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The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions
Applied Mathematical Modelling 37 (2013) 1451–1467
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The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions R. Ellahi ⇑ Department of Mechanical Engineering, Bourns Hall, University of California Riverside, CA 92521, USA Department of Mathematics & Statistics, FBAS, IIUI, Islamabad 44000, Pakistan
a r t i c l e
i n f o
Article history: Received 10 January 2012 Received in revised form 22 March 2012 Accepted 2 April 2012 Available online 17 April 2012 Keywords: Non-Newtonian nanofluid Variable viscosities MHD Heat transfer analysis Analytical solutions
a b s t r a c t This article examines the magnetohydrodynamic (MHD) flow of non-Newtonian nanofluid in a pipe. The temperature of the pipe is assumed to be higher than the temperature of the fluid. In particular two temperature dependent viscosity models, have been considered. The nonlinear partial differential equations along with the boundary conditions are first cast into a dimensionless form and then the equations are solved by homotopy analysis method (HAM). Explicit analytical expressions for the velocity field, the temperature distribution and nano concentration have been derived analytically. The effects of various physical parameters on velocity, temperature and nano concentration are discussed by using graphical approach. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Magnetohydrodynamic (MHD) deals with the motion of conducting fluids. The applications of MHD cover a wide range of physical areas from liquid metals to cosmic plasmas; for instance, MHD pumps, MHD power generators, electrostatic precipitation, petroleum industry, electrostatic precipitation, purification of crude oil, aerodynamics heating, geophysics, plasma physics and fluid droplets sprays [1–6]. Moreover, non-Newtonian fluids [7–13] have been found much important and useful for technological point of view such as multi-grade oils, liquid detergents, paints, polymer solutions and polymer melts. Furthermore, recent advances in nanotechnology have led to the development of a new innovative class of heat transfer called nanofluids created by dispersing nanoparticles [14]. Non-Newtonian nanofluids are widely encountered in many industrial and technology applications, for example, melts of polymers, biological solutions, paints, tars, asphalts and glues etc. Nanofluids appear to have the potential to significantly increase heat transfer rates in a variety of areas such as industrial cooling applications, nuclear reactors, transportation industry, micro-electromechanical systems, electronics and instrumentation, and biomedical applications. Nanofluid has also been found to possess enhanced thermophysical properties such as thermal conductivity, thermal diffusively, viscosity and convective heat transfer coefficients compared to those of base fluids like oil or water. A careful review of the literature reveals that a very little efforts are devoted to examine the non Newtonian nanofluid. Some relevant studies on the topic can be found from the list of Refs. [15,16]. Motivated by these facts, in the present study we have investigated the effects of MHD and variable viscosities on nonNewtonian nanofluid in a pipe. The flow is generated by constant pressure gradient. To derive the solutions of nonlinear ⇑ Address: Department of Mechanical Engineering, Bourns Hall, University of California Riverside, CA 92521, USA. Tel.: +1 951 2756747. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.04.004
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
governing equations, we have used an efficient method, homotopy analysis method (HAM) [17–21], which is particularly suitable for strongly nonlinear problems. After the introduction in Section 1, the outlines of this paper are as follows. Section 2 contains mathematical formulation. In Section 3 solutions of the problems are presented by using HAM. Convergence and discussion are given in Sections 4 and 5 respectively. Finally Section 6 summaries the concluding remarks. 2. Formulation of the problem The governing equations of the fluid motion are the conservation of momentum
q
dV ¼ div T þ qb þ J B; dt
ð1Þ
where q is the density, d=dt is the material time derivative, V is the velocity field, T is the Cauchy stress tensor, J is the electric current density, B is the total magnetic field, b ¼ q e g k, is the body force, k being the unit vector in the z-direction, and e g the acceleration due to gravity. The fact that the fluid undergoes only isochoric motion, therefore, the law of conservation of mass is defined by
div V ¼ 0:
ð2Þ
In view of the principle of conservation of heat energy, the energy equation for nanofluid is given by
q
de Dt ¼ div Q ðqcÞp Db ru rh þ rh rh ; dt h
ð3Þ
where e is specific internal energy, h is temperature, cp is specific heat, Db is Brownian diffusion coefficient, Dt is thermophoretic diffusion coefficient and Q is heat flux. According to Fourier’s law of heat transfer
Q ¼ k grad h
ð4Þ
div Q ¼ kr ðrhÞ;
ð5Þ
and
k is thermal conductivity. Due to complexity of non-Newtonian nanofluids, there is no single model which describes all of their properties. Therefore, several constitutive equations have been proposed which can describe all the behaviors of non-Newtonian nanofluids; for example, stress differences, shear thinning or shear thickening, stress relaxation, elastic effects and memory effects. Amongst the many models, there is a grade three model which is the most popular. This is particularly due to the fact that one can reasonably explain the shear thinning/shear thickening properties even for steady and unidirectional flows. The stress in a third grade fluid is given by
T ¼ pI þ lA1 þ a1 A2 þ a2 A21 þ b1 A3 þ b2 ðA1 A2 þ A2 A1 Þ þ b3 ðtrA21 ÞA1 ;
ð6Þ
where l is the coefficient of viscosity, p is hydrostatic pressure, T is Cauchy stress tensor, pI is the spherical stress due to the constraint of incompressibility, ai ði ¼ 1; 2Þ are material constants, bj ðj ¼ 1; 2Þ are grade three parameters and first three Rivlin–Ericksen kinematical tensors A1 ; A2 and A3 are defined by
A1 ¼ ðgrad VÞ þ ðgrad VÞt ; An ¼
ð7Þ
dAn1 t þ An1 ðgrad VÞ þ ðgrad VÞ An1 ; dt
for n > 1;
ð8Þ
where V ¼ ½0; 0; v ðrÞ denotes the velocity vector. If all the motions of the fluid are to be compatible with thermodynamics in the sense that these motions satisfy the Clausius–Duhem inequality and if it is assumed that the specific Helmholtz free energy is a minimum when the fluid is locally at rest, then thermodynamics imposes the following constraints [22]
l P 0; a1 P 0; ja1 þ a2 j 6
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24lb3 ;
b1 ¼ b2 ¼ 0;
b3 P 0:
ð9Þ
It is noted that this constitutive relation not only predicts the normal stress differences, but can also predict the ‘‘shearthickening’’ phenomenon (since b3 > 0) which is the increase in viscosity with increasing shear rate. In the present analysis we assume that the fluid is thermodynamically compatible, and therefore, Eq. (6) reduces to
T ¼ p1 I þ
h
i
l þ b3 ðtrA21 Þ A1 þ a1 A2 þ a2 A21 :
ð10Þ
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
1453
We render the above equations dimensionless by setting
v ¼ vv ;
w / ¼ /// ; m /w
2b3 v 20
K¼
^ w l ¼ ll0 ; h ¼ hhh ; c1 ¼ @@zp ; m hw
r ¼ Rr ;
0
l0 R2
;
Nb ¼ Db ð/m /w Þ; Gr ¼
ðhm hw Þqfw R2 ð1/w Þg ; l 0 u0
Nt ¼ Dt ðhhmwhw Þ ; Br ¼
9 2 > c ¼ vc10 Rl > 0 > rffiffiffiffiffiffiffiffiffiffi > > =
N¼
ðqp qw ÞR2 ð/m /w Þg l0 u0
rB20 R2 l0 >;
> > > > ;
ð11Þ
where R is the radius of pipe, hw is the pipe temperature, hm is the fluid temperature, v is velocity, h is temperature, / is concentration, qf is the density of the base fluid, v 0 , is the reference velocity and l0 is the reference viscosity. Substituting Eq. (10) in the balance of linear momentum and using the non-dimensional quantities given in Eq. (11), the dimensionless form of Eqs. (1)–(3), after dropping the over bars for convenience, we obtain the following non-dimensional coupled equations
3 2 2 2 dl dv l dv d v K dv dv d v þ 3K ¼ c þ N2 v Gr h Br /; þ þl 2 þ 2 dr dr r dr r dr dr dr dr 2
a
d h dr
2
þ
! 2 1 dh dh d/ dh þ Nb ¼ 0; þ a1 Nt r dr dr dr dr
ð12Þ
ð13Þ
! ! 2 1 dh d / 1 d/ þ Nt ¼ 0; þ þ 2 2 r dr r dr dr dr 2
Nb
d h
ð14Þ
along with the corresponding boundary conditions
v ð1Þ ¼ hð1Þ ¼ /ð1Þ ¼ 0;
dv ð0Þ dhð0Þ d/ð0Þ ¼ ¼ ¼ 0; dr dr dr
ð15Þ
Here K, N, c, Gr ; Br ; N t and N b are non-Newtonian parameter, MHD parameter, constant pressure gradient, thermophoresis diffusion constant, Brownian diffusion constant, thermophoresis parameter and Brownian diffusion coefficient respectively. 3. Solution of the problems We now apply the HAM to establish analytical solutions to determine for the velocity, temperature and nano-concentration distributions by using Reynolds and Vogel’s models of viscosity. Case-I: Reynolds’ model Here viscosity is taken in the form
l ¼ eMh ;
ð16Þ
which after using the Maclaurin’s series can be written
l ¼ 1 hM þ Oðh2 Þ; M ¼ nðhm hw Þ;
ð17Þ
where M is related to how the viscosity of the Reynolds model varies with respect to temperature. For HAM solution we select
u0 ðrÞ ¼
cðr 2 1Þ ; 2
h0 ðrÞ ¼
cðr 2 1Þ ; 2
/0 ¼
cðr 2 1Þ ; 2
ð18Þ
as the initial approximations of v, h and /, respectively which satisfy the linear operator and corresponding boundary conditions. We use the method of higher order differential mapping, to choose the auxiliary linear operator L [23] 2
L¼
d
dr
2
;
ð19Þ
which satisfies the following relation
L½C 1 þ C 2 ln r ¼ 0;
ð20Þ
where C 1 and C 2 are the arbitrary constants. We now construct the homotopy
Hv ½v ðr; pÞ ¼ ð1 pÞL½v ðr; pÞ v 0 ðrÞ phN v ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð21Þ
Hh ½h ðr; pÞ ¼ ð1 pÞL½h ðr; pÞ h0 ðrÞ phN h ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð22Þ
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
H/ ½v ðr; pÞ ¼ ð1 pÞL½/ ðr; pÞ /0 ðrÞ phN / ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð23Þ
where embedding parameter p 2 ½0; 1, and h is a nonzero auxiliary parameter. Setting Hv ½v ðr; pÞ ¼ Hh ½h ðr; pÞ ¼ H/ ½/ ðr; pÞ ¼ 0, the zeroth order deformation equations are given by the following relations
ð1 pÞL½v ðr; pÞ v 0 ðrÞ ¼ phN v ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð24Þ
ð1 pÞL½h ðr; pÞ h0 ðrÞ ¼ p hN h ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð25Þ
ð1 pÞL½/ ðr; pÞ /0 ðrÞ ¼ p hN / ½/ ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð26Þ
v ð1; pÞ ¼ 0;
h ð1; pÞ ¼ 0;
/ ð1; pÞ ¼ 0;
dv ð0; pÞ dh ð0; pÞ d/ ð0; pÞ ¼ ¼ : dr dr dr
ð27Þ
The mth order deformation problems for Reynolds models are
L v m vm v m1 ¼ hR1m ðrÞ;
ð28Þ
L hm vm hm1 ¼ hR2m ðrÞ;
ð29Þ
L /m vm /m1 ¼ hR3m ðrÞ;
ð30Þ
along with the boundary conditions
v m ð1Þ ¼ hm ð1Þ ¼ /m ð1Þ ¼ 0;
dv m ð0Þ dhm ð0Þ d/m ð0Þ ¼ ¼ ¼ 0; dr dr dr
ð31Þ
where
9 > > > þM M > dr > > > i¼0 i¼0 j¼0 > > > > m1 m1 X X > 2 2 > dv mi1 d v mi1 d v m1 1 dv mi1 = þ r dr M h þ M h i i 2 2 dr dr dr R1m ðrÞ ¼ ; i¼0 i¼0 > " # > > m1 i m1 i > X X X X 2 dv ij dv ij d v j > dv mi1 dv mi1 > > þ 3K þ Kr dr dr dr dr dr2 > > > > i¼0 j¼0 i¼0 j¼0 > > ; 2 cð1 vm Þ N v m1 þ Gr hm1 þ Br /m1 ¼ 0 m1 X dh
2
R2m ðrÞ ¼ a
d hm1 dr
2
mi1
1 dhm1 þ r dr
2
d hm1
R3m ðrÞ ¼ Nb
dr
2
dv i dr
þ
2
m1 X
dhmi1 dr
!
1 dhm1 r dr
þ Nb
i X dv hij drj
2 m1 X dhmi1 d/i dhm1 ¼ 0; þ a1 N t dr dr dr i¼0
!
2
þ Nt
d /m1 dr
þ
2
1 d/m1 r dr
ð32Þ
ð33Þ
! ¼0
ð34Þ
and the non-linear operators N v ; N h and N / are
9 3 > M dhdr ddrv þ ð1hr MÞ ddrv þ Kr ddrv > =
N v ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ ¼ þð1 h MÞ d2 v2 þ 3K dv 2 d2 v2 þ Gr h ; dr > dr dr > ; þBr / c Nv 2
N h ½v
1 dh ðr; pÞ; h ðr; pÞ; / ðr; pÞ ¼ a þ 2 r dr dr
d h
2
N / ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ ¼ N b
vm ¼
0;
m 6 1;
1;
m > 1:
d h
!
1 dh þ 2 r dr dr
þ Nb
2 dh d/ dh ; þ a1 Nt dr dr dr
!
! 1 d/ ; þ 2 r dr dr
ð35Þ
ð36Þ
2
þ Nt
d /
ð37Þ
ð38Þ
By using the widely applied symbolic computation software MATHEMATICA to solve Eqs. (28)–(31), we obtain the analytical expressions for velocity, temperature and nano concentration for Reynolds model (up to m ¼ 2 ) of the following form:
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
v ¼ c2 245 cBr þ Mc6
2
2
r 5cG þ 5cN þ 24 48 2
2 3 r r þ Mc cB cG þ cN c24K r 4 12 24 24 48
h ¼ ca þ
c 2
9 2 2 r = þ cB4r Mc4 þ cG cN8 r 2 > 24 ; > ;
c2 c c c Nb r4 ; Nb þ a1 Nt car2 þ cðr 2 1Þ a1 Nt r 2 2 2 12 12
/ ¼ cðNb þ Nt Þ cðNb þ N t Þr 2 þ cðr 2 1Þ:
1455
ð39Þ
ð40Þ ð41Þ
Case-II: Vogel’s model We choose viscosity given by
l ¼ l0 eðBþhhw Þ ; A
ð42Þ
or Eq. (42) can be approximated [24] as
l¼
c hA 1 2 ; S B
ð43Þ
where A S ¼ lo eðBh0 Þ ;
A¼
C3 ; hm hw
B¼
C 4 þ hw ; hm hw
ð44Þ
A and B indicate how the viscosity of Vogel’s model varies and C 3 and C 4 are constants. For HAM solution here we select the same initial guess and linear operator given in Eqs. (19) and (20), the non-dimensionless form of the problem is
2 3 9 dv d v þ rSc 1 hA þ Sc 1 hA þ Kr ddrv = B2 dr B2 dr2 ; 2 2 ; þ3K ddrv ddrv2 c N2 v þ Gr h þ Br / ¼ 0
cA dh dv SB2 dr dr
2
a
d h dr
2
þ
! 2 1 dh dh d/ dh þ Nb þ a1 Nt ¼ 0; r dr dr dr dr
2
Nb
d h dr
2
þ
ð45Þ
ð46Þ
! ! 2 1 dh d / 1 d/ þ Nt ¼ 0; þ 2 r dr r dr dr
ð47Þ
along with the boundary conditions
v ð1Þ ¼ hð1Þ ¼ /ð1Þ ¼ 0;
dv ð0Þ dhð0Þ d/ð0Þ ¼ ¼ ¼ 0; dr dr dr
ð48Þ
With the same contrast as given for Reynolds’ model, the zeroth order deformation problem is given by
ð1 pÞL½v ðr; pÞ v 0 ðrÞ ¼ phN v ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð49Þ
ð1 pÞL½h ðr; pÞ h0 ðrÞ ¼ phN h ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð50Þ
ð1 pÞL½/ ðr; pÞ /0 ðrÞ ¼ phN / ½/ ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð51Þ
v ð1; pÞ ¼ 0;
h ð1; pÞ ¼ 0;
/ ð1; pÞ ¼ 0;
dv ð0; pÞ dh ð0; pÞ d/ ð0; pÞ ¼ ¼ : dr dr dr
ð52Þ
The mth order deformation problems for Vogel’s models are
L v m vm v m1 ¼ hR4m ðrÞ;
ð53Þ
L hm vm hm1 ¼ hR5m ðrÞ;
ð54Þ
L /m vm /m1 ¼ hR6m ðrÞ;
ð55Þ
v m ð1Þ ¼ hm ð1Þ ¼ /m ð1Þ ¼ 0;
dv m ð0Þ dhm ð0Þ d/m ð0Þ ¼ ¼ ¼ 0; dr dr dr
ð56Þ
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
where cA SB2
R4m ðrÞ ¼
m1 X
dhmi1 dv i dr dr
i¼0
BcA2
m1 i¼0
m1 X d2 v i¼0
3K
X dv cA mi1 þ rSc dvdrm1 rSB hi þ Sc 2 dr
m1 X i¼0
mi1
dr 2
dv mi1 dr
hi þ Kr
m1 X dv
dr i¼0
i X j¼0
mi1
i X dv
dv ij d2 v j dr dr2
þGr hm1 þ Br /m1 ¼ 0
ij
dr
dv j dr
j¼0
cð1 vm Þ N2 v
d2 v m1 dr2
9 > > > > > > > > > > > > > = ; > > > > > > > > > > > > > ;
Fig. 1. h-curve for velocity profile in case of Reynold’s model at 20th order approximation.
Fig. 2. h-curve for temperature profile in case of Reynold’s model at 20th order approximation.
Fig. 3. h-curve for the nanoparticles concentration profile in case of Reynold’s model at 20th order approximation.
ð57Þ
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467 2
R5m ðrÞ ¼ a
d hm1 dr
2
2
R6m ðrÞ ¼ Nb
þ
d hm1 dr
2
1 dhm1 r dr
!
1 dhm1 þ r dr
þ Nb
2 m1 X dhmi1 d/i dhm1 ¼ 0; þ a1 Nt dr dr dr i¼0
!
2
þ Nt
d /m1 dr
2
1 d/m1 þ r dr
1457
ð58Þ
! ¼ 0;
Fig. 4. h-curve for velocity profile in case of Vogel’s model at 20th order approximation.
Fig. 5. h-curve for temperature profile in case of Vogel’s model at 20th order approximation.
Fig. 6. h-curve for the nanoparticles concentration profile in case of Vogel’s model at 20th order approximation.
ð59Þ
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
9 cA dh dv c h A dv > SB 2 dr dr þ rS 1 2 dr > B > >
> > c h A d2 v K dv 3 = þ S 1 B2 dr2 þ r dr N v ½v ðr; qÞ; h ðr; qÞ; / ðr; qÞ ¼ ; > 2 2 > > þ3K ddrv ddrv2 c þ Gr h > > > ; þBr / Nv 2
N h ½v ðr; qÞ; h ðr; qÞ; / ðr; qÞ ¼ a
d h
1 dh þ 2 r dr dr
! þ Nb
2 dh d/ dh ; þ a1 Nt dr dr dr
Fig. 7. Residual error curve for velocity in case of Reynold’s model .
Fig. 8. Residual error curve for temperature in case of Reynold’s model.
Fig. 9. Residual error curve for nano concentraion in case of Reynold’s model.
ð60Þ
ð61Þ
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467 2
N / ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ ¼ Nb
d h dr
2
þ
1 dh r dr
!
2
þ Nt
d / dr
2
þ
! 1 d/ : r dr
1459
ð62Þ
Thus the analytical solutions of Eqs. (53)–(56) up to m ¼ 2 for Vogel’s model are of the following form
9 2 3 5Ac3 c2 Ac3 þ 5cN þ 24B þ c 3K > > 2 þ S þ 24 > 8B2 S >
= cBr cGr 3c cN2 Ac3 c2 Ac3 2 þ 2 þ 4 þ 4 4 4B2 S 4B2 S r ; > 2
> > 3 3 3 > Ac Ac r r ; þ cN cB cG þ 24B c 3K r 4 2 þ 24 24 24 8B2 S
v ¼ 3c2 245 cBr 5cG 24
r
c a2 N 2
2
ð63Þ
c2 N2
þ 3ca 2ca2 þ 3ca21 Nt 2caa1 Nt 12 t þ c 4Nb 6 b c h ¼ 3c 2 2
2 2 2 2 2 c2 N 2 5c 18aNb c a16Nb Nt þ c 4Nb þ 6 b þ c N6b Nt þ 5c 18aNb þ c a16Nb Nt r2
ca2 N2 c3 N 2 þ 3c 3ca þ 2ca2 3ca21 Nt þ 2caa1 N t þ 12 t r 4 þ 90b r 6 2
2N
b Nt
6
9 > > > > = ; > > > > ;
Fig. 10. Residual error curve for velocity in case of Vogel’s model.
Fig. 11. Residual error curve for temperature in case of Vogel’s model.
Fig. 12. Residual error curve for nano concentraion in case of Vogel’s model.
ð64Þ
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
9 c2 N 2 = / ¼ 3c þ 3cN b þ 3cN t 2cN b Nt 2cN 2t 2caNb ca1 Nb Nt 9 b 2
: 2 N2 c þ 3c 3cN b 3cN t þ 2cN b Nt þ 2cN 2t þ 2caNb þ ca1 Nb Nt r 2 þ 9 b r 4 ; 2
ð65Þ
The mth order solutions For p ¼ 0 and p ¼ 1, we have
v ðr; 0Þ ¼ v 0 ðrÞ; v ðr; 1Þ ¼ v ðrÞ;
h ðr; 0Þ ¼ h0 ðyÞ; h ðr; 1Þ ¼ hðrÞ;
/ ðr; 0Þ ¼ /0 ðrÞ / ðr; 1Þ ¼ /ðrÞ
:
Fig. 13. Effects of N on the velocity profile for Reynolds model.
Fig. 14. Effects of M on the velocity profile for Reynolds model.
Fig. 15. Effects of N b on the velocity profile for Reynolds model.
ð66Þ
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
1461
When p increases from 0 to 1; v ðr; pÞ; h ðr; pÞ / ðr; pÞ varies from v 0 ðrÞ; h0 ðrÞ /0 ðrÞ to v ðrÞ; hðrÞ and /ðrÞ respectively. By Taylor’s theorem Eq. (66) takes the following form
9 > v ðr; pÞ ¼ v 0 ðrÞ þ v m ðrÞp > > > > > m¼1 > > > 1 = X m h ðr; pÞ ¼ h0 ðrÞ þ hm ðrÞp ; > > m¼1 > > > 1 X > > > / ðr; pÞ ¼ /0 ðrÞ þ /m ðrÞpm > ;
1 X
m
m¼1
Fig. 16. Effects of N t on the velocity profile for Reynolds model.
Fig. 17. Effects of M on the temperature distribution for Reynolds model.
Fig. 18. Effects of N b on the temperature distribution for Reynolds model.
ð67Þ
1462
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
where
v m ðrÞ ¼
1 @ m v ðr; pÞ ; m! @pm p¼0
hm ðrÞ ¼
1 @ m h ðr; pÞ 1 @ m / ðr; pÞ / ðrÞ ¼ : m m! @pm p¼0 m! @pm p¼0
ð68Þ
The convergence of Eq. (67) depends upon h, therefore, we choose h in such a way that it should be convergent at p ¼ 1. Finally in view of Eq. (67), the mth order solution for both models can be written as
v ðrÞ ¼ v 0 ðrÞ þ
1 X
v m ðrÞ;
m¼1
hðrÞ ¼ h0 ðrÞ þ
1 X m¼1
hm ðrÞ /ðrÞ ¼ /0 ðrÞ þ
1 X /m ðrÞ: m¼1
Fig. 19. Effects of N t on the temperature distribution for Reynolds model.
Fig. 20. Effects of N b on the nanoparticle concentration for Reynolds model.
Fig. 21. Effects of N t on the nanoparticle concentration for Reynolds model.
ð69Þ
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
1463
4. Convergence of the solutions Here we discuss the convergence of solutions. The analytical series solutions (28)–(30) and (53)–(55) contain the nonzero auxiliary parameter h which can adjust and control the convergence of the series solutions. As pointed out by Liao [25], the appropriate region for h is a horizontal line segment. It is found that in all cases the convergence is achieved at 20th order of approximations. Figs. 1–3 potray the h-curves of velocity, temperature and nanoparticle concentration respectively to find the range of h in case of Reynold’s model. The range for admissible values of h for velocity is 0:4 6 h 6 0:1, for temperature is 1:4 6 h 6 0:1 and for nanoparticle concentration profile is 0:8 6 h 6 0:2. Figs. 4–6 represent the hcurves for Vogel’s model. The admissible ranges for velocity profile, temperature profile and nanoparticle concentration are 0:2 6 h 6 0:05; 1 6 h 6 0 and 1 6 h 6 0 respectively. In Figs. 7–12, the graphs of residual errors for velocity, temperature and nanoparticle concentration are plotted respectively. The error of norm 2 for successive approximations over ½0; 1 with HAM by 20th-order approximations are calculated by
Fig. 22. Effects of N on the velocity profile for Vogel’s model.
Fig. 23. Effects of A on the velocity profile for Vogel’s model.
Fig. 24. Effects of B on the velocity profile for Vogel’s model.
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 u 20 > u X > 2 1 > E2 ¼ t21 ðv 20 ði=20ÞÞ ¼ f1 > > > > > i¼0 > > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > u 20 = u X 2 1 t E2 ¼ 21 ðh20 ði=20ÞÞ ¼ f2 : > > i¼0 > > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > u 20 > > u X > 2 1 t > ð/20 ði=20ÞÞ ¼ f3 > E2 ¼ 21 > ; i¼0
It is seen that in all cases the error is minimum in the admissible range of h.
Fig. 25. Effects of N b on the velocity profile for Vogel’s model.
Fig. 26. Effects of N t on the velocity profile for Vogel’s model.
Fig. 27. Effects of A on the temperature distribution for Vogel’s model.
ð70Þ
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
1465
5. Results and discussion In this section a parametric study has been performed. The independent dimensionless numbers are N, M; A; B; N b and N t . The dimensionless number N is related to the MHD parameter; M is related to how the viscosity of the Reynolds model varies with respect to temperature; A and B indicate how the viscosity of Vogel’s model varies; N b and N t are the nano concentration parameters. Fig. 13 indicates the effects of MHD parameter N on the velocity profile. The effect of MHD parameter is to decrease the velocity of the fluid. Obviously, velocity decreases by increasing the MHD parameter N. This behavior holds even in the presence of variable viscosity. The same behavior is observed in Fig. 16 although the profiles are flatter and this due to the constants C 3 and C 4 involved in viscosity indexes A and B respectively. Figs. 14, 17, 22–24, and 26–28, show the influence of the
Fig. 28. Effects of B on the temperature distribution for Vogel’s model.
Fig. 29. Effects of N b on the temperature distribution for Vogel’s model.
Fig. 30. Effects of N t on the temperature distribution for Vogel’s model.
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
Fig. 31. Effects of N b on the nanoparticle concentration for Vogel’s model.
Fig. 32. Effects of N t on the nanoparticle concentration for Vogel’s model.
viscosity indexes M, A and B on the solutions. These figures show the strong dependence of velocity and temperature on viscosity indexes. It is revealed that velocity and temperature profiles for both Reynolds and Vogel’s models decrease by increasing the values of M and A respectively while the velocity and temperature profiles increase by increasing B. Figs. 15, 16, 18–21, 25, 26 and 29–32 have been displayed for non Newtonian nanofluid parameters namely thermophoresis parameter N t and Brownian parameter N t . These figures elucidate that the velocity, temperature and nano-concentration profiles decrease by increasing the thermophoresis parameter when Brownian parameter is fixed. The behavior in an opposite manner is observed when one varies Brownian parameter keeping thermophoresis parameter fixed. This is accordance with the fact that for thermal boundary the effects of thermophoresis parameter and Brownian parameter are different. The results show that, the velocity profile is qualitatively larger than that of temperature and concentration profiles. It is also observed that the maximum velocity, temperature and nano- concentration of the fluid is at the center of the pipe.
6. Conclusion The non-linear differential equations for the magnetohydrodynamic non-Newtonian nanofluid flow for variable viscosities are derived in this article. The flow is generated by the constant pressure gradient. The analytical solutions are developed by using the homotopy analysis method (HAM) and the convergence is discussed properly. Undoubtedly, the HAM provides us with a convenient way to control the convergence of approximation series; that is fundamental difference between the HAM and other methods for finding approximate solution. Also, the results are presented graphically and the effects of nondimensional parameters on the flow field are analyzed. It is observed that the MHD parameter N decreases the fluid motion and the velocity profile is larger than that of temperature profile even in the presence of variable viscosities. The effects of variable viscosity indexes and nano-concentration parameters are carefully examined. Here it is seen that the effect of viscosity indexes M and A is quite opposite to that of B. These figures show the strong dependence of velocity and temperature on viscosity indexes. The behavior in an opposite manner is observed in case of Brownian parameter N b and thermophoresis parameter N t . The results obtained reveal many interesting behaviors that also warrant further study on the equations related to non-Newtonian nanofluids, especially for the shear-thinning and shear thinking phenomena.
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
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Acknowledgments R. Ellahi thanks to United State Education Foundation Pakistan and CIES USA to honored him by Fulbright Scholar Award for the year 2011–2012. R.E. also grateful to the Higher Education Commission and PCST of Pakistan to award him with the awards of NRPU and Productive Scientist respectively. References [1] I. Hartmann, Hg-. Dynamics, I. Theory of laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Kg 1. Danske Videnskabernes Selskab, Math. Fys. Medd. 15 (1937). [2] V.J. Rossow, On Flow of Electrically Conducting Fluids over a Flat Plate in the Presence of a Transverse Magnetic Field, NACA TN 3971, 1957. [3] J.H. Davidson, F.A Kulacki, P.F. Dun, Convective heat transfer with electric and magnetic fields, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987, pp. 9.1–9.49. [4] F. Shakeri, M. Dehghan, A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations, Appl. Numer. Math. 61 (2011) 1–23. [5] M. Dehghan, D. Mirzaei, Meshless local Petrov–Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl. Numer. Math. 59 (2009) 1043–1058. [6] M. Dehghan, D. Mirzaei, Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes, Comput. Phys. Commun. 180 (2009) 1458–1466. [7] W.C. Tan, T. Masuoka, Stokes’ first problem for a second grade fluid in a porous half space with heated boundary, Int. J. Non-linear Mech. 40 (2005) 512–522. [8] W.C. Tan, T. Masuoka, Stability analysis of a Maxwell fluid in a porous medium heated from below, Phys. Lett. A 360 (2007) 454–460. [9] F.M. Mahomed, T. Hayat, Note on an exact solution for the pipe flow of a third grade fluid, Acta Mech. 190 (2007) 233–236. [10] M. Hameed, S. Nadeem, Unsteady MHD flow of a non-Newtonian fluid on a porous plate, J. Math. Anal. Appl. 325 (2007) 724–733. [11] M. Dehghan, F. Shakeri, The numerical solution of second Painlevé equation, Numer. Methods Part. Diff. Equat. 25 (2009) 1238–1259. [12] M.Y. Malik, A. Hussain, S. Nadeem, T. Hayat, Flow of a third grade fluid between coaxial cylinders with variable viscosity, Z. Naturforsch. 64a (2009) 588–596. [13] C. Fetecau, C. Fetecau, On some axial Couette flows of non-Newtonian fluids, Z. Angew. Math. Phys. (ZAMP) 56 (2005) 1098–1106. [14] S.U.S. Choi, Nanofluids: from vision to reality through research, J. Heat Transfer 131 (2009) 1–9. [15] S.U.S. Choi, J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, The Proceedings of the ASME International Mechanical Engineering Congress and Exposition, vol. 66, ASME, San Francisco, 1995, pp. 99–105. [16] K. Khanafer, K. Vafai, A critical synthesis of thermophysical characteristics of nanofluids, Int. J. Heat Mass Transfer 54 (2011) 4410–4428. [17] S.J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992. [18] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004) 499–513. [19] S.J. Liao, A. Campo, Analytic solutions of the temperature distribution in Blasius viscous flow problems, J. Fluid Mech. 453 (2002) 411–425. [20] M. Dehghan, J. Manaan, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Part. Diff. Equat. 26 (2010) 448–479. [21] M. Dehghan, J. Manaan, A. Saadatmandi, Application of semi-analytic methods for the Fitzhugh–Nagumo equation which models the transmission of nerve impulses, Math. Methods Appl. Sci. 33 (2010) 1384–1398. [22] R.L. Fosdick, K.R. Rajagopal, Thermodynamics and stability of fluids of third grade, Proc. Roy. Soc. Lond. A 339 (1980) 351–377. [23] R.A. Van Gorder, K. Vajravelu, On the selection of auxiliary functions operators and con-vergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 4078–4089. [24] M. Pakdemirli, B.S. Yilbas, Entropy generation for pipe flow of a third grade fluid with Vogel’s model viscosity, Int. J. Non-linear Mech. 41 (2006) 432– 437. [25] S.J. Liao, in: Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall, Boca Raton, 2003.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions R. Ellahi ⇑ Department of Mechanical Engineering, Bourns Hall, University of California Riverside, CA 92521, USA Department of Mathematics & Statistics, FBAS, IIUI, Islamabad 44000, Pakistan
a r t i c l e
i n f o
Article history: Received 10 January 2012 Received in revised form 22 March 2012 Accepted 2 April 2012 Available online 17 April 2012 Keywords: Non-Newtonian nanofluid Variable viscosities MHD Heat transfer analysis Analytical solutions
a b s t r a c t This article examines the magnetohydrodynamic (MHD) flow of non-Newtonian nanofluid in a pipe. The temperature of the pipe is assumed to be higher than the temperature of the fluid. In particular two temperature dependent viscosity models, have been considered. The nonlinear partial differential equations along with the boundary conditions are first cast into a dimensionless form and then the equations are solved by homotopy analysis method (HAM). Explicit analytical expressions for the velocity field, the temperature distribution and nano concentration have been derived analytically. The effects of various physical parameters on velocity, temperature and nano concentration are discussed by using graphical approach. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Magnetohydrodynamic (MHD) deals with the motion of conducting fluids. The applications of MHD cover a wide range of physical areas from liquid metals to cosmic plasmas; for instance, MHD pumps, MHD power generators, electrostatic precipitation, petroleum industry, electrostatic precipitation, purification of crude oil, aerodynamics heating, geophysics, plasma physics and fluid droplets sprays [1–6]. Moreover, non-Newtonian fluids [7–13] have been found much important and useful for technological point of view such as multi-grade oils, liquid detergents, paints, polymer solutions and polymer melts. Furthermore, recent advances in nanotechnology have led to the development of a new innovative class of heat transfer called nanofluids created by dispersing nanoparticles [14]. Non-Newtonian nanofluids are widely encountered in many industrial and technology applications, for example, melts of polymers, biological solutions, paints, tars, asphalts and glues etc. Nanofluids appear to have the potential to significantly increase heat transfer rates in a variety of areas such as industrial cooling applications, nuclear reactors, transportation industry, micro-electromechanical systems, electronics and instrumentation, and biomedical applications. Nanofluid has also been found to possess enhanced thermophysical properties such as thermal conductivity, thermal diffusively, viscosity and convective heat transfer coefficients compared to those of base fluids like oil or water. A careful review of the literature reveals that a very little efforts are devoted to examine the non Newtonian nanofluid. Some relevant studies on the topic can be found from the list of Refs. [15,16]. Motivated by these facts, in the present study we have investigated the effects of MHD and variable viscosities on nonNewtonian nanofluid in a pipe. The flow is generated by constant pressure gradient. To derive the solutions of nonlinear ⇑ Address: Department of Mechanical Engineering, Bourns Hall, University of California Riverside, CA 92521, USA. Tel.: +1 951 2756747. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.04.004
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
governing equations, we have used an efficient method, homotopy analysis method (HAM) [17–21], which is particularly suitable for strongly nonlinear problems. After the introduction in Section 1, the outlines of this paper are as follows. Section 2 contains mathematical formulation. In Section 3 solutions of the problems are presented by using HAM. Convergence and discussion are given in Sections 4 and 5 respectively. Finally Section 6 summaries the concluding remarks. 2. Formulation of the problem The governing equations of the fluid motion are the conservation of momentum
q
dV ¼ div T þ qb þ J B; dt
ð1Þ
where q is the density, d=dt is the material time derivative, V is the velocity field, T is the Cauchy stress tensor, J is the electric current density, B is the total magnetic field, b ¼ q e g k, is the body force, k being the unit vector in the z-direction, and e g the acceleration due to gravity. The fact that the fluid undergoes only isochoric motion, therefore, the law of conservation of mass is defined by
div V ¼ 0:
ð2Þ
In view of the principle of conservation of heat energy, the energy equation for nanofluid is given by
q
de Dt ¼ div Q ðqcÞp Db ru rh þ rh rh ; dt h
ð3Þ
where e is specific internal energy, h is temperature, cp is specific heat, Db is Brownian diffusion coefficient, Dt is thermophoretic diffusion coefficient and Q is heat flux. According to Fourier’s law of heat transfer
Q ¼ k grad h
ð4Þ
div Q ¼ kr ðrhÞ;
ð5Þ
and
k is thermal conductivity. Due to complexity of non-Newtonian nanofluids, there is no single model which describes all of their properties. Therefore, several constitutive equations have been proposed which can describe all the behaviors of non-Newtonian nanofluids; for example, stress differences, shear thinning or shear thickening, stress relaxation, elastic effects and memory effects. Amongst the many models, there is a grade three model which is the most popular. This is particularly due to the fact that one can reasonably explain the shear thinning/shear thickening properties even for steady and unidirectional flows. The stress in a third grade fluid is given by
T ¼ pI þ lA1 þ a1 A2 þ a2 A21 þ b1 A3 þ b2 ðA1 A2 þ A2 A1 Þ þ b3 ðtrA21 ÞA1 ;
ð6Þ
where l is the coefficient of viscosity, p is hydrostatic pressure, T is Cauchy stress tensor, pI is the spherical stress due to the constraint of incompressibility, ai ði ¼ 1; 2Þ are material constants, bj ðj ¼ 1; 2Þ are grade three parameters and first three Rivlin–Ericksen kinematical tensors A1 ; A2 and A3 are defined by
A1 ¼ ðgrad VÞ þ ðgrad VÞt ; An ¼
ð7Þ
dAn1 t þ An1 ðgrad VÞ þ ðgrad VÞ An1 ; dt
for n > 1;
ð8Þ
where V ¼ ½0; 0; v ðrÞ denotes the velocity vector. If all the motions of the fluid are to be compatible with thermodynamics in the sense that these motions satisfy the Clausius–Duhem inequality and if it is assumed that the specific Helmholtz free energy is a minimum when the fluid is locally at rest, then thermodynamics imposes the following constraints [22]
l P 0; a1 P 0; ja1 þ a2 j 6
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 24lb3 ;
b1 ¼ b2 ¼ 0;
b3 P 0:
ð9Þ
It is noted that this constitutive relation not only predicts the normal stress differences, but can also predict the ‘‘shearthickening’’ phenomenon (since b3 > 0) which is the increase in viscosity with increasing shear rate. In the present analysis we assume that the fluid is thermodynamically compatible, and therefore, Eq. (6) reduces to
T ¼ p1 I þ
h
i
l þ b3 ðtrA21 Þ A1 þ a1 A2 þ a2 A21 :
ð10Þ
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
1453
We render the above equations dimensionless by setting
v ¼ vv ;
w / ¼ /// ; m /w
2b3 v 20
K¼
^ w l ¼ ll0 ; h ¼ hhh ; c1 ¼ @@zp ; m hw
r ¼ Rr ;
0
l0 R2
;
Nb ¼ Db ð/m /w Þ; Gr ¼
ðhm hw Þqfw R2 ð1/w Þg ; l 0 u0
Nt ¼ Dt ðhhmwhw Þ ; Br ¼
9 2 > c ¼ vc10 Rl > 0 > rffiffiffiffiffiffiffiffiffiffi > > =
N¼
ðqp qw ÞR2 ð/m /w Þg l0 u0
rB20 R2 l0 >;
> > > > ;
ð11Þ
where R is the radius of pipe, hw is the pipe temperature, hm is the fluid temperature, v is velocity, h is temperature, / is concentration, qf is the density of the base fluid, v 0 , is the reference velocity and l0 is the reference viscosity. Substituting Eq. (10) in the balance of linear momentum and using the non-dimensional quantities given in Eq. (11), the dimensionless form of Eqs. (1)–(3), after dropping the over bars for convenience, we obtain the following non-dimensional coupled equations
3 2 2 2 dl dv l dv d v K dv dv d v þ 3K ¼ c þ N2 v Gr h Br /; þ þl 2 þ 2 dr dr r dr r dr dr dr dr 2
a
d h dr
2
þ
! 2 1 dh dh d/ dh þ Nb ¼ 0; þ a1 Nt r dr dr dr dr
ð12Þ
ð13Þ
! ! 2 1 dh d / 1 d/ þ Nt ¼ 0; þ þ 2 2 r dr r dr dr dr 2
Nb
d h
ð14Þ
along with the corresponding boundary conditions
v ð1Þ ¼ hð1Þ ¼ /ð1Þ ¼ 0;
dv ð0Þ dhð0Þ d/ð0Þ ¼ ¼ ¼ 0; dr dr dr
ð15Þ
Here K, N, c, Gr ; Br ; N t and N b are non-Newtonian parameter, MHD parameter, constant pressure gradient, thermophoresis diffusion constant, Brownian diffusion constant, thermophoresis parameter and Brownian diffusion coefficient respectively. 3. Solution of the problems We now apply the HAM to establish analytical solutions to determine for the velocity, temperature and nano-concentration distributions by using Reynolds and Vogel’s models of viscosity. Case-I: Reynolds’ model Here viscosity is taken in the form
l ¼ eMh ;
ð16Þ
which after using the Maclaurin’s series can be written
l ¼ 1 hM þ Oðh2 Þ; M ¼ nðhm hw Þ;
ð17Þ
where M is related to how the viscosity of the Reynolds model varies with respect to temperature. For HAM solution we select
u0 ðrÞ ¼
cðr 2 1Þ ; 2
h0 ðrÞ ¼
cðr 2 1Þ ; 2
/0 ¼
cðr 2 1Þ ; 2
ð18Þ
as the initial approximations of v, h and /, respectively which satisfy the linear operator and corresponding boundary conditions. We use the method of higher order differential mapping, to choose the auxiliary linear operator L [23] 2
L¼
d
dr
2
;
ð19Þ
which satisfies the following relation
L½C 1 þ C 2 ln r ¼ 0;
ð20Þ
where C 1 and C 2 are the arbitrary constants. We now construct the homotopy
Hv ½v ðr; pÞ ¼ ð1 pÞL½v ðr; pÞ v 0 ðrÞ phN v ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð21Þ
Hh ½h ðr; pÞ ¼ ð1 pÞL½h ðr; pÞ h0 ðrÞ phN h ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð22Þ
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
H/ ½v ðr; pÞ ¼ ð1 pÞL½/ ðr; pÞ /0 ðrÞ phN / ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð23Þ
where embedding parameter p 2 ½0; 1, and h is a nonzero auxiliary parameter. Setting Hv ½v ðr; pÞ ¼ Hh ½h ðr; pÞ ¼ H/ ½/ ðr; pÞ ¼ 0, the zeroth order deformation equations are given by the following relations
ð1 pÞL½v ðr; pÞ v 0 ðrÞ ¼ phN v ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð24Þ
ð1 pÞL½h ðr; pÞ h0 ðrÞ ¼ p hN h ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð25Þ
ð1 pÞL½/ ðr; pÞ /0 ðrÞ ¼ p hN / ½/ ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð26Þ
v ð1; pÞ ¼ 0;
h ð1; pÞ ¼ 0;
/ ð1; pÞ ¼ 0;
dv ð0; pÞ dh ð0; pÞ d/ ð0; pÞ ¼ ¼ : dr dr dr
ð27Þ
The mth order deformation problems for Reynolds models are
L v m vm v m1 ¼ hR1m ðrÞ;
ð28Þ
L hm vm hm1 ¼ hR2m ðrÞ;
ð29Þ
L /m vm /m1 ¼ hR3m ðrÞ;
ð30Þ
along with the boundary conditions
v m ð1Þ ¼ hm ð1Þ ¼ /m ð1Þ ¼ 0;
dv m ð0Þ dhm ð0Þ d/m ð0Þ ¼ ¼ ¼ 0; dr dr dr
ð31Þ
where
9 > > > þM M > dr > > > i¼0 i¼0 j¼0 > > > > m1 m1 X X > 2 2 > dv mi1 d v mi1 d v m1 1 dv mi1 = þ r dr M h þ M h i i 2 2 dr dr dr R1m ðrÞ ¼ ; i¼0 i¼0 > " # > > m1 i m1 i > X X X X 2 dv ij dv ij d v j > dv mi1 dv mi1 > > þ 3K þ Kr dr dr dr dr dr2 > > > > i¼0 j¼0 i¼0 j¼0 > > ; 2 cð1 vm Þ N v m1 þ Gr hm1 þ Br /m1 ¼ 0 m1 X dh
2
R2m ðrÞ ¼ a
d hm1 dr
2
mi1
1 dhm1 þ r dr
2
d hm1
R3m ðrÞ ¼ Nb
dr
2
dv i dr
þ
2
m1 X
dhmi1 dr
!
1 dhm1 r dr
þ Nb
i X dv hij drj
2 m1 X dhmi1 d/i dhm1 ¼ 0; þ a1 N t dr dr dr i¼0
!
2
þ Nt
d /m1 dr
þ
2
1 d/m1 r dr
ð32Þ
ð33Þ
! ¼0
ð34Þ
and the non-linear operators N v ; N h and N / are
9 3 > M dhdr ddrv þ ð1hr MÞ ddrv þ Kr ddrv > =
N v ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ ¼ þð1 h MÞ d2 v2 þ 3K dv 2 d2 v2 þ Gr h ; dr > dr dr > ; þBr / c Nv 2
N h ½v
1 dh ðr; pÞ; h ðr; pÞ; / ðr; pÞ ¼ a þ 2 r dr dr
d h
2
N / ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ ¼ N b
vm ¼
0;
m 6 1;
1;
m > 1:
d h
!
1 dh þ 2 r dr dr
þ Nb
2 dh d/ dh ; þ a1 Nt dr dr dr
!
! 1 d/ ; þ 2 r dr dr
ð35Þ
ð36Þ
2
þ Nt
d /
ð37Þ
ð38Þ
By using the widely applied symbolic computation software MATHEMATICA to solve Eqs. (28)–(31), we obtain the analytical expressions for velocity, temperature and nano concentration for Reynolds model (up to m ¼ 2 ) of the following form:
R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
v ¼ c2 245 cBr þ Mc6
2
2
r 5cG þ 5cN þ 24 48 2
2 3 r r þ Mc cB cG þ cN c24K r 4 12 24 24 48
h ¼ ca þ
c 2
9 2 2 r = þ cB4r Mc4 þ cG cN8 r 2 > 24 ; > ;
c2 c c c Nb r4 ; Nb þ a1 Nt car2 þ cðr 2 1Þ a1 Nt r 2 2 2 12 12
/ ¼ cðNb þ Nt Þ cðNb þ N t Þr 2 þ cðr 2 1Þ:
1455
ð39Þ
ð40Þ ð41Þ
Case-II: Vogel’s model We choose viscosity given by
l ¼ l0 eðBþhhw Þ ; A
ð42Þ
or Eq. (42) can be approximated [24] as
l¼
c hA 1 2 ; S B
ð43Þ
where A S ¼ lo eðBh0 Þ ;
A¼
C3 ; hm hw
B¼
C 4 þ hw ; hm hw
ð44Þ
A and B indicate how the viscosity of Vogel’s model varies and C 3 and C 4 are constants. For HAM solution here we select the same initial guess and linear operator given in Eqs. (19) and (20), the non-dimensionless form of the problem is
2 3 9 dv d v þ rSc 1 hA þ Sc 1 hA þ Kr ddrv = B2 dr B2 dr2 ; 2 2 ; þ3K ddrv ddrv2 c N2 v þ Gr h þ Br / ¼ 0
cA dh dv SB2 dr dr
2
a
d h dr
2
þ
! 2 1 dh dh d/ dh þ Nb þ a1 Nt ¼ 0; r dr dr dr dr
2
Nb
d h dr
2
þ
ð45Þ
ð46Þ
! ! 2 1 dh d / 1 d/ þ Nt ¼ 0; þ 2 r dr r dr dr
ð47Þ
along with the boundary conditions
v ð1Þ ¼ hð1Þ ¼ /ð1Þ ¼ 0;
dv ð0Þ dhð0Þ d/ð0Þ ¼ ¼ ¼ 0; dr dr dr
ð48Þ
With the same contrast as given for Reynolds’ model, the zeroth order deformation problem is given by
ð1 pÞL½v ðr; pÞ v 0 ðrÞ ¼ phN v ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð49Þ
ð1 pÞL½h ðr; pÞ h0 ðrÞ ¼ phN h ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð50Þ
ð1 pÞL½/ ðr; pÞ /0 ðrÞ ¼ phN / ½/ ðr; pÞ; h ðr; pÞ; / ðr; pÞ;
ð51Þ
v ð1; pÞ ¼ 0;
h ð1; pÞ ¼ 0;
/ ð1; pÞ ¼ 0;
dv ð0; pÞ dh ð0; pÞ d/ ð0; pÞ ¼ ¼ : dr dr dr
ð52Þ
The mth order deformation problems for Vogel’s models are
L v m vm v m1 ¼ hR4m ðrÞ;
ð53Þ
L hm vm hm1 ¼ hR5m ðrÞ;
ð54Þ
L /m vm /m1 ¼ hR6m ðrÞ;
ð55Þ
v m ð1Þ ¼ hm ð1Þ ¼ /m ð1Þ ¼ 0;
dv m ð0Þ dhm ð0Þ d/m ð0Þ ¼ ¼ ¼ 0; dr dr dr
ð56Þ
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467
where cA SB2
R4m ðrÞ ¼
m1 X
dhmi1 dv i dr dr
i¼0
BcA2
m1 i¼0
m1 X d2 v i¼0
3K
X dv cA mi1 þ rSc dvdrm1 rSB hi þ Sc 2 dr
m1 X i¼0
mi1
dr 2
dv mi1 dr
hi þ Kr
m1 X dv
dr i¼0
i X j¼0
mi1
i X dv
dv ij d2 v j dr dr2
þGr hm1 þ Br /m1 ¼ 0
ij
dr
dv j dr
j¼0
cð1 vm Þ N2 v
d2 v m1 dr2
9 > > > > > > > > > > > > > = ; > > > > > > > > > > > > > ;
Fig. 1. h-curve for velocity profile in case of Reynold’s model at 20th order approximation.
Fig. 2. h-curve for temperature profile in case of Reynold’s model at 20th order approximation.
Fig. 3. h-curve for the nanoparticles concentration profile in case of Reynold’s model at 20th order approximation.
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R. Ellahi / Applied Mathematical Modelling 37 (2013) 1451–1467 2
R5m ðrÞ ¼ a
d hm1 dr
2
2
R6m ðrÞ ¼ Nb
þ
d hm1 dr
2
1 dhm1 r dr
!
1 dhm1 þ r dr
þ Nb
2 m1 X dhmi1 d/i dhm1 ¼ 0; þ a1 Nt dr dr dr i¼0
!
2
þ Nt
d /m1 dr
2
1 d/m1 þ r dr
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ð58Þ
! ¼ 0;
Fig. 4. h-curve for velocity profile in case of Vogel’s model at 20th order approximation.
Fig. 5. h-curve for temperature profile in case of Vogel’s model at 20th order approximation.
Fig. 6. h-curve for the nanoparticles concentration profile in case of Vogel’s model at 20th order approximation.
ð59Þ
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9 cA dh dv c h A dv > SB 2 dr dr þ rS 1 2 dr > B > >
> > c h A d2 v K dv 3 = þ S 1 B2 dr2 þ r dr N v ½v ðr; qÞ; h ðr; qÞ; / ðr; qÞ ¼ ; > 2 2 > > þ3K ddrv ddrv2 c þ Gr h > > > ; þBr / Nv 2
N h ½v ðr; qÞ; h ðr; qÞ; / ðr; qÞ ¼ a
d h
1 dh þ 2 r dr dr
! þ Nb
2 dh d/ dh ; þ a1 Nt dr dr dr
Fig. 7. Residual error curve for velocity in case of Reynold’s model .
Fig. 8. Residual error curve for temperature in case of Reynold’s model.
Fig. 9. Residual error curve for nano concentraion in case of Reynold’s model.
ð60Þ
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N / ½v ðr; pÞ; h ðr; pÞ; / ðr; pÞ ¼ Nb
d h dr
2
þ
1 dh r dr
!
2
þ Nt
d / dr
2
þ
! 1 d/ : r dr
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ð62Þ
Thus the analytical solutions of Eqs. (53)–(56) up to m ¼ 2 for Vogel’s model are of the following form
9 2 3 5Ac3 c2 Ac3 þ 5cN þ 24B þ c 3K > > 2 þ S þ 24 > 8B2 S >
= cBr cGr 3c cN2 Ac3 c2 Ac3 2 þ 2 þ 4 þ 4 4 4B2 S 4B2 S r ; > 2
> > 3 3 3 > Ac Ac r r ; þ cN cB cG þ 24B c 3K r 4 2 þ 24 24 24 8B2 S
v ¼ 3c2 245 cBr 5cG 24
r
c a2 N 2
2
ð63Þ
c2 N2
þ 3ca 2ca2 þ 3ca21 Nt 2caa1 Nt 12 t þ c 4Nb 6 b c h ¼ 3c 2 2
2 2 2 2 2 c2 N 2 5c 18aNb c a16Nb Nt þ c 4Nb þ 6 b þ c N6b Nt þ 5c 18aNb þ c a16Nb Nt r2
ca2 N2 c3 N 2 þ 3c 3ca þ 2ca2 3ca21 Nt þ 2caa1 N t þ 12 t r 4 þ 90b r 6 2
2N
b Nt
6
9 > > > > = ; > > > > ;
Fig. 10. Residual error curve for velocity in case of Vogel’s model.
Fig. 11. Residual error curve for temperature in case of Vogel’s model.
Fig. 12. Residual error curve for nano concentraion in case of Vogel’s model.
ð64Þ
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9 c2 N 2 = / ¼ 3c þ 3cN b þ 3cN t 2cN b Nt 2cN 2t 2caNb ca1 Nb Nt 9 b 2
: 2 N2 c þ 3c 3cN b 3cN t þ 2cN b Nt þ 2cN 2t þ 2caNb þ ca1 Nb Nt r 2 þ 9 b r 4 ; 2
ð65Þ
The mth order solutions For p ¼ 0 and p ¼ 1, we have
v ðr; 0Þ ¼ v 0 ðrÞ; v ðr; 1Þ ¼ v ðrÞ;
h ðr; 0Þ ¼ h0 ðyÞ; h ðr; 1Þ ¼ hðrÞ;
/ ðr; 0Þ ¼ /0 ðrÞ / ðr; 1Þ ¼ /ðrÞ
:
Fig. 13. Effects of N on the velocity profile for Reynolds model.
Fig. 14. Effects of M on the velocity profile for Reynolds model.
Fig. 15. Effects of N b on the velocity profile for Reynolds model.
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When p increases from 0 to 1; v ðr; pÞ; h ðr; pÞ / ðr; pÞ varies from v 0 ðrÞ; h0 ðrÞ /0 ðrÞ to v ðrÞ; hðrÞ and /ðrÞ respectively. By Taylor’s theorem Eq. (66) takes the following form
9 > v ðr; pÞ ¼ v 0 ðrÞ þ v m ðrÞp > > > > > m¼1 > > > 1 = X m h ðr; pÞ ¼ h0 ðrÞ þ hm ðrÞp ; > > m¼1 > > > 1 X > > > / ðr; pÞ ¼ /0 ðrÞ þ /m ðrÞpm > ;
1 X
m
m¼1
Fig. 16. Effects of N t on the velocity profile for Reynolds model.
Fig. 17. Effects of M on the temperature distribution for Reynolds model.
Fig. 18. Effects of N b on the temperature distribution for Reynolds model.
ð67Þ
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where
v m ðrÞ ¼
1 @ m v ðr; pÞ ; m! @pm p¼0
hm ðrÞ ¼
1 @ m h ðr; pÞ 1 @ m / ðr; pÞ / ðrÞ ¼ : m m! @pm p¼0 m! @pm p¼0
ð68Þ
The convergence of Eq. (67) depends upon h, therefore, we choose h in such a way that it should be convergent at p ¼ 1. Finally in view of Eq. (67), the mth order solution for both models can be written as
v ðrÞ ¼ v 0 ðrÞ þ
1 X
v m ðrÞ;
m¼1
hðrÞ ¼ h0 ðrÞ þ
1 X m¼1
hm ðrÞ /ðrÞ ¼ /0 ðrÞ þ
1 X /m ðrÞ: m¼1
Fig. 19. Effects of N t on the temperature distribution for Reynolds model.
Fig. 20. Effects of N b on the nanoparticle concentration for Reynolds model.
Fig. 21. Effects of N t on the nanoparticle concentration for Reynolds model.
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4. Convergence of the solutions Here we discuss the convergence of solutions. The analytical series solutions (28)–(30) and (53)–(55) contain the nonzero auxiliary parameter h which can adjust and control the convergence of the series solutions. As pointed out by Liao [25], the appropriate region for h is a horizontal line segment. It is found that in all cases the convergence is achieved at 20th order of approximations. Figs. 1–3 potray the h-curves of velocity, temperature and nanoparticle concentration respectively to find the range of h in case of Reynold’s model. The range for admissible values of h for velocity is 0:4 6 h 6 0:1, for temperature is 1:4 6 h 6 0:1 and for nanoparticle concentration profile is 0:8 6 h 6 0:2. Figs. 4–6 represent the hcurves for Vogel’s model. The admissible ranges for velocity profile, temperature profile and nanoparticle concentration are 0:2 6 h 6 0:05; 1 6 h 6 0 and 1 6 h 6 0 respectively. In Figs. 7–12, the graphs of residual errors for velocity, temperature and nanoparticle concentration are plotted respectively. The error of norm 2 for successive approximations over ½0; 1 with HAM by 20th-order approximations are calculated by
Fig. 22. Effects of N on the velocity profile for Vogel’s model.
Fig. 23. Effects of A on the velocity profile for Vogel’s model.
Fig. 24. Effects of B on the velocity profile for Vogel’s model.
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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 u 20 > u X > 2 1 > E2 ¼ t21 ðv 20 ði=20ÞÞ ¼ f1 > > > > > i¼0 > > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > u 20 = u X 2 1 t E2 ¼ 21 ðh20 ði=20ÞÞ ¼ f2 : > > i¼0 > > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > u 20 > > u X > 2 1 t > ð/20 ði=20ÞÞ ¼ f3 > E2 ¼ 21 > ; i¼0
It is seen that in all cases the error is minimum in the admissible range of h.
Fig. 25. Effects of N b on the velocity profile for Vogel’s model.
Fig. 26. Effects of N t on the velocity profile for Vogel’s model.
Fig. 27. Effects of A on the temperature distribution for Vogel’s model.
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5. Results and discussion In this section a parametric study has been performed. The independent dimensionless numbers are N, M; A; B; N b and N t . The dimensionless number N is related to the MHD parameter; M is related to how the viscosity of the Reynolds model varies with respect to temperature; A and B indicate how the viscosity of Vogel’s model varies; N b and N t are the nano concentration parameters. Fig. 13 indicates the effects of MHD parameter N on the velocity profile. The effect of MHD parameter is to decrease the velocity of the fluid. Obviously, velocity decreases by increasing the MHD parameter N. This behavior holds even in the presence of variable viscosity. The same behavior is observed in Fig. 16 although the profiles are flatter and this due to the constants C 3 and C 4 involved in viscosity indexes A and B respectively. Figs. 14, 17, 22–24, and 26–28, show the influence of the
Fig. 28. Effects of B on the temperature distribution for Vogel’s model.
Fig. 29. Effects of N b on the temperature distribution for Vogel’s model.
Fig. 30. Effects of N t on the temperature distribution for Vogel’s model.
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Fig. 31. Effects of N b on the nanoparticle concentration for Vogel’s model.
Fig. 32. Effects of N t on the nanoparticle concentration for Vogel’s model.
viscosity indexes M, A and B on the solutions. These figures show the strong dependence of velocity and temperature on viscosity indexes. It is revealed that velocity and temperature profiles for both Reynolds and Vogel’s models decrease by increasing the values of M and A respectively while the velocity and temperature profiles increase by increasing B. Figs. 15, 16, 18–21, 25, 26 and 29–32 have been displayed for non Newtonian nanofluid parameters namely thermophoresis parameter N t and Brownian parameter N t . These figures elucidate that the velocity, temperature and nano-concentration profiles decrease by increasing the thermophoresis parameter when Brownian parameter is fixed. The behavior in an opposite manner is observed when one varies Brownian parameter keeping thermophoresis parameter fixed. This is accordance with the fact that for thermal boundary the effects of thermophoresis parameter and Brownian parameter are different. The results show that, the velocity profile is qualitatively larger than that of temperature and concentration profiles. It is also observed that the maximum velocity, temperature and nano- concentration of the fluid is at the center of the pipe.
6. Conclusion The non-linear differential equations for the magnetohydrodynamic non-Newtonian nanofluid flow for variable viscosities are derived in this article. The flow is generated by the constant pressure gradient. The analytical solutions are developed by using the homotopy analysis method (HAM) and the convergence is discussed properly. Undoubtedly, the HAM provides us with a convenient way to control the convergence of approximation series; that is fundamental difference between the HAM and other methods for finding approximate solution. Also, the results are presented graphically and the effects of nondimensional parameters on the flow field are analyzed. It is observed that the MHD parameter N decreases the fluid motion and the velocity profile is larger than that of temperature profile even in the presence of variable viscosities. The effects of variable viscosity indexes and nano-concentration parameters are carefully examined. Here it is seen that the effect of viscosity indexes M and A is quite opposite to that of B. These figures show the strong dependence of velocity and temperature on viscosity indexes. The behavior in an opposite manner is observed in case of Brownian parameter N b and thermophoresis parameter N t . The results obtained reveal many interesting behaviors that also warrant further study on the equations related to non-Newtonian nanofluids, especially for the shear-thinning and shear thinking phenomena.
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Acknowledgments R. Ellahi thanks to United State Education Foundation Pakistan and CIES USA to honored him by Fulbright Scholar Award for the year 2011–2012. R.E. also grateful to the Higher Education Commission and PCST of Pakistan to award him with the awards of NRPU and Productive Scientist respectively. References [1] I. Hartmann, Hg-. Dynamics, I. Theory of laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Kg 1. Danske Videnskabernes Selskab, Math. Fys. Medd. 15 (1937). [2] V.J. Rossow, On Flow of Electrically Conducting Fluids over a Flat Plate in the Presence of a Transverse Magnetic Field, NACA TN 3971, 1957. [3] J.H. Davidson, F.A Kulacki, P.F. Dun, Convective heat transfer with electric and magnetic fields, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987, pp. 9.1–9.49. [4] F. Shakeri, M. Dehghan, A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations, Appl. Numer. Math. 61 (2011) 1–23. [5] M. Dehghan, D. Mirzaei, Meshless local Petrov–Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl. Numer. Math. 59 (2009) 1043–1058. [6] M. Dehghan, D. Mirzaei, Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes, Comput. Phys. Commun. 180 (2009) 1458–1466. [7] W.C. Tan, T. Masuoka, Stokes’ first problem for a second grade fluid in a porous half space with heated boundary, Int. J. Non-linear Mech. 40 (2005) 512–522. [8] W.C. Tan, T. Masuoka, Stability analysis of a Maxwell fluid in a porous medium heated from below, Phys. Lett. A 360 (2007) 454–460. [9] F.M. Mahomed, T. Hayat, Note on an exact solution for the pipe flow of a third grade fluid, Acta Mech. 190 (2007) 233–236. [10] M. Hameed, S. Nadeem, Unsteady MHD flow of a non-Newtonian fluid on a porous plate, J. Math. Anal. Appl. 325 (2007) 724–733. [11] M. Dehghan, F. Shakeri, The numerical solution of second Painlevé equation, Numer. Methods Part. Diff. Equat. 25 (2009) 1238–1259. [12] M.Y. Malik, A. Hussain, S. Nadeem, T. Hayat, Flow of a third grade fluid between coaxial cylinders with variable viscosity, Z. Naturforsch. 64a (2009) 588–596. [13] C. Fetecau, C. Fetecau, On some axial Couette flows of non-Newtonian fluids, Z. Angew. Math. Phys. (ZAMP) 56 (2005) 1098–1106. [14] S.U.S. Choi, Nanofluids: from vision to reality through research, J. Heat Transfer 131 (2009) 1–9. [15] S.U.S. Choi, J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, The Proceedings of the ASME International Mechanical Engineering Congress and Exposition, vol. 66, ASME, San Francisco, 1995, pp. 99–105. [16] K. Khanafer, K. Vafai, A critical synthesis of thermophysical characteristics of nanofluids, Int. J. Heat Mass Transfer 54 (2011) 4410–4428. [17] S.J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992. [18] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004) 499–513. [19] S.J. Liao, A. Campo, Analytic solutions of the temperature distribution in Blasius viscous flow problems, J. Fluid Mech. 453 (2002) 411–425. [20] M. Dehghan, J. Manaan, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Part. Diff. Equat. 26 (2010) 448–479. [21] M. Dehghan, J. Manaan, A. Saadatmandi, Application of semi-analytic methods for the Fitzhugh–Nagumo equation which models the transmission of nerve impulses, Math. Methods Appl. Sci. 33 (2010) 1384–1398. [22] R.L. Fosdick, K.R. Rajagopal, Thermodynamics and stability of fluids of third grade, Proc. Roy. Soc. Lond. A 339 (1980) 351–377. [23] R.A. Van Gorder, K. Vajravelu, On the selection of auxiliary functions operators and con-vergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 4078–4089. [24] M. Pakdemirli, B.S. Yilbas, Entropy generation for pipe flow of a third grade fluid with Vogel’s model viscosity, Int. J. Non-linear Mech. 41 (2006) 432– 437. [25] S.J. Liao, in: Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall, Boca Raton, 2003.