The problem of fully developed steady, laminar, incompressible flow in a vertical channel is studied analytically, one region is filled with water based copper nanofluid and the other region is filled with clear viscous fluid. The resulting coupled nonlinear ordinary differential equations (ODEs) are solved by optimal homotopy analysis method (OHAM). The convergence of our results is discussed by the so-called total average squared residual error. Analytical results are presented for different values of the physical parameters, such as the mixed convection parameters, the Brownian motion parameter, and thermophoresis parameter. Reversed flow is observed for sufficiently high buoyancy (mixed convection parameter). Further we investigate the effects of the Brownian motion parameter and thermophoresis parameter on the fluid flow and heat transfer at the interface of the two regions.

References

References
1.
Tao
,
L. N.
,
1960
, “
On Combined Free and Forced Convection in Channels
,”
ASME J. Heat Transfer
,
82
, pp.
233
238
.10.1115/1.3679915
2.
Aung
,
W.
, and
Worku
,
G.
,
1986
, “
Developing Flow and Flow Reversal in a Vertical Channel With Asymmetric Wall Temperature
,”
ASME J. Heat Transfer
,
108
, pp.
299
304
.10.1115/1.3246919
3.
Aung
,
W.
, and
Worku
,
G.
,
1986
, “
Theory of Fully Developed
,
Combined Convection Including Flow Reversal
,”
ASME J. Heat Transfer
,
108
, pp.
485
488
.10.1115/1.3246958
4.
Kimura
,
T.
,
Heya
,
N.
,
Takeuchi
,
M.
, and
Isomi
,
H.
,
1986
, “
Natural Convection Heat Transfer Phenomena in an Enclosure Filled With Two Stratified Fluids
,”
Trans. Jpn. Soc. Mech. Eng.
,
Ser. B
,
52
, pp.
617
625
.10.1299/kikaib.52.617
5.
Malashetty
,
M. S.
,
Umavathi
,
J. C.
, and
Kumar
,
J. P.
,
2006
, “
Magnetoconvection of Two-Immiscible Fluids in a Vertical Enclosure
,”
J. Heat Mass Transfer
,
42
, pp.
977
993
.10.1007/s00231-005-0062-x
6.
Nikodijevic
,
D.
,
Stamenkovic
,
Z.
,
Milenkovic
,
D.
,
Lagojevic
,
B.
, and
Nikodikevic
,
J.
,
2011
, “
Flow and Heat Transfer of Two Immiscible Fluids in the Presence of Uniform Inclined Magnetic Field
,”
Math. Probl. Eng.
,
2011
, p.
132302
.10.1155/2011/132302
7.
Kumar
,
J. P.
,
Umavathi
,
J. C.
,
Chamkha
,
A. J.
, and
Pop
,
I.
,
2010
, “
Fully-Developed Free Convective Flow of Micropolar and Viscous Fluids in a Vertical Channel
,”
Appl. Math. Model.
,
34
, pp.
1175
1186
.10.1016/j.apm.2009.08.007
8.
Umavathi
,
J. C.
,
Liu
,
I. C.
,
Kumar
,
J. P.
, and
Meera
,
D. S.
,
2010
, “
Unsteady Flow and Heat Transfer of Porous Media Sandwiched Between Viscous Fluids
,”
Appl. Math. Mech.
,
31
(
12
), pp.
1497
1516
.10.1007/s10483-010-1379-6
9.
Choi
,
S. U.
,
1995
, “
Enhancing Thermal Conductivity of Fluids With Nanoparticle
,”
Conference on International Mechanical Engineering Congress and Exhibition
, San Francisco, CA, Nov. 12–17.
10.
Choi
,
S. U.
,
Zhang
,
Z. G.
,
Lockwood
,
W.
, and
Grulke
,
F. E.
,
2001
, “
Anomalously Thermal Conductivity Enhancement in Nanotube Suspension
,”
J. Appl. Phys. Lett.
,
79
, pp.
2252
2254
.10.1063/1.1408272
11.
Das
,
S. K.
,
Choi
,
S. U.
,
Yu
,
W.
, and
Pardeep
,
T.
,
2007
,
Nanofluids: Science and Technology
,
Wiley
,
Hoboken, NJ
.
12.
Xu
,
H.
, and
Pop
,
I.
,
2012
, “
Fully Developed Mixed Convection Flow in a Vertical Channel Filled With Nanofluids
,”
Int. Commun. Heat Mass Transfer
,
39
, pp.
1086
1092
.10.1016/j.icheatmasstransfer.2012.06.003
13.
Grosan
,
T.
, and
Pop
,
I.
,
2012
, “
Fully Developed Mixed Convection in a Vertical Channel Filled by a Nanofluid
,”
ASME J. Heat Transfer
,
134
, pp.
1
5
.10.1115/1.4006159
14.
Xu
,
H.
,
Fan
,
T.
, and
Pop
,
I.
,
2013
, “
Analysis of Mixed Convection Flow of a Nanofluid in a Vertical Channel With the Buongiorno Mathematical Model
,”
J. Int. Commun. Heat Mass Transfer
,
44
, pp.
15
22
.10.1016/j.icheatmasstransfer.2013.03.015
15.
Gorder
,
R. A. V.
,
Prasad
,
K. V.
, and
Vajravelu
,
K.
,
2012
, “
Convective Heat Transfer in the Vertical Channel Flow of a Clear Fluid Adjacent to a Nanofluid Layer: A Two-Fluid Model
,”
Heat and Mass Transfer
,
48
, pp.
1247
1255
.10.1007/s00231-012-0973-2
16.
Liao
,
S.
,
1997
, “
A Kind of Approximate Solution Technique Which Does not Depend Upon Small Parameters (II)—An Application in Fluid Mechanics
,”
Int. J. Nonlinear Mech.
,
32
, pp.
815
822
.10.1016/S0020-7462(96)00101-1
17.
Liao
,
S.
,
2010
, “
An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations
,”
J. Commun. Nonlinear Sci. Numer. Simul.
,
15
, pp.
2003
2016
.10.1016/j.cnsns.2009.09.002
18.
Khanafer
,
K.
,
Vafai
,
K.
, and
Lightstone
,
M.
,
2003
, “
Buoyancy-Driven Heat Transfer Enhancement in a Two-Dimensional Enclosure Utilizing Nanofluids
,”
Int. J. Heat Mass Transfer
,
46
, pp.
3639
3653
.10.1016/S0017-9310(03)00156-X
19.
Buongiorno
,
J.
,
2006
, “
Convective Transport in Nano fluids
,”
ASME J. Heat Transfer
,
128
, pp.
240
250
.10.1115/1.2150834
20.
Liao
,
S.
,
1999
, “
An Explicit
,
Totally Analytic Approximation of Blasius Viscous Flow Problems
,”
Int. J. Nonlinear Mech.
,
34
, pp.
759
778
.10.1016/S0020-7462(98)00056-0
21.
Liao
,
S.
,
1999
, “
A Uniformly Valid Solutions of 2D Viscous Flow Past a Semi-Infinite Flat Plate
,”
J. Fluid Mech.
,
385
, pp.
101
128
.10.1017/S0022112099004292
22.
Liao
,
S.
, and
Campo
,
A.
,
2002
, “
Analytic Solutions of the Temperature Distribution in Blasius Viscous Flow Problems
,”
J. Fluid Mech.
,
453
, pp.
411
425
.10.1017/S0022112001007169
23.
Liao
,
S.
,
2003
, “
On the Analytic Solution of Magnetohydrodynamic Flows of Non-Newtonian Fluids Over a Stretching Sheet
,”
J. Fluid Mech.
,
488
, pp.
189
212
.10.1017/S0022112003004865
24.
Liao
,
S.
,
2003
,
Beyond Perturbation. Introduction to Homotopy Analysis Method
,
Chapman and Hall/CRC Press
,
Boca Raton, FL
.
25.
Liao
,
S.
,
2012
,
Homotopy Analysis Method in Nonlinear Differential Equations
,
Higher Education Press
,
Beijing
.
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