Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 11 (2017), pp. 8025-8038
© Research India Publications
http://www.ripublication.com
Numerical Investigation of a Channel Flow with Third
Grade Fluid in the Presence of Magnetic Field
Manju Agarwal1, Vivek Joseph2, Ramesh Yadav2 & Parul Saxena1
1Department of Mathematics & Astronomy, University of Lucknow, Lucknow U.P 2Department of Mathematics, Babu Banarasi Das Northern India Institute of
Technology Lucknow U.P, India.
Abstract
In this paper an investigation on the flow of a third grade fluid bounded by
two parallel porous plates given by using Numerical analysis has been done by
using Matlab Software. The main focus of this study is to analyze the effect of
magnetic field, on the third grade fluid in a channel flow. The effects of
Reynolds number (Re), third grade fluid parameter (T) and magnetic
parameter (M) on the flow of fluid have been tabulated and analyzed
graphically. Different set of parameters have been taken to critically measure
the effect of all the parameters simultaneously on the channel flow with
suitable boundary conditions.
Keywords: - Channel flow, Hartmann Number M, Reynolds number Re.
Third grade parameter.
INTRODUCTION
Non-Newtonian fluids have many applications in chemicals, cosmetics,
pharmaceuticals Industries. Number of non-Newtonian fluid models is solved to
explain the characteristics of such fluids. Some of them are power law, second grade,
third grade, fourth grade, Brinkman type, micro polar, Jeffery, Walters’s B, Maxwell,
Oldroyd-B, Burgers and generalized Burgers fluid models. The earliest model of
viscoelastic fluids was proposed by Rivlin and Ericksen (1955), who studied stress
deformation relation for isotropic material, which included two parameters 𝛼1 and 𝛼2
8026 Manju Agarwal, Vivek Joseph, Ramesh Yadav & Parul Saxena
in the constitutive equation besides the Newtonian viscosity𝜇. The inclusion of the
parameter𝛼1, in particular, leads to some spectacular ramifications in the solution of
the fluid problems. In most of the flows, the order of the differential equations
governing the motion is raised by at least one, while there is no corresponding
increase in the number of boundary conditions. Though there have been several
proposals regarding the extra boundary conditions, at the time of writing, there is no
consensus amongst the researchers on the acceptability of any of the proposed
boundary condition. Under the circumstances, the solutions of the flow problems have
to be obtained on the basis of some plausible assumptions regarding the behavior of
the solution for values of 𝛼1 close to zero. Fosdick and Rajagopal (1979) studied
anomalous features in the model of second order fluids.
The solution for the stagnation point flow was facilitated by the fact that the boundary
value problem describing the motion, even though is of higher order, was singular at
the boundary. Such an aid is not available when the flow takes place between parallel
porous plates. Now there is an increase in the order of differential equation, but the
highest derivative is multiplied by𝛼1. Because of this, a different approach must be
chosen that involves the pruning of the spurious solution introduced on account of the
𝛼1-term in the differential equation. Ariel (1992) derived the solutions for the flow of
a viscoelastic fluid between parallel boundaries when there is an injection of the fluid
at one boundary and an equal suction at the other boundary.
The inclusion of some more parameters in the constitutive equations, known as the
third grade fluid parameters, makes the model more realistic. However it adds a new
dimension in the solution processes of the flow problem, namely, non-linearity. For
the flow between parallel plates, Ariel (2003) derived an interesting method for
computing the flow by seeking the solution in a series of exponential terms. He was
able to obtain the solution for a combination of values of physical parameter, but the
performance of the series solution degraded sharply as the value of the third grade
fluid parameter was increased. This warranted the search of alternate methods for
computing the flow of the third grade fluids. Ayaz (2004) studied Solution of the
systems of differential equations by differential transform method. Akhildiz et al.
(2004) studied exact solution of nonlinear differential equations arising in third grade
fluid flows. They compared the exact solutions with numerical ones. It is observed
that the difference between the exact and the numerical solutions is about 1% for
small R (the non-dimensional distance between the cylinders) and is about 3% when
R = 100. This difference increases with an increasing R. Moreover, for large R it is
not easy to obtain meaningful results numerically. Hence, these exact solutions for
various values of the parameters R and 𝜔 (rotating parameter) are useful for
experimental and numerical studies, and warrant further study. Ganji and Rajabi
Numerical Investigation of a Channel Flow With Third Grade Fluid in.... 8027
(2006) studied Assessment of homotopy-perturbation and perturbation methods in
heat radiation equations.
Mukhopadhyay (2009) studied Effects of radiation and variable fluid viscosity on
flow and heat transfer along a symmetric wedge. He has found that increase of
temperature-dependent fluid viscosity parameter (i.e. with decreasing viscosity), the
fluid velocity increases up to the cross-over points (η0 ≈ 0.90 is the nearest numerical
value of the cross-over point) and after the crossing over the point the fluid velocity is
found to decrease but the temperature increases at a particular point. Hayat et al.
(2006) studied Homotopy solution for the channel flow of a third grade fluid. They
compared the exact numerical results and HAM solution and get very close to results.
Hayat et al. (2010) studied Heat transfer for flow of a third grade fluid between two
porous plates. They found the dependency of the viscoelastic parameter on the
velocity and heat transfer of the fluid.
Shafiq et al. (2013) studied Magneto hydrodynamic axisymmetric flow of a third
grade fluid between two porous desks. They found the effects of dimensionless
parameter on the radial and axial components of the velocity and skin-friction
coefficients at upper and lower disks are tabulated for various values of the
dimensionless physical parameters. Aiyesimi et al. (2014) studied analysis of
unsteady magneto hydrodynamic thin film flow of a third grade fluid with heat
transfer down an inclined plane. They found that the variation of magnetic parameter,
gravitational parameter on the velocity and temperature profile of the fluid. Azimi et
al. (2014) studied Investigation of the film flow of a third grade fluid on an inclined
plane using Homotopy perturbation method (HPM). They compared their result with
previous result Runge Kutta method, numerical method and other. Taza et al. (2014)
studied thin film flow in MHD third grade fluid on a vertical belt with temperature
dependent viscosity. They have solved the problem, using optimal asymptotic method
(OHAM), they discussed the physical characteristics of the problem. Hayat et al.
(2015) studied MHD axisymmetric flow of third grade fluid by a stretching cylinder.
The main focus on the analytic solution is steady boundary layer axisymmetric flow
of third grade fluid over a continuously stretching cylinder in the presence of
magnetic field. They have found the effect of the emerging parameter such as third
grade parameter, second grade parameter and Reynolds number on the velocity of
third grade fluid.
In this present paper the laminar flow of a third grade fluid through a flat porous
channel has been investigated, when the rate of injection at one wall is equal to the
rate of suction at other wall. The flow is caused by the external pressure gradient. The
numerical solution is expressed in the parameter of third grade fluid (T), magnetic
Field (M), second grade fluid parameter (K), and Reynolds number (R).
8028 Manju Agarwal, Vivek Joseph, Ramesh Yadav & Parul Saxena
2. MATHEMATICAL FORMULATION
Let us assume the steady flow of a third grade fluid between two porous walls at 𝑦 =
𝑎 and 𝑦 = 𝑏. The flow of fluid is due to a constant pressure gradient. Also, there is
cross flow because of uniform injection of the fluid flow at lower wall with velocity
𝜐0 and an equal suction at the upper wall. For third grade fluids, physical
considerations were taken into account by Fosdick and Rajagopal [6], in order to
obtain the following form for the constitutive law:
𝑇 = −𝑝 𝐼 + 𝜇 𝐴1 + 𝛼1 𝐴2 + 𝛼2 𝐴12 + 𝛽3(𝑡𝑟 𝐴1
2)𝐴1 (1)
Which, when introduced in the equation of conservation of momentum in the
presence of magnetic field leads to the following equation
𝜇𝑑2𝑢
𝑑𝑦2 + 𝛼1𝜐0𝑑3𝑢
𝑑𝑦3− 𝜌𝜐0
𝑑𝑢
𝑑𝑦+ 6 𝛽3 (
𝑑𝑢
𝑑𝑦)
2 𝑑2𝑢
𝑑𝑦2 −𝑠 B0
𝜌 𝑢 =
𝜕𝑝
𝜕𝑥 (2)
where 𝐵0 is magnetic field, 𝑠 is electrical conductivity of the field, 𝜌 is the viscosity
of the fluid, moreover, the coefficients 𝜇, 𝛼1, 𝛼2 and 𝛽3 must be satisfy the following
inequalities:
𝜇 ≥ 0, 𝛼1 ≥ 0, 𝛽3 ≥ 0, and |𝛼1 + 𝛼2| ≤ √24 𝜇 𝛽3 (3)
The boundary conditions are
𝑢(𝑎) = 𝑢(𝑏) = 0 . (4)
Now defining the non-dimensional variables
𝜆 =𝑦
𝑏 and 𝑈 = −
𝜇𝑢
𝑏2 (𝜕𝑝
𝜕𝑥)
−1
. (5)
Putting these values in equation (2) & (4), we get
𝜇 (−𝑏2
𝜇) (
𝜕𝑝
𝜕𝑥)
𝑑2𝑈
𝑏2𝑑𝜆2 + 𝛼1𝜐0 (−𝑏2
𝜇) (
𝜕𝑝
𝜕𝑥)
𝑑3𝑈
𝑏3𝑑𝜆3 − 𝜌𝜐0 (−𝑏2
𝜇) (
𝜕𝑝
𝜕𝑥)
𝑑𝑈
𝑏 𝑑𝜆+
6 𝛽3 {(−𝑏2
𝜇) (
𝜕𝑝
𝜕𝑥)
𝑑𝑈
𝑏𝑑𝜆}
2
(−𝑏2
𝜇) (
𝜕𝑝
𝜕𝑥)
𝑑2𝑢
𝑏2𝑑𝜆2 −𝑠 B0
𝜌(−
𝑏2
𝜇) (
𝜕𝑝
𝜕𝑥) 𝑈 =
𝜕𝑝
𝜕𝑥 , (6)
or
−𝑑2𝑈
𝑑𝜆2 − (𝛼1
𝜌𝑏2) (𝜌𝜐0 𝑏
𝜇)
𝑑3𝑈
𝑑𝜆3 + (𝜌𝜐0𝑏
𝜇)
𝑑𝑈
𝑑𝜆+
6 𝛽3𝑏2
𝜇3 (𝜕𝑝
𝜕𝑥)
2
(𝑑𝑈
𝑑𝜆)
2𝑑2𝑢
𝑑𝜆2 + (𝑠 B0𝑏2
𝜌𝜇) 𝑈 = 1 (7)
or
𝑑2𝑈
𝑑𝜆2 + 𝐾𝑅𝑒𝑑3𝑈
𝑑𝜆3 − 𝑅𝑒𝑑𝑈
𝑑𝜆+ 𝑇 (
𝑑𝑈
𝑑𝜆)
2 𝑑2𝑢
𝑑𝜆2 − 𝑀2𝑈 = −1 (8)
𝑈(𝜎) = 𝑢(1) = 𝑢(0) = 0 , (9)
Numerical Investigation of a Channel Flow With Third Grade Fluid in.... 8029
in the above equation is,
𝜎 =𝑎
𝑏 , 𝐾 = (
𝛼1
𝜌𝑏2) , 𝑅𝑒 = (𝜌𝜐0 𝑏
𝜇) , 𝑇 =
6 𝛽3𝑏2
𝜇3 (𝜕𝑝
𝜕𝑥)
2
& 𝑀2 = (𝑠 B0𝑏2
𝜌𝜇), (10)
3. METHOD OF SOLUTION
In this study the solution of the above differential equation (8) with boundary
conditions (9) has been calculated numerically using ode 45 solver in Matlab. For the
purpose the time interval (0, 10) with initial condition vector (0, 0, 0) has been taken
and for convergence criteria, options has been chosen ('RelTol',1e-4,'AbsTol',[1e-4
1e-4 1e-5]). The range of dimensionless variable 𝜆 (0 ≤ 𝜆 ≤ 10), the value
Reynolds No. has been taken {1, 3, 6, 8, 12}, third grade fluid parameter T has been
taken {1, 2, 3, 4, 5, 6}, Magnetic parameter M has been taken {1, 2, 3, 4, 5}. The
second grade parameter K has been taken {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 0.9, 1.0}.
The transformed set of differential equations for using ode45 solver is dy(1) = y(2);
dy(2) = y(3); dy(3) = (-1+𝑀2*y(1)+R*y(2)-y(3)-T*(y(1)^2)*y(3))/(K*R). Various
graphs have been plotted with described set of parameters and discussed in detail in
the next section.
4. RESULTS AND DISCUSSION
The non-linear differential equation (9) subject to (10) must be integrated by
numerical procedure to use ode45 solver and get the results. In fig. 1, U is plotted
against 𝜆 for T = 2, M = 2, Re = 2, K = 0.5, for these value of parameters we get the
velocity of fluid 𝑈(𝜆), 𝑈′(𝜆), 𝑈′′(𝜆) increases sharply with increase of 𝜆. From fig.
2, velocity of fluid plotted against 𝜆 for Re = 1, T = 1, K = 0.1, for these parameter
velocity of fluid decreases with increase of dimensionless variable 𝜆 (0 ≤ 𝜆 ≤ 10); it
also seen that increase of magnetic parameter M (1, 2, 3, 4) velocity of fluid
decreases. From fig. 3, 𝑈′ plotted against 𝜆 for Re = 1, T = 1, K = 0.1, for these
parameter velocity of fluid decreases sharply between 0 ≤ 𝜆 ≤ 2 , and decreases
slowly 2 ≤ 𝜆 ≤ 10; in this graph show that increase of magnetic parameter M (1, 2, 3,
4, 5) velocity of fluid decreases
From fig. 4, velocity of fluid plotted against 𝜆 for M = 2, Re = 1, K = 0.1, for these
parameter velocity of fluid decreases with increase of dimensionless variable 𝜆 (0 ≤
𝜆 ≤ 10); it also seen that increase of third grade parameter T (1, 2, 3, 4, 5) velocity of
fluid increases. From fig. 5, 𝑈′ plotted against 𝜆 for M = 2, Re = 1, K = 0.1, for these
parameter velocity of fluid decreases sharply between 0 ≤ 𝜆 ≤ 2 , and then decreases
slowly between 2 ≤ 𝜆 ≤ 10; in this graph show that increase of third grade parameter
8030 Manju Agarwal, Vivek Joseph, Ramesh Yadav & Parul Saxena
T (1, 2, 3, 4, 5, 6) velocity of fluid increases. From fig. 6, velocity of fluid 𝑈 plotted
against 𝜆 for M = 2, T = 2, K = 0.1, for these parameter velocity of fluid decreases
slowly with increase of dimensionless variable 𝜆 (0 ≤ 𝜆 ≤ 10); it also seen that
increase of Reynolds no R (1, 3, 6, 8, 12) velocity of fluid decreases.
From fig. 7, 𝑈′ plotted against 𝜆 for M = 2, T = 2, K = 0.1, for these parameter
velocity of fluid decreases sharply between 0 ≤ 𝜆 ≤ 2 , and decreases slowly 2 ≤
𝜆 ≤ 10; in this graph show that increase of Reynolds number Re (1, 3, 6, 8, 12)
velocity of fluid decreases. From fig. 8, velocity of fluid plotted against 𝜆 for M = 2,
Re = 2, T = 2, for these parameter velocity of fluid decreases with increase of
dimensionless variable 𝜆 (0 ≤ 𝜆 ≤ 10); it also seen that increase of parameter K
(0.2, 0.4, 0.6, 0.8, 1.0) velocity of fluid increases.
From fig. 9, 𝑈′ plotted against 𝜆 for M = 2, Re = 2, T = 2, for these parameter velocity
of fluid decreases sharply between 0 ≤ 𝜆 ≤ 2 , and then decreases slowly between
2 ≤ 𝜆 ≤ 10; in this graph show that increase of parameter K (0.2, 0.4, 0.6, 0.8, 1.0)
velocity of fluid increases. From fig. 10, velocity of fluid plotted against 𝜆 for M = 2,
Re = 1, T = 5, for these parameter velocity of fluid decreases sharply with increase of
dimensionless variable 𝜆 (0 ≤ 𝜆 ≤ 1); it also seen that increase of parameter K (0.2,
0.5, 0.9) velocity of fluid increases. From fig. 11, velocity of fluid plotted against 𝜆
for M = 0, Re = 1, T = 5, for these parameter velocity of fluid decreases sharply with
increase of dimensionless variables 𝜆 (0 ≤ 𝜆 ≤ 1. ) ; in this graph show that increase
of parameter K (0.2, 0.5, 0.9) velocity of fluid increases.
.
Figure 1: Variation of the velocity of fluid 𝑈(𝜆), 𝑈′(𝜆), 𝑈′′ (𝜆) .
0 1 2 3 4 5 6 -20
-15
-10
-5
0
U
M = 2, Re = 2, T = 2, K = 0 .5, U (
M = 2, Re = 2, T = 2, K = 0 .5, U ' (
M = 2, Re = 2, T = 2, K = 0 .5, U ''(
Numerical Investigation of a Channel Flow With Third Grade Fluid in.... 8031
Figure 2: Variation of velocity profile 𝑈 (𝜆) for different Magnetic field (M).
Figure 3: Variation of the radial velocity of fluid 𝑈′(𝜆) for different
Magnetic field (M)
0 1 2 3 4 5 6 7 8 9 10-18
-16
-14
-12
-10
-8
-6
-4
-2
0
U '
(
)
M = 1
M = 2
M = 3
M = 4
M = 5
0 1 2 3 4 5 6 7 8 9 10 -120
-100
-80
-60
-40
-20
0
U (𝜆)
Re = 1, T = 1, K = 0 .1, M = 1
Re= 1, T = 1, K = 0 .1, M = 2
Re = 1, T = 1, K = 0 .1, M = 3
Re = 1, T = 1, K = 0 .1, M = 4
8032 Manju Agarwal, Vivek Joseph, Ramesh Yadav & Parul Saxena
Figure 4: Variation of the velocity of fluid 𝑈 (𝜆) with different third grade parameter
(T)
Figure 5: Variation of the velocity of fluid 𝑈′ (𝜆) with different third
grade parameter (T)
0 1 2 3 4 5 6 7 8 9 10-7
-6
-5
-4
-3
-2
-1
0
U '
(
)
T = 1
T = 2
T = 3
T = 4
T = 5
T = 6
0 1 2 3 4 5 6 7 8 9 10
0
-60
60
-50
50
-40
40
-30
30
-20
20
-10
10
0
U(𝜆)
M = 2, Re = 1, K = 0 .1, T = 1 M = 2, Re = 1, K = 0 .1, T = 2 M = 2, Re = 1, K = 0 .1, T = 3 M = 2, Re = 1, K = 0 .1, T = 4 M = 2, Re = 1, K = 0 .1, T = 5
Numerical Investigation of a Channel Flow With Third Grade Fluid in.... 8033
Figure 6: Variation of the velocity of fluid 𝑈 (𝜆) for different Reynolds no (R)
Figure 7: Variation of the velocity of fluid 𝑈 ′(𝜆) for different Reynolds no (R)
0 2 4 6 8 10 -10
-8
-6
-4
-2
0
U ( 𝜆)
M = 2, T = 2, K = 0 .1, Re = 1 M = 2, T = 2, K = 0 .1, Re = 3 M = 2, T = 2, K = 0 .1, Re = 6 M = 2, T = 2, K = 0 .1, Re = 8 M = 2, T = 2, K = 0 .1, Re = 12
0 1 2 3 4 5 6 7 8 9 10 -80
-70
-60
-50
-40
-30
-20
-10
0
U (𝜆)
M = 2, T = 2, K = 0 .1, Re = 1 M = 2, T = 2, K = 0 .1, Re = 3 M = 2, T = 2, K = 0 .1, Re = 6 M = 2, T = 2, K = 0 .1, Re = 8 M = 2, T = 2, K = 0 .1, Re = 12
8034 Manju Agarwal, Vivek Joseph, Ramesh Yadav & Parul Saxena
Figure 8. Variation of the velocity of fluid 𝑈 (𝜆) for different Parameter (K)
Figure 9: Variation of the velocity of fluid 𝑈′ (𝜆) for different Parameter (K)
0 2 4 6 8 10 -6
-5
-4
-3
-2
-1
0
U ‘(𝜆)
M = 2, Re = 2, T = 2, K = 0 .2 M = 2, Re = 2, T = 2, K = 0 .4 M = 2, Re = 2, T = 2, K = 0 .6 M = 2, Re = 2, T = 2, K = 0 .8 M = 2, Re = 2, T = 2, K = 1 .0
0 1 2 3 4 5 6 7 8 9 10 -45
-40
-35
-30
-25
-20
-15
-10
-5
0
U U
M = 2, Re = 2, T = 2, K = 0 .2 M = 2, Re = 2, T = 2, K = 0 .4 M = 2, Re = 2, T = 2, K = 0 .6 M = 2, Re = 2, T = 2, K = 0 .8 M = 2, Re = 2, T = 2, K = 1 .0
Numerical Investigation of a Channel Flow With Third Grade Fluid in.... 8035
Figure 10: Variation of the velocity of fluid 𝑈 (𝜆) for different Parameter (K)
Figure 11: Variation of the velocity of fluid 𝑈 (𝜆) for different Parameter (K)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
U (𝜆)
M = 0, T = 5, Re = 1, K = 0 .2
M = 0, T = 5, Re = 1, K = 0 .5
M = 0, T = 5, Re = 1, K = 0 .9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
U(𝜆)
M = 2, Re = 1, T = 5, K = 0 .2
M = 2, Re = 1, T = 5, K = 0 .5
M = 2, Re = 1, T = 5, K = 0 .9
8036 Manju Agarwal, Vivek Joseph, Ramesh Yadav & Parul Saxena
Figure 12: Variation of the velocity of fluid 𝑈′′ (𝜆) for different third
grade parameter (T)
Figure 13: Variation of the velocity of fluid 𝑈′′ (𝜆) for different third
grade parameter (T)
From fig. 12, 𝑈′′ plotted against 𝜆 for M = 0, Re = 5, K = 0.5, for these parameter
velocity of fluid decreases with increase of dimensionless variable(−3 ≤ 𝜆 ≤ 0); it
also show that increase of parameter T (1, 2, 3,) velocity of fluid increases sharply.
-3 -2.5 -2 -1.5 -1 -0.5 0 0
50
100
150
200
250
U ''(𝜆)
M = 2, Re = 5, K = 0 .5, T = 1
M = 2, Re = 5, K = 0 .5, T = 2
M = 2, Re = 5, K = 0 .5, T = 3
-3 -2.5 -2 -1.5 -1 -0.5 0 0
200
400
600
800
1000
U ''(𝜆)
M = 0, Re = 5. K = 0 .5, T = 1
M = 0, Re = 5. K = 0 .5, T = 2
M = 0, Re = 5. K = 0 .5, T = 3
Numerical Investigation of a Channel Flow With Third Grade Fluid in.... 8037
From fig. 13, 𝑈′′ plotted against 𝜆 for M = 2, Re = 5, K = 0.5, for these parameter
velocity of fluid decreases with increase of dimensionless variables 𝜆 (−3 ≤ 𝜆 ≤ 0) ;
in this graph show that increase of parameter T (1, 2, 3) velocity of fluid increases
sharply.
5. CONCLUSIONS
The main objective it to investigate the combined effect of magnetic field (M),
Reynolds number (Re), Third grade parameter (T), and parameter K on the velocity of
fluid and for the purpose the numerical technique is used. The enhancement of
Magnetic field (M), sharply decreases the velocity of fluid and reciprocally effect on
the velocity with the different third grade parameter (T) and parameter (K). This study
have practical applications in nuclear engineering control, plasma aerodynamics,
mechanical engineering manufacturing processes, astrophysical fluid dynamics, and
Magneto hydrodynamic (MHD) energy system.
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