research article flow of an eyring-powell model fluid
TRANSCRIPT
Hindawi Publishing CorporationChinese Journal of EngineeringVolume 2013 Article ID 808342 7 pageshttpdxdoiorg1011552013808342
Research ArticleFlow of an Eyring-Powell Model Fluid betweenCoaxial Cylinders with Variable Viscosity
Azad Hussain M Y Malik and Farzana Khan
Department of Mathematics Quaid-i-Azam University Islamabad 45320 Pakistan
Correspondence should be addressed to Azad Hussain azadhussainsamoteyahoocom
Received 21 July 2013 Accepted 18 August 2013
Academic Editors G Chen and S Wei-dong
Copyright copy 2013 Azad Hussain et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We consider the flow of Eyring-Powell model fluid in the annulus between two cylinders whose viscosity depends upon thetemperature We consider the steady flow in the annulus due to the motion of inner cylinder and constant pressure gradient In theproblem considered the flow is found to be remarkedly different from that for the incompressible Navier-Stokes fluid with constantviscosity An analytical solution of the nonlinear problem is obtained using homotopy analysis method The behavior of pertinentparameters is analyzed and depicted through graphs
1 Introduction
The analysis of the behaviour of the fluid motion of thenon-Newtonian fluids becomesmuch complicated and subtleas compared to Newtonian fluids due to the fact that non-Newtonian fluids do not exhibit the linear relationshipbetween stress and strain Rivlin and Ericksen [1] and Trues-dell and Noll [2] classified viscoelastic fluids with the helpof constitutive relations for the stress tensor as a function ofthe symmetric part of the velocity gradient and its higher(total) derivatives In recent years there have been severalstudies [3ndash12] on flows of non-Newtonian fluids It is awell-known fact that it is not possible to obtain a singleconstitutive equation exhibiting all properties of all non-Newtonian fluids from the available literature That is whyseveral models of non-Newtonian fluids have been proposedin the literature Eyring-Powell model fluid is one of thesemodels Eyring-Powell model was first introduced by Powelland Eyring in 1944 However the literature survey indicatesthat very low energy has been devoted to the flows of Eyring-Powell model fluid with variable viscosity Massoudi andChristie [13] have considered the effects of variable viscosityand viscous dissipation on the flow of a third grade fluidin a uniform pipe Massoudi and Christie [13] found thenumerical solutions with the help of straight forward finite
difference method They also discussed that the flow of afluid-solid mixture is very complicated and may depend onmany variables such as physical properties of each phaseand size and shape of solid particles Later on the influenceof constant and space dependent viscosity on the flow of athird grade fluid in a pipe has been discussed analyticallyby Hayat et al [14] The approximate and analytical solutionof non-Newtonian fluid with variable viscosity has beenanalyzed by Yurusoy and Pakdermirli [15] and Pakdemirliand Yilbas [16] The pipe flow of non-Newtonian fluid withvariable viscosity keeping no slip and partial slip has beendiscussed analytically by Nadeem and Ali [17] and Nadeemet al [18] More recently Nadeem and Akbar [19] studiedthe effects of temperature dependent viscosity on peristalticflow of a Jeffrey-six constant fluid in a uniform vertical tubeThe main aim of the present study is to venture furtherin the regime of Eyring-Powell model fluid with variableviscosity To the best of the authors knowledge no attempthas been made to investigate Eyring-Powell model fluid inthe annulus between two cylinders whose viscosity dependsupon the temperature The governing equations for Eyring-Powell model fluid are formulated considering cylindricalcoordinates system The equations are simplified using theassumptions of long wave length and low Reynolds numberapproximation The obtained non-linear problem is solved
2 Chinese Journal of Engineering
using homotopy analysis method [20ndash28] The effects ofthe emerging parameters are analyzed and depicted throughgraphs
2 Mathematical Model
The constitutive equation for a Cauchy stress in an Eyring-Powell model fluid is given by
S = 120583nabla119881 +
1
120573
sinhminus1 (1
119888
nabla119881)
sinhminus1 (1
119888
nabla119881) sim
1
119888
nabla119881 minus
1
6
(
1
119888
nabla119881)
3
1003816100381610038161003816100381610038161003816
1
119888
nabla119881
1003816100381610038161003816100381610038161003816
≪ 1
(1)
where 119881 is the velocity S is the Cauchy stress tensor 120583 isthe coefficient of shear viscosity and 120573 and 119888 are the materialconstants We take the velocity and stress as
V (119903) = (
0
0
V) S (119903) =
[
[
119878119903119903
119878119903120579
119878119903119911
119878120579119903
119878120579120579
119878120579119911
119878119911119903
119878119911120579
119878119911119911
]
]
(2)
3 Physical Model
Consider the steady flow of an Eyring-Powell model fluidwith variable temperature dependent viscosity between coax-ial cylindersThemotion is caused due to a constant pressuregradient and by themotion of the inner cylinder parallel to itslength whereas the outer cylinder is kept stationaryThe heattransfer analysis is also taken into accountThe dimensionlessproblem which can describe the flow is
120583
119903
119889V119889119903
+
119872
119903
119889V119889119903
+ 120583
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861 = 0
120583Γ(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
= 0
(3)
V (119903) = 1 120579 (119903) = 1 119903 = 1
V (119903) = 0 120579 (119903) = 0 119903 = 119887
(4)
whence
119903 =
119903
119877
Γ =
120583lowast
119881
2
0
119896 (120579119898
minus 120579119908
)
120579 =
(120579 minus 120579119908
)
(120579119898
minus 120579119908
)
119911 =
119881
2
0
119877
2
119888
2
1198621
=
120597119901
120597119911
119872 =
1
120573119888120583lowast
119870 =
119881
2
0
119877
2
119888
2
119861 =
1198621
119877
2
120583lowast
1198810
V =
VV0
120583 =
120583
120583lowast
(5)
where 120583lowast
120579119898
1198810
and Γ are respectively the reference viscos-ity a reference temperature (the bulk mean fluid tempera-ture) and reference velocity Γ is related to the Prandtl numberand Eckert number
4 Series Solutions for Reynoldsrsquo Model
Here the viscosity is expressed in the form
120583 = 119890
minus119875120579 (6)
which by Maclaurinrsquos series can be written as
120583 = 1 minus 119875120579 + 119874 (120579
2
) (7)
Note that119872 = 0 corresponds to the case of constant viscosityInvoking the above equation into (3) one has
119872
119903
119889V119889119903
+
1
119903
119889V119889119903
minus
119875120579
119903
119889V119889119903
+
119889
2V119889119903
2
minus 119875120579
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861 = 0
Γ(
119889V119889119903
)
2
minus Γ119875120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
= 0
(8)
For HAM solution we choose the following initial guesses
V0
(119903) =
(119903 minus 119887)
(1 minus 119887)
1205790
(119903) =
(119903 minus 119887)
(1 minus 119887)
(9)
The auxiliary linear operators are in the form
poundV119903 (V) = V10158401015840 (10)
pound120579119903
(120579) = 120579
10158401015840 (11)
which satisfy
LV119903 (1198601 + 1198611
119903) = 0 L120579119903
(1198602
+ 1198612
119903) = 0 (12)
where 1198601
1198602
1198611
and 1198612
are the constantsIf 119901 isin [0 1] is an embedding parameter and ℎV and ℎ
120579
areauxiliary parameters then the problems at the zero and 119898thorder are respectively given by
(1 minus 119901)LV [V (119903 119901) minus V0
(119903)] = 119901ℎV119873V [V (119903 119901) 120579 (119903 119901)]
(13)
(1 minus 119901)L120579
[120579 (119903 119901) minus 1205790
(119903)] = 119901ℎ120579
119873120579
[V (119903 119901) 120579 (119903 119901)]
(14)
LV [V119898 (119903) minus 120594119898
V119898minus1
(119903)] = ℎV119877V (119903) (15)
L120579
[120579119898
(119903) minus 120594119898
120579119898minus1
(119903)] = ℎ120579
119877120579
(119903) (16)
V (119903 119901) = 120579 (119903 119901) = 1 119903 = 1 (17)
V (119903 119901) = 120579 (119903 119901) = 0 119903 = 119887 (18)
The boundary conditions at the 119898th order are
V119898
(119903 119901) = 120579119898
(119903 119901) = 0 119903 = 1
V119898
(119903 119901) = 120579119898
(119903 119901) = 0 119903 = 119887
(19)
Chinese Journal of Engineering 3
In (11)ndash(13)
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119875120579
119903
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119889
2V119889119903
2
minus 119875120579
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ(
119889V119889119903
)
2
minus Γ119875120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
(20)
119877V = minus
119875
119903
119898minus1
sum
119896=0
V1015840119898minus1minus119896
120579119896
+
1
119903
V1015840119898minus1
+
119872
119903
V1015840119898minus1
+ V10158401015840119898minus1
minus 119875
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
120579119896
+ 119872V10158401015840119898minus1
minus 3119870
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V10158401015840119897
minus
119870
119903
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897
minus 119861
119877120579
= Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119875
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
120579
1015840
119897
+ 119872Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119870
119898minus1
sum
119896=0
119896
sum
119897=0
119897
sum
119904=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897minus119904
V1015840119904
minus
1
119903
120579
1015840
119898minus1
minus 120579
10158401015840
119898minus1
(21)
By Mathematica the solutions of (21) can be written as
V119898
(119903) =
3119898
sum
119899=0
119886119898119899
119903
119899
119898 ge 0
120579119898
(119903) =
3119898+1
sum
119899=0
119889119898119899
119903
119899
119898 ge 0
(22)
where 119886119898119899
and 119889119898119899
are constants which can be determinedon substituting (22) into (15) and (16)
5 Series Solutions for Vogelrsquos Model
Here
120583 = 120583lowast
exp [
119860
(119905 + 120579)
minus 120579119908
] (23)
0
minus02
minus04
minus06
minus08
minus12 minus1 minus08 minus06 minus04 minus02 0
V998400998400
h
Γ = 01 K = 01M = 01 P = 001 and B = minus1
Figure 1 ℎ-curve for Reynoldsrsquo model for velocity profile
15
125
1
075
05
025
0 02 04 06 08 1 12 14
Q998400998400
h
Γ = 01 K = 01M = 01 P = 001 and B = minus1
Figure 2 ℎ-curve for Reynoldsrsquo model for temperature profile
V998400998400
h
3
2
1
minus1
minus2
minus2 minus15 minus1 minus05 0
0
Γ = 01 K = 01M = 01 s = minus01 A = 00001 t = 01 and B = minus01
Figure 3 ℎ-curve for Vogelrsquos model for velocity profile
which by Maclaurinrsquos series reduces to
120583 =
119861
119904
(1 minus
119860120579
119905
2
) (24)
4 Chinese Journal of EngineeringQ998400998400
h
175
125
15
1
075
05
025
0 02 04 06 08 1 12
B = minus01 q = minus01M = 01 A = 01 s = 01
Γ = 01 t = 01 z = 01 and u = 01
Figure 4 ℎ-curve for Vogelrsquos model for temperature profile
B = 100
B = 105
B = 110
40
30
20
10
0
Q(r)
1 12 14 16 18 2
r
Γ = 01 K = 01M = 01 and P = 001
Figure 5 Temperature profile for Reynoldsrsquo model for differentvalues of 119861
r
1
08
06
04
02
1 12 14 16 18 2
V(r)
B = minus1
B = minus2
B = minus3
Γ = 01 K = 01M = 01 and P = 001
Figure 6 Velocity profile for Reynoldsrsquo model for different values of119861
r
1
08
06
04
02
V(r)
1 12 14 16 18 2
P = 010
P = 020
P = 030
B = minus1 K = 01M = 01 and Γ = 01
Figure 7 Velocity profile for Reynoldsrsquo model for different values of119875
r
1
08
06
04
02
Q(r)
Γ = minus1Γ = minus2
Γ = minus3
1 12 14 16 18 2
B = minus1 K = 01M = 01 and P = 001
Figure 8 Temperature profile for Reynoldsrsquo model for differentvalues of Γ
Invoking the above expressions (1) become
minus
119860119861120579
119903119904119905
2
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119861
119904
119889
2V119889119903
2
minus
119860119861
119904119905
2
120579
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861 = 0
Γ(
119889V119889119903
)
2
minus Γ
119860119861
119904119905
2
120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
= 0
(25)
Chinese Journal of Engineering 5
r
Q(r)
1
08
06
04
02
1 12 14 16 18 2
t = 00065t = 00070
t = 00085
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 9 Temperature profile for Vogelrsquos model for different valuesof 119905
t = 00083
t = 00084
t = 00085
12
1
08
06
04
02
1 12 14 16 18 2
V(r)
r
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 10 Velocity profile for Vogelrsquos model for different values of 119905
With the following initial guesses and auxiliary linear opera-tors
V0V (119903) =
(119903 minus 119887)
(1 minus 119887)
1205790V (119903) =
(119903 minus 119887)
(1 minus 119887)
poundVV (V) = V10158401015840 pound120579V (120579) = 120579
10158401015840
(26)
Γ = minus100
Γ = minus200Γ = minus300
1
08
06
04
02
1 12 14 16 18 2
Q(r)
r
B = minus01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 11 Temperature profile for Vogelrsquos model for different valuesof Γ
the 119898th-order deformation problems are
poundV119903 [V119898 (119903) minus 120594119898
V119898minus1
(119903)] = ℏV119877V119903 (119903)
pound120579119903
[120579119898
(119903) minus 120594119898
120579119898minus1
(119903)] = ℏ120579
119877120579119903
(119903)
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861120579
119903119904119905
2
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119861
119904
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus
119860119861
119904119905
2
120579
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ(
119889V119889119903
)
2
minus Γ
119860119861
119904119905
2
120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861
119903119904119905
2
119898minus1
sum
119896=0
V1015840119898minus1minus119896
120579119896
+
1
119903
V1015840119898minus1
+
119872
119903
V1015840119898minus1
+
119861
119904
V10158401015840119898minus1
minus
119860119861
119904119905
2
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
120579119896
+ 119872V10158401015840119898minus1
minus 3119870
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V10158401015840119897
6 Chinese Journal of EngineeringV(r)
r
3
25
2
15
1
05
1 12 14 16 18 2
s = 300
s = 500
s = 900
Γ = 01 K = 01M = 01 B = minus01 A = 00001 and t = 01
Figure 12 Velocity profile for Vogelrsquos model for different values of119904
minus
119870
119903
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
V1015840119896
minus Γ
119860119861
119904119905
2
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
120579
1015840
119897
+ 119872Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119870
119898minus1
sum
119896=0
119896
sum
119897=0
119897
sum
119904=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897minus119904
V1015840119904
minus
1
119903
120579
1015840
119898minus1
minus 120579
10158401015840
119898minus1
(27)
The expressions of V119898
and 120579119898
are finally given by
V119898
(119903) =
3119898
sum
119899=0
119886
1015840
119898119899
119903
119899
119898 ge 0
120579119898
(119903) =
3119898+1
sum
119899=0
119889
1015840
119898119899
119903
119899
119898 ge 0
(28)
6 Graphical Results and Discussion
In order to report the convergence of the obtained seriessolutions and the effects of sundry parameters in the present
1
08
06
04
02
1 12 14 16 18 2
r
V(r)
B = minus011
B = minus012B = minus013
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 13 Velocity profile for Vogelrsquos model for different values of119861
investigation we plotted Figures 1ndash13 Figures 1ndash4 are pre-pared to see the convergence region Figures 1 and 2 cor-respond to Reynoldsrsquo model whereas Figures 3 and 4 relateto Vogelrsquos model Figure 5 shows the temperature variationfor different values of 119861 for Reynoldsrsquo model It can be seenthat temperature decreases as 119861 increases Figure 6 depictsthe velocity variation for Reynoldsrsquo model for different valuesof 119861 Velocity also decreases as 119861 increases Figure 7 showsthe velocity variation for different values of 119875 for Reynoldsrsquomodel It can be seen that velocity increases as 119875 increasesFigure 8 is plotted in order to see the temperature variationfor Reynoldsrsquomodel for different values of Γ it is depicted thattemperature increases as Γ increases Figures 9ndash13 correspondto Vogelrsquos model Figure 9 depicts temperature variationfor Vogelrsquos model for different values of 119905 It is seen thattemperature increases as 119905 increases Figure 10 shows thevelocity variation for Vogelrsquos model for different values of 119905It is observed that velocity decreases as 119905 increases Figure 11is prepared to observe the temperature variation for Vogelrsquosmodel for different values of Γ It is observed that temperaturedecreases as Γ increases Figure 12 is plotted to see thethe velocity variation for Vogelrsquos model for different valuesof 119904 It is observed that velocity decreases as 119904 increasesFigure 13 depicts the velocity variation for Vogelrsquos model fordifferent values of 119861 It is observed that velocity decreases as119861 increases
7 Conclusions
In this paper we consider the flow of Eyring-Powell modelfluid in the annulus between two cylinders whose viscositydepends upon the temperature We discussed the steadyflow in the annulus due to the motion of inner cylinderand constant pressure gradient In the problem consideredthe flow is found to be remarkedly different from thatfor the incompressible Navier-Stokes fluid with constantviscosity The behavior of pertinent parameters is analyzed
Chinese Journal of Engineering 7
and depicted through graphs Using usual similarity transfor-mations the governing equations have been transformed intonon-linear ordinary differential equations The highly non-linear problem is then solved by homotopy analysis methodEffects of the various parameters on velocity and temperatureprofiles are examined
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R S Rivlin and J L Ericksen ldquoStress deformation relations forisotropic materialsrdquo Rational Mechanics and Analysis Journalvol 4 pp 323ndash425 1955
[2] C Truesdell and W Noll The Non-Linear Field Theories ofMechanics Springer New York NY USA 2nd edition 1992
[3] K R Rajagopal ldquoA note on unsteady unidirectional flows ofa non-Newtonian fluidrdquo International Journal of Non-LinearMechanics vol 17 no 5-6 pp 369ndash373 1982
[4] K R Rajagopal and A S Gupta ldquoAn exact solution for theflow of a non-newtonian fluid past an infinite porous platerdquoMeccanica vol 19 no 2 pp 158ndash160 1984
[5] K R Rajagopal and R K Bhatnagar ldquoExact solutions for somesimple flows of an Oldroyd-B fluidrdquo Acta Mechanica vol 113no 1ndash4 pp 233ndash239 1995
[6] K R Rajagopal ldquoOn the creeping flow of the second-orderfluidrdquo Journal of Non-Newtonian Fluid Mechanics vol 15 no2 pp 239ndash246 1984
[7] K R Rajagopal ldquoLongitudinal and torsional oscillations of arod in a non-Newtonian fluidrdquoActaMechanica vol 49 no 3-4pp 281ndash285 1983
[8] A M Benharbit and A M Siddiqui ldquoCertain solutions of theequations of the planarmotion of a second grade fluid for steadyand unsteady casesrdquo Acta Mechanica vol 94 no 1-2 pp 85ndash961992
[9] T Hayat S Asghar and A M Siddiqui ldquoPeriodic unsteadyflows of a non-Newtonian fluidrdquo Acta Mechanica vol 131 no3-4 pp 169ndash175 1998
[10] A M Siddiqui T Hayat and S Asghar ldquoPeriodic flows of anon-Newtonian fluid between two parallel platesrdquo InternationalJournal of Non-Linear Mechanics vol 34 no 5 pp 895ndash8991999
[11] T Hayat S Asghar and A M Siddiqui ldquoOn the moment of aplane disk in a non-Newtonian fluidrdquo Acta Mechanica vol 136no 3 pp 125ndash131 1999
[12] T Hayat S Asghar and A M Siddiqui ldquoSome unsteadyunidirectional flows of a non-Newtonian fluidrdquo InternationalJournal of Engineering Science vol 38 no 3 pp 337ndash346 2000
[13] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[14] T Hayat R Ellahi and S Asghar ldquoThe influence of variableviscosity and viscous dissipation on the non-Newtonian flowan analytical solutionrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 12 no 3 pp 300ndash313 2007
[15] M Yurusoy and M Pakdermirli ldquoApproximate analytical solu-tions for flow of a third grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[16] M Pakdemirli and B S Yilbas ldquoEntropy generation for pipeflow of a third grade fluid with Vogel model viscosityrdquo Interna-tional Journal of Non-Linear Mechanics vol 41 no 3 pp 432ndash437 2006
[17] S Nadeem and M Ali ldquoAnalytical solutions for pipe flow ofa fourth grade fluid with Reynold and Vogelrsquos models of vis-cositiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2073ndash2090 2009
[18] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[19] S Nadeem and N S Akbar ldquoEffects of temperature dependentviscosity on peristaltic flow of a Jeffrey-six constant fluid ina non-uniform vertical tuberdquo Communications in NonlinearScience andNumerical Simulation vol 15 no 12 pp 3950ndash39642010
[20] M Y Malik A Hussain and S Nadeem ldquoAnalytical treatmentof an oldroyd 8-constant fluid between coaxial cylinders withvariable viscosityrdquo Communications in Theoretical Physics vol56 no 5 pp 933ndash938 2011
[21] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[22] M Y Malik A Hussain S Nadeem and T Hayat ldquoFlow ofa third grade fluid between coaxial cylinders with variableviscosityrdquo Zeitschrift fur Naturforschung A vol 64 no 9-10 pp588ndash596 2009
[23] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method CRC Press Boca Raton Fla USA 2003
[24] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[25] S J Liao ldquoAn analytic solution of unsteady boundary-layerflows caused by an impulsively stretching platerdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 11 no3 pp 326ndash339 2006
[26] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006
[27] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[28] S Abbasbandy Y Tan and S J Liao ldquoNewton-homotopyanalysis method for nonlinear equationsrdquo Applied Mathematicsand Computation vol 188 no 2 pp 1794ndash1800 2007
International Journal of
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DistributedSensor Networks
International Journal of
2 Chinese Journal of Engineering
using homotopy analysis method [20ndash28] The effects ofthe emerging parameters are analyzed and depicted throughgraphs
2 Mathematical Model
The constitutive equation for a Cauchy stress in an Eyring-Powell model fluid is given by
S = 120583nabla119881 +
1
120573
sinhminus1 (1
119888
nabla119881)
sinhminus1 (1
119888
nabla119881) sim
1
119888
nabla119881 minus
1
6
(
1
119888
nabla119881)
3
1003816100381610038161003816100381610038161003816
1
119888
nabla119881
1003816100381610038161003816100381610038161003816
≪ 1
(1)
where 119881 is the velocity S is the Cauchy stress tensor 120583 isthe coefficient of shear viscosity and 120573 and 119888 are the materialconstants We take the velocity and stress as
V (119903) = (
0
0
V) S (119903) =
[
[
119878119903119903
119878119903120579
119878119903119911
119878120579119903
119878120579120579
119878120579119911
119878119911119903
119878119911120579
119878119911119911
]
]
(2)
3 Physical Model
Consider the steady flow of an Eyring-Powell model fluidwith variable temperature dependent viscosity between coax-ial cylindersThemotion is caused due to a constant pressuregradient and by themotion of the inner cylinder parallel to itslength whereas the outer cylinder is kept stationaryThe heattransfer analysis is also taken into accountThe dimensionlessproblem which can describe the flow is
120583
119903
119889V119889119903
+
119872
119903
119889V119889119903
+ 120583
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861 = 0
120583Γ(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
= 0
(3)
V (119903) = 1 120579 (119903) = 1 119903 = 1
V (119903) = 0 120579 (119903) = 0 119903 = 119887
(4)
whence
119903 =
119903
119877
Γ =
120583lowast
119881
2
0
119896 (120579119898
minus 120579119908
)
120579 =
(120579 minus 120579119908
)
(120579119898
minus 120579119908
)
119911 =
119881
2
0
119877
2
119888
2
1198621
=
120597119901
120597119911
119872 =
1
120573119888120583lowast
119870 =
119881
2
0
119877
2
119888
2
119861 =
1198621
119877
2
120583lowast
1198810
V =
VV0
120583 =
120583
120583lowast
(5)
where 120583lowast
120579119898
1198810
and Γ are respectively the reference viscos-ity a reference temperature (the bulk mean fluid tempera-ture) and reference velocity Γ is related to the Prandtl numberand Eckert number
4 Series Solutions for Reynoldsrsquo Model
Here the viscosity is expressed in the form
120583 = 119890
minus119875120579 (6)
which by Maclaurinrsquos series can be written as
120583 = 1 minus 119875120579 + 119874 (120579
2
) (7)
Note that119872 = 0 corresponds to the case of constant viscosityInvoking the above equation into (3) one has
119872
119903
119889V119889119903
+
1
119903
119889V119889119903
minus
119875120579
119903
119889V119889119903
+
119889
2V119889119903
2
minus 119875120579
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861 = 0
Γ(
119889V119889119903
)
2
minus Γ119875120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
= 0
(8)
For HAM solution we choose the following initial guesses
V0
(119903) =
(119903 minus 119887)
(1 minus 119887)
1205790
(119903) =
(119903 minus 119887)
(1 minus 119887)
(9)
The auxiliary linear operators are in the form
poundV119903 (V) = V10158401015840 (10)
pound120579119903
(120579) = 120579
10158401015840 (11)
which satisfy
LV119903 (1198601 + 1198611
119903) = 0 L120579119903
(1198602
+ 1198612
119903) = 0 (12)
where 1198601
1198602
1198611
and 1198612
are the constantsIf 119901 isin [0 1] is an embedding parameter and ℎV and ℎ
120579
areauxiliary parameters then the problems at the zero and 119898thorder are respectively given by
(1 minus 119901)LV [V (119903 119901) minus V0
(119903)] = 119901ℎV119873V [V (119903 119901) 120579 (119903 119901)]
(13)
(1 minus 119901)L120579
[120579 (119903 119901) minus 1205790
(119903)] = 119901ℎ120579
119873120579
[V (119903 119901) 120579 (119903 119901)]
(14)
LV [V119898 (119903) minus 120594119898
V119898minus1
(119903)] = ℎV119877V (119903) (15)
L120579
[120579119898
(119903) minus 120594119898
120579119898minus1
(119903)] = ℎ120579
119877120579
(119903) (16)
V (119903 119901) = 120579 (119903 119901) = 1 119903 = 1 (17)
V (119903 119901) = 120579 (119903 119901) = 0 119903 = 119887 (18)
The boundary conditions at the 119898th order are
V119898
(119903 119901) = 120579119898
(119903 119901) = 0 119903 = 1
V119898
(119903 119901) = 120579119898
(119903 119901) = 0 119903 = 119887
(19)
Chinese Journal of Engineering 3
In (11)ndash(13)
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119875120579
119903
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119889
2V119889119903
2
minus 119875120579
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ(
119889V119889119903
)
2
minus Γ119875120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
(20)
119877V = minus
119875
119903
119898minus1
sum
119896=0
V1015840119898minus1minus119896
120579119896
+
1
119903
V1015840119898minus1
+
119872
119903
V1015840119898minus1
+ V10158401015840119898minus1
minus 119875
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
120579119896
+ 119872V10158401015840119898minus1
minus 3119870
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V10158401015840119897
minus
119870
119903
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897
minus 119861
119877120579
= Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119875
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
120579
1015840
119897
+ 119872Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119870
119898minus1
sum
119896=0
119896
sum
119897=0
119897
sum
119904=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897minus119904
V1015840119904
minus
1
119903
120579
1015840
119898minus1
minus 120579
10158401015840
119898minus1
(21)
By Mathematica the solutions of (21) can be written as
V119898
(119903) =
3119898
sum
119899=0
119886119898119899
119903
119899
119898 ge 0
120579119898
(119903) =
3119898+1
sum
119899=0
119889119898119899
119903
119899
119898 ge 0
(22)
where 119886119898119899
and 119889119898119899
are constants which can be determinedon substituting (22) into (15) and (16)
5 Series Solutions for Vogelrsquos Model
Here
120583 = 120583lowast
exp [
119860
(119905 + 120579)
minus 120579119908
] (23)
0
minus02
minus04
minus06
minus08
minus12 minus1 minus08 minus06 minus04 minus02 0
V998400998400
h
Γ = 01 K = 01M = 01 P = 001 and B = minus1
Figure 1 ℎ-curve for Reynoldsrsquo model for velocity profile
15
125
1
075
05
025
0 02 04 06 08 1 12 14
Q998400998400
h
Γ = 01 K = 01M = 01 P = 001 and B = minus1
Figure 2 ℎ-curve for Reynoldsrsquo model for temperature profile
V998400998400
h
3
2
1
minus1
minus2
minus2 minus15 minus1 minus05 0
0
Γ = 01 K = 01M = 01 s = minus01 A = 00001 t = 01 and B = minus01
Figure 3 ℎ-curve for Vogelrsquos model for velocity profile
which by Maclaurinrsquos series reduces to
120583 =
119861
119904
(1 minus
119860120579
119905
2
) (24)
4 Chinese Journal of EngineeringQ998400998400
h
175
125
15
1
075
05
025
0 02 04 06 08 1 12
B = minus01 q = minus01M = 01 A = 01 s = 01
Γ = 01 t = 01 z = 01 and u = 01
Figure 4 ℎ-curve for Vogelrsquos model for temperature profile
B = 100
B = 105
B = 110
40
30
20
10
0
Q(r)
1 12 14 16 18 2
r
Γ = 01 K = 01M = 01 and P = 001
Figure 5 Temperature profile for Reynoldsrsquo model for differentvalues of 119861
r
1
08
06
04
02
1 12 14 16 18 2
V(r)
B = minus1
B = minus2
B = minus3
Γ = 01 K = 01M = 01 and P = 001
Figure 6 Velocity profile for Reynoldsrsquo model for different values of119861
r
1
08
06
04
02
V(r)
1 12 14 16 18 2
P = 010
P = 020
P = 030
B = minus1 K = 01M = 01 and Γ = 01
Figure 7 Velocity profile for Reynoldsrsquo model for different values of119875
r
1
08
06
04
02
Q(r)
Γ = minus1Γ = minus2
Γ = minus3
1 12 14 16 18 2
B = minus1 K = 01M = 01 and P = 001
Figure 8 Temperature profile for Reynoldsrsquo model for differentvalues of Γ
Invoking the above expressions (1) become
minus
119860119861120579
119903119904119905
2
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119861
119904
119889
2V119889119903
2
minus
119860119861
119904119905
2
120579
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861 = 0
Γ(
119889V119889119903
)
2
minus Γ
119860119861
119904119905
2
120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
= 0
(25)
Chinese Journal of Engineering 5
r
Q(r)
1
08
06
04
02
1 12 14 16 18 2
t = 00065t = 00070
t = 00085
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 9 Temperature profile for Vogelrsquos model for different valuesof 119905
t = 00083
t = 00084
t = 00085
12
1
08
06
04
02
1 12 14 16 18 2
V(r)
r
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 10 Velocity profile for Vogelrsquos model for different values of 119905
With the following initial guesses and auxiliary linear opera-tors
V0V (119903) =
(119903 minus 119887)
(1 minus 119887)
1205790V (119903) =
(119903 minus 119887)
(1 minus 119887)
poundVV (V) = V10158401015840 pound120579V (120579) = 120579
10158401015840
(26)
Γ = minus100
Γ = minus200Γ = minus300
1
08
06
04
02
1 12 14 16 18 2
Q(r)
r
B = minus01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 11 Temperature profile for Vogelrsquos model for different valuesof Γ
the 119898th-order deformation problems are
poundV119903 [V119898 (119903) minus 120594119898
V119898minus1
(119903)] = ℏV119877V119903 (119903)
pound120579119903
[120579119898
(119903) minus 120594119898
120579119898minus1
(119903)] = ℏ120579
119877120579119903
(119903)
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861120579
119903119904119905
2
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119861
119904
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus
119860119861
119904119905
2
120579
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ(
119889V119889119903
)
2
minus Γ
119860119861
119904119905
2
120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861
119903119904119905
2
119898minus1
sum
119896=0
V1015840119898minus1minus119896
120579119896
+
1
119903
V1015840119898minus1
+
119872
119903
V1015840119898minus1
+
119861
119904
V10158401015840119898minus1
minus
119860119861
119904119905
2
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
120579119896
+ 119872V10158401015840119898minus1
minus 3119870
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V10158401015840119897
6 Chinese Journal of EngineeringV(r)
r
3
25
2
15
1
05
1 12 14 16 18 2
s = 300
s = 500
s = 900
Γ = 01 K = 01M = 01 B = minus01 A = 00001 and t = 01
Figure 12 Velocity profile for Vogelrsquos model for different values of119904
minus
119870
119903
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
V1015840119896
minus Γ
119860119861
119904119905
2
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
120579
1015840
119897
+ 119872Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119870
119898minus1
sum
119896=0
119896
sum
119897=0
119897
sum
119904=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897minus119904
V1015840119904
minus
1
119903
120579
1015840
119898minus1
minus 120579
10158401015840
119898minus1
(27)
The expressions of V119898
and 120579119898
are finally given by
V119898
(119903) =
3119898
sum
119899=0
119886
1015840
119898119899
119903
119899
119898 ge 0
120579119898
(119903) =
3119898+1
sum
119899=0
119889
1015840
119898119899
119903
119899
119898 ge 0
(28)
6 Graphical Results and Discussion
In order to report the convergence of the obtained seriessolutions and the effects of sundry parameters in the present
1
08
06
04
02
1 12 14 16 18 2
r
V(r)
B = minus011
B = minus012B = minus013
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 13 Velocity profile for Vogelrsquos model for different values of119861
investigation we plotted Figures 1ndash13 Figures 1ndash4 are pre-pared to see the convergence region Figures 1 and 2 cor-respond to Reynoldsrsquo model whereas Figures 3 and 4 relateto Vogelrsquos model Figure 5 shows the temperature variationfor different values of 119861 for Reynoldsrsquo model It can be seenthat temperature decreases as 119861 increases Figure 6 depictsthe velocity variation for Reynoldsrsquo model for different valuesof 119861 Velocity also decreases as 119861 increases Figure 7 showsthe velocity variation for different values of 119875 for Reynoldsrsquomodel It can be seen that velocity increases as 119875 increasesFigure 8 is plotted in order to see the temperature variationfor Reynoldsrsquomodel for different values of Γ it is depicted thattemperature increases as Γ increases Figures 9ndash13 correspondto Vogelrsquos model Figure 9 depicts temperature variationfor Vogelrsquos model for different values of 119905 It is seen thattemperature increases as 119905 increases Figure 10 shows thevelocity variation for Vogelrsquos model for different values of 119905It is observed that velocity decreases as 119905 increases Figure 11is prepared to observe the temperature variation for Vogelrsquosmodel for different values of Γ It is observed that temperaturedecreases as Γ increases Figure 12 is plotted to see thethe velocity variation for Vogelrsquos model for different valuesof 119904 It is observed that velocity decreases as 119904 increasesFigure 13 depicts the velocity variation for Vogelrsquos model fordifferent values of 119861 It is observed that velocity decreases as119861 increases
7 Conclusions
In this paper we consider the flow of Eyring-Powell modelfluid in the annulus between two cylinders whose viscositydepends upon the temperature We discussed the steadyflow in the annulus due to the motion of inner cylinderand constant pressure gradient In the problem consideredthe flow is found to be remarkedly different from thatfor the incompressible Navier-Stokes fluid with constantviscosity The behavior of pertinent parameters is analyzed
Chinese Journal of Engineering 7
and depicted through graphs Using usual similarity transfor-mations the governing equations have been transformed intonon-linear ordinary differential equations The highly non-linear problem is then solved by homotopy analysis methodEffects of the various parameters on velocity and temperatureprofiles are examined
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R S Rivlin and J L Ericksen ldquoStress deformation relations forisotropic materialsrdquo Rational Mechanics and Analysis Journalvol 4 pp 323ndash425 1955
[2] C Truesdell and W Noll The Non-Linear Field Theories ofMechanics Springer New York NY USA 2nd edition 1992
[3] K R Rajagopal ldquoA note on unsteady unidirectional flows ofa non-Newtonian fluidrdquo International Journal of Non-LinearMechanics vol 17 no 5-6 pp 369ndash373 1982
[4] K R Rajagopal and A S Gupta ldquoAn exact solution for theflow of a non-newtonian fluid past an infinite porous platerdquoMeccanica vol 19 no 2 pp 158ndash160 1984
[5] K R Rajagopal and R K Bhatnagar ldquoExact solutions for somesimple flows of an Oldroyd-B fluidrdquo Acta Mechanica vol 113no 1ndash4 pp 233ndash239 1995
[6] K R Rajagopal ldquoOn the creeping flow of the second-orderfluidrdquo Journal of Non-Newtonian Fluid Mechanics vol 15 no2 pp 239ndash246 1984
[7] K R Rajagopal ldquoLongitudinal and torsional oscillations of arod in a non-Newtonian fluidrdquoActaMechanica vol 49 no 3-4pp 281ndash285 1983
[8] A M Benharbit and A M Siddiqui ldquoCertain solutions of theequations of the planarmotion of a second grade fluid for steadyand unsteady casesrdquo Acta Mechanica vol 94 no 1-2 pp 85ndash961992
[9] T Hayat S Asghar and A M Siddiqui ldquoPeriodic unsteadyflows of a non-Newtonian fluidrdquo Acta Mechanica vol 131 no3-4 pp 169ndash175 1998
[10] A M Siddiqui T Hayat and S Asghar ldquoPeriodic flows of anon-Newtonian fluid between two parallel platesrdquo InternationalJournal of Non-Linear Mechanics vol 34 no 5 pp 895ndash8991999
[11] T Hayat S Asghar and A M Siddiqui ldquoOn the moment of aplane disk in a non-Newtonian fluidrdquo Acta Mechanica vol 136no 3 pp 125ndash131 1999
[12] T Hayat S Asghar and A M Siddiqui ldquoSome unsteadyunidirectional flows of a non-Newtonian fluidrdquo InternationalJournal of Engineering Science vol 38 no 3 pp 337ndash346 2000
[13] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[14] T Hayat R Ellahi and S Asghar ldquoThe influence of variableviscosity and viscous dissipation on the non-Newtonian flowan analytical solutionrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 12 no 3 pp 300ndash313 2007
[15] M Yurusoy and M Pakdermirli ldquoApproximate analytical solu-tions for flow of a third grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[16] M Pakdemirli and B S Yilbas ldquoEntropy generation for pipeflow of a third grade fluid with Vogel model viscosityrdquo Interna-tional Journal of Non-Linear Mechanics vol 41 no 3 pp 432ndash437 2006
[17] S Nadeem and M Ali ldquoAnalytical solutions for pipe flow ofa fourth grade fluid with Reynold and Vogelrsquos models of vis-cositiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2073ndash2090 2009
[18] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[19] S Nadeem and N S Akbar ldquoEffects of temperature dependentviscosity on peristaltic flow of a Jeffrey-six constant fluid ina non-uniform vertical tuberdquo Communications in NonlinearScience andNumerical Simulation vol 15 no 12 pp 3950ndash39642010
[20] M Y Malik A Hussain and S Nadeem ldquoAnalytical treatmentof an oldroyd 8-constant fluid between coaxial cylinders withvariable viscosityrdquo Communications in Theoretical Physics vol56 no 5 pp 933ndash938 2011
[21] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[22] M Y Malik A Hussain S Nadeem and T Hayat ldquoFlow ofa third grade fluid between coaxial cylinders with variableviscosityrdquo Zeitschrift fur Naturforschung A vol 64 no 9-10 pp588ndash596 2009
[23] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method CRC Press Boca Raton Fla USA 2003
[24] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[25] S J Liao ldquoAn analytic solution of unsteady boundary-layerflows caused by an impulsively stretching platerdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 11 no3 pp 326ndash339 2006
[26] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006
[27] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[28] S Abbasbandy Y Tan and S J Liao ldquoNewton-homotopyanalysis method for nonlinear equationsrdquo Applied Mathematicsand Computation vol 188 no 2 pp 1794ndash1800 2007
International Journal of
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Active and Passive Electronic Components
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
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DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 3
In (11)ndash(13)
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119875120579
119903
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119889
2V119889119903
2
minus 119875120579
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ(
119889V119889119903
)
2
minus Γ119875120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
(20)
119877V = minus
119875
119903
119898minus1
sum
119896=0
V1015840119898minus1minus119896
120579119896
+
1
119903
V1015840119898minus1
+
119872
119903
V1015840119898minus1
+ V10158401015840119898minus1
minus 119875
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
120579119896
+ 119872V10158401015840119898minus1
minus 3119870
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V10158401015840119897
minus
119870
119903
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897
minus 119861
119877120579
= Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119875
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
120579
1015840
119897
+ 119872Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119870
119898minus1
sum
119896=0
119896
sum
119897=0
119897
sum
119904=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897minus119904
V1015840119904
minus
1
119903
120579
1015840
119898minus1
minus 120579
10158401015840
119898minus1
(21)
By Mathematica the solutions of (21) can be written as
V119898
(119903) =
3119898
sum
119899=0
119886119898119899
119903
119899
119898 ge 0
120579119898
(119903) =
3119898+1
sum
119899=0
119889119898119899
119903
119899
119898 ge 0
(22)
where 119886119898119899
and 119889119898119899
are constants which can be determinedon substituting (22) into (15) and (16)
5 Series Solutions for Vogelrsquos Model
Here
120583 = 120583lowast
exp [
119860
(119905 + 120579)
minus 120579119908
] (23)
0
minus02
minus04
minus06
minus08
minus12 minus1 minus08 minus06 minus04 minus02 0
V998400998400
h
Γ = 01 K = 01M = 01 P = 001 and B = minus1
Figure 1 ℎ-curve for Reynoldsrsquo model for velocity profile
15
125
1
075
05
025
0 02 04 06 08 1 12 14
Q998400998400
h
Γ = 01 K = 01M = 01 P = 001 and B = minus1
Figure 2 ℎ-curve for Reynoldsrsquo model for temperature profile
V998400998400
h
3
2
1
minus1
minus2
minus2 minus15 minus1 minus05 0
0
Γ = 01 K = 01M = 01 s = minus01 A = 00001 t = 01 and B = minus01
Figure 3 ℎ-curve for Vogelrsquos model for velocity profile
which by Maclaurinrsquos series reduces to
120583 =
119861
119904
(1 minus
119860120579
119905
2
) (24)
4 Chinese Journal of EngineeringQ998400998400
h
175
125
15
1
075
05
025
0 02 04 06 08 1 12
B = minus01 q = minus01M = 01 A = 01 s = 01
Γ = 01 t = 01 z = 01 and u = 01
Figure 4 ℎ-curve for Vogelrsquos model for temperature profile
B = 100
B = 105
B = 110
40
30
20
10
0
Q(r)
1 12 14 16 18 2
r
Γ = 01 K = 01M = 01 and P = 001
Figure 5 Temperature profile for Reynoldsrsquo model for differentvalues of 119861
r
1
08
06
04
02
1 12 14 16 18 2
V(r)
B = minus1
B = minus2
B = minus3
Γ = 01 K = 01M = 01 and P = 001
Figure 6 Velocity profile for Reynoldsrsquo model for different values of119861
r
1
08
06
04
02
V(r)
1 12 14 16 18 2
P = 010
P = 020
P = 030
B = minus1 K = 01M = 01 and Γ = 01
Figure 7 Velocity profile for Reynoldsrsquo model for different values of119875
r
1
08
06
04
02
Q(r)
Γ = minus1Γ = minus2
Γ = minus3
1 12 14 16 18 2
B = minus1 K = 01M = 01 and P = 001
Figure 8 Temperature profile for Reynoldsrsquo model for differentvalues of Γ
Invoking the above expressions (1) become
minus
119860119861120579
119903119904119905
2
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119861
119904
119889
2V119889119903
2
minus
119860119861
119904119905
2
120579
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861 = 0
Γ(
119889V119889119903
)
2
minus Γ
119860119861
119904119905
2
120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
= 0
(25)
Chinese Journal of Engineering 5
r
Q(r)
1
08
06
04
02
1 12 14 16 18 2
t = 00065t = 00070
t = 00085
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 9 Temperature profile for Vogelrsquos model for different valuesof 119905
t = 00083
t = 00084
t = 00085
12
1
08
06
04
02
1 12 14 16 18 2
V(r)
r
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 10 Velocity profile for Vogelrsquos model for different values of 119905
With the following initial guesses and auxiliary linear opera-tors
V0V (119903) =
(119903 minus 119887)
(1 minus 119887)
1205790V (119903) =
(119903 minus 119887)
(1 minus 119887)
poundVV (V) = V10158401015840 pound120579V (120579) = 120579
10158401015840
(26)
Γ = minus100
Γ = minus200Γ = minus300
1
08
06
04
02
1 12 14 16 18 2
Q(r)
r
B = minus01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 11 Temperature profile for Vogelrsquos model for different valuesof Γ
the 119898th-order deformation problems are
poundV119903 [V119898 (119903) minus 120594119898
V119898minus1
(119903)] = ℏV119877V119903 (119903)
pound120579119903
[120579119898
(119903) minus 120594119898
120579119898minus1
(119903)] = ℏ120579
119877120579119903
(119903)
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861120579
119903119904119905
2
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119861
119904
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus
119860119861
119904119905
2
120579
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ(
119889V119889119903
)
2
minus Γ
119860119861
119904119905
2
120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861
119903119904119905
2
119898minus1
sum
119896=0
V1015840119898minus1minus119896
120579119896
+
1
119903
V1015840119898minus1
+
119872
119903
V1015840119898minus1
+
119861
119904
V10158401015840119898minus1
minus
119860119861
119904119905
2
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
120579119896
+ 119872V10158401015840119898minus1
minus 3119870
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V10158401015840119897
6 Chinese Journal of EngineeringV(r)
r
3
25
2
15
1
05
1 12 14 16 18 2
s = 300
s = 500
s = 900
Γ = 01 K = 01M = 01 B = minus01 A = 00001 and t = 01
Figure 12 Velocity profile for Vogelrsquos model for different values of119904
minus
119870
119903
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
V1015840119896
minus Γ
119860119861
119904119905
2
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
120579
1015840
119897
+ 119872Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119870
119898minus1
sum
119896=0
119896
sum
119897=0
119897
sum
119904=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897minus119904
V1015840119904
minus
1
119903
120579
1015840
119898minus1
minus 120579
10158401015840
119898minus1
(27)
The expressions of V119898
and 120579119898
are finally given by
V119898
(119903) =
3119898
sum
119899=0
119886
1015840
119898119899
119903
119899
119898 ge 0
120579119898
(119903) =
3119898+1
sum
119899=0
119889
1015840
119898119899
119903
119899
119898 ge 0
(28)
6 Graphical Results and Discussion
In order to report the convergence of the obtained seriessolutions and the effects of sundry parameters in the present
1
08
06
04
02
1 12 14 16 18 2
r
V(r)
B = minus011
B = minus012B = minus013
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 13 Velocity profile for Vogelrsquos model for different values of119861
investigation we plotted Figures 1ndash13 Figures 1ndash4 are pre-pared to see the convergence region Figures 1 and 2 cor-respond to Reynoldsrsquo model whereas Figures 3 and 4 relateto Vogelrsquos model Figure 5 shows the temperature variationfor different values of 119861 for Reynoldsrsquo model It can be seenthat temperature decreases as 119861 increases Figure 6 depictsthe velocity variation for Reynoldsrsquo model for different valuesof 119861 Velocity also decreases as 119861 increases Figure 7 showsthe velocity variation for different values of 119875 for Reynoldsrsquomodel It can be seen that velocity increases as 119875 increasesFigure 8 is plotted in order to see the temperature variationfor Reynoldsrsquomodel for different values of Γ it is depicted thattemperature increases as Γ increases Figures 9ndash13 correspondto Vogelrsquos model Figure 9 depicts temperature variationfor Vogelrsquos model for different values of 119905 It is seen thattemperature increases as 119905 increases Figure 10 shows thevelocity variation for Vogelrsquos model for different values of 119905It is observed that velocity decreases as 119905 increases Figure 11is prepared to observe the temperature variation for Vogelrsquosmodel for different values of Γ It is observed that temperaturedecreases as Γ increases Figure 12 is plotted to see thethe velocity variation for Vogelrsquos model for different valuesof 119904 It is observed that velocity decreases as 119904 increasesFigure 13 depicts the velocity variation for Vogelrsquos model fordifferent values of 119861 It is observed that velocity decreases as119861 increases
7 Conclusions
In this paper we consider the flow of Eyring-Powell modelfluid in the annulus between two cylinders whose viscositydepends upon the temperature We discussed the steadyflow in the annulus due to the motion of inner cylinderand constant pressure gradient In the problem consideredthe flow is found to be remarkedly different from thatfor the incompressible Navier-Stokes fluid with constantviscosity The behavior of pertinent parameters is analyzed
Chinese Journal of Engineering 7
and depicted through graphs Using usual similarity transfor-mations the governing equations have been transformed intonon-linear ordinary differential equations The highly non-linear problem is then solved by homotopy analysis methodEffects of the various parameters on velocity and temperatureprofiles are examined
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R S Rivlin and J L Ericksen ldquoStress deformation relations forisotropic materialsrdquo Rational Mechanics and Analysis Journalvol 4 pp 323ndash425 1955
[2] C Truesdell and W Noll The Non-Linear Field Theories ofMechanics Springer New York NY USA 2nd edition 1992
[3] K R Rajagopal ldquoA note on unsteady unidirectional flows ofa non-Newtonian fluidrdquo International Journal of Non-LinearMechanics vol 17 no 5-6 pp 369ndash373 1982
[4] K R Rajagopal and A S Gupta ldquoAn exact solution for theflow of a non-newtonian fluid past an infinite porous platerdquoMeccanica vol 19 no 2 pp 158ndash160 1984
[5] K R Rajagopal and R K Bhatnagar ldquoExact solutions for somesimple flows of an Oldroyd-B fluidrdquo Acta Mechanica vol 113no 1ndash4 pp 233ndash239 1995
[6] K R Rajagopal ldquoOn the creeping flow of the second-orderfluidrdquo Journal of Non-Newtonian Fluid Mechanics vol 15 no2 pp 239ndash246 1984
[7] K R Rajagopal ldquoLongitudinal and torsional oscillations of arod in a non-Newtonian fluidrdquoActaMechanica vol 49 no 3-4pp 281ndash285 1983
[8] A M Benharbit and A M Siddiqui ldquoCertain solutions of theequations of the planarmotion of a second grade fluid for steadyand unsteady casesrdquo Acta Mechanica vol 94 no 1-2 pp 85ndash961992
[9] T Hayat S Asghar and A M Siddiqui ldquoPeriodic unsteadyflows of a non-Newtonian fluidrdquo Acta Mechanica vol 131 no3-4 pp 169ndash175 1998
[10] A M Siddiqui T Hayat and S Asghar ldquoPeriodic flows of anon-Newtonian fluid between two parallel platesrdquo InternationalJournal of Non-Linear Mechanics vol 34 no 5 pp 895ndash8991999
[11] T Hayat S Asghar and A M Siddiqui ldquoOn the moment of aplane disk in a non-Newtonian fluidrdquo Acta Mechanica vol 136no 3 pp 125ndash131 1999
[12] T Hayat S Asghar and A M Siddiqui ldquoSome unsteadyunidirectional flows of a non-Newtonian fluidrdquo InternationalJournal of Engineering Science vol 38 no 3 pp 337ndash346 2000
[13] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[14] T Hayat R Ellahi and S Asghar ldquoThe influence of variableviscosity and viscous dissipation on the non-Newtonian flowan analytical solutionrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 12 no 3 pp 300ndash313 2007
[15] M Yurusoy and M Pakdermirli ldquoApproximate analytical solu-tions for flow of a third grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[16] M Pakdemirli and B S Yilbas ldquoEntropy generation for pipeflow of a third grade fluid with Vogel model viscosityrdquo Interna-tional Journal of Non-Linear Mechanics vol 41 no 3 pp 432ndash437 2006
[17] S Nadeem and M Ali ldquoAnalytical solutions for pipe flow ofa fourth grade fluid with Reynold and Vogelrsquos models of vis-cositiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2073ndash2090 2009
[18] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[19] S Nadeem and N S Akbar ldquoEffects of temperature dependentviscosity on peristaltic flow of a Jeffrey-six constant fluid ina non-uniform vertical tuberdquo Communications in NonlinearScience andNumerical Simulation vol 15 no 12 pp 3950ndash39642010
[20] M Y Malik A Hussain and S Nadeem ldquoAnalytical treatmentof an oldroyd 8-constant fluid between coaxial cylinders withvariable viscosityrdquo Communications in Theoretical Physics vol56 no 5 pp 933ndash938 2011
[21] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[22] M Y Malik A Hussain S Nadeem and T Hayat ldquoFlow ofa third grade fluid between coaxial cylinders with variableviscosityrdquo Zeitschrift fur Naturforschung A vol 64 no 9-10 pp588ndash596 2009
[23] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method CRC Press Boca Raton Fla USA 2003
[24] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[25] S J Liao ldquoAn analytic solution of unsteady boundary-layerflows caused by an impulsively stretching platerdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 11 no3 pp 326ndash339 2006
[26] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006
[27] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[28] S Abbasbandy Y Tan and S J Liao ldquoNewton-homotopyanalysis method for nonlinear equationsrdquo Applied Mathematicsand Computation vol 188 no 2 pp 1794ndash1800 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Chinese Journal of EngineeringQ998400998400
h
175
125
15
1
075
05
025
0 02 04 06 08 1 12
B = minus01 q = minus01M = 01 A = 01 s = 01
Γ = 01 t = 01 z = 01 and u = 01
Figure 4 ℎ-curve for Vogelrsquos model for temperature profile
B = 100
B = 105
B = 110
40
30
20
10
0
Q(r)
1 12 14 16 18 2
r
Γ = 01 K = 01M = 01 and P = 001
Figure 5 Temperature profile for Reynoldsrsquo model for differentvalues of 119861
r
1
08
06
04
02
1 12 14 16 18 2
V(r)
B = minus1
B = minus2
B = minus3
Γ = 01 K = 01M = 01 and P = 001
Figure 6 Velocity profile for Reynoldsrsquo model for different values of119861
r
1
08
06
04
02
V(r)
1 12 14 16 18 2
P = 010
P = 020
P = 030
B = minus1 K = 01M = 01 and Γ = 01
Figure 7 Velocity profile for Reynoldsrsquo model for different values of119875
r
1
08
06
04
02
Q(r)
Γ = minus1Γ = minus2
Γ = minus3
1 12 14 16 18 2
B = minus1 K = 01M = 01 and P = 001
Figure 8 Temperature profile for Reynoldsrsquo model for differentvalues of Γ
Invoking the above expressions (1) become
minus
119860119861120579
119903119904119905
2
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119861
119904
119889
2V119889119903
2
minus
119860119861
119904119905
2
120579
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861 = 0
Γ(
119889V119889119903
)
2
minus Γ
119860119861
119904119905
2
120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
= 0
(25)
Chinese Journal of Engineering 5
r
Q(r)
1
08
06
04
02
1 12 14 16 18 2
t = 00065t = 00070
t = 00085
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 9 Temperature profile for Vogelrsquos model for different valuesof 119905
t = 00083
t = 00084
t = 00085
12
1
08
06
04
02
1 12 14 16 18 2
V(r)
r
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 10 Velocity profile for Vogelrsquos model for different values of 119905
With the following initial guesses and auxiliary linear opera-tors
V0V (119903) =
(119903 minus 119887)
(1 minus 119887)
1205790V (119903) =
(119903 minus 119887)
(1 minus 119887)
poundVV (V) = V10158401015840 pound120579V (120579) = 120579
10158401015840
(26)
Γ = minus100
Γ = minus200Γ = minus300
1
08
06
04
02
1 12 14 16 18 2
Q(r)
r
B = minus01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 11 Temperature profile for Vogelrsquos model for different valuesof Γ
the 119898th-order deformation problems are
poundV119903 [V119898 (119903) minus 120594119898
V119898minus1
(119903)] = ℏV119877V119903 (119903)
pound120579119903
[120579119898
(119903) minus 120594119898
120579119898minus1
(119903)] = ℏ120579
119877120579119903
(119903)
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861120579
119903119904119905
2
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119861
119904
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus
119860119861
119904119905
2
120579
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ(
119889V119889119903
)
2
minus Γ
119860119861
119904119905
2
120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861
119903119904119905
2
119898minus1
sum
119896=0
V1015840119898minus1minus119896
120579119896
+
1
119903
V1015840119898minus1
+
119872
119903
V1015840119898minus1
+
119861
119904
V10158401015840119898minus1
minus
119860119861
119904119905
2
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
120579119896
+ 119872V10158401015840119898minus1
minus 3119870
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V10158401015840119897
6 Chinese Journal of EngineeringV(r)
r
3
25
2
15
1
05
1 12 14 16 18 2
s = 300
s = 500
s = 900
Γ = 01 K = 01M = 01 B = minus01 A = 00001 and t = 01
Figure 12 Velocity profile for Vogelrsquos model for different values of119904
minus
119870
119903
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
V1015840119896
minus Γ
119860119861
119904119905
2
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
120579
1015840
119897
+ 119872Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119870
119898minus1
sum
119896=0
119896
sum
119897=0
119897
sum
119904=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897minus119904
V1015840119904
minus
1
119903
120579
1015840
119898minus1
minus 120579
10158401015840
119898minus1
(27)
The expressions of V119898
and 120579119898
are finally given by
V119898
(119903) =
3119898
sum
119899=0
119886
1015840
119898119899
119903
119899
119898 ge 0
120579119898
(119903) =
3119898+1
sum
119899=0
119889
1015840
119898119899
119903
119899
119898 ge 0
(28)
6 Graphical Results and Discussion
In order to report the convergence of the obtained seriessolutions and the effects of sundry parameters in the present
1
08
06
04
02
1 12 14 16 18 2
r
V(r)
B = minus011
B = minus012B = minus013
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 13 Velocity profile for Vogelrsquos model for different values of119861
investigation we plotted Figures 1ndash13 Figures 1ndash4 are pre-pared to see the convergence region Figures 1 and 2 cor-respond to Reynoldsrsquo model whereas Figures 3 and 4 relateto Vogelrsquos model Figure 5 shows the temperature variationfor different values of 119861 for Reynoldsrsquo model It can be seenthat temperature decreases as 119861 increases Figure 6 depictsthe velocity variation for Reynoldsrsquo model for different valuesof 119861 Velocity also decreases as 119861 increases Figure 7 showsthe velocity variation for different values of 119875 for Reynoldsrsquomodel It can be seen that velocity increases as 119875 increasesFigure 8 is plotted in order to see the temperature variationfor Reynoldsrsquomodel for different values of Γ it is depicted thattemperature increases as Γ increases Figures 9ndash13 correspondto Vogelrsquos model Figure 9 depicts temperature variationfor Vogelrsquos model for different values of 119905 It is seen thattemperature increases as 119905 increases Figure 10 shows thevelocity variation for Vogelrsquos model for different values of 119905It is observed that velocity decreases as 119905 increases Figure 11is prepared to observe the temperature variation for Vogelrsquosmodel for different values of Γ It is observed that temperaturedecreases as Γ increases Figure 12 is plotted to see thethe velocity variation for Vogelrsquos model for different valuesof 119904 It is observed that velocity decreases as 119904 increasesFigure 13 depicts the velocity variation for Vogelrsquos model fordifferent values of 119861 It is observed that velocity decreases as119861 increases
7 Conclusions
In this paper we consider the flow of Eyring-Powell modelfluid in the annulus between two cylinders whose viscositydepends upon the temperature We discussed the steadyflow in the annulus due to the motion of inner cylinderand constant pressure gradient In the problem consideredthe flow is found to be remarkedly different from thatfor the incompressible Navier-Stokes fluid with constantviscosity The behavior of pertinent parameters is analyzed
Chinese Journal of Engineering 7
and depicted through graphs Using usual similarity transfor-mations the governing equations have been transformed intonon-linear ordinary differential equations The highly non-linear problem is then solved by homotopy analysis methodEffects of the various parameters on velocity and temperatureprofiles are examined
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R S Rivlin and J L Ericksen ldquoStress deformation relations forisotropic materialsrdquo Rational Mechanics and Analysis Journalvol 4 pp 323ndash425 1955
[2] C Truesdell and W Noll The Non-Linear Field Theories ofMechanics Springer New York NY USA 2nd edition 1992
[3] K R Rajagopal ldquoA note on unsteady unidirectional flows ofa non-Newtonian fluidrdquo International Journal of Non-LinearMechanics vol 17 no 5-6 pp 369ndash373 1982
[4] K R Rajagopal and A S Gupta ldquoAn exact solution for theflow of a non-newtonian fluid past an infinite porous platerdquoMeccanica vol 19 no 2 pp 158ndash160 1984
[5] K R Rajagopal and R K Bhatnagar ldquoExact solutions for somesimple flows of an Oldroyd-B fluidrdquo Acta Mechanica vol 113no 1ndash4 pp 233ndash239 1995
[6] K R Rajagopal ldquoOn the creeping flow of the second-orderfluidrdquo Journal of Non-Newtonian Fluid Mechanics vol 15 no2 pp 239ndash246 1984
[7] K R Rajagopal ldquoLongitudinal and torsional oscillations of arod in a non-Newtonian fluidrdquoActaMechanica vol 49 no 3-4pp 281ndash285 1983
[8] A M Benharbit and A M Siddiqui ldquoCertain solutions of theequations of the planarmotion of a second grade fluid for steadyand unsteady casesrdquo Acta Mechanica vol 94 no 1-2 pp 85ndash961992
[9] T Hayat S Asghar and A M Siddiqui ldquoPeriodic unsteadyflows of a non-Newtonian fluidrdquo Acta Mechanica vol 131 no3-4 pp 169ndash175 1998
[10] A M Siddiqui T Hayat and S Asghar ldquoPeriodic flows of anon-Newtonian fluid between two parallel platesrdquo InternationalJournal of Non-Linear Mechanics vol 34 no 5 pp 895ndash8991999
[11] T Hayat S Asghar and A M Siddiqui ldquoOn the moment of aplane disk in a non-Newtonian fluidrdquo Acta Mechanica vol 136no 3 pp 125ndash131 1999
[12] T Hayat S Asghar and A M Siddiqui ldquoSome unsteadyunidirectional flows of a non-Newtonian fluidrdquo InternationalJournal of Engineering Science vol 38 no 3 pp 337ndash346 2000
[13] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[14] T Hayat R Ellahi and S Asghar ldquoThe influence of variableviscosity and viscous dissipation on the non-Newtonian flowan analytical solutionrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 12 no 3 pp 300ndash313 2007
[15] M Yurusoy and M Pakdermirli ldquoApproximate analytical solu-tions for flow of a third grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[16] M Pakdemirli and B S Yilbas ldquoEntropy generation for pipeflow of a third grade fluid with Vogel model viscosityrdquo Interna-tional Journal of Non-Linear Mechanics vol 41 no 3 pp 432ndash437 2006
[17] S Nadeem and M Ali ldquoAnalytical solutions for pipe flow ofa fourth grade fluid with Reynold and Vogelrsquos models of vis-cositiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2073ndash2090 2009
[18] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[19] S Nadeem and N S Akbar ldquoEffects of temperature dependentviscosity on peristaltic flow of a Jeffrey-six constant fluid ina non-uniform vertical tuberdquo Communications in NonlinearScience andNumerical Simulation vol 15 no 12 pp 3950ndash39642010
[20] M Y Malik A Hussain and S Nadeem ldquoAnalytical treatmentof an oldroyd 8-constant fluid between coaxial cylinders withvariable viscosityrdquo Communications in Theoretical Physics vol56 no 5 pp 933ndash938 2011
[21] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[22] M Y Malik A Hussain S Nadeem and T Hayat ldquoFlow ofa third grade fluid between coaxial cylinders with variableviscosityrdquo Zeitschrift fur Naturforschung A vol 64 no 9-10 pp588ndash596 2009
[23] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method CRC Press Boca Raton Fla USA 2003
[24] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[25] S J Liao ldquoAn analytic solution of unsteady boundary-layerflows caused by an impulsively stretching platerdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 11 no3 pp 326ndash339 2006
[26] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006
[27] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[28] S Abbasbandy Y Tan and S J Liao ldquoNewton-homotopyanalysis method for nonlinear equationsrdquo Applied Mathematicsand Computation vol 188 no 2 pp 1794ndash1800 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 5
r
Q(r)
1
08
06
04
02
1 12 14 16 18 2
t = 00065t = 00070
t = 00085
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 9 Temperature profile for Vogelrsquos model for different valuesof 119905
t = 00083
t = 00084
t = 00085
12
1
08
06
04
02
1 12 14 16 18 2
V(r)
r
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and B = minus01
Figure 10 Velocity profile for Vogelrsquos model for different values of 119905
With the following initial guesses and auxiliary linear opera-tors
V0V (119903) =
(119903 minus 119887)
(1 minus 119887)
1205790V (119903) =
(119903 minus 119887)
(1 minus 119887)
poundVV (V) = V10158401015840 pound120579V (120579) = 120579
10158401015840
(26)
Γ = minus100
Γ = minus200Γ = minus300
1
08
06
04
02
1 12 14 16 18 2
Q(r)
r
B = minus01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 11 Temperature profile for Vogelrsquos model for different valuesof Γ
the 119898th-order deformation problems are
poundV119903 [V119898 (119903) minus 120594119898
V119898minus1
(119903)] = ℏV119877V119903 (119903)
pound120579119903
[120579119898
(119903) minus 120594119898
120579119898minus1
(119903)] = ℏ120579
119877120579119903
(119903)
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861120579
119903119904119905
2
119889V119889119903
+
1
119903
119889V119889119903
+
119872
119903
119889V119889119903
+
119861
119904
119889
2V119889119903
2
+ 119872
119889
2V119889119903
2
minus
119860119861
119904119905
2
120579
119889
2V119889119903
2
minus 3119870(
119889V119889119903
)
2
119889
2V119889119903
2
minus
119870
119903
(
119889V119889119903
)
3
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ(
119889V119889119903
)
2
minus Γ
119860119861
119904119905
2
120579(
119889V119889119903
)
2
+ 119872Γ(
119889V119889119903
)
2
minus Γ119870(
119889V119889119903
)
4
minus
1
119903
119889120579
119889119903
minus
119889
2
120579
119889119903
2
119873V [V (119903 119901) 120579 (119903 119901)]
= minus
119860119861
119903119904119905
2
119898minus1
sum
119896=0
V1015840119898minus1minus119896
120579119896
+
1
119903
V1015840119898minus1
+
119872
119903
V1015840119898minus1
+
119861
119904
V10158401015840119898minus1
minus
119860119861
119904119905
2
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
120579119896
+ 119872V10158401015840119898minus1
minus 3119870
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V10158401015840119897
6 Chinese Journal of EngineeringV(r)
r
3
25
2
15
1
05
1 12 14 16 18 2
s = 300
s = 500
s = 900
Γ = 01 K = 01M = 01 B = minus01 A = 00001 and t = 01
Figure 12 Velocity profile for Vogelrsquos model for different values of119904
minus
119870
119903
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
V1015840119896
minus Γ
119860119861
119904119905
2
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
120579
1015840
119897
+ 119872Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119870
119898minus1
sum
119896=0
119896
sum
119897=0
119897
sum
119904=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897minus119904
V1015840119904
minus
1
119903
120579
1015840
119898minus1
minus 120579
10158401015840
119898minus1
(27)
The expressions of V119898
and 120579119898
are finally given by
V119898
(119903) =
3119898
sum
119899=0
119886
1015840
119898119899
119903
119899
119898 ge 0
120579119898
(119903) =
3119898+1
sum
119899=0
119889
1015840
119898119899
119903
119899
119898 ge 0
(28)
6 Graphical Results and Discussion
In order to report the convergence of the obtained seriessolutions and the effects of sundry parameters in the present
1
08
06
04
02
1 12 14 16 18 2
r
V(r)
B = minus011
B = minus012B = minus013
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 13 Velocity profile for Vogelrsquos model for different values of119861
investigation we plotted Figures 1ndash13 Figures 1ndash4 are pre-pared to see the convergence region Figures 1 and 2 cor-respond to Reynoldsrsquo model whereas Figures 3 and 4 relateto Vogelrsquos model Figure 5 shows the temperature variationfor different values of 119861 for Reynoldsrsquo model It can be seenthat temperature decreases as 119861 increases Figure 6 depictsthe velocity variation for Reynoldsrsquo model for different valuesof 119861 Velocity also decreases as 119861 increases Figure 7 showsthe velocity variation for different values of 119875 for Reynoldsrsquomodel It can be seen that velocity increases as 119875 increasesFigure 8 is plotted in order to see the temperature variationfor Reynoldsrsquomodel for different values of Γ it is depicted thattemperature increases as Γ increases Figures 9ndash13 correspondto Vogelrsquos model Figure 9 depicts temperature variationfor Vogelrsquos model for different values of 119905 It is seen thattemperature increases as 119905 increases Figure 10 shows thevelocity variation for Vogelrsquos model for different values of 119905It is observed that velocity decreases as 119905 increases Figure 11is prepared to observe the temperature variation for Vogelrsquosmodel for different values of Γ It is observed that temperaturedecreases as Γ increases Figure 12 is plotted to see thethe velocity variation for Vogelrsquos model for different valuesof 119904 It is observed that velocity decreases as 119904 increasesFigure 13 depicts the velocity variation for Vogelrsquos model fordifferent values of 119861 It is observed that velocity decreases as119861 increases
7 Conclusions
In this paper we consider the flow of Eyring-Powell modelfluid in the annulus between two cylinders whose viscositydepends upon the temperature We discussed the steadyflow in the annulus due to the motion of inner cylinderand constant pressure gradient In the problem consideredthe flow is found to be remarkedly different from thatfor the incompressible Navier-Stokes fluid with constantviscosity The behavior of pertinent parameters is analyzed
Chinese Journal of Engineering 7
and depicted through graphs Using usual similarity transfor-mations the governing equations have been transformed intonon-linear ordinary differential equations The highly non-linear problem is then solved by homotopy analysis methodEffects of the various parameters on velocity and temperatureprofiles are examined
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R S Rivlin and J L Ericksen ldquoStress deformation relations forisotropic materialsrdquo Rational Mechanics and Analysis Journalvol 4 pp 323ndash425 1955
[2] C Truesdell and W Noll The Non-Linear Field Theories ofMechanics Springer New York NY USA 2nd edition 1992
[3] K R Rajagopal ldquoA note on unsteady unidirectional flows ofa non-Newtonian fluidrdquo International Journal of Non-LinearMechanics vol 17 no 5-6 pp 369ndash373 1982
[4] K R Rajagopal and A S Gupta ldquoAn exact solution for theflow of a non-newtonian fluid past an infinite porous platerdquoMeccanica vol 19 no 2 pp 158ndash160 1984
[5] K R Rajagopal and R K Bhatnagar ldquoExact solutions for somesimple flows of an Oldroyd-B fluidrdquo Acta Mechanica vol 113no 1ndash4 pp 233ndash239 1995
[6] K R Rajagopal ldquoOn the creeping flow of the second-orderfluidrdquo Journal of Non-Newtonian Fluid Mechanics vol 15 no2 pp 239ndash246 1984
[7] K R Rajagopal ldquoLongitudinal and torsional oscillations of arod in a non-Newtonian fluidrdquoActaMechanica vol 49 no 3-4pp 281ndash285 1983
[8] A M Benharbit and A M Siddiqui ldquoCertain solutions of theequations of the planarmotion of a second grade fluid for steadyand unsteady casesrdquo Acta Mechanica vol 94 no 1-2 pp 85ndash961992
[9] T Hayat S Asghar and A M Siddiqui ldquoPeriodic unsteadyflows of a non-Newtonian fluidrdquo Acta Mechanica vol 131 no3-4 pp 169ndash175 1998
[10] A M Siddiqui T Hayat and S Asghar ldquoPeriodic flows of anon-Newtonian fluid between two parallel platesrdquo InternationalJournal of Non-Linear Mechanics vol 34 no 5 pp 895ndash8991999
[11] T Hayat S Asghar and A M Siddiqui ldquoOn the moment of aplane disk in a non-Newtonian fluidrdquo Acta Mechanica vol 136no 3 pp 125ndash131 1999
[12] T Hayat S Asghar and A M Siddiqui ldquoSome unsteadyunidirectional flows of a non-Newtonian fluidrdquo InternationalJournal of Engineering Science vol 38 no 3 pp 337ndash346 2000
[13] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[14] T Hayat R Ellahi and S Asghar ldquoThe influence of variableviscosity and viscous dissipation on the non-Newtonian flowan analytical solutionrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 12 no 3 pp 300ndash313 2007
[15] M Yurusoy and M Pakdermirli ldquoApproximate analytical solu-tions for flow of a third grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[16] M Pakdemirli and B S Yilbas ldquoEntropy generation for pipeflow of a third grade fluid with Vogel model viscosityrdquo Interna-tional Journal of Non-Linear Mechanics vol 41 no 3 pp 432ndash437 2006
[17] S Nadeem and M Ali ldquoAnalytical solutions for pipe flow ofa fourth grade fluid with Reynold and Vogelrsquos models of vis-cositiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2073ndash2090 2009
[18] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[19] S Nadeem and N S Akbar ldquoEffects of temperature dependentviscosity on peristaltic flow of a Jeffrey-six constant fluid ina non-uniform vertical tuberdquo Communications in NonlinearScience andNumerical Simulation vol 15 no 12 pp 3950ndash39642010
[20] M Y Malik A Hussain and S Nadeem ldquoAnalytical treatmentof an oldroyd 8-constant fluid between coaxial cylinders withvariable viscosityrdquo Communications in Theoretical Physics vol56 no 5 pp 933ndash938 2011
[21] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[22] M Y Malik A Hussain S Nadeem and T Hayat ldquoFlow ofa third grade fluid between coaxial cylinders with variableviscosityrdquo Zeitschrift fur Naturforschung A vol 64 no 9-10 pp588ndash596 2009
[23] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method CRC Press Boca Raton Fla USA 2003
[24] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[25] S J Liao ldquoAn analytic solution of unsteady boundary-layerflows caused by an impulsively stretching platerdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 11 no3 pp 326ndash339 2006
[26] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006
[27] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[28] S Abbasbandy Y Tan and S J Liao ldquoNewton-homotopyanalysis method for nonlinear equationsrdquo Applied Mathematicsand Computation vol 188 no 2 pp 1794ndash1800 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Chinese Journal of EngineeringV(r)
r
3
25
2
15
1
05
1 12 14 16 18 2
s = 300
s = 500
s = 900
Γ = 01 K = 01M = 01 B = minus01 A = 00001 and t = 01
Figure 12 Velocity profile for Vogelrsquos model for different values of119904
minus
119870
119903
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897
minus 119861
119873120579
[V (119903 119901) 120579 (119903 119901)]
= Γ
119898minus1
sum
119896=0
V10158401015840119898minus1minus119896
V1015840119896
minus Γ
119860119861
119904119905
2
119898minus1
sum
119896=0
119896
sum
119897=0
V1015840119898minus1minus119896
V1015840119896minus119897
120579
1015840
119897
+ 119872Γ
119898minus1
sum
119896=0
V1015840119898minus1minus119896
V1015840119896
minus Γ119870
119898minus1
sum
119896=0
119896
sum
119897=0
119897
sum
119904=0
V1015840119898minus1minus119896
V1015840119896minus119897
V1015840119897minus119904
V1015840119904
minus
1
119903
120579
1015840
119898minus1
minus 120579
10158401015840
119898minus1
(27)
The expressions of V119898
and 120579119898
are finally given by
V119898
(119903) =
3119898
sum
119899=0
119886
1015840
119898119899
119903
119899
119898 ge 0
120579119898
(119903) =
3119898+1
sum
119899=0
119889
1015840
119898119899
119903
119899
119898 ge 0
(28)
6 Graphical Results and Discussion
In order to report the convergence of the obtained seriessolutions and the effects of sundry parameters in the present
1
08
06
04
02
1 12 14 16 18 2
r
V(r)
B = minus011
B = minus012B = minus013
Γ = 01 K = 01M = 01 s = minus01 A = 00001 and t = 01
Figure 13 Velocity profile for Vogelrsquos model for different values of119861
investigation we plotted Figures 1ndash13 Figures 1ndash4 are pre-pared to see the convergence region Figures 1 and 2 cor-respond to Reynoldsrsquo model whereas Figures 3 and 4 relateto Vogelrsquos model Figure 5 shows the temperature variationfor different values of 119861 for Reynoldsrsquo model It can be seenthat temperature decreases as 119861 increases Figure 6 depictsthe velocity variation for Reynoldsrsquo model for different valuesof 119861 Velocity also decreases as 119861 increases Figure 7 showsthe velocity variation for different values of 119875 for Reynoldsrsquomodel It can be seen that velocity increases as 119875 increasesFigure 8 is plotted in order to see the temperature variationfor Reynoldsrsquomodel for different values of Γ it is depicted thattemperature increases as Γ increases Figures 9ndash13 correspondto Vogelrsquos model Figure 9 depicts temperature variationfor Vogelrsquos model for different values of 119905 It is seen thattemperature increases as 119905 increases Figure 10 shows thevelocity variation for Vogelrsquos model for different values of 119905It is observed that velocity decreases as 119905 increases Figure 11is prepared to observe the temperature variation for Vogelrsquosmodel for different values of Γ It is observed that temperaturedecreases as Γ increases Figure 12 is plotted to see thethe velocity variation for Vogelrsquos model for different valuesof 119904 It is observed that velocity decreases as 119904 increasesFigure 13 depicts the velocity variation for Vogelrsquos model fordifferent values of 119861 It is observed that velocity decreases as119861 increases
7 Conclusions
In this paper we consider the flow of Eyring-Powell modelfluid in the annulus between two cylinders whose viscositydepends upon the temperature We discussed the steadyflow in the annulus due to the motion of inner cylinderand constant pressure gradient In the problem consideredthe flow is found to be remarkedly different from thatfor the incompressible Navier-Stokes fluid with constantviscosity The behavior of pertinent parameters is analyzed
Chinese Journal of Engineering 7
and depicted through graphs Using usual similarity transfor-mations the governing equations have been transformed intonon-linear ordinary differential equations The highly non-linear problem is then solved by homotopy analysis methodEffects of the various parameters on velocity and temperatureprofiles are examined
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R S Rivlin and J L Ericksen ldquoStress deformation relations forisotropic materialsrdquo Rational Mechanics and Analysis Journalvol 4 pp 323ndash425 1955
[2] C Truesdell and W Noll The Non-Linear Field Theories ofMechanics Springer New York NY USA 2nd edition 1992
[3] K R Rajagopal ldquoA note on unsteady unidirectional flows ofa non-Newtonian fluidrdquo International Journal of Non-LinearMechanics vol 17 no 5-6 pp 369ndash373 1982
[4] K R Rajagopal and A S Gupta ldquoAn exact solution for theflow of a non-newtonian fluid past an infinite porous platerdquoMeccanica vol 19 no 2 pp 158ndash160 1984
[5] K R Rajagopal and R K Bhatnagar ldquoExact solutions for somesimple flows of an Oldroyd-B fluidrdquo Acta Mechanica vol 113no 1ndash4 pp 233ndash239 1995
[6] K R Rajagopal ldquoOn the creeping flow of the second-orderfluidrdquo Journal of Non-Newtonian Fluid Mechanics vol 15 no2 pp 239ndash246 1984
[7] K R Rajagopal ldquoLongitudinal and torsional oscillations of arod in a non-Newtonian fluidrdquoActaMechanica vol 49 no 3-4pp 281ndash285 1983
[8] A M Benharbit and A M Siddiqui ldquoCertain solutions of theequations of the planarmotion of a second grade fluid for steadyand unsteady casesrdquo Acta Mechanica vol 94 no 1-2 pp 85ndash961992
[9] T Hayat S Asghar and A M Siddiqui ldquoPeriodic unsteadyflows of a non-Newtonian fluidrdquo Acta Mechanica vol 131 no3-4 pp 169ndash175 1998
[10] A M Siddiqui T Hayat and S Asghar ldquoPeriodic flows of anon-Newtonian fluid between two parallel platesrdquo InternationalJournal of Non-Linear Mechanics vol 34 no 5 pp 895ndash8991999
[11] T Hayat S Asghar and A M Siddiqui ldquoOn the moment of aplane disk in a non-Newtonian fluidrdquo Acta Mechanica vol 136no 3 pp 125ndash131 1999
[12] T Hayat S Asghar and A M Siddiqui ldquoSome unsteadyunidirectional flows of a non-Newtonian fluidrdquo InternationalJournal of Engineering Science vol 38 no 3 pp 337ndash346 2000
[13] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[14] T Hayat R Ellahi and S Asghar ldquoThe influence of variableviscosity and viscous dissipation on the non-Newtonian flowan analytical solutionrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 12 no 3 pp 300ndash313 2007
[15] M Yurusoy and M Pakdermirli ldquoApproximate analytical solu-tions for flow of a third grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[16] M Pakdemirli and B S Yilbas ldquoEntropy generation for pipeflow of a third grade fluid with Vogel model viscosityrdquo Interna-tional Journal of Non-Linear Mechanics vol 41 no 3 pp 432ndash437 2006
[17] S Nadeem and M Ali ldquoAnalytical solutions for pipe flow ofa fourth grade fluid with Reynold and Vogelrsquos models of vis-cositiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2073ndash2090 2009
[18] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[19] S Nadeem and N S Akbar ldquoEffects of temperature dependentviscosity on peristaltic flow of a Jeffrey-six constant fluid ina non-uniform vertical tuberdquo Communications in NonlinearScience andNumerical Simulation vol 15 no 12 pp 3950ndash39642010
[20] M Y Malik A Hussain and S Nadeem ldquoAnalytical treatmentof an oldroyd 8-constant fluid between coaxial cylinders withvariable viscosityrdquo Communications in Theoretical Physics vol56 no 5 pp 933ndash938 2011
[21] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[22] M Y Malik A Hussain S Nadeem and T Hayat ldquoFlow ofa third grade fluid between coaxial cylinders with variableviscosityrdquo Zeitschrift fur Naturforschung A vol 64 no 9-10 pp588ndash596 2009
[23] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method CRC Press Boca Raton Fla USA 2003
[24] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[25] S J Liao ldquoAn analytic solution of unsteady boundary-layerflows caused by an impulsively stretching platerdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 11 no3 pp 326ndash339 2006
[26] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006
[27] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[28] S Abbasbandy Y Tan and S J Liao ldquoNewton-homotopyanalysis method for nonlinear equationsrdquo Applied Mathematicsand Computation vol 188 no 2 pp 1794ndash1800 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Chinese Journal of Engineering 7
and depicted through graphs Using usual similarity transfor-mations the governing equations have been transformed intonon-linear ordinary differential equations The highly non-linear problem is then solved by homotopy analysis methodEffects of the various parameters on velocity and temperatureprofiles are examined
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R S Rivlin and J L Ericksen ldquoStress deformation relations forisotropic materialsrdquo Rational Mechanics and Analysis Journalvol 4 pp 323ndash425 1955
[2] C Truesdell and W Noll The Non-Linear Field Theories ofMechanics Springer New York NY USA 2nd edition 1992
[3] K R Rajagopal ldquoA note on unsteady unidirectional flows ofa non-Newtonian fluidrdquo International Journal of Non-LinearMechanics vol 17 no 5-6 pp 369ndash373 1982
[4] K R Rajagopal and A S Gupta ldquoAn exact solution for theflow of a non-newtonian fluid past an infinite porous platerdquoMeccanica vol 19 no 2 pp 158ndash160 1984
[5] K R Rajagopal and R K Bhatnagar ldquoExact solutions for somesimple flows of an Oldroyd-B fluidrdquo Acta Mechanica vol 113no 1ndash4 pp 233ndash239 1995
[6] K R Rajagopal ldquoOn the creeping flow of the second-orderfluidrdquo Journal of Non-Newtonian Fluid Mechanics vol 15 no2 pp 239ndash246 1984
[7] K R Rajagopal ldquoLongitudinal and torsional oscillations of arod in a non-Newtonian fluidrdquoActaMechanica vol 49 no 3-4pp 281ndash285 1983
[8] A M Benharbit and A M Siddiqui ldquoCertain solutions of theequations of the planarmotion of a second grade fluid for steadyand unsteady casesrdquo Acta Mechanica vol 94 no 1-2 pp 85ndash961992
[9] T Hayat S Asghar and A M Siddiqui ldquoPeriodic unsteadyflows of a non-Newtonian fluidrdquo Acta Mechanica vol 131 no3-4 pp 169ndash175 1998
[10] A M Siddiqui T Hayat and S Asghar ldquoPeriodic flows of anon-Newtonian fluid between two parallel platesrdquo InternationalJournal of Non-Linear Mechanics vol 34 no 5 pp 895ndash8991999
[11] T Hayat S Asghar and A M Siddiqui ldquoOn the moment of aplane disk in a non-Newtonian fluidrdquo Acta Mechanica vol 136no 3 pp 125ndash131 1999
[12] T Hayat S Asghar and A M Siddiqui ldquoSome unsteadyunidirectional flows of a non-Newtonian fluidrdquo InternationalJournal of Engineering Science vol 38 no 3 pp 337ndash346 2000
[13] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[14] T Hayat R Ellahi and S Asghar ldquoThe influence of variableviscosity and viscous dissipation on the non-Newtonian flowan analytical solutionrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 12 no 3 pp 300ndash313 2007
[15] M Yurusoy and M Pakdermirli ldquoApproximate analytical solu-tions for flow of a third grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[16] M Pakdemirli and B S Yilbas ldquoEntropy generation for pipeflow of a third grade fluid with Vogel model viscosityrdquo Interna-tional Journal of Non-Linear Mechanics vol 41 no 3 pp 432ndash437 2006
[17] S Nadeem and M Ali ldquoAnalytical solutions for pipe flow ofa fourth grade fluid with Reynold and Vogelrsquos models of vis-cositiesrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 5 pp 2073ndash2090 2009
[18] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[19] S Nadeem and N S Akbar ldquoEffects of temperature dependentviscosity on peristaltic flow of a Jeffrey-six constant fluid ina non-uniform vertical tuberdquo Communications in NonlinearScience andNumerical Simulation vol 15 no 12 pp 3950ndash39642010
[20] M Y Malik A Hussain and S Nadeem ldquoAnalytical treatmentof an oldroyd 8-constant fluid between coaxial cylinders withvariable viscosityrdquo Communications in Theoretical Physics vol56 no 5 pp 933ndash938 2011
[21] S Nadeem T Hayat S Abbasbandy and M Ali ldquoEffects ofpartial slip on a fourth-grade fluid with variable viscosity ananalytic solutionrdquo Nonlinear Analysis Real World Applicationsvol 11 no 2 pp 856ndash868 2010
[22] M Y Malik A Hussain S Nadeem and T Hayat ldquoFlow ofa third grade fluid between coaxial cylinders with variableviscosityrdquo Zeitschrift fur Naturforschung A vol 64 no 9-10 pp588ndash596 2009
[23] S J Liao Beyond Perturbation Introduction to HomotopyAnalysis Method CRC Press Boca Raton Fla USA 2003
[24] S J Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[25] S J Liao ldquoAn analytic solution of unsteady boundary-layerflows caused by an impulsively stretching platerdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 11 no3 pp 326ndash339 2006
[26] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006
[27] S Abbasbandy ldquoHomotopy analysis method for heat radiationequationsrdquo International Communications in Heat and MassTransfer vol 34 no 3 pp 380ndash387 2007
[28] S Abbasbandy Y Tan and S J Liao ldquoNewton-homotopyanalysis method for nonlinear equationsrdquo Applied Mathematicsand Computation vol 188 no 2 pp 1794ndash1800 2007
International Journal of
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RoboticsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
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International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of