numerical investigation of a channel flow with third grade fluid … · 2018-02-19 · explain the...

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).


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)


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


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 ''(


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


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


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


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


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


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


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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