Abstract The flow of a non-Newtonian fluid through a porous media in between twoparallel plates at different temperatures is considered. The governing momentum equation ofthird-grade fluid with modified Darcy’s law and energy equation have been derived. Approxi-mate analytical solutions of momentum and energy equations are obtained by using perturba-tion techniques. Constant viscosity, Reynold’s model viscosity, and Vogel’s model viscositycases are treated separately. The criteria for validity of approximate solutions are derived.A numerical residual error analysis is performed for the solutions. Within the validity range,analytical and numerical solutions are in good agreement.
Research on non-Newtonian fluids is intensified in the past few decades. In order to mathe-matically describe the properties of non-Newtonian fluids, various constitutive equations areproposed. Among many proposed models, Rivlin–Ericksen fluids received much attention(Rivlin and Ericksen 1955). A special Rivlin–Ericksen fluid, namely, the third-grade fluidmodel is considered in this study. The fluid has been thermodynamically analyzed in detailby Fosdick and Rajagopal (1980) and Dunn and Rajagopal (1995). The pipe flow of thethird-grade fluid with temperature-dependent viscosity problem has been extensively ana-lyzed using perturbation techniques by Yürüsoy and Pakdemirli (2002). They showed thatwithin the validity range, perturbation solutions are in excellent agreement with the numer-ical solutions of the same problem presented by Massoudi and Christie (1995). Momentumand energy equations of the third-grade fluid past an infinite porous plate with suction andblowing are derived and solved numerically by Maneschy et al. (1993). In the case of the
Y. Aksoy · M. Pakdemirli (B)Department of Mechanical Engineering, Celal Bayar University, 45140 Muradiye, Manisa, Turkeye-mail: [email protected]
123
376 Y. Aksoy, M. Pakdemirli
flow between parallel plates, Hayat et al. (2006) have applied homotopy analysis method,and numerical scheme to obtain solutions of a pressure-driven third-grade fluid past a porouschannel with suction and injection at walls. Homotopy and perturbation solutions of a third-grade fluid past a porous plate are obtained by Ayub et al. (2003). Furthermore, Siddiquiet al. (2008) applied homotopy perturbation technique to the parallel plate flow of third-grade fluid for constant viscosity. Later, Yürüsoy et al. (2008a) obtained solutions of thethird-grade fluid with variable viscosity flowing between parallel plates at different temper-atures via perturbation techniques. Entropy analysis of the third-grade fluid in an annularpipe has been studied for temperature-dependent viscosity by Yürüsoy et al. (2008b). Elec-tro-osmotic flow of a third-grade fluid between parallel plates was treated by Akgül andPakdemirli (2008). Third-grade fluid models can be used in describing polymer flows. Byemploying non-equilibrium molecular dynamics simulations, Daivis et al. (2007) computedcoefficients of the third-grade fluid. Using kinetic theory of polymeric fluids, Degond et al.(2002) derived some of the visco-elastic models which include a third-grade fluid as a specialcase.
Flow through porous media received substantial attention due to widespread applicationssuch as oil recovery, paper and textile coating, composite manufacturing processes, mix-ture theory, filtration processes, geothermal engineering, insulation systems etc. For New-tonian fluids, Yang et al. (2002) developed a mathematical model for the flow of waterthrough a channel impregnated with a polymer gel that is treated as an elastic and deform-able porous medium. Kumar et al. (1991) investigated the permeability of a rock fracture.The attempts to include porous media in the flows of complex fluids need some new phys-ical parameters besides non-Newtonian fluid parameters. Thus, Darcy’s equations or somegeneralization of it depending on pressure field, not neglecting porosity, are appropriateto study this type of flows through the porous media which is rigid or nearly rigid solid(Subramanian and Rajagopal 2007; Kannan and Rajagopal 2008). Liu (2005) examinedflow of an electrically conducting fluid of second grade through porous media subject toa transverse magnetic field over a stretching sheet with power-law surface temperature.Hayat et al. (2007) applied homotopy analysis method to solve the third-grade fluid flowproblem immersed in a porous media over a moving flat plate. In the case of the suc-tion and injection flows of a third-grade fluid through porous media, extensive study hasbeen performed by Sajid and Hayat (2008). They investigated the effect of porosity param-eter on the velocity profile and the boundary layer thickness. Channel flow with porousmedium of a MHD-driven second-grade fluid have been considered by Hayat and Abbas(2008). MHD flow of a Sisko Fluid occupying porous media have been investigated byKhan et al. (2008). The momentum equations based upon a modified Darcy’s law of a fourth-grade fluid were derived and solved analytically by Hayat et al. (2009). Ahmad (2009)proposed a simple analytical solution for a flow of third-grade fluid through porous media.Ellahi and Afzal (2009) studied pipe flow of a third-grade fluid through porous media withheat transfer.
In this study, parallel plate flow of a third-grade fluid through porous media is treatedfor the first time. The governing equations are written, cast in a nondimensional form andsolved approximately using perturbation techniques. This study is an extension of the analy-sis performed by Yürüsoy et al. (2008a) i.e. it depicts the effect of porous media on the flowadditionally. Constant and temperature-dependent viscosity cases are investigated and valid-ity criteria for the analytical solutions are derived. Solutions are contrasted with numericalsolutions. It is found that within the validity range, there is a good agreement between theresults. An application of this study may be the extrusion of polymeric melts in the entranceof a die with filtering.
Flow of a Third-Grade Fluid Through a Porous Medium 377
Fig. 1 Schematic view of the parallel plates filled with porous media
2 Governing Equations
In this study, a subclass of Rivlin–Ericksen fluids, namely, a third-grade fluid-filling porousmedia is considered between two parallel plates at different temperatures (Fig. 1). The Cauchystress tensor for the third-grade fluid can be written as (Fosdick and Rajagopal 1980)
T = −pI + μA1 + α1A2 + α2A21 + β(trA2
1)A1, (1)
where p is the pressure, μ is the viscosity coefficient, I is the identity tensor and α1, α2, β
are material constants. The Rivlin–Ericksen tensors are defined as
A1 = L + LT, A2 = dA1
dt+ A1L + LTA1, L = ∇V (2)
with V being the velocity vector. When the fluid is locally rest, specific Helmholtz free energymust be a minimum (Fosdick and Rajagopal 1980), then
μ ≥ 0, |α1 + α2| <√
24μβ, α1 ≥ 0, β ≥ 0 (3)
Mass conservation, linear momentum and energy equations are
divV = 0 (4)
ρdVdt
= divT + R (5)
ρcpdθ∗
dt= T · I − kth∇2θ∗ (6)
where ρ is the density of fluid, cp is the specific heat, kth is the thermal conductivity, θ∗ isthe temperature and R is the Darcy’s resistance. Because of the geometry of flow, the bodyforces acting on the fluid as well as radiation effects in the energy equation are neglected.Darcy’s resistance can be interpreted as a measure of the resistance to the flow in the porousmedia. The Darcy’s resistance is (Hayat et al. 2007)
R = −[
μ + 2β
(du∗
dy∗
)2]
ϕVk
(7)
in which ϕ is the porosity of porous space and k is the permeability. For a channel flow inCartesian coordinates, a straightforward calculation yields the x-component of R as follows
rx = −[
μ + 2β
(du∗
dy∗
)2]
ϕu∗
k(8)
123
378 Y. Aksoy, M. Pakdemirli
3 Perturbation Solutions
In the case of the steady channel flow, velocity is assumed to be only in the x direction
V = (u∗(y∗), 0, 0
)(9)
with a variation in y∗-direction. The temperature also depends on the y∗ coordinate only
θ∗ = θ∗(y∗) (10)
Two different cases, namely, the constant viscosity and variable viscosity cases will be treatedseparately.
3.1 Constant Viscosity Case
For this case, two different approximate solutions will be presented. In the first approach,both non-Newtonian and porous media effects will be assumed small. In the second approach,however, only the non-Newtonian effects will be assumed to be small.
3.1.1 Both Non-Newtonian and Porous Media Effects Small
Using the velocity components, the momentum equation is given by
μd2u∗
dy∗2 + 6β
(du∗
dy∗
)2 d2u∗
dy∗2 − ϕu∗
k
[
μ + 2β
(du∗
dy∗
)2]
= ∂p∗
∂x∗ (11)
d
dy∗
[
(2α1 + α2)
(du∗
dy∗
)2]
= ∂p∗
∂y∗ (12)
If a new pressure is defined as
p∗ = p∗ − (2α1 + α2)
(∂u∗
∂y∗
)2
(13)
the above equation reduces to ∂ p∗/∂y∗ = 0 and hence p∗ = p∗(x). Eq. 11 is
μd2u∗
dy∗2 + 6β
(du∗
dy∗
)2 d2u∗
dy∗2 − ϕu∗
k
[
μ + 2β
(du∗
dy∗
)2]
= ∂ p∗
∂x∗ (14)
The pressure gradient is assumed to be a constant and Eq. 14 is
μd2u∗
dy∗2 + 6β
(du∗
dy∗
)2 d2u∗
dy∗2 − ϕu∗
k
[
μ + 2β
(du∗
dy∗
)2]
= C0 (15)
where ϕ is the porosity of the porous media and C0 is the constant pressure gradient. Thenon-dimensional form of Eq. 15 is as follows:
d2u
dy2 + 6�1
(du
dy
)2 d2u
dy2 − �2u
[
1 + 2�1
(du
dy
)2]
= �3 (16)
The non-dimensional parameters are
�1 = βU 2
μh2 , �2 = ϕh2
k, �3 = C0h2
μU, u = u∗
U, y = y∗
h(17)
123
Flow of a Third-Grade Fluid Through a Porous Medium 379
where U is an arbitrary reference velocity, h is the distance between parallel plates, �1 is thedimensionless parameter related to the non-newtonian behavior, and �2 is the dimensionlessparameter for the porosity of media. This term represents the ratio of the hydraulic resistancedue to the porous medium to the hydraulic resistance due to the parallel plates (Yang et al.2002). The boundary conditions for Eq. 17 in a non-dimensional form is given by
u(0) = 0, u(1) = 0 (18)
The approximate solutions can be obtained by
�1 = ελ1, �2 = ελ2 (19)
where ε is our perturbation parameter selected as a book-keeping small parameter. Approx-imate velocity profile
u = u0 + εu1 + · · · (20)
is substituted into Eqs. 16 and 18 together with (19) and terms are separated at each order ofε yielding finallyOrder 1
u′′0 = �3 (21)
u0(0) = 0, u0(1) = 0 (22)
Order ε
u′′1 + 6λ1
(u′
0
)2u′′
0 − λ2u0 = 0 (23)
u1(0) = 0, u1(1) = 0 (24)
where prime denotes derivative with respect to y. For the first order, solution satisfying theboundary conditions is
u0 = �3
2
(y2 − y
)(25)
Substituting this solution to order ε equation, one finally obtains
u1 = λ2�3
24
(y4 − 2y3 + y
) + λ1�33
4
(−2y4 + 4y3 − 3y2 + y)
(26)
Substituting solutions found at each order to Eq. 20 and returning back to the original dimen-sionless parameters, one finally has the following approximate solution of velocity profile:
u = �3
2
(y2 − y
) + �2�3
24
(y4 − 2y3 + y
) + �1�33
4
(−2y4 + 4y3 − 3y2 + y)
(27)
From the restrictions of perturbation theory, for a uniformly valid solution, correction termshould be much smaller than the leading term (Nayfeh 1981). In (27), the first term is theleading term, and the following two terms are the correction terms of order ε. Thus restrictionsfor the expansion to be valid requires
�2�3
24� �3
2,
�1�33
4� �3
2or
Cr1 = �2
6� 1, Cr2 = 2�1�
23 � 1 (28)
123
380 Y. Aksoy, M. Pakdemirli
Under the steady state assumption and assumptions for velocity and temperature profiles, theenergy equation given in Eq. 6 is finally
kthd2θ∗
dy∗2 + μ
(du∗
dy∗
)2
+ 2β
(du∗
dy∗
)4
= 0 (29)
where kth is thermal conductivity of fluid. The non-dimensional form of Eq. 29 is given by
d2θ
dy2 + �
(du
dy
)2
+ 2��1
(du
dy
)4
= 0 (30)
The non-dimensional parameters are
� = μU 2
kth (θ2 − θ1), θ = θ∗ − θ1
θ2 − θ1(31)
where θ is the dimensionless temperature, θ1 and θ2 are lower and upper plate temperatures,respectively. The boundary conditions for Eq. 31 in a non-dimensional form is given by
θ(0) = 0, θ(1) = 1 (32)
Perturbation expansion for temperature profile can be written as follows:
θ = θ0 + εθ1 + · · · (33)
Substituting Eq. 33 into Eq. 30 and separating terms at each other of ε, one hasOrder 1
θ′′0 = −�u′2
0 (34)
θ0(0) = 0, θ0(1) = 1 (35)
Order ε
θ ′′1 = −2�u′
0u′1 − 2λ1�
(u′
0
)4 (36)
θ1(0) = 0, θ1(1) = 0 (37)
The first-order solution, applying the boundary conditions is
θ0 = y + ��23
24(−2y4 + 4y3 − 3y2 + y) (38)
Substituting this solution to order ε equations and applying boundary conditions (37), onefinally obtains
θ1 = �λ2�23
720(−8y6 + 24y5 − 15y4 − 10y3 + 15y2 − 6y)
+ �λ1�43
240(16y6 − 48y5 + 60y4 − 40y3 + 15y2 − 3y) (39)
Combining the solutions at each order of approximation and returning back to the originaldimensionless parameters, one finally has:
θ = y + ��23
24(−2y4 + 4y3 − 3y2 + y) + ��2�
23
720(−8y6 + 24y5 − 15y4
− 10y3 + 15y2 − 6y) + ��1�43
240(16y6 − 48y5 + 60y4 − 40y3 + 15y2 − 3y) (40)
123
Flow of a Third-Grade Fluid Through a Porous Medium 381
Criteria for temperature profiles to be valid are
�2
5� 1,
3�1�23
2� 1 (41)
3.1.2 Only Non-Newtonian Effect Small
In fact for a perturbation solution, selecting only the non-Newtonian effects (�1) small issufficient. Selecting further �2 small means that hydraulic resistance is due mainly to theparallel plates, rather than to the porous medium. The analysis in Sect. 3.1.1 is presentedbecause the analytical solutions are much simpler and compact than the solutions presentedhere, which require extensive algebra and can only be represented with many more terms.With the help of Mathematica, rigorous calculations can be made for both velocity andtemperature profiles. The final solutions are
u(y) =(−1 + Cosh
[√�22 (−1 + 2y)
]Sech
[√�22
])�3
�2
− �1�33
6(1 + e√
�2)4�22
e−3y√
�2(−1 + ey�2) (−3e 3√
�2 − 3e4√
�2 + 7e4y√
�2
+ 3e5y√
�2 + 24e2(1+y)√
�2 − 7e3(1+y)√
�2 − 4e(2+y)√
�2 − 11e(3+y)√
�2
− 7e(4+y)√
�2 + 7e(1+2y)√
�2 + 4e2(1+2y)√
�2 + 17e(3+2y)√
�2 − 17e(1+3y)√
�2
− 24e(2+3y)√
�2 + 11e(1+4y)√
�2 + 3e(1+5y)√
�2)
+ 2�1�33
(1 + e√
�2)4�3/22
e−(1+5y)√
�2[e3(1+2y)
√�2 (−1 + y) − e(3+4y)
√�2(−1 + y)
− e4(1+y)√
�2 y + e 2(1+3y)√
�2 y]
(42)
θ(y) = y + �(Cosh
[√�2
] − Cosh[(1 − 2y)
√�2
] + 2 (−1 + y) y�2)�2
3
4(1 + Cosh
[√�2
])�2
2
+ ��1�43(
10Cosh[√
�22
]+ 5Cosh
[3√
�22
]+ Cosh
[5√
�22
])
{1
�32
(253
27Cosh
[√�2
2
]
+ 1337
432Cosh
[3√
�2
2
]+ 97
432Cosh
[5√
�2
2
]+ 1
16Cosh
[1
2(3 − 8y)
√�2
]
+ 1
16Cosh
[1
2(5 − 8y)
√�2
]+ 8
27Cosh
[1
2(1 − 6y)
√�2
]
+ 8
27Cosh
[1
2(5 − 6y)
√�2
]− 13
12Cosh
[1
2(1 − 4y)
√�2
]
− 13
12Cosh
[1
2(3 − 4y)
√�2
]− 7
12Cosh
[1
2(5 − 4y)
√�2
]
− 16
3Cosh
[1
2(1 − 2y)
√�2
]+ 16
27Cosh
[3
2(1 − 2y)
√�2
]
− 8
3Cosh
[1
2(3 − 2y)
√�2
]− 8
3Cosh
[1
2(1 + 2y)
√�2
]
123
382 Y. Aksoy, M. Pakdemirli
− 7
12Cosh
[1
2(1 + 4y)
√�2
])+ 1
�5/22
(Sinh
[√�2
2
]
+ (−1 + y) Sinh
[1
2(1 − 4y)
√�2
]+ ySinh
[1
2(3 − 4y)
√�2
])
+(−1 + y) yCosh
[√�22
] (15 + 14Cosh
[√�22
])
3�22
−2 (−1 + y) ySinh
[√�22
]
�3/22
⎫⎬
⎭
(43)
The first term in Eq. 42 and the terms in the first line in (43) are the O(1) solutionswhich represent Newtonian case corresponding to �1 = 0. Starting from (42) with �1 = 0,re-substituting dimensional quantities from (17)
u∗ = C0
μ
k
ϕ
⎛
⎜⎝−1 +
Cosh[√
ϕk
(y∗ − h
2
)]
Cosh[√
ϕk
h2
]
⎞
⎟⎠ (44)
and making the coordinate transformation z = y∗ + h/2, adjusting the notations
k = K2, ϕ = φ2, C0 = φ2 dH
dx
the final Newtonian solution is
u∗ = − K2
μ
dH
dx
(
1 − Cosh[φz/
√K2
]
Cosh[φh/
√K2
]
)
(45)
which is exactly the solution given by Kumar et al. (1991).
3.2 Temperature-Dependent Viscosity
In the case of temperature-dependent viscosity, two different models are employed. The firstmodel is the Reynold’s model and the second is the Vogel’s model. In this section, porosityeffects are considered to be small, because the algebra involved is tremendous and presentinganalytical results with pages of equations will not be practical and easy to implement.
3.2.1 Reynold’s Model
In this case, Reynold’s model is used to account for the temperature-dependent viscosity.Non-dimensional momentum and energy equations are
dμ
dy
du
dy+ μ
d2u
dy2 + 6�1
(du
dy
)2 d2u
dy2 − �2u
[
μ + 2�1
(du
dy
)2]
= �3 (46)
d2θ
dy2 + μ�
(du
dy
)2
+ 2��1
(du
dy
)4
= 0 (47)
The non-dimensional parameters are
�1 = βU 2
μ0h2 , �2 = ϕh2
kμ0, �3 = C0h2
μ0U, � = μ0U 2
kth (θ2 − θ1), μ = μ∗
μ0(48)
123
Flow of a Third-Grade Fluid Through a Porous Medium 383
where μ0 is a reference viscosity. For Reynold’s model, the dimensionless viscosity
μ = exp(−Mθ) (49)
can be used to represent the variation of viscosity with temperature. The boundary conditionsfor Eq. 46 and 47 in a dimensionless form is given by
u(0) = 0, u(1) = 0, θ(0) = 0, θ(1) = 1 (50)
Approximate solutions can be obtained by selecting non-Newtonian parameter, porosity, andviscosity index small, i.e., �1 = ελ1,�2 = ελ2, M = εm. ε is a small parameter artificiallyintroduced which will be eliminated at the end by returning to the original parameters. Theapproximate velocity and temperature profiles are as follows:
u = u0 + εu1 + · · · (51)
θ = θ0 + εθ1 + · · · (52)
Using Taylor Series expansion, one may represent viscosity and its derivative
μ ∼= 1 − εmθ,dμ
dy∼= −εm
dθ
dy(53)
in an approximate form. Inserting expansions into the dimensionless equations (i.e. Eqs. 46and 47) and separating at each order of approximation yieldsOrder 1
u′′0 = �3 (54)
θ′′0 = −�u′2
0 (55)
u0(0) = 0, u0(1) = 0, θ0(0) = 0, θ0(1) = 1 (56)
Order ε
u′′1 − mθ ′
0u′0 − mθ0u′′
0 + 6λ1(u′
0
)2u′′
0 − λ2u0 = 0 (57)
θ ′′1 − m�θ0
(u′
0
)2 + 2�u′0u′
1 + 2λ1�(u′
0
)4 = 0 (58)
u1(0) = 0, u1(1) = 0, θ1(0) = 0, θ1(1) = 0 (59)
For the first order, the solutions satisfying boundary conditions are
u0 = �3
2
(y2 − y
)(60)
θ0 = y + ��23
24(−2y4 + 4y3 − 3y2 + y) (61)
Substituting these solutions to order ε equations one finally obtains
u1 = m�3
12(4y3 − 3y2 − y) + m��3
3
288(−4y6 + 12y5 − 15y4 + 10y3 − 3y2)
− λ1�33
4(2y4 − 4y3 + 3y2 − y) + λ2�3
24(y4 − 2y3 + y) (62)
θ1 = m��23
360(−18y5 + 30y4 − 5y3 − 15y2 + 8y) + �λ2�
23
720(−8y6 + 24y5 − 15y4
123
384 Y. Aksoy, M. Pakdemirli
− 10y3 + 15y2 − 6y) + m�2�43
8064(12y8 − 48y7 + 84y6 − 84y5 + 49y4 − 14y3 + y)
+ �λ1�43
240(16y6 − 48y5 + 60y4 − 40y3 + 15y2 − 3y) (63)
Substituting (60)–(63) into the expansions given in (51) and (52) and returning back to theoriginal parameters since ε is artificially introduced, one has
The first term in (64) and the first two terms in (65) are the O(1) leading solutions. For anexpansion up to O(ε) to be valid, the correction terms should be much smaller than theseterms. The final criteria then for uniform solutions are
Cr1 = 2M � 1, Cr2 = 4�2
5� 1, Cr3 = M��2
3
4� 1, Cr4 = 6�1�
23 � 1 (66)
For reliable solutions, all of the four criteria should be met. For M = 0, Eqs. 64 and 65 reduceto those of constant viscosity case with small porosity.
3.2.2 Vogel’s Model
For Vogel’s Model, viscosity is taken as (Massoudi and Christie 1995)
μ = μ∗ exp
(A
B + θ− θw
)(67)
Using Taylor series expansion, Eq. 67 can be written as
μ = α
(1 − Aθ
B2
)(68)
where α = μ∗(exp A
B − θw
)and A, B are viscosity parameters related to Vogel’s model.
Required expansions to perform perturbation analysis in this case are as follows:
u = u0 + εu1 + · · · (69)
θ = θ0 + εθ1 + · · · (70)
Selecting �1 = ελ1,�2 = ελ2, A = εa yields simple equations at the first order. Substitut-ing all into Eqs. 46 and 47, one finally obtains
123
Flow of a Third-Grade Fluid Through a Porous Medium 385
Order 1
αu′′0 = �3 (71)
θ′′0 = −α�u′2
0 (72)
u0(0) = 0, u0(1) = 0, θ0(0) = 0, θ0(1) = 1 (73)
Order ε
αu′′1 − αa
B2 θ ′0u′
0 − αa
B2 θ0u′′0 + 6λ1
(u′
0
)2u′′
0 − αλ2u0 = 0 (74)
θ ′′1 − αa�
B2 θ0(u′
0
)2 + 2α�u′0u′
1 + 2λ1�(u′
0
)4 = 0 (75)
u1(0) = 0, u1(1) = 0, θ1(0) = 0, θ1(1) = 0 (76)
Solving Eq. 71 and 72 subject to the boundary conditions (73), one obtains
u0 = �3
2α
(y2 − y
)(77)
θ0 = y + ��23
24α(−2y4 + 4y3 − 3y2 + y) (78)
Substituting these solutions into (74) and (75), one has
Fig. 2 a Dimensionless velocity profiles for different non-Newtonian parameters, �1. (�3 = −1, � =1, �2 = 0.1). b Residual errors of dimensionless velocity profiles for different non-Newtonian parameters,�1. (�3 = −1, � = 1, �2 = 0.1)
4 Results and Discussions
In this section, approximate temperature and velocity profiles obtained in the previous sec-tions will be calculated and contrasted with the numerical solutions. If the validity criteriadeveloped in the previous section are met, numerical and analytical solutions are in very goodagreement. Sample graphs are also drawn to show the deviation of analytical and numericalresults when the validity range of the criteria are violated. Two cases of constant viscosityand two cases of temperature-dependent viscosity are treated separately.
4.1 Constant Viscosity Case
In Sect. 3, first simple approximate analytical solutions are presented for small non-Newtonianand porosity parameters. Then a more involved solution is presented for small non-Newtonianparameter only. Both cases are contrasted with numerical simulations.
4.1.1 Both Non-Newtonian and Porous Media Effects Small
For this case, the velocity and temperature profiles are given in Eqs. 27 and 40. Figures 2aand 3a show the velocity profiles across the channel for different non-Newtonian param-eters and dimensionless parameters related to the porosity. Figure 2a indicates that mag-
123
Flow of a Third-Grade Fluid Through a Porous Medium 387
Fig. 3 a Dimensionless velocity profiles for different porous media parameters, �2. (�1 = 0.01,�3 =−1, � = 1). b Residual errors of dimensionless velocity profiles for different porous media parameters, �2.(�1 = 0.01,�3 = −1, � = 1)
nitude of maximum velocity, and rate of strain reduces as the non-Newtonian behaviorincreases. Consequently, increasing the non-Newtonian parameter leads to lower fluid veloc-ity.Figure 2b shows the residuals between approximate solutions and numerical results. Residualerror increases with increasing non-Newtonian parameter. Figure 3a illustrates the effect ofporous media parameter on the velocity profiles. Velocity decreases for large values of �2.Increasing porosity increases the residual error as shown in Fig. 3b.
Figure 4a shows the temperature distribution for various non-Newtonian parameters. Tem-perature profiles become lower for increasing non-Newtonian parameter albeit the differenceis substantially small. Residual errors increase with increasing non-Newtonian parameter(Fig. 4b). Effect of porosity on the temperature profiles is appreciably small (Fig. 5a), andthe error increases with increasing porosity (Fig. 5b). In Fig. 6, numerical and approximateanalytical velocity profiles are contrasted for different porosity values. As porosity increases,numerical solutions deviate from the analytical ones.
4.1.2 Only Non-Newtonian Effect Small
In the case the approximate analytical solutions where only non-Newtonian effects are takenas small, the velocity and temperature profiles are given in (42) and (43). In Fig. 7 for smallnon-Newtonian effect, approximate and numerical solutions are contrasted. As can be seen,
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Fig. 4 a Dimensionless temperature distribution for different non-Newtonian parameters, �1. (�3 =−2, � = 10, �2 = 0.01). b Residual errors of dimensionless temperature distribution for different non-Newtonian parameters, �1. (�3 = −2, � = 10, �2 = 0.01)
Fig. 5 a Dimensionless temperature distribution for different porous media parameters, �2. (�1 =0.01,�3 = −2, � = 10). b Residual errors of dimensionless temperature distribution for different porousmedia parameters, �2. (�1 = 0.01,�3 = −2, � = 10)
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Flow of a Third-Grade Fluid Through a Porous Medium 389
Fig. 6 Comparison of the approximate analytical velocity profiles with numerical velocity profiles for the caseof non-Newtonian effect and porosity both being small. �2 values indicated on the graphs (�3 = −1, � =10, �1 = 0.1)
Fig. 7 Comparison of the approximate analytical velocity profiles with numerical solutions for the case ofonly small non-Newtonian effects. �2 values indicated on the graphs (�3 = −1, � = 100, �1 = 0.1)
for porosity being large, the match is excellent since in deriving (42) and (43) porosity isnot assumed to be small. The same conclusions apply to temperature profiles as given inFig. 8. In Fig. 9, the effect of non-Newtonian parameter on the velocity profiles are pre-sented under the small non-Newtonian parameter assumption. With non-Newtonian param-eter increasing from 0.1 to 1, the analytical and numerical solutions deviate from each other,as expected, since these solutions are derived under the small non-Newtonian parameterassumption.
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Fig. 8 Comparison of the approximate analytical temperature profiles with numerical temperature profiles forthe case of only small non-Newtonian effects. �2 values indicated on the graphs (�3 = −1, � = 100,�1 =0.1)
Fig. 9 Comparison of the approximate analytical velocity profiles with numerical velocity profiles for thecase of only small non-Newtonian effects. �1 values indicated on the graphs. (�3 = −1, � = 10,�2 = 1)
4.2 Temperature-Dependent Viscosity
In Sect. 3.2, solutions of two different temperature-dependent viscosity models are presented.For each model, velocity and temperature profiles are shown.
4.2.1 Reynold’s Model Case
Solutions for this case are given in Eqs. 64 and 65. Effect of viscosity index on the velocityprofiles are depicted in Fig. 10. As the viscosity index increases, viscosity drops which in
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Fig. 10 Dimensionless velocity profiles for different viscosity index values, M. (�3 = −1, � = 1,�1 =0.01,�2 = 0.01)
Fig. 11 Dimensionless temperature distribution for different viscosity index values, M. (�3 = −2, � =10, �1 = 0.01,�2 = 0.01)
turn results in a higher velocity profile. The residual errors are within 10−4 in the wholedomain which is appreciably small. The influence of the viscosity index on the temperaturedistribution between the plates can be seen from Fig. 11. As the viscosity index increases,maximum temperature and temperature profiles become higher.
4.2.2 Vogel’s Model Case
Solutions for this case are given in Eqs. 81 and 82. Figure 12 depicts the influence of Vogel’sviscosity index A on velocity profiles. Higher values of A lead to decreasing velocity profiles.
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Fig. 12 Dimensionless velocity profiles for different Vogel’s viscosity index values, A. (�3 = −1, � =1, �1 = 0.01,�2 = 0.01, θw = 1, μ∗ = 2, B = 2)
Fig. 13 Dimensionless velocity profiles for different Vogel’s viscosity index values, B. (�3 = −2, � =1, �1 = 0.01,�2 = 0.01, θw = 1, μ∗ = 2, A = 0.1)
If the other viscosity parameter B is increased, an increase in velocity profiles is observedfrom Fig. 13. As shown in Fig. 14, higher θw values cause higher velocity profiles.
Effects of Vogel’s viscosity parameters on temperature profiles are presented in Figs. 15,16 and 17. The influence of Vogel’s viscosity indexes A and B are appreciably small and thedifferences can be seen on the magnified figures. On the contrary, effect of θw on the tem-peratures is much more significant as can be observed from Fig. 17. As θw values increase,temperatures rise.
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Flow of a Third-Grade Fluid Through a Porous Medium 393
Fig. 14 Dimensionless velocity profiles for different θw values. (�3 = −1, � = 1, �1 = 0.01,�2 =0.01, B = 2, μ∗ = 2, A = 0.1)
Fig. 15 Dimensionless temperature distribution for different Vogel’s viscosity index values, A. (�3 =−1, � = 10, �1 = 0.01,�2 = 0.01, θw = 0.1, μ∗ = 1, B = 2)
5 Concluding Remarks
Approximate analytical solutions are presented for the first time for parallel plate flow of athird-grade fluid with constant and temperature-dependent viscosity in porous media. Per-turbation techniques are applied to solve the momentum and energy equations. Constantviscosity and temperature-dependent viscosity cases are treated separately. In the constantviscosity case, two different approximate solutions are presented. In the first solution whichis in a simple polynomial type, both non-Newtonian and porosity effects are considered tobe small. In the second solution, which is more involved, only the non-Newtonian effect is
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Fig. 16 Dimensionless temperature distribution for different Vogel’s viscosity index values, B. (�3 =−1, � = 10, �1 = 0.01,�2 = 0.01, θw = 0.1, μ∗ = 1, A = 0.1)
Fig. 17 Dimensionless temperature distribution for different θw values. (�3 = −1, � = 10,�1 =0.01,�2 = 0.01, B = 2, μ∗ = 1, A = 0.1)
considered to be small. In the cases of temperature-dependent viscosity, two different mod-els, namely, Reynold’s model and Vogel’s models are considered. Approximate analyticalsolutions and validity criteria of solutions for each case are derived. It is found that withinthe validity range, maximum value of residual errors are less than 10−3 with errors increas-ing in the middle parts of the domain. The effects of non-Newtonian parameter, porosity,viscosity indexes of both models on the velocity, and temperature profiles are depicted infigures.
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References
Ahmad, F.: A simple analytical solution for the steady flow of a third grade fluid in a porous half space. Com-mun. Nonlinear Sci. Numer. Simul. 14, 2848–2852 (2009)
Akgül, M.B., Pakdemirli, M.: Analytical and numerical solutions of electro-osmotically driven flow of a thirdgrade fluid between micro-parallel plates. Int. J. Nonlinear Mech. 43, 985–992 (2008)
Ayub, M., Rasheed, A., Hayat, T.: Exact flow of a third grade fluid past a porous plate using homotopy analysismethod. Int. J. Nonlinear Mech. 41, 2091–2103 (2003)
Daivis, P.J., Matin, M.L., Todd, B.D.: Nonlinear shear and elongational rheology of model polymer melts atlow strain rates. J. Non-Newtonian Fluid Mech. 147, 35–44 (2007)
Degond, P., Lemou, M., Picasso, M.: Viscoelastic fluid models derived from kinetic equations for poly-mers. SIAM J. Appl. Math. 62, 1501–1519 (2002)
Dunn, J.E., Rajagopal, K.R.: Fluids of differential type: critical review and thermodynamic analysis. Int.J. Eng. Sci. 33, 689–729 (1995)
Ellahi, R., Afzal, S.: Effects of variable viscosity in a third grade fluid with porous medium: an analyticsolution. Commun. Nonlinear Sci. Numer. Simul. 14, 2056–2072 (2009)
Fosdick, R.L., Rajagopal, K.R.: Thermodynamics and stability of fluids of third grade. Proc. R. Soc. Lond.A 339, 351–377 (1980)
Hayat, T., Abbas, Z.: Heat transfer analysis on the MHD flow of a second grade fluid in a channel with porousmedium. Chaos Solitons Fractals 38, 556–567 (2008)
Hayat, T., Ellahi, R., Ariel, P.D., Asghar, S.: Homotopy solution for the channel flow of a third grade fluid. Non-linear Dyn. 45, 55–64 (2006)
Hayat, T., Shahzad, F., Ayub, M.: Analytical solution for the steady flow of the third grade fluid in a poroushalf space. Appl. Math. Model. 31, 2424–2432 (2007)
Hayat, T., Mambili-Mamboundou, H., Mohamed, F.M.: A note on some solutions for the flow of a fourthgrade fluid in a porous space. Nonlinear Anal. Real World Appl. 10, 368–374 (2009)
Kannan, K., Rajagopal, K.R.: Flow through porous media due to high pressure gradients. Appl. Math. Com-put. 199, 748–759 (2008)
Khan, M., Abbas, Z., Hayat, T.: Analytic solution for flow of sisko fluid through a porous medium. Transp.Porous Media 71, 23–37 (2008)
Kumar, S., Zimmerman, R.W., Bodvarsson, G.S.: Permeability of a fracture with cylindrical asperities. FluidDyn. Res. 7, 131–137 (1991)
Liu, I.: Flow and heat transfer of an electrically conducting fluid of second grade in porous medium over astretching sheet subject to a transverse magnetic field. Int. J. Nonlinear Mech. 40, 465–474 (2005)
Maneschy, C.E., Massoudi, M., Ghoneimy, A.: Heat transfer analysis of a non-Newtonian fluid past a porousplate. Int. J. Nonlinear Mech. 28, 131–143 (1993)
Massoudi, M., Christie, I.: Effects of variable viscosity and viscous dissipation on the flow of a third gradefluid in a pipe. Int. J. Nonlinear Mech. 30, 687–699 (1995)
Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)Rivlin, R.S., Ericksen, J.L.: Stress deformation relation for isotropic materials. J. Ration. Mech. Anal. 4,
323–329 (1955)Sajid, M., Hayat, T.: Series solution for steady flow of a third grade fluid through porous space. Transp. Porous
Media 71, 173–183 (2008)Siddiqui, A.M., Zeb, A., Ghori, Q.K., Benharbit, A.M.: Homotopy perturbation method for heat transfer flow
of a third grade fluid between parallel plates. Chaos Solitons Fractals 36, 182–192 (2008)Subramanian, S.C., Rajagopal, K.R.: A note on the flow through porous solids at high pressures. Comput.
Math. Appl. 53, 260–275 (2007)Yang, C., Grattoni, C.A., Muggeridge, A.N., Zimmerman, R.W.: Flow of water through channels filled with
deformable polymer gels. J. Colloid Interface Sci. 250, 466–470 (2002)Yürüsoy, M., Pakdemirli, M.: Approximate analytical solutions for the flow of a third grade fluid in a pipe. Int.
J. Nonlinear Mech. 37, 187–195 (2002)Yürüsoy, M., Pakdemirli, M., Yilbas, B.S.: Perturbation solution for a third-grade fluid flowing between parallel
plates. Proc. IMechE, Part C, J. Mech. Eng. Sci. 222, 653–656 (2008a)Yürüsoy, M., Bayrakçeken, H., Kapucu, M., Aksoy, F.: Entropy analysis for third grade fluid flow with Vogel
model viscosity in annular pipe. Int. J. Nonlinear Mech. 43, 588–599 (2008b)
