The heat transfer characteristics of a viscous liquid film flow over an unsteady stretching sheet subject to variable heat flux are investigated numerically. The effect of thermal radiation applying to an optically thick medium is also considered. The governing boundary layer equations are transformed into a set of nonlinear ordinary differential equations using an efficient fifth-order step-adapted Runge–Kutta integration scheme together with Newton–Raphson method. The dimensionless temperature is plotted for various governing parameters; say, unsteadiness parameter, effective Prandtl number, distance index, as well as time index. It is found that the heat transfer aspects are strongly influenced by the relevant parameters.

References

1.
Crane
,
L. J.
,
1970
, “
Flow Past a Stretching Plane
,”
ZAMP
,
21
, pp.
645
647
.10.1007/BF01587695
2.
Gupta
,
A. S.
, and
Gupta
,
P. S.
,
1977
, “
Heat and Mass Transfer on a Stretching Sheet With Suction or Blowing
,”
Can. J. Chem. Eng.
,
55
, pp.
744
746
.10.1002/cjce.5450550619
3.
Carragher
,
P.
, and
Crane
,
L. J.
,
1982
, “
Heat Transfer on a Continuous Stretching Sheet
,”
ZAMM
,
62
, pp.
564
565
.10.1002/zamm.19820621009
4.
Jeng
,
D. R.
,
Chang
,
T. C. A.
, and
DeWitt
,
K. J.
,
1986
, “
Momentum and Heat Transfer on a Continuous Moving Surface
,”
ASME J. Heat Trans.
,
108
, pp.
532
539
.10.1115/1.3246967
5.
Wang
,
C. Y.
,
1990
, “
Liquid Film on an Unsteady Stretching Surface
,”
Quart. Appl. Math.
,
48
, pp.
601
610
.
6.
Wang
,
C.
,
2006
, “
Analytic Solutions for a Liquid Film on an Unsteady Stretching Surface
,”
Heat Mass Transfer
,
42
, pp.
759
766
.10.1007/s00231-005-0027-0
7.
Andersson
,
H. I.
,
Aarseth
,
J. B.
,
Braud
,
N.
, and
Dandapat
,
B. S.
,
1996
, “
Flow of a Power-Law Fluid on an Unsteady Stretching Surface
,”
J. Non-Newtonian Fluid Mech.
,
62
, pp.
1
8
.10.1016/0377-0257(95)01392-X
8.
Wang
,
C.
, and
Pop
,
I.
,
2006
, “
Analysis of the Flow of a Power-Law Fluid Film on an Unsteady Stretching Surface by Means of Homotopy Analysis Method
,”
J. Non-Newtonian Fluid Mech.
,
138
, pp.
161
172
.10.1016/j.jnnfm.2006.05.011
9.
Abbas
,
Z.
,
Hayat
,
T.
,
Sajid
,
M.
, and
Asghar
,
S.
,
2008
, “
Unsteady Flow of a Second Grade Fluid Film Over an Unsteady Stretching Sheet
,”
Math. Comput. Modelling
,
48
, pp.
518
526
.10.1016/j.mcm.2007.09.015
10.
Hayat
,
T.
,
Saif
,
S.
, and
Abbas
,
Z.
,
2008
, “
The Influence of Heat Transfer in an MHD Second Grade Fluid Film Over an Unsteady Stretching Sheet
,”
Phys. Lett. A
,
372
, pp.
5037
5045
.10.1016/j.physleta.2008.03.066
11.
Elgazery
,
N. S.
, and
Hassan
,
M. A.
,
2008
, “
The Effects of Variable Fluid Properties and Magnetic Field on the Flow of Non-Newtonian Fluid Film on an Unsteady Stretching Sheet Through a Porous Medium
,”
Commun. Numer. Methods Eng.
,
24
, pp.
2113
2129
.10.1002/cnm.1099
12.
Andersson
,
H. I.
,
Aarseth
,
J. B.
, and
Dandapat
,
B. S.
,
2000
, “
Heat Transfer in a Liquid Film on an Unsteady Stretching Surface
,”
Int. J. Heat Mass Transfer
,
43
, pp.
69
74
.10.1016/S0017-9310(99)00123-4
13.
Chen
,
C. H.
,
2003
, “
Heat Transfer in a Power-Law Fluid Film Over a Unsteady Stretching Sheet
,”
Heat Mass Transfer
,
39
, pp.
791
796
.10.1007/s00231-002-0363-2
14.
Dandapat
,
B. S.
,
Santra
,
B.
, and
Andersson
,
H. I.
,
2003
, “
Thermocapillarity in a Liquid Film on an Unsteady Stretching Surface
,”
Int. J. Heat Mass Transfer
,
46
, pp.
3009
3015
.10.1016/S0017-9310(03)00078-4
15.
Dandapat
,
B. S.
,
Santra
,
B.
, and
Andersson
,
H. I.
,
2007
, “
The Effect of Variable Properties and Thermocapillarity on the Flow of a Thin Film on an Unsteady Stretching Sheet
,”
Int. J. Heat Mass Transfer
,
50
, pp.
991
996
.10.1016/j.ijheatmasstransfer.2006.08.007
16.
Noor
,
N. F. M.
, and
Hashim
,
I.
,
2010
, “
Thermocapillarity and Magnetic Field Effects in a Thin Liquid Film on an Unsteady Stretching Surface
,”
Int. J. Heat Mass Transfer
,
53
, pp.
2044
2051
.10.1016/j.ijheatmasstransfer.2009.12.052
17.
Chen
,
C. H.
,
2007
, “
Marangoni Effects on Forced Convection of Power-Law Liquids in a Thin Film Over a Stretching Surface
,”
Phys. Lett. A
,
370
, pp.
51
57
.10.1016/j.physleta.2007.05.024
18.
Mahmoud
,
M. A. A.
, and
Megahed
,
A. M.
,
2009
, “
MHD Flow and Heat Transfer in a Non-Newtonian Liquid Film Over an Unsteady Stretching Sheet With Variable Fluid Properties
,”
Can. J. Phys.
,
87
, pp.
1065
1071
.10.1139/P09-066
19.
Chen
,
C. H.
,
2006
, “
Effect of Viscous Dissipation on Heat Transfer in a Non-Newtonian Liquid Film Over an Unsteady Stretching Sheet
,”
J. Non-Newtonian Fluid Mech.
,
135
, pp.
128
135
.10.1016/j.jnnfm.2006.01.009
20.
Abel
,
M. S.
,
Mahesha
,
N.
, and
Tawade
,
J.
,
2009
, “
Heat Transfer in a Liquid Film Over an Unsteady Stretching Surface With Viscous Dissipation in the Presence of External Magnetic Field
,”
Appl. Math. Modelling
,
33
, pp.
3430
3441
.10.1016/j.apm.2008.11.021
21.
Abel
,
M. S.
,
Tawade
,
J.
, and
Nandeppanavar
,
M. M.
,
2009
, “
Effect of Non-Uniform Heat Source on MHD Heat Transfer in a Liquid Film Over an Unsteady Stretching Sheet
,”
Int. J. Nonlinear Mech.
,
44
, pp.
990
998
.10.1016/j.ijnonlinmec.2009.07.004
22.
Kiwan
,
S.
, and
Al-Nimr
,
M. A.
,
2009
, “
Flow and Heat Transfer Over a Stretched Mirco-Surface
,”
ASME J. Heat Transfer
,
131
(
6
), pp.
1
8
.10.1115/1.3090811
23.
Dandapat
,
B. S.
, and
Maity
,
S.
,
2006
, “
Flow of a Thin Film on an Unsteady Stretching Sheet
,”
Phys. Fluids
,
18
, p.
102101
.10.1063/1.2360256
24.
Santra
,
B.
, and
Dandapat
,
B. S.
,
2009
, “
Unsteady Thin Film on an Unsteady Stretching Sheet
,”
Int. J. Heat Mass Transfer
,
52
, pp.
1956
1970
.10.1016/j.ijheatmasstransfer.2008.09.036
25.
Dandapat
,
B. S.
, and
Chakraborty
,
S.
,
2010
, “
Effect of Variable Fluid Properties on Unsteady Thin-Film Flow Over a Non-Linear Stretching Sheet
,”
Int. J. Heat Mass Transfer
,
53
, pp.
5757
5763
.10.1016/j.ijheatmasstransfer.2010.08.007
26.
Dandapat
,
B. S.
, and
Singh
,
S. K.
,
2011
, “
Thin Film Flow Over a Heated Nonlinear Stretching Sheet in Presence of Uniform Transverse Magnetic Field
,”
Int. Commun. Heat Mass Transfer
,
38
, pp.
324
328
.10.1016/j.icheatmasstransfer.2010.11.009
27.
Liu
,
I.-C.
, and
Andersson
,
H. I.
,
2008
, “
Heat Transfer in a Liquid Film on an Unsteady Stretching Sheet
,”
Int. J. Therm. Sci.
,
47
, pp.
766
772
.
28.
Dutta
,
B. K.
,
Roy
,
P.
, and
Gupta
,
A. S.
,
1985
, “
Temperature Field in Flow Over a Stretching Sheet With Uniform Heat Flux
,”
Int. Commun. Heat Mass Transfer
,
12
, pp.
89
94
.10.1016/0735-1933(85)90010-7
29.
Elbashbeshy
,
E. M. A.
,
1998
, “
Heat Transfer Over a Stretching Surface With Variable Surface Heat Flux
,”
J. Phys. D: Appl. Phys.
,
31
, pp.
1951
1954
.10.1088/0022-3727/31/16/002
30.
Surma Devi
,
C. D.
,
Takhar
,
H. S.
, and
Nath
,
G.
,
1986
, “
Unsteady Three-Dimensional Boundary-Layer Flow Due to a Stretching Surface
,”
Int. J. Heat Mass Transfer
,
29
, pp.
1996
1999
.10.1016/0017-9310(86)90020-7
31.
Lakshmisha
,
K. N.
,
Venkateswaran
,
S.
, and
Nath
,
G.
,
1998
, “
Three-Dimensional Unsteady Flow With Heat and Mass Transfer Over a Continuous Stretching Surface
,”
ASME J. Heat Mass Transfer
,
110
, pp.
590
595
.10.1115/1.3250533
32.
Abd El-Aziz
,
M.
,
2009
, “
Radiation Effect on the Flow and Heat Transfer Over an Unsteady Stretching Sheet
,”
Int. Commun. Heat Mass Transfer
,
36
, pp.
521
524
.10.1016/j.icheatmasstransfer.2009.01.016
33.
Mukhopadhyay
,
S.
,
2009
, “
Effect of Thermal Radiation on Unsteady Mixed Convection Flow and Heat Transfer Over a Porous Stretching Surface in Porous Medium
,”
Int. J. Heat Mass Transfer
,
52
, pp.
3261
3265
.10.1016/j.ijheatmasstransfer.2008.12.029
34.
Pal
,
D.
,
2011
, “
Combined Effects of Non-Uniform Heat Source/Sink and Thermal Radiation on Heat Transfer Over an Unsteady Stretching Permeable Surface
,”
Commun. Nonlin. Sci. Numer. Simul.
,
16
, pp.
1890
1904
.10.1016/j.cnsns.2010.08.023
35.
Rosseland
,
S.
,
1931
,
Astrophysik und Atom-Theoretische Grundlagen
,
Springer-Verlag
,
Berlin
, pp.
41
44
.
36.
Christen
,
T.
,
Kassubek
,
F.
, and
Gati
,
R.
,
2011
, “
Radiative Heat Transfer and Effective Transport Coefficients
,”
Heat Transfer—Mathematically Modelling, Numerical Methods and Information Technology
,
A.
Belmiloudi
, ed.,
InTech
,
New York
, pp.
102
120
.
37.
Magyari
,
E.
, and
Pantokratoras
,
A.
,
2011
, “
Note on the Effect of Thermal Radiation in the Linearized Rosseland Approximation on the Heat Transfer Characteristics of Various Boundary Layer Flows
,”
Int. Commun. Heat Mass Trans.
,
38
, pp.
554
556
.10.1016/j.icheatmasstransfer.2011.03.006
38.
Abramowitz
,
M.
, and
Stegun
,
I. A.
,
1965
,
Handbook of Mathematical Functions
, Chap. 13,
Dover Publishers
,
New York
.
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