3450571
11.
Sesták
,
J.
, and
Rieger
,
F.
,
1969
, “
Laminar Heat Transfer to a Steady Couette Flow Between Parallel Plates
,”
Int. J. Heat Mass Transfer
,
12
, pp.
71
80
.10.1016/0017-9310(69)90079-9
12.
Schamberg
,
R.
,
1947
, “
The Fundamental Differential Equations and the Boundary Conditions for High Speed Slip-Flow, and Their Application to Several Specific Problems
,” Ph.D. thesis, California Institute of Technology, Pasadena, CA.
13.
Marques
,
W.
, Jr.
,
Kremer
,
G.
, and
Sharipov
,
F.
,
2000
, “
Couette Flow With Slip and Jump Boundary Conditions
,”
Continuum Mech. Thermodyn.
,
12
(
6
), pp.
379
386
.10.1007/s001610050143
14.
Fang
,
Y.
, and
Liou
,
W. W.
,
2002
, “
Computations of the Flow and Heat Transfer in Microdevices Using DSMC With Implicit Boundary Conditions
,”
ASME J. Heat Transfer
,
124
(
2
), pp.
338
345
.10.1115/1.1447933
15.
Sharipov
,
F.
, and
Strapasson
,
J. L.
,
2013
, “
Benchmark Problems for Mixtures of Rarefied Gases. I. Couette Flow
,”
Phys. Fluids
,
25
(
2
), p.
027101
.10.1063/1.4791604
16.
Milicev
,
S. S.
, and
Stevanovic
,
N. D.
,
2013
, “
A Non-Isothermal Couette Slip Gas Flow
,”
Sci. Chin. Phys., Mech. Astron.
,
56
(
9
), pp.
1782
1797
.10.1007/s11433-013-5120-7
17.
Navier
,
C.
,
1823
, “
Mémoire sur les du mouvement des fluids
,”
Mémoires de l'Académie Royale des Sciences de l'Institut de France
,
6
, pp.
389
440
.
18.
Maxwell
,
J. C.
,
1965
,
Scientific Papers
,
Dover Publications
,
New York
.
19.
Kennard
,
E. H.
,
1938
,
Kinetic Theory of Gases
, 1st ed.,
McGraw-Hill
,
New York
.
20.
Sparrow
,
E.
, and
Lin
,
S.
,
1962
, “
Laminar Heat Transfer in Tubes Under Slip Flow Conditions
,”
ASME J. Heat Transfer
, pp.
362
369
.
21.
Barron
,
R. F.
,
Wang
,
X.
,
Ameel
,
T. A.
, and
Warrington
,
R. O.
,
1997
, “
The Graetz Problem Extended to Slip-Flow
,”
Int. J. Heat Mass Transfer
,
40
(
8
), pp.
1817
1823
.10.1016/S0017-9310(96)00256-6
22.
Jiji
,
L. M.
,
2009
,
Heat Convection
, 2nd ed.,
Springer-Verlag
, Berlin.
23.
Colin
,
S.
,
2012
, “
Gas Microflows in the Slip Flow Regime: A Critical Review on Convective Heat Transfer
,”
ASME J. Heat Transfer
,
134
(
2
), p.
020908
.10.1115/1.4005063
24.
Cheng
,
Y.
,
Teo
,
C.
, and
Khoo
,
B.
,
2009
, “
Microchannel Flows With Superhydrophobic Surfaces: Effects of Reynolds Number and Pattern Width to Channel Height Ratio
,”
Phys. Fluids
,
21
, p.
122004
.10.1063/1.3281130
25.
Rothstein
,
J. P.
,
2010
, “
Slip on Superhydrophobic Surfaces
,”
Annu. Rev. Fluid Mech.
,
42
(
1
), pp.
89
109
.10.1146/annurev-fluid-121108-145558
26.
Philip
,
J.
,
1972
, “
Flows Satisfying Mixed No-Slip and No-Shear Conditions
,”
J. Appl. Math. Phys.
,
23
, pp.
353
372
.10.1007/BF01595477
27.
Philip
,
J.
,
1972
, “
Integral Properties of Flows Satisfying Mixed No-Slip and No-Shear Conditions
,”
J. Appl. Math. Phys.
,
23
, pp.
960
968
.10.1007/BF01596223
28.
Lauga
,
E.
, and
Stone
,
H.
,
2003
, “
Effective Slip in Pressure-Driven Stokes Flow
,”
J. Fluid Mech.
,
489
, pp.
55
77
.10.1017/S0022112003004695
29.
Ou
,
J.
,
Perot
,
B.
, and
Rothstein
,
J. P.
,
2004
, “
Laminar Drag Reduction in Microchannels Using Ultrahydrophobic Surfaces
,”
Phys. Fluids
,
16
(
12
), pp.
4635
4643
.10.1063/1.1812011
30.
Ou
,
J.
, and
Rothstein
,
J.
,
2005
, “
Direct Velocity Measurements of the Flow Past Drag-Reducing Ultrahydrophobic Surfaces
,”
Phys. Fluids
,
17
, p.
103606
.10.1063/1.2109867
31.
Priezjev
,
N. V.
,
Darhuber
,
A. A.
, and
Troian
,
S. M.
,
2005
, “
Slip Behavior in Liquid Films on Surfaces of Patterned Wettability
,”
Phys. Rev. E
,
71
, p.
041608
.10.1103/PhysRevE.71.041608
32.
Truesdell
,
R.
,
Mammoli
,
A.
,
Vorobieff
,
P.
,
van Swol
,
F.
, and
Brinker
,
C.
,
2006
, “
Drag Reduction on a Patterned Superhydrophobic Surface
,”
Phys. Rev. Lett.
,
97
(
4
), pp.
1
4
.10.1103/PhysRevLett.97.044504
33.
Lee
,
C.
,
Choi
,
C.-H.
, and
Kim
,
C.-J.
,
2008
, “
Structured Surfaces for a Giant Liquid Slip
,”
Phys. Rev. Lett.
,
101
(
6
), pp.
1
4
.
34.
Ybert
,
C.
,
Barentin
,
C.
,
Cottin-Bizonne
,
C.
,
Joseph
,
P.
, and
Bocquet
,
L.
,
2007
, “
Achieving Large Slip With Superhydrophobic Surfaces: Scaling Laws for Generic Geometries
,”
Phys. Fluids
,
19
(
12
), p.
123601
.10.1063/1.2815730
35.
Salamon
,
T.
,
Lee
,
W.
,
Hodes
,
M.
,
Kolodner
,
P.
,
Enright
,
R.
, and
Salinger
,
A.
,
2005
, “
Numerical Simulation of Fluid Flow in Microchannels With Superhydrophobic Walls
,”
IMECE Conference Proceedings
, ASME, pp.
819
829
, Paper No. 42215.
36.
Maynes
,
D.
,
Jeffs
,
K.
,
Woolford
,
B.
, and
Webb
,
B. W.
,
2007
, “
Laminar Flow in a Microchannel With Hydrophobic Surface Patterned Microribs Oriented Parallel to the Flow Direction
,”
Phys. Fluids
,
19
(
9
), p.
093603
.10.1063/1.2772880
37.
Davies
,
J.
,
Maynes
,
D.
,
Webb
,
B. W.
, and
Woolford
,
B.
,
2006
, “
Laminar Flow in a Microchannel With Superhydrophobic Walls Exhibiting Transverse Ribs
,”
Phys. Fluids
,
18
(
8
), p.
087110
.10.1063/1.2336453
38.
Teo
,
C. J.
, and
Khoo
,
B. C.
,
2010
, “
Flow Past Superhydrophobic Surfaces Containing Longitudinal Grooves: Effects of Interface Curvature
,”
Microfluidics Nanofluidics
,
9
(
2–3
), pp.
499
511
.10.1007/s10404-010-0566-7
39.
Steinberger
,
A.
,
Cottin-Bizonne
,
C.
,
Kleimann
,
P.
, and
Charlaix
,
E.
,
2007
, “
High Friction on a Bubble Mattress
,”
Nature Mater.
,
6
(
9
), pp.
665
668
.10.1038/nmat1962
40.
Enright
,
R.
,
Hodes
,
M.
,
Salamon
,
T.
, and
Muzychka
,
Y.
,
2014
, “
Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels
,”
ASME J. Heat Transfer
,
136
, p.
012402
.10.1115/1.4024837
41.
Enright
,
R.
,
Hodes
,
M.
,
Salamon
,
T.
,
Krupenkin
,
T.
,
Kolodner
,
P.
,
Dalton
,
T.
, and
Eason
,
C.
,
2006
, “
Friction Factors and Nusselt Numbers in Microchannels With Superhydrophobic Walls
,”
Proceedings of the Fourth International Conference on Nanochannels, Microchannels and Minichannels
, Limerick Ireland, ASME, New York, pp.
599
609
, Paper No. ICNMM2006-96134.
42.
Williams
,
A. D.
,
Vorobieff
,
P.
, and
Mammoli
,
A.
,
2012
, “
Effect of Slip Flow on Heat Transfer: Numerical Analysis
,”
50th AIAA Aerospace Sciences Meeting
, p.
7726
.
43.
Maynes
,
D.
,
Webb
,
B. W.
, and
Davies
,
J.
,
2008
, “
Thermal Transport in a Microchannel Exhibiting Ultrahydrophobic Microribs Maintained at Constant Temperature
,”
ASME J. Heat Transfer
,
130
(
2
), p.
022402
.10.1115/1.2789715
44.
Maynes
,
D.
,
Webb
,
B.
,
Crockett
,
J.
, and
Solovjov
,
V.
,
2013
, “
Analysis of Laminar Slip-Flow Thermal Transport in Microchannels With Transverse Rib and Cavity Structured Superhydrophobic Walls at Constant Heat Flux
,”
ASME J. Heat Transfer
,
135
(
2
), p.
021701
.10.1115/1.4007429
45.
Maynes
,
D.
, and
Crockett
,
J.
,
2014
, “
Apparent Temperature Jump and Thermal Transport in Channels With Streamwise Rib and Cavity Featured Superhydrophobic Walls at Constant Heat Flux
,”
ASME J. Heat Transfer
,
136
, p.
011701
10.1115/1.4025045.
46.
Inman
,
R. M.
,
1964
, “
Laminar Slip Flow Heat Transfer in a Parallel-Plate Channel or a round Tube With Uniform Wall Heating
,” NASA Technical Note D-2393.
47.
Teo
,
C. J.
, and
Khoo
,
B. C.
,
2009
, “
Analysis of Stokes Flow in microchannels With Superhydrophobic Surfaces Containing a Periodic Array of Micro-Grooves
,”
Microfluid. Nanofluid.
,
7
, pp.
352
382
.10.1007/s10404-008-0387-0
48.
Weigand
,
B.
,
2004
,
Analytical Methods for Heat Transfer and Fluid Flow Problems
,
Springer
,
New York
.
49.
Haberman
,
R.
,
2004
,
Applied Partial Differential Equations With Fourier Series and Boundary Value Problems
, 4th ed.,
Prentice-Hall
, Englewood Cliffs, NJ.
50.
Greenberg
,
M.
,
1998
,
Advanced Engineering Mathematics
, 2nd ed.,
Prentice-Hall
,
Englewood Cliffs, NJ
.
You do not currently have access to this content.