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MHD flow of a third grade fluid in a porous half space with plate suction or injection: An analytical approach
Applied Mathematics and Computation 218 (2012) 10443–10453
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MHD flow of a third grade fluid in a porous half space with plate suction or injection: An analytical approach Asim Aziz a,⇑, Taha Aziz b a
NUST College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Rawalpindi 46070, Pakistan Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa
b
a r t i c l e
i n f o
Keywords: Third grade fluid MHD flow Suction/blowing Symmetry approach Porous medium
a b s t r a c t The present work deals with the modeling and solution of the unsteady flow of an incompressible third grade fluid over a porous plate within a porous medium. The flow is generated due to an arbitrary velocity of the porous plate. The fluid is electrically conducting in the presence of a uniform magnetic field applied transversely to the flow. Lie group theory is employed to find symmetries of the modeled equation. These symmetries have been applied to transform the original third order partial differential equation into third order ordinary differential equations. These third order ordinary differential equations are then solved analytically and numerically. The manner in which various emerging parameters have an effect on the structure of the velocity is discussed with the help of several graphs. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Fluid flow through a porous media is a common phenomena in nature and in many fields of science and engineering. Important flow phenomenons include transport of water in living plants, trees and fertilizers or wastes in soil. In industry fluid flow through porous media has attracted much attention due to its importance in several technological processes, for example, filtration, catalysis, chromatography, petroleum exploration and recovery, cooling of electronic equipment, etc. In recent years non-Newtonian fluid flow models have been studied extensively because of their relevance to many engineering and industrial applications, particularly in the extraction of crude oil from petroleum products, synthetic fibers, food stuffs, etc. The non-Newtonian models exhibit a nonlinear relationship between stress and rate of strain. A special differential type fluid, namely the third grade fluid model, is considered in this study. This model is known to capture the nonNewtonian affects such as shear thinning or shear thickening as well as normal stresses. The third grade fluid model is thermodynamically analyzed in detail by Fosdick and Rajagopal [1]. Some recent contributions dealing with the flows of third grade fluid are [2–5]. Few authors have dealt with the flows of a third grade fluid over a porous plate [6] or within a porous medium [7]. The governing equations for non-Newtonain fluid flow models are higher order nonlinear partial differential equations, whose analysis would present a particular challenge for researchers and scientists. Due to the nonlinear terms in partial differential equations the exact solutions are difficult to be obtained for the flow of non-Newtonian fluids. Some recent studies in finding the analytical solutions for such problems can be found in [8–13]. To the best of our knowledge no attempt is available in the literature where the unsteady MHD flow of a third grade fluid over a porous plate within a porous medium is discussed. The purpose of present investigation is to discuss the analytical solutions for the unsteady MHD flow of a third grade fluid over a porous plate within a porous medium. The flow is induced due to arbitrary velocity VðtÞ of the plate. The fluid is ⇑ Corresponding author. E-mail address: [email protected] (A. Aziz). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.04.006
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considered as electrically conducting in the presence of a uniform magnetic field. Fluid flow is studied under conditions of low Reynolds number to ensure that the contributions from inertial terms (convection) does not prevail over the viscous mechanisms of momentum transfer and thus the induced magnetic field is neglected. The applied and induced electric fields are negligible. Analytical solutions of the model equation are obtained with the help of Lie symmetry methods. The paper is organized as follows: In the next section, the problem formulation is developed. Section 3 contains the Lie point symmetries of the governing equation. Section 4 deals in obtaining the similarity solutions for the modeled equation and the numerical solution of the reduced ordinary differential equation. In Section 5, the results and discussion of velocity for various interesting parameters is given and Section 6 synthesizes the results with concluding remarks. 2. Problem formulation Let us consider a third grade fluid which occupies the porous half space y > 0 and is in contact with an infinite porous plate at y ¼ 0. The plate is moving in its own plane with an arbitrary velocity. The fluid is electrically conducting and the magnetic field strength is taken as constant. The unsteady motion of the conducting fluid through a porous medium is governed by the conservation laws of momentum and mass, that is
q
dV ¼ div T þ R rB20 V; dt
ð1Þ
div V ¼ 0:
ð2Þ
In the above relations V is the velocity vector, q the density, p the pressure, T the Cauchy stress tensor, r the electrical conductivity, B0 the uniform applied magnetic field and R represents the Darcy’s resistance due to porous medium. In this work we have considered a third grade fluid and its Cauchy stress tensor (in a thermodynamically compatible form [1,2,5]) is given by
T ¼ pI þ lA1 þ a1 A2 þ a2 A21 þ b3 ðtr A21 ÞA1 :
ð3Þ
where I is the identity tensor, l the dynamic viscosity, ai ði ¼ 1; 2Þ; b3 are the material constants, and Ai ði ¼ 1; 2Þ are the Rivlin–Ericksen tensors which are defined by the following equations:
A1 ¼ ðLÞ þ ðLÞT ; An ¼
ð4Þ
dAn1 þ An1 ðLÞ þ ðLÞT An1 ; dt
ðn > 1Þ;
ð5Þ
where L ¼ $V and $ is the gradient operator. The Darcy’s resistance can be interpreted as a measure of the resistance to the flow in the porous media. For unidirectional flow of a third grade fluid over a rigid plate, the expression (for x-component of R) has been proposed by Hayat et al. [7]. Following the methodology of Ref. [7], a straight-forward calculation yields the x-component of R for unidirectional flow over a porous plate as follows:
Rx ¼
/
j
"
l þ a1
2 # @ @ @u u; a1 W 0 þ 2b3 @t @y @y
ð6Þ
in which / is the porosity and j is the permeability of the porous medium. For unidirectional flow the velocity field is given by
V ¼ ½uðy; tÞ; W 0 ; 0;
ð7Þ
where W 0 > 0 indicates suction velocity and W 0 < 0 indicates blowing or injection velocity. The above definition of velocity satisfies the law of conservation of mass for incompressible fluid. Substituting Eqs. (3)–(7) into Eq. (1), one obtains the following governing equation in the absence of the modified pressure gradient
q
" 2 2 2 # @u @u @2u @3u @3u @u @ u / @ @ @u u rB20 u: W0 ¼ l 2 þ a1 2 a1 W 0 3 þ 6b3 þ 2b l þ a a W 1 1 0 3 @t @y @y @y @t @y @y @y2 j @t @y @y ð8Þ
The relevant boundary and initial conditions are specified as follows:
uð0; tÞ ¼ U 0 VðtÞ;
t > 0;
uðy; tÞ ! 0 as y ! 1; uðy; 0Þ ¼ f ðyÞ;
y > 0;
ð9Þ t > 0;
ð10Þ ð11Þ
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where U 0 is the characteristic velocity, VðtÞ and f ðyÞ are unspecified functions. Introducing the non-dimensional variables
¼ u
u ; U0
¼ b
2b3 U 40 3
qm
¼ y
U0 y
m
2
t ¼ U 0 t ;
;
m
¼ /m ; / jU 20 2
;
a ¼
a1 U 20 ; qm2
ð12Þ
rB20 m W0 ; W0 ¼ : U0 qU 20
M2 ¼
Therefore, momentum equation (8) becomes
" 2 2 2 # @u @u @2u @3u @3u @u @ u @u @u @u ¼ 2 þ a 2 aW 0 3 þ 3b M 2 u: W0 aW 0 þ bu / uþa @t @y @y @y @t @y @y @y2 @t @y @y
ð13Þ
The corresponding initial and the boundary conditions (9)–(11) reduce to
uð0; tÞ ¼ VðtÞ;
t > 0;
ð14Þ
uðy; tÞ ! 0 as y ! 1; uðy; 0Þ ¼ f ðyÞ;
t > 0;
ð15Þ
y > 0:
ð16Þ
For simplicity we omit the bars in all the non-dimensional variables. Rewriting Eq. (13) as
2 2 2 @u @2u @3u @3u @u @ u @u @u c u þ W0 ¼ l 2 þ a 2 a W 0 3 þ c / u M2 u; @t @y @y @t @y @y @y2 @y @y
ð17Þ
where
l ¼
1 ; ð1 þ a/Þ
a ¼
c ¼
b/ ; ð1 þ a/Þ
/ ¼
a ð1 þ a/Þ
;
/ ; ð1 þ a/Þ
c¼
3b ; ð1 þ a/Þ
M2 ¼
ð18Þ
M2 : ð1 þ a/Þ
3. Lie symmetry analysis Here we present a complete Lie point symmetry analysis [14,15] of the nonlinear partial differential equation (17). We make use of these symmetry generators to solve Eq. (17) analytically. According to the Lie theory, an operator of the form
X ¼ sðt; y; uÞ
@ @ @ þ nðt; y; uÞ þ gðt; y; uÞ ; @t @y @u
ð19Þ
where s; n and g are functions of t; y and u, is a Lie point symmetry generator of Eq. (17) if and only if
h i X ½3 ut l uyy a uyyt þ a W 0 uyyy cðuy Þ2 uyy þ c uðuy Þ2 W 0 uy þ ð/ þ M 2 Þu
17
¼ 0:
ð20Þ
The operator X ½3 is the third prolongation of the operator X which is
X ½3 ¼ X þ ft
@ @ @ @ @ þ fy þ fyy þ fyyy þ ftyy ; @ut @uy @uyy @uyyy @utyy
ð21Þ
where the coefficients f’s are given by the formulae
ft ¼ Dt g ut Dt s uy Dt n; fy ¼ Dy g ut Dy s uy Dy n; fyy ¼ Dy fy uty Dy s uyy Dy n;
ð22Þ
fyyy ¼ Dy fyy utyy Dy s uyyy Dy n; ftyy ¼ Dt fyy utyy Dt s uyyy Dt n: The total derivative operators are given as
@ @ @ @ þ ut þ utt þ uty þ ; @t @u @ut @uy @ @ @ @ þ uy þ uyy þ uty þ Dy ¼ @y @u @uy @ut Dt ¼
ð23Þ
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Substituting Eqs. (21) and (22) in the symmetry condition (20) and separating it by powers of the derivative of u, yield an overdetermined system of linear homogeneous partial differential equations called determining equations. The following determining equations were generated manually and are given as
su ¼ sy ¼ 0; nu ¼ ny ¼ 0;
guu ¼ gy ¼ 0; a gtu þ l st ¼ 0;
ð24Þ
W 0 st þ nt ¼ 0; ð/ þ M 2 Þg þ uð/ þ M2 Þst þ gt uð/ þ M 2 Þgu ¼ 0;
g þ ust þ ugu ¼ 0;
st þ 2gu ¼ 0: The solution of the above system gives rise to two cases: Case 1. when / þ M 2 – l =a For this case, we find a two-dimensional Lie algebra generated by
X1 ¼
@ ; @t
X2 ¼
@ : @y
ð25Þ
Case 2. when / þ M 2 ¼ l =a Here we obtain three-dimensional Lie algebra generated by
@ ; @y 2 2 2 @ @ @ X 3 ¼ e2ð/ þM Þt þ W 0 e2ð/ þM Þt þ uð/ þ M2 Þe2ð/ þM Þt ; @t @y @u X1 ¼
@ ; @t
X2 ¼
ð26Þ
where X 1 is the time translational symmetry generator, X 2 is the space translational symmetry generator and X 3 has path curves which are equivalent to a combination of translation and scaling in u.
4. Similarity solutions In this section, similarity solutions (group invariant solutions) of the partial differential equation (17) corresponding to symmetries (25) and (26) will be constructed. 4.1. Traveling wave solutions Traveling wave solutions are the one special kind of group invariant solutions which are invariant under a linear combination of time-translation and space-translation generator. 4.1.1. Forward type We look for the invariant solution under the operator X 1 þ cX 2 (with c > 0) which represents a forward wave-front type traveling wave solutions with constant wave speed c, in this case the waves are propagating away from the plate. These are solutions of the form
uðy; tÞ ¼ G1 ðx1 Þ;
where x1 ¼ y ct:
ð27Þ
Using Eq. (27) into Eq. (17) results in a third-order ordinary differential equation for Gðx1 Þ,
c
2 2 2 2 3 3 dG1 d G1 d G1 d G1 dG1 d G1 dG1 dG1 ¼ l ca a W 0 þc c G1 þ W0 ð/ þ M 2 ÞG1 ; 2 3 3 2 dx1 dx dx dx1 1 1 dx1 dx1 dx1 dx1
ð28Þ
hence the above equation admits the exact solution of the form (which is required to be zero at infinity due to the second boundary condition)
pffiffiffiffiffi c G1 ðx1 Þ ¼ A exp pffiffiffi x1
c
ð29Þ
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provided that
rffiffiffiffiffi
rffiffiffiffiffi
c c c l þ ca þ a W 0 c c c
rffiffiffiffiffi c ½c þ W 0 ð/ þ M2 Þ ¼ 0:
ð30Þ
c
Thus Eq. (17) subject to Eq. (30) gives the solution of the form
pffiffiffiffiffi c ðy ctÞ : uðy; tÞ ¼ exp pffiffiffi
ð31Þ
c
The solution (31) satisfies the initial and boundary conditions (14)–(16) for the particular values of the unspecified functions VðtÞ and f ðyÞ. Using Eq. (31) into Eqs. (14)–(16) gives
pffiffiffiffiffi c c t pffiffiffi ; c pffiffiffiffiffi cy uðy; 0Þ ¼ f ðyÞ ¼ exp pffiffiffi :
uð0; tÞ ¼ VðtÞ ¼ exp
ð32aÞ ð32bÞ
c
Here VðtÞ and f ðyÞ depend on the physical parameters of the flow. The solution given in Eq. (31) is plotted in Figs. 1 and 2 for various values of the emerging parameters which satisfies the condition (30). 4.1.2. Backward type We now look for the invariant solution under the operator X 1 cX 2 (with c > 0) which represents a backward wave-front type traveling wave solutions with constant wave speed c, in this case the waves are propagating towards the plate. These are solutions of the form
uðy; tÞ ¼ G2 ðx2 Þ;
where x2 ¼ y þ ct:
ð33Þ
Using Eq. (33) into Eq. (17) results in a third-order ordinary differential equation for Gðx1 Þ,
c
2 2 2 2 3 3 dG2 d G2 d G2 d G2 dG2 d G2 dG2 dG2 ¼ l þ c a a W þ c c G þ W0 ð/ þ M 2 ÞG2 ; 0 2 2 3 3 2 dx2 dx2 dx2 dx2 dx2 dx2 dx2 dx2
ð34Þ
and hence the above equation admits the exact solution of the form (which we require to be zero at infinity due to the second boundary condition)
pffiffiffiffiffi c G2 ðx2 Þ ¼ A exp pffiffiffi x2 ;
ð35Þ
c
provided that
rffiffiffiffiffi
rffiffiffiffiffi
c c c l ca þ a W 0 þ c c c
rffiffiffiffiffi c ½c W 0 ð/ þ M2 Þ ¼ 0:
ð36Þ
c
Thus Eq. (17) subject to Eq. (36) gives the solution
pffiffiffiffiffi c ðy þ ctÞ : uðy; tÞ ¼ exp pffiffiffi
ð37Þ
c
7
t=0 t=0.5 t=1 t=1.5 t=1.9 t=2.2
6 5
u
4 3 2 1 0
0
1
2
3
4
5
y Fig. 1. Plot of the velocity field (31) for varying values of t when c ¼ 1:5; c ¼ c ¼ 1 are fixed.
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5
c=0.1 c=0.5 c=0.8 c=1 c=1.3
4
u
3
2
1
0
0
1
2
3
4
5
y Fig. 2. Plot of the velocity field (31) for varying values of c when c ¼ 1:5; c ¼ 1; t ¼ p=2 are fixed.
We note that this solution satisfies the initial and boundary conditions (14)–(16) for the specific values of the functions VðtÞ and f ðyÞ, where
pffiffiffiffiffi c ct uð0; tÞ ¼ VðtÞ ¼ exp pffiffiffi ; c pffiffiffiffiffi cy uðy; 0Þ ¼ f ðyÞ ¼ exp pffiffiffi :
ð38aÞ ð38bÞ
c
The solution given in Eq. (37) is plotted in Figs. 3 and 4 for various values of the emerging parameters which satisfies the condition (36). Moreover, we can also find group invariant solution corresponding to the operator X 1 and X 3 , which provides other meaningful solution as well. 4.2. Solution via subgroup generated by X 3 Consider the invariant solution under the operator X 3 given as 2
X 3 ¼ e2ð/ þM Þt
2 2 @ @ @ þ W 0 e2ð/ þM Þt þ uð/ þ M 2 Þe2ð/ þM Þt : @t @y @u
ð39Þ
The similarity solution corresponding to the above operator X 3 assumes the form
h i uðy; tÞ ¼ hðhÞ exp ð/ þ M 2 Þt ;
h ¼ W 0 t þ y;
ð40Þ
where hðhÞ is an undetermined function. Substituting Eq. (40) into Eq. (17) gives a second order linear ordinary differential equation
dh
2
c h ¼ 0: c
ð41Þ 1
t=0 t=0.5 t=1.5 t=2.3 t=3.8 t=4.5
0.8
0.6
u
2
d h
0.4
0.2
0
0
1
2
3
4
5
y Fig. 3. Plot of the velocity field (37) for varying values of t when c ¼ 1:5; c ¼ 1; t ¼ p=2 are fixed.
A. Aziz, T. Aziz / Applied Mathematics and Computation 218 (2012) 10443–10453
0.35
10449
c=1 c=1.2 c=2 c=2.5 c=3
0.3 0.25
u
0.2 0.15 0.1 0.05 0
0
1
2
3
4
5
y Fig. 4. Plot of the velocity field (37) for varying values of c when c ¼ 1:5; c ¼ 1; t ¼ p=2 are fixed.
Using condition (15), one can write the boundary condition for Eq. (41) as
hðlÞ ¼ 0;
l ! 1:
ð42Þ
This is only part of the boundary conditions. One can not obtain another boundary condition on h from condition (16). However, we utilize this to deduce the exact solution
rffiffiffiffiffi c hðhÞ ¼ a exp h :
ð43Þ
c
Substituting hðhÞ into Eq. (40) and simplifying we obtain the solution for uðy; tÞ of the form
rffiffiffiffiffi rffiffiffiffiffi c c uðy; tÞ ¼ exp W 0 þ / þ M2 t þ y :
c
ð44Þ
c
We observe that this solution satisfies the initial and boundary conditions (14)–(16). Making use of Eq. (44) into Eqs. (14)– (16) we get
rffiffiffiffiffi c W 0 þ / þ M2 t ; uð0; tÞ ¼ VðtÞ ¼ exp c pffiffiffiffiffi c y uðy; 0Þ ¼ f ðyÞ ¼ exp pffiffiffi ;
ð45aÞ ð45bÞ
c
where VðtÞ and f ðyÞ once again depend on the physical parameters of the flow. The solution in Eq. (44) is plotted in Figs. 5–9 for various values of emerging parameters.
1
t=0 t=0.2 t=0.5 t=0.9 t=1.2 t=1.9
0.8
u
0.6
0.4
0.2
0
0
1
2
3
4
5
y Fig. 5. Plot of the velocity field (44) for varying values of t when c ¼ 1:5; c ¼ 1; W 0 ¼ 0:5; / ¼ 0:5; M ¼ W 0 ¼ 1 are fixed.
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0.25
W0=-0.2 W0=-0.7
0.2
W0=0 W0=0.2 W0=0.7
u
0.15
0.1
0.05
0
0
1
2
3
4
5
y Fig. 6. Plot of the velocity field (44) for varying values of W 0 when c ¼ 1:5; c ¼ 1; / ¼ 0:5; M ¼ 1; t ¼ p=2 are fixed.
0.12
M*=0.2 M*=0.6
0.1
M*=0.8 M*=1.0
0.08
u
M*=1.2 0.06 0.04 0.02 0
0
1
2
y
3
4
5
Fig. 7. Plot of the velocity field (44) for varying values of M when c ¼ 1:5; c ¼ 1; / ¼ 0:5; W 0 ¼ 1; t ¼ p=2 are fixed.
0.09
γ =0.5
0.08
γ =0.8 γ =1.2
0.07
γ =1.7 γ =2.1
0.06
u
0.05 0.04 0.03 0.02 0.01 0
0
1
2
y
3
4
5
Fig. 8. Plot of the velocity field (44) for varying values of c when c ¼ 1:3; / ¼ 0:2; W 0 ¼ 0:5; M ¼ 1; t ¼ p=2 are fixed.
4.3. Solution via subgroup generated by X 1 The invariant solution admitted by X 1 is the steady-state solution
uðy; tÞ ¼ FðyÞ:
ð46Þ
A. Aziz, T. Aziz / Applied Mathematics and Computation 218 (2012) 10443–10453
0.1
10451
γ *=0.5 γ *=0.8
0.08
γ *=1.2 γ *=1.7 γ *=2.1
u
0.06
0.04
0.02
0
0
1
2
y
3
4
5
Fig. 9. Plot of the velocity field (44) for varying values of c when c ¼ 1:3; / ¼ 0:2; W 0 ¼ 0:5; M ¼ 1; t ¼ p=2 are fixed.
Substituting Eq. (46) into Eq. (17) yield the third-order ordinary differential equation for FðyÞ 2
l
d F 2
dy
2 2 2 dF d F dF dF ð/ þ M2 ÞF ¼ 0 þc c F þ W0 2 dy dy dy dy dy 3
a W 0
d F 3
ð47Þ
with the boundary conditions
Fð0Þ ¼ v 0 ; FðlÞ ¼ 0;
l > 0;
ð48Þ
where l can take a sufficiently large value with V ¼ v 0 a constant. Remark 1. We present numerical solution of reduced equation (47) subject to the boundary conditions (48) using Mathematica solver NDSolve.
5. Graphical results and discussion In this section, velocity profiles obtained in Section 4 are plotted. The traveling wave solutions over an infinite porous plate are plotted in Figs. 1–4 for different values of time t and wave speed c. These solutions does not predict the effects of suction or injection as well as the effects of magnetic field on the flow. Figs. 1 and 2 describe the effects of time t and wave speed c on the forward wave-front type traveling wave solution (31). These figures show that with the increase of time t and wave speed c the velocity profile is increasing. Figs. 3 and 4 predict the effects of time t and wave speed c on the backward wave-front type traveling wave solution (37). From these figures, it is clear that velocity decreases by increasing the time t and the wave speed c. In this way we can conclude that both time and the wave speed have an opposite effect on the forward and the backward wave-front type traveling wave solutions, respectively. In order to see the effects of plate suction or injection and the effects of magnetic field on the flow, the similarity solution (44) is plotted in Figs. 5–9 for different values of emerging parameters. Fig. 5 shows the velocity profiles for various values of time t at fixed values of other parameters. As expected, the velocity profile will decay with time. In order to examine the effects of the plate suction or injection, the velocity profiles versus y for various values of suction parameter W 0 are depicted in Fig. 6. Curve with W 0 ¼ 0 corresponds to the case of rigid plate. The positive values of W 0 correspond to the suction occurring at the plate surface whereas the negative values of W 0 correspond to the injection or blowing at the plate surface. From this figure, we can clearly observe the effects of suction parameter. It is observed that by increasing W 0 ð> 0Þ the velocity profile and the boundary layer thickness decrease. This is in accordance with the fact that suction causes reduction in the boundary layer thickness. The effects of increasing injection W 0 ð< 0Þ are opposite to those of suction. Fig. 7 shows the effects of magnetic field on the flow. It shows that velocity decreases when the magnitude of the magnetic field is increased. The effects of the fluid parameters c and c in the presence of plate suction or injection and magnetic field are shown in Figs. 8 and 9. These figures reveal that c and c have opposite roles on the velocity even in the presence of plate suction or injection. From these figures, it is noted that velocity increases for large values of c whereas it decreases for increasing c . Qualitatively, the effects of c and c on the velocity profiles for porous plate are similar to those of rigid plate, however, these effects are not similar quantitatively. The numerical solution of the reduced ordinary differential equation (47) subject to boundary conditions (48) for different values of suction parameter W 0 and the magnetic field parameter M are shown in Figs. 10 and 11, respectively. It is observed that the effects of W 0 and M on the velocity profiles are the same as noticed previously for the analytical solutions.
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1.2
W0=-1.5 W0=0
1
W0=1.5
u
0.8 0.6 0.4 0.2 0
0
1
2
y
3
4
5
Fig. 10. Numerical solution of (47) subject to conditions (48) for varying values of W 0 when c ¼ 1:5; c ¼ 1; fixed.
l ¼ 0:5; M ¼ 0:5; / ¼ 0:75; a ¼ 1 are
1 M*=0.5
0.9
M*=1.0 M*=2.0
0.8 0.7
u
0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
1.5
2
2.5
y
3
3.5
4
4.5
Fig. 11. Numerical solution of (47) subject to conditions (48) for varying values of M when c ¼ 1:5; c ¼ 0:5;
5
l ¼ 0:5; W 0 ¼ 0:75; / ¼ a ¼ 1 are fixed.
6. Concluding remarks In this paper we have studied the unsteady MHD flow of an incompressible third grade fluid occupying the porous half space over an infinite porous plate which moves with an arbitrary velocity in its own plane. By employing Lie symmetry method, symmetries of the modeled equation were calculated. The principal Lie algebra was two-dimensional which was then increased to three-dimensional for the restricted cases of the parameters. With the help of these symmetries, we obtain two types of analytical solutions. The similarity solution corresponding to the non-trivial symmetry generator X 3 shows the effects of suction (or injection) and the magnetic field on the physical system directly. Whereas the traveling wave solutions are satisfied subject to particular conditions. These solutions did not directly contain the term which is responsible for showing the affects of suction (or injection) and the magnetic field but the imposing conditions contains the suction (or injection) and the magnetic field term. Thus, this solution is valid for the particular value of the suction (or injection) and the magnetic field parameter. To emphasize, we say that the similarity solution of the non-traveling type generated by X 3 is better than the traveling wave solution in the sense that it represents the physics in a much better way to the problem considered. References [1] [2] [3] [4]
R.L. Fosdick, K.R. Rajagopal, Thermodynamics and stability of fluids of third grade, Proc. R. Soc. Lond. A. 339 (1980) 351–377. P.D. Ariel, Flow of a third grade fluid through a porous flat channel, Int. J. Eng. Sci. 41 (2003) 1267–1285. B. Sahoo, Hiemenz flow and heat transfer of a third grade fluid, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 811–826. K. Fakhar, Z.-C. Chen, Steady flow of third grade fluid subject to suction, Int. J. Appl. Math. 15 (2004) 387.
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[5] T. Hayat, A.H. Kara, E. Momoniat, Exact flow of a third grade fluid on a porous wall, Int. J. Non-Linear Mech. 38 (2003) 1533–1537. [6] K. Fakhar, Z. Xu, C. Yi, Exact solutions of a third grade fluid flow on a porous plate, Appl. Math. Comput. 202 (2008) 376–382. [7] T. Hayat, H.M. Mamboundou, F.M. Mahomed, Unsteady solutions in a third grade fluid filling the porous space, Math. Prob. Eng. 208 (2008) 13. Article ID 139560. [8] T. Hayat, H.M. Mamboundou, F.M. Mahomed, A note on some solutions for the flow of a fourth grade fluid in a porous space, Nonlinear Anal.: Real World Appl. 10 (2009) 368–374. [9] M. Molati, T. Hayat, F. Mahomed, Rayleigh problem for a MHD Sisko fluid, Nonlinear Anal.: Real World Appl. 10 (2009) 3428–3434. [10] T. Hayat, R.J. Moitsheki, S. Abelman, Stokes’ first problem for Sisko fluid over a porous wall, Appl. Math. Comput. 217 (2010) 622–628. [11] T. Hayat, H.M. Mamboundou, C.M. Khalique, F.M. Mahomed, Effect of magnetic field on the flow of a fourth grade fluid, Nonlinear Anal.: Real World Appl. 10 (2009) 3413–3419. [12] C. Wafo Soh, Invariant solutions of the unidirectional flow of an electrically charged power law magnetic field, Commun. Nonlinear Sci. Numer. Simul. 10 (2005) 537–548. [13] T. Hayat, A.H. Kara, E. Momoniat, The unsteady flow of a fourth-grade fluid past a porous plate, Math. Comput. Model. 41 (2005) 1347–1353. [14] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, NewYork, 1989. [15] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.
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MHD flow of a third grade fluid in a porous half space with plate suction or injection: An analytical approach Asim Aziz a,⇑, Taha Aziz b a
NUST College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Rawalpindi 46070, Pakistan Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa
b
a r t i c l e
i n f o
Keywords: Third grade fluid MHD flow Suction/blowing Symmetry approach Porous medium
a b s t r a c t The present work deals with the modeling and solution of the unsteady flow of an incompressible third grade fluid over a porous plate within a porous medium. The flow is generated due to an arbitrary velocity of the porous plate. The fluid is electrically conducting in the presence of a uniform magnetic field applied transversely to the flow. Lie group theory is employed to find symmetries of the modeled equation. These symmetries have been applied to transform the original third order partial differential equation into third order ordinary differential equations. These third order ordinary differential equations are then solved analytically and numerically. The manner in which various emerging parameters have an effect on the structure of the velocity is discussed with the help of several graphs. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Fluid flow through a porous media is a common phenomena in nature and in many fields of science and engineering. Important flow phenomenons include transport of water in living plants, trees and fertilizers or wastes in soil. In industry fluid flow through porous media has attracted much attention due to its importance in several technological processes, for example, filtration, catalysis, chromatography, petroleum exploration and recovery, cooling of electronic equipment, etc. In recent years non-Newtonian fluid flow models have been studied extensively because of their relevance to many engineering and industrial applications, particularly in the extraction of crude oil from petroleum products, synthetic fibers, food stuffs, etc. The non-Newtonian models exhibit a nonlinear relationship between stress and rate of strain. A special differential type fluid, namely the third grade fluid model, is considered in this study. This model is known to capture the nonNewtonian affects such as shear thinning or shear thickening as well as normal stresses. The third grade fluid model is thermodynamically analyzed in detail by Fosdick and Rajagopal [1]. Some recent contributions dealing with the flows of third grade fluid are [2–5]. Few authors have dealt with the flows of a third grade fluid over a porous plate [6] or within a porous medium [7]. The governing equations for non-Newtonain fluid flow models are higher order nonlinear partial differential equations, whose analysis would present a particular challenge for researchers and scientists. Due to the nonlinear terms in partial differential equations the exact solutions are difficult to be obtained for the flow of non-Newtonian fluids. Some recent studies in finding the analytical solutions for such problems can be found in [8–13]. To the best of our knowledge no attempt is available in the literature where the unsteady MHD flow of a third grade fluid over a porous plate within a porous medium is discussed. The purpose of present investigation is to discuss the analytical solutions for the unsteady MHD flow of a third grade fluid over a porous plate within a porous medium. The flow is induced due to arbitrary velocity VðtÞ of the plate. The fluid is ⇑ Corresponding author. E-mail address: [email protected] (A. Aziz). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.04.006
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considered as electrically conducting in the presence of a uniform magnetic field. Fluid flow is studied under conditions of low Reynolds number to ensure that the contributions from inertial terms (convection) does not prevail over the viscous mechanisms of momentum transfer and thus the induced magnetic field is neglected. The applied and induced electric fields are negligible. Analytical solutions of the model equation are obtained with the help of Lie symmetry methods. The paper is organized as follows: In the next section, the problem formulation is developed. Section 3 contains the Lie point symmetries of the governing equation. Section 4 deals in obtaining the similarity solutions for the modeled equation and the numerical solution of the reduced ordinary differential equation. In Section 5, the results and discussion of velocity for various interesting parameters is given and Section 6 synthesizes the results with concluding remarks. 2. Problem formulation Let us consider a third grade fluid which occupies the porous half space y > 0 and is in contact with an infinite porous plate at y ¼ 0. The plate is moving in its own plane with an arbitrary velocity. The fluid is electrically conducting and the magnetic field strength is taken as constant. The unsteady motion of the conducting fluid through a porous medium is governed by the conservation laws of momentum and mass, that is
q
dV ¼ div T þ R rB20 V; dt
ð1Þ
div V ¼ 0:
ð2Þ
In the above relations V is the velocity vector, q the density, p the pressure, T the Cauchy stress tensor, r the electrical conductivity, B0 the uniform applied magnetic field and R represents the Darcy’s resistance due to porous medium. In this work we have considered a third grade fluid and its Cauchy stress tensor (in a thermodynamically compatible form [1,2,5]) is given by
T ¼ pI þ lA1 þ a1 A2 þ a2 A21 þ b3 ðtr A21 ÞA1 :
ð3Þ
where I is the identity tensor, l the dynamic viscosity, ai ði ¼ 1; 2Þ; b3 are the material constants, and Ai ði ¼ 1; 2Þ are the Rivlin–Ericksen tensors which are defined by the following equations:
A1 ¼ ðLÞ þ ðLÞT ; An ¼
ð4Þ
dAn1 þ An1 ðLÞ þ ðLÞT An1 ; dt
ðn > 1Þ;
ð5Þ
where L ¼ $V and $ is the gradient operator. The Darcy’s resistance can be interpreted as a measure of the resistance to the flow in the porous media. For unidirectional flow of a third grade fluid over a rigid plate, the expression (for x-component of R) has been proposed by Hayat et al. [7]. Following the methodology of Ref. [7], a straight-forward calculation yields the x-component of R for unidirectional flow over a porous plate as follows:
Rx ¼
/
j
"
l þ a1
2 # @ @ @u u; a1 W 0 þ 2b3 @t @y @y
ð6Þ
in which / is the porosity and j is the permeability of the porous medium. For unidirectional flow the velocity field is given by
V ¼ ½uðy; tÞ; W 0 ; 0;
ð7Þ
where W 0 > 0 indicates suction velocity and W 0 < 0 indicates blowing or injection velocity. The above definition of velocity satisfies the law of conservation of mass for incompressible fluid. Substituting Eqs. (3)–(7) into Eq. (1), one obtains the following governing equation in the absence of the modified pressure gradient
q
" 2 2 2 # @u @u @2u @3u @3u @u @ u / @ @ @u u rB20 u: W0 ¼ l 2 þ a1 2 a1 W 0 3 þ 6b3 þ 2b l þ a a W 1 1 0 3 @t @y @y @y @t @y @y @y2 j @t @y @y ð8Þ
The relevant boundary and initial conditions are specified as follows:
uð0; tÞ ¼ U 0 VðtÞ;
t > 0;
uðy; tÞ ! 0 as y ! 1; uðy; 0Þ ¼ f ðyÞ;
y > 0;
ð9Þ t > 0;
ð10Þ ð11Þ
A. Aziz, T. Aziz / Applied Mathematics and Computation 218 (2012) 10443–10453
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where U 0 is the characteristic velocity, VðtÞ and f ðyÞ are unspecified functions. Introducing the non-dimensional variables
¼ u
u ; U0
¼ b
2b3 U 40 3
qm
¼ y
U0 y
m
2
t ¼ U 0 t ;
;
m
¼ /m ; / jU 20 2
;
a ¼
a1 U 20 ; qm2
ð12Þ
rB20 m W0 ; W0 ¼ : U0 qU 20
M2 ¼
Therefore, momentum equation (8) becomes
" 2 2 2 # @u @u @2u @3u @3u @u @ u @u @u @u ¼ 2 þ a 2 aW 0 3 þ 3b M 2 u: W0 aW 0 þ bu / uþa @t @y @y @y @t @y @y @y2 @t @y @y
ð13Þ
The corresponding initial and the boundary conditions (9)–(11) reduce to
uð0; tÞ ¼ VðtÞ;
t > 0;
ð14Þ
uðy; tÞ ! 0 as y ! 1; uðy; 0Þ ¼ f ðyÞ;
t > 0;
ð15Þ
y > 0:
ð16Þ
For simplicity we omit the bars in all the non-dimensional variables. Rewriting Eq. (13) as
2 2 2 @u @2u @3u @3u @u @ u @u @u c u þ W0 ¼ l 2 þ a 2 a W 0 3 þ c / u M2 u; @t @y @y @t @y @y @y2 @y @y
ð17Þ
where
l ¼
1 ; ð1 þ a/Þ
a ¼
c ¼
b/ ; ð1 þ a/Þ
/ ¼
a ð1 þ a/Þ
;
/ ; ð1 þ a/Þ
c¼
3b ; ð1 þ a/Þ
M2 ¼
ð18Þ
M2 : ð1 þ a/Þ
3. Lie symmetry analysis Here we present a complete Lie point symmetry analysis [14,15] of the nonlinear partial differential equation (17). We make use of these symmetry generators to solve Eq. (17) analytically. According to the Lie theory, an operator of the form
X ¼ sðt; y; uÞ
@ @ @ þ nðt; y; uÞ þ gðt; y; uÞ ; @t @y @u
ð19Þ
where s; n and g are functions of t; y and u, is a Lie point symmetry generator of Eq. (17) if and only if
h i X ½3 ut l uyy a uyyt þ a W 0 uyyy cðuy Þ2 uyy þ c uðuy Þ2 W 0 uy þ ð/ þ M 2 Þu
17
¼ 0:
ð20Þ
The operator X ½3 is the third prolongation of the operator X which is
X ½3 ¼ X þ ft
@ @ @ @ @ þ fy þ fyy þ fyyy þ ftyy ; @ut @uy @uyy @uyyy @utyy
ð21Þ
where the coefficients f’s are given by the formulae
ft ¼ Dt g ut Dt s uy Dt n; fy ¼ Dy g ut Dy s uy Dy n; fyy ¼ Dy fy uty Dy s uyy Dy n;
ð22Þ
fyyy ¼ Dy fyy utyy Dy s uyyy Dy n; ftyy ¼ Dt fyy utyy Dt s uyyy Dt n: The total derivative operators are given as
@ @ @ @ þ ut þ utt þ uty þ ; @t @u @ut @uy @ @ @ @ þ uy þ uyy þ uty þ Dy ¼ @y @u @uy @ut Dt ¼
ð23Þ
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Substituting Eqs. (21) and (22) in the symmetry condition (20) and separating it by powers of the derivative of u, yield an overdetermined system of linear homogeneous partial differential equations called determining equations. The following determining equations were generated manually and are given as
su ¼ sy ¼ 0; nu ¼ ny ¼ 0;
guu ¼ gy ¼ 0; a gtu þ l st ¼ 0;
ð24Þ
W 0 st þ nt ¼ 0; ð/ þ M 2 Þg þ uð/ þ M2 Þst þ gt uð/ þ M 2 Þgu ¼ 0;
g þ ust þ ugu ¼ 0;
st þ 2gu ¼ 0: The solution of the above system gives rise to two cases: Case 1. when / þ M 2 – l =a For this case, we find a two-dimensional Lie algebra generated by
X1 ¼
@ ; @t
X2 ¼
@ : @y
ð25Þ
Case 2. when / þ M 2 ¼ l =a Here we obtain three-dimensional Lie algebra generated by
@ ; @y 2 2 2 @ @ @ X 3 ¼ e2ð/ þM Þt þ W 0 e2ð/ þM Þt þ uð/ þ M2 Þe2ð/ þM Þt ; @t @y @u X1 ¼
@ ; @t
X2 ¼
ð26Þ
where X 1 is the time translational symmetry generator, X 2 is the space translational symmetry generator and X 3 has path curves which are equivalent to a combination of translation and scaling in u.
4. Similarity solutions In this section, similarity solutions (group invariant solutions) of the partial differential equation (17) corresponding to symmetries (25) and (26) will be constructed. 4.1. Traveling wave solutions Traveling wave solutions are the one special kind of group invariant solutions which are invariant under a linear combination of time-translation and space-translation generator. 4.1.1. Forward type We look for the invariant solution under the operator X 1 þ cX 2 (with c > 0) which represents a forward wave-front type traveling wave solutions with constant wave speed c, in this case the waves are propagating away from the plate. These are solutions of the form
uðy; tÞ ¼ G1 ðx1 Þ;
where x1 ¼ y ct:
ð27Þ
Using Eq. (27) into Eq. (17) results in a third-order ordinary differential equation for Gðx1 Þ,
c
2 2 2 2 3 3 dG1 d G1 d G1 d G1 dG1 d G1 dG1 dG1 ¼ l ca a W 0 þc c G1 þ W0 ð/ þ M 2 ÞG1 ; 2 3 3 2 dx1 dx dx dx1 1 1 dx1 dx1 dx1 dx1
ð28Þ
hence the above equation admits the exact solution of the form (which is required to be zero at infinity due to the second boundary condition)
pffiffiffiffiffi c G1 ðx1 Þ ¼ A exp pffiffiffi x1
c
ð29Þ
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provided that
rffiffiffiffiffi
rffiffiffiffiffi
c c c l þ ca þ a W 0 c c c
rffiffiffiffiffi c ½c þ W 0 ð/ þ M2 Þ ¼ 0:
ð30Þ
c
Thus Eq. (17) subject to Eq. (30) gives the solution of the form
pffiffiffiffiffi c ðy ctÞ : uðy; tÞ ¼ exp pffiffiffi
ð31Þ
c
The solution (31) satisfies the initial and boundary conditions (14)–(16) for the particular values of the unspecified functions VðtÞ and f ðyÞ. Using Eq. (31) into Eqs. (14)–(16) gives
pffiffiffiffiffi c c t pffiffiffi ; c pffiffiffiffiffi cy uðy; 0Þ ¼ f ðyÞ ¼ exp pffiffiffi :
uð0; tÞ ¼ VðtÞ ¼ exp
ð32aÞ ð32bÞ
c
Here VðtÞ and f ðyÞ depend on the physical parameters of the flow. The solution given in Eq. (31) is plotted in Figs. 1 and 2 for various values of the emerging parameters which satisfies the condition (30). 4.1.2. Backward type We now look for the invariant solution under the operator X 1 cX 2 (with c > 0) which represents a backward wave-front type traveling wave solutions with constant wave speed c, in this case the waves are propagating towards the plate. These are solutions of the form
uðy; tÞ ¼ G2 ðx2 Þ;
where x2 ¼ y þ ct:
ð33Þ
Using Eq. (33) into Eq. (17) results in a third-order ordinary differential equation for Gðx1 Þ,
c
2 2 2 2 3 3 dG2 d G2 d G2 d G2 dG2 d G2 dG2 dG2 ¼ l þ c a a W þ c c G þ W0 ð/ þ M 2 ÞG2 ; 0 2 2 3 3 2 dx2 dx2 dx2 dx2 dx2 dx2 dx2 dx2
ð34Þ
and hence the above equation admits the exact solution of the form (which we require to be zero at infinity due to the second boundary condition)
pffiffiffiffiffi c G2 ðx2 Þ ¼ A exp pffiffiffi x2 ;
ð35Þ
c
provided that
rffiffiffiffiffi
rffiffiffiffiffi
c c c l ca þ a W 0 þ c c c
rffiffiffiffiffi c ½c W 0 ð/ þ M2 Þ ¼ 0:
ð36Þ
c
Thus Eq. (17) subject to Eq. (36) gives the solution
pffiffiffiffiffi c ðy þ ctÞ : uðy; tÞ ¼ exp pffiffiffi
ð37Þ
c
7
t=0 t=0.5 t=1 t=1.5 t=1.9 t=2.2
6 5
u
4 3 2 1 0
0
1
2
3
4
5
y Fig. 1. Plot of the velocity field (31) for varying values of t when c ¼ 1:5; c ¼ c ¼ 1 are fixed.
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A. Aziz, T. Aziz / Applied Mathematics and Computation 218 (2012) 10443–10453
5
c=0.1 c=0.5 c=0.8 c=1 c=1.3
4
u
3
2
1
0
0
1
2
3
4
5
y Fig. 2. Plot of the velocity field (31) for varying values of c when c ¼ 1:5; c ¼ 1; t ¼ p=2 are fixed.
We note that this solution satisfies the initial and boundary conditions (14)–(16) for the specific values of the functions VðtÞ and f ðyÞ, where
pffiffiffiffiffi c ct uð0; tÞ ¼ VðtÞ ¼ exp pffiffiffi ; c pffiffiffiffiffi cy uðy; 0Þ ¼ f ðyÞ ¼ exp pffiffiffi :
ð38aÞ ð38bÞ
c
The solution given in Eq. (37) is plotted in Figs. 3 and 4 for various values of the emerging parameters which satisfies the condition (36). Moreover, we can also find group invariant solution corresponding to the operator X 1 and X 3 , which provides other meaningful solution as well. 4.2. Solution via subgroup generated by X 3 Consider the invariant solution under the operator X 3 given as 2
X 3 ¼ e2ð/ þM Þt
2 2 @ @ @ þ W 0 e2ð/ þM Þt þ uð/ þ M 2 Þe2ð/ þM Þt : @t @y @u
ð39Þ
The similarity solution corresponding to the above operator X 3 assumes the form
h i uðy; tÞ ¼ hðhÞ exp ð/ þ M 2 Þt ;
h ¼ W 0 t þ y;
ð40Þ
where hðhÞ is an undetermined function. Substituting Eq. (40) into Eq. (17) gives a second order linear ordinary differential equation
dh
2
c h ¼ 0: c
ð41Þ 1
t=0 t=0.5 t=1.5 t=2.3 t=3.8 t=4.5
0.8
0.6
u
2
d h
0.4
0.2
0
0
1
2
3
4
5
y Fig. 3. Plot of the velocity field (37) for varying values of t when c ¼ 1:5; c ¼ 1; t ¼ p=2 are fixed.
A. Aziz, T. Aziz / Applied Mathematics and Computation 218 (2012) 10443–10453
0.35
10449
c=1 c=1.2 c=2 c=2.5 c=3
0.3 0.25
u
0.2 0.15 0.1 0.05 0
0
1
2
3
4
5
y Fig. 4. Plot of the velocity field (37) for varying values of c when c ¼ 1:5; c ¼ 1; t ¼ p=2 are fixed.
Using condition (15), one can write the boundary condition for Eq. (41) as
hðlÞ ¼ 0;
l ! 1:
ð42Þ
This is only part of the boundary conditions. One can not obtain another boundary condition on h from condition (16). However, we utilize this to deduce the exact solution
rffiffiffiffiffi c hðhÞ ¼ a exp h :
ð43Þ
c
Substituting hðhÞ into Eq. (40) and simplifying we obtain the solution for uðy; tÞ of the form
rffiffiffiffiffi rffiffiffiffiffi c c uðy; tÞ ¼ exp W 0 þ / þ M2 t þ y :
c
ð44Þ
c
We observe that this solution satisfies the initial and boundary conditions (14)–(16). Making use of Eq. (44) into Eqs. (14)– (16) we get
rffiffiffiffiffi c W 0 þ / þ M2 t ; uð0; tÞ ¼ VðtÞ ¼ exp c pffiffiffiffiffi c y uðy; 0Þ ¼ f ðyÞ ¼ exp pffiffiffi ;
ð45aÞ ð45bÞ
c
where VðtÞ and f ðyÞ once again depend on the physical parameters of the flow. The solution in Eq. (44) is plotted in Figs. 5–9 for various values of emerging parameters.
1
t=0 t=0.2 t=0.5 t=0.9 t=1.2 t=1.9
0.8
u
0.6
0.4
0.2
0
0
1
2
3
4
5
y Fig. 5. Plot of the velocity field (44) for varying values of t when c ¼ 1:5; c ¼ 1; W 0 ¼ 0:5; / ¼ 0:5; M ¼ W 0 ¼ 1 are fixed.
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A. Aziz, T. Aziz / Applied Mathematics and Computation 218 (2012) 10443–10453
0.25
W0=-0.2 W0=-0.7
0.2
W0=0 W0=0.2 W0=0.7
u
0.15
0.1
0.05
0
0
1
2
3
4
5
y Fig. 6. Plot of the velocity field (44) for varying values of W 0 when c ¼ 1:5; c ¼ 1; / ¼ 0:5; M ¼ 1; t ¼ p=2 are fixed.
0.12
M*=0.2 M*=0.6
0.1
M*=0.8 M*=1.0
0.08
u
M*=1.2 0.06 0.04 0.02 0
0
1
2
y
3
4
5
Fig. 7. Plot of the velocity field (44) for varying values of M when c ¼ 1:5; c ¼ 1; / ¼ 0:5; W 0 ¼ 1; t ¼ p=2 are fixed.
0.09
γ =0.5
0.08
γ =0.8 γ =1.2
0.07
γ =1.7 γ =2.1
0.06
u
0.05 0.04 0.03 0.02 0.01 0
0
1
2
y
3
4
5
Fig. 8. Plot of the velocity field (44) for varying values of c when c ¼ 1:3; / ¼ 0:2; W 0 ¼ 0:5; M ¼ 1; t ¼ p=2 are fixed.
4.3. Solution via subgroup generated by X 1 The invariant solution admitted by X 1 is the steady-state solution
uðy; tÞ ¼ FðyÞ:
ð46Þ
A. Aziz, T. Aziz / Applied Mathematics and Computation 218 (2012) 10443–10453
0.1
10451
γ *=0.5 γ *=0.8
0.08
γ *=1.2 γ *=1.7 γ *=2.1
u
0.06
0.04
0.02
0
0
1
2
y
3
4
5
Fig. 9. Plot of the velocity field (44) for varying values of c when c ¼ 1:3; / ¼ 0:2; W 0 ¼ 0:5; M ¼ 1; t ¼ p=2 are fixed.
Substituting Eq. (46) into Eq. (17) yield the third-order ordinary differential equation for FðyÞ 2
l
d F 2
dy
2 2 2 dF d F dF dF ð/ þ M2 ÞF ¼ 0 þc c F þ W0 2 dy dy dy dy dy 3
a W 0
d F 3
ð47Þ
with the boundary conditions
Fð0Þ ¼ v 0 ; FðlÞ ¼ 0;
l > 0;
ð48Þ
where l can take a sufficiently large value with V ¼ v 0 a constant. Remark 1. We present numerical solution of reduced equation (47) subject to the boundary conditions (48) using Mathematica solver NDSolve.
5. Graphical results and discussion In this section, velocity profiles obtained in Section 4 are plotted. The traveling wave solutions over an infinite porous plate are plotted in Figs. 1–4 for different values of time t and wave speed c. These solutions does not predict the effects of suction or injection as well as the effects of magnetic field on the flow. Figs. 1 and 2 describe the effects of time t and wave speed c on the forward wave-front type traveling wave solution (31). These figures show that with the increase of time t and wave speed c the velocity profile is increasing. Figs. 3 and 4 predict the effects of time t and wave speed c on the backward wave-front type traveling wave solution (37). From these figures, it is clear that velocity decreases by increasing the time t and the wave speed c. In this way we can conclude that both time and the wave speed have an opposite effect on the forward and the backward wave-front type traveling wave solutions, respectively. In order to see the effects of plate suction or injection and the effects of magnetic field on the flow, the similarity solution (44) is plotted in Figs. 5–9 for different values of emerging parameters. Fig. 5 shows the velocity profiles for various values of time t at fixed values of other parameters. As expected, the velocity profile will decay with time. In order to examine the effects of the plate suction or injection, the velocity profiles versus y for various values of suction parameter W 0 are depicted in Fig. 6. Curve with W 0 ¼ 0 corresponds to the case of rigid plate. The positive values of W 0 correspond to the suction occurring at the plate surface whereas the negative values of W 0 correspond to the injection or blowing at the plate surface. From this figure, we can clearly observe the effects of suction parameter. It is observed that by increasing W 0 ð> 0Þ the velocity profile and the boundary layer thickness decrease. This is in accordance with the fact that suction causes reduction in the boundary layer thickness. The effects of increasing injection W 0 ð< 0Þ are opposite to those of suction. Fig. 7 shows the effects of magnetic field on the flow. It shows that velocity decreases when the magnitude of the magnetic field is increased. The effects of the fluid parameters c and c in the presence of plate suction or injection and magnetic field are shown in Figs. 8 and 9. These figures reveal that c and c have opposite roles on the velocity even in the presence of plate suction or injection. From these figures, it is noted that velocity increases for large values of c whereas it decreases for increasing c . Qualitatively, the effects of c and c on the velocity profiles for porous plate are similar to those of rigid plate, however, these effects are not similar quantitatively. The numerical solution of the reduced ordinary differential equation (47) subject to boundary conditions (48) for different values of suction parameter W 0 and the magnetic field parameter M are shown in Figs. 10 and 11, respectively. It is observed that the effects of W 0 and M on the velocity profiles are the same as noticed previously for the analytical solutions.
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1.2
W0=-1.5 W0=0
1
W0=1.5
u
0.8 0.6 0.4 0.2 0
0
1
2
y
3
4
5
Fig. 10. Numerical solution of (47) subject to conditions (48) for varying values of W 0 when c ¼ 1:5; c ¼ 1; fixed.
l ¼ 0:5; M ¼ 0:5; / ¼ 0:75; a ¼ 1 are
1 M*=0.5
0.9
M*=1.0 M*=2.0
0.8 0.7
u
0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.5
1
1.5
2
2.5
y
3
3.5
4
4.5
Fig. 11. Numerical solution of (47) subject to conditions (48) for varying values of M when c ¼ 1:5; c ¼ 0:5;
5
l ¼ 0:5; W 0 ¼ 0:75; / ¼ a ¼ 1 are fixed.
6. Concluding remarks In this paper we have studied the unsteady MHD flow of an incompressible third grade fluid occupying the porous half space over an infinite porous plate which moves with an arbitrary velocity in its own plane. By employing Lie symmetry method, symmetries of the modeled equation were calculated. The principal Lie algebra was two-dimensional which was then increased to three-dimensional for the restricted cases of the parameters. With the help of these symmetries, we obtain two types of analytical solutions. The similarity solution corresponding to the non-trivial symmetry generator X 3 shows the effects of suction (or injection) and the magnetic field on the physical system directly. Whereas the traveling wave solutions are satisfied subject to particular conditions. These solutions did not directly contain the term which is responsible for showing the affects of suction (or injection) and the magnetic field but the imposing conditions contains the suction (or injection) and the magnetic field term. Thus, this solution is valid for the particular value of the suction (or injection) and the magnetic field parameter. To emphasize, we say that the similarity solution of the non-traveling type generated by X 3 is better than the traveling wave solution in the sense that it represents the physics in a much better way to the problem considered. References [1] [2] [3] [4]
R.L. Fosdick, K.R. Rajagopal, Thermodynamics and stability of fluids of third grade, Proc. R. Soc. Lond. A. 339 (1980) 351–377. P.D. Ariel, Flow of a third grade fluid through a porous flat channel, Int. J. Eng. Sci. 41 (2003) 1267–1285. B. Sahoo, Hiemenz flow and heat transfer of a third grade fluid, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 811–826. K. Fakhar, Z.-C. Chen, Steady flow of third grade fluid subject to suction, Int. J. Appl. Math. 15 (2004) 387.
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[5] T. Hayat, A.H. Kara, E. Momoniat, Exact flow of a third grade fluid on a porous wall, Int. J. Non-Linear Mech. 38 (2003) 1533–1537. [6] K. Fakhar, Z. Xu, C. Yi, Exact solutions of a third grade fluid flow on a porous plate, Appl. Math. Comput. 202 (2008) 376–382. [7] T. Hayat, H.M. Mamboundou, F.M. Mahomed, Unsteady solutions in a third grade fluid filling the porous space, Math. Prob. Eng. 208 (2008) 13. Article ID 139560. [8] T. Hayat, H.M. Mamboundou, F.M. Mahomed, A note on some solutions for the flow of a fourth grade fluid in a porous space, Nonlinear Anal.: Real World Appl. 10 (2009) 368–374. [9] M. Molati, T. Hayat, F. Mahomed, Rayleigh problem for a MHD Sisko fluid, Nonlinear Anal.: Real World Appl. 10 (2009) 3428–3434. [10] T. Hayat, R.J. Moitsheki, S. Abelman, Stokes’ first problem for Sisko fluid over a porous wall, Appl. Math. Comput. 217 (2010) 622–628. [11] T. Hayat, H.M. Mamboundou, C.M. Khalique, F.M. Mahomed, Effect of magnetic field on the flow of a fourth grade fluid, Nonlinear Anal.: Real World Appl. 10 (2009) 3413–3419. [12] C. Wafo Soh, Invariant solutions of the unidirectional flow of an electrically charged power law magnetic field, Commun. Nonlinear Sci. Numer. Simul. 10 (2005) 537–548. [13] T. Hayat, A.H. Kara, E. Momoniat, The unsteady flow of a fourth-grade fluid past a porous plate, Math. Comput. Model. 41 (2005) 1347–1353. [14] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, NewYork, 1989. [15] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.