In this paper, we use the homotopy analysis method as a tool to obtain analytic approximations to the nonlinear problem of the cooling of turbine disks with a non-Newtonian viscoelastic fluid. The application of this method is executed via a polynomial exponential basis. The effects on velocity and temperature profiles with variations of the cross viscosity parameter, the Reynolds number, and the Prandtl number are discussed. A comparison with corresponding results of the perturbation method is illustrated and also, as a result of application of the homotopy analysis method, an analytic evaluation for the Nusselt number compared to the perturbation method is achieved.

References

References
1.
Bohme
,
G.
, 1987,
Non-Newtonian Fluid Mechanics (North-Holland Series in Applied Mathematics and Mechanics)
, Vol.
31
,
North-Holland
,
Amsterdam
.
2.
Akcay
,
M.
, and
Yukselen
,
M. A.
, 1999, “
Drag Reduction of a Non-Newtonian Fluid by Fluid Injection on a Moving Wall
,”
Arch. Appl. Mech.
,
69
, pp.
215
225
.
3.
Berman
,
A. S.
, 1953, “
Laminar Flow in Channels With Porous Walls
,”
J. Appl. Phys.
,
24
, pp.
1232
1235
.
4.
Yuan
,
S. W.
, and
Finkelstein
,
A. B.
, 1956, “
Laminar Pipe Flow With Injection and Suction Through a Porous Wall
,”
ASME J. Heat Transfer
,
78
, pp.
719
724
.
5.
White
,
F. M.,
Jr.
,
Barfield
,
B. F.
, and
Coglia
,
M. J.
, 1958, “
Laminar Flow in Uniformly Porous Channel
,”
ASME J. Appl. Mech.
,
80
, pp.
613
617
.
6.
Terrill
,
R. M.
, 1965, “
Laminar Flow in Uniformly Porous Channel With Large Injection
,”
Aeronaut. Q.
,
16
, pp.
320
332
.
7.
White
,
J. L.
, and
Metzner
,
A. B.
, 1965, “
Constitutive Equations for Viscoelastic Fluids With Application to Rapid External Flows
,”
AIChE J.
,
11
, pp.
324
330
.
8.
Debruge
,
L. L.
, and
Han
,
L. S.
, 1972, “
Heat Transfer in a Channel With a Porous Wall for Turbine Cooling Application
,”
ASME J. Heat Transfer
,
11
, pp.
385
390
.
9.
Kurtcebe
,
C.
, and
Erim
,
M. Z.
, 2002, “
Heat Transfer of a Non-Newtonian Viscoinelastic Fluid in an Axisymmetric Channel With a Porous Wall for Turbine Cooling Application
,”
Int. Commun. Heat Mass Transfer
,
29
, pp.
971
982
.
10.
Shin
,
S.
, 1996, “
The Effect of the Shear Rate Dependent Thermal Conductivity of Non-Newtonian Fluids on the Heat Transfer in a Pipe Flow
,”
Int. Commun. Heat Mass Transfer
,
23
, pp.
665
678
.
11.
Goldstein
,
R. J.
,
Eckert
,
E. R. G.
,
Ibele
,
W. E.
,
Patankar
,
S. V.
,
Simon
,
T. W.
,
Kuehn
,
T. H.
,
Strykowski
,
P. J.
,
Tamma
,
K. K.
,
Bar-Cohen
,
A.
,
Heberlein
,
J. V. R.
,
Davidson
,
J. H.
,
Bischof
,
J.
,
Kulacki
,
F. A.
,
Kortshagen
,
U.
, and
Garrick
,
S.
, 2001, “
Heat Transfer-A Review of 1999 Literature
,”
Int. J. Heat Mass Transfer
,
44
, pp.
3579
3699
.
12.
Goldstein
,
R. J.
,
Eckert
,
E. R. G.
,
Ibele
,
W. E.
,
Patankar
,
S. V.
,
Simon
,
T. W.
,
Kuehn
,
T. H.
,
Strykowski
,
P. J.
,
Tamma
,
K. K.
,
Heberlein
,
J. V. R.
,
Davidson
,
J. H.
,
Bishof
,
J.
,
Kulacki
,
F. A.
,
Kortshagen
,
U.
, and
Garrick
,
S.
, 2003, “
Heat Transfer-A Review of 2001 Literature
,”
Int. J. Heat Mass Transfer
,
46
, pp.
1887
1992
.
13.
Goldstein
,
R. J.
,
Eckert
,
E. R. G.
,
Ibele
,
W. E.
,
Patankar
,
S. V.
,
Simon
,
T. W.
,
Kuehn
,
T. H.
,
Strykowski
,
P. J.
,
Tamma
,
K. K.
,
Bar-Cohen
,
A.
,
Heberlein
,
J. V. R.
,
Davidson
,
J. H.
,
Bishof
,
J.
,
Kulacki
,
F. A.
,
Kortshagen
,
U.
,
Garrick
,
S.
, and
Srinivasan
,
V.
, 2005, “
Heat Transfer-A Review of 2002 Literature
,”
Int. J. Heat Mass Transfer
,
48
, pp.
819
927
.
14.
Goldstein
,
R. J.
,
Ibele
,
W. E.
,
Patankar
,
S. V.
,
Simon
,
T. W.
,
Kuehn
,
T. H.
,
Strykowski
,
P. J.
,
Tamma
,
K. K.
,
Heberlein
,
J. V. R.
,
Davidson
,
J. H.
,
Bischof
,
J.
,
Kulacki
,
F. A.
,
Kortshagen
,
U.
,
Garrick
,
S.
, and
Srinivasan
,
V.
, 2006, “
Heat Transfer-A Review of 2003 Literature
,”
Int. J. Heat Mass Transfer
,
49
, pp.
451
534
.
15.
Cleeton
,
J. P. E.
,
Kavanagh
,
R. M.
, and
Parks
,
G. T.
, 2009, “
Blade Cooling Optimisation in Humid-Air and Steam-Injected Gas Turbines
,”
Appl. Therm. Eng.
,
29
, pp.
3274
3283
.
16.
Kim
,
K. M.
,
Park
,
J. S.
,
Lee
,
D. H.
,
Lee
,
T. W.
, and
Cho
,
H. H.
, 2011, “
Analysis of Conjugated Heat Transfer, Stress and Failure in a Gas Turbine Blade With Circular Cooling Passages
,”
Eng. Failure Anal.
,
18
, pp.
1212
1222
.
17.
Liao
,
S. J.
, 1992, “
The Proposed Homotopy Analysis Technique for the Solutions of Nonlinear Problems
,” Ph.D. thesis, Shanghai Jiao Tong University, Shanghai.
18.
Liao
,
S. J.
, 1999, “
An Explicit Totally Analytic Approximate Solution for Blasius Viscous Flow Problems
,”
Int. J. Non-Linear Mech.
,
34
, pp.
759
769
.
19.
Liao
,
S. J.
, 2002, “
An Analytic Approximation of the Drag Coefficient for the Viscous Flow Past a Sphere
,”
Int. J. Non-Linear Mech.
,
37
, pp.
1
18
.
20.
Liao
,
S. J.
, 2003,
Beyond Perturbation: Introduction to Homotopy Analysis Method
,
Chapman and Hall/CRC
,
Boca Raton
.
21.
Cole
,
J. D.
, 1968,
Perturbation Methods in Applied Mathematics
,
Blaisdell
,
Waltham
.
22.
Bush
,
A. W.
, 1992,
Perturbation Methods for Engineers and Scientists
,
CRC, Library of Engineering Mathematics
,
Boca Raton
.
23.
Nayfeh
,
A. H.
, 2000,
Perturbation Methods
,
Wiley
,
New York
.
24.
Lyapunov
,
A. M.
, 1992,
General Problem on Stability of Motion
,
Taylor & Francis
,
London
.
25.
Adomian
,
G.
, 1988, “
A Review of the Decomposition Method in Applied Mathematics
,”
J. Math. Anal. Appl.
,
135
, pp.
501
544
.
26.
Adomian
,
G.
, 1991, “
A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations
,”
Comput. Math. Appl.
,
21
, pp.
101
127
.
27.
Adomian
,
G.
, 1994, “
Solution of Physical Problems by Decomposition
,”
Comput. Math. Appl.
,
27
, pp.
145
154
.
28.
Adomian
,
G.
, 1994,
Solving Frontier Problems of Physics: The Decomposition Method
,
Kluwer Academic
,
Boston
.
29.
Liao
,
S. J.
, 2003, “
An Explicit Analytic Solution to the Thomas-Fermi Equation
,”
Appl. Math. Comput.
,
144
, pp.
495
506
.
30.
Liao
,
S. J.
, 2005, “
Comparison Between the Homotopy Analysis Method and Homotopy Perturbation Method
,”
Appl. Math. Comput.
,
169
, pp.
1186
1194
.
31.
Liao
,
S. J.
,
Su
,
J.
, and
Chwang
,
A. T.
, 2006, “
Series Solutions for a Nonlinear Model of Combined Convective and Radiative Cooling of a Spherical Body
,”
Int. J. Heat Mass Transfer
,
49
, pp.
2437
2445
.
32.
Liao
,
S. J.
, and
Tan
,
Y.
, 2007, “
A General Approach to Obtain Series Solutions of Nonlinear Differential Equations
,”
Stud. Appl. Math.
,
119
, pp.
297
355
.
33.
Cheng
,
J.
,
Liao
,
S. J.
,
Mohapatra
,
R. N.
, and
Vajravelu
,
K.
, 2008, “
Series Solutions of Nano Boundary Layer Flows by Means of the Homotopy Analysis Method
,”
J. Math. Anal. Appl.
,
343
, pp.
233
245
.
34.
Liao
,
S. J.
, 2010, “
An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations
,”
Commun. Nonlinear Sci. Numer. Simulat.
,
15
, pp.
2003
2016
.
35.
Kousar
,
N.
, and
Liao
,
S. J.
, 2011, “
Unsteady Non-Similarity Boundary-Layer Flows Caused by an Impulsively Stretching Flat Sheet
,”
Nonlinear Anal.: Real World Appl.
,
12
, pp.
333
342
.
36.
Rivlin
,
R. S.
, and
Ericksen
,
J. L.
, 1955, “
Stress-Deformation Relations for Isotropic Materials
,”
J. Ration. Mech. Anal.
,
4
, pp.
323
425
.
You do not currently have access to this content.