MHD Flow of A Non-Newtonian Fluid With Temperature Dependent Viscosity Past A Vertical Plate In The Presence Of Radiative Heat Flux And Chemical Reaction

MHD Flow of A Non-Newtonian Fluid With Temperature Dependent Viscosity Past A Vertical Plate In The Presence Of Radiative Heat Flux And Chemical Reaction

MHD Flow of A Non-Newtonian Fluid With Temperature Dependent Viscosity Past A Vertical Plate In The Presence Of Radiative Heat Flux And Chemical Reaction.

Department of Mathematical Sciences Federal University of Technology, Akure

Abstract

We investigated in this research a hydromagnetic heat and mass transfer in MHD flow of an incompressible, electrically conducting, Non- Newionian viscous fluid past a continuously moving vertical infinite plate under suction velocity and heat flux. The non-linear coupled equations were solved analytically using an asymptotic series expansion . Numerical solution was also obtained using finite difference implicit scheme method to snable comparison with the approximate analytical solution The results were displayed graphically and compared with the previous works in the literature.Our approximate analytical solution and Numerical solution are in good agreement.

Key words: MHD, temperature dependent viscosity, radiative heat flux, chemical reaction.

1. Literature review

   Many research works have been done and a lot of studies are still going on, on the hydromagnetic heat and mass transfer in MHD flow of a viscous fluid. A number of researchers have considered works in relation to geophysics, astrophysics, engineering, soil sciences, and industrial processes.

   Vast majority of these researchers such as Raptis and Massalas[1], Chamkha [2], Okedoye et al [3,4],Ibrahim and Makinde studied MHD flow of a viscous fluid on the supposition that viscosity was constant. However, this fluid property varies with temperature, time or the space variable. The results have significant applications in space technology, solar power technology, space vehicle re-entry, nuclear engineering and so on. Based on this, Hossain and Munir [5] in their research, analysed a two dimensional mixed convection flow of a viscous incompressible fluid of temperature dependent viscosity past a vertical plate. Fang [6] studied the influence of fluid property variation on the boundary layer of a stretching surface. Hossain et al [7] discussed the effect of radiation of free convection flow of a fluid with variable viscosity from a porous vertical plate. Okedoye et al [8] investigated the effects of variable viscosity on MHD boundary layer flow on a continuously moving vertical plate in the presence of radiation and a chemical reaction of order one. In all these studies the solutions were obtained numerically.

   Most of the aforementioned references dealt with constant viscosity. In this paper, we propose the effect of variable viscosity (temperature dependent) and suction velocity on hydromagnetic heat and mass transfer in MHD flow.

2. Nomenclature

u : Velocity along x coordinate T ' : Non dimensional fluid temperature

v : Velocity along y coordinate C ' : Non dimensional species concentration

g : Acceleration due to gravity T : Fluid temperature

U ' : Non dimensional fluid velocity : Reaction parameter

Tw : Ambient temperature : Stefan- Boltzmann constant

*

C : Species concentration B : Coefficient of mass expansion

Cw : Ambient species concentration

B0 : Transverse magnetic field

: Skin-friction coefficient

q : Heat flux

B : Coefficient of thermal expansion

w : Ambient density

: Electrical conductivity

: Density of the fluid

k : Thermal conductivity

cp : Specific heat at constant pressure

Sc : Schmidt number

R : Radiation parameter

qr : Radiative heat flux Grc : Mass grashof number

v0 : Normal velocity at the plate Gr : Thermal grashof number

k * : Mean absorption coefficient

: Delta, 0 1

Pr : Prandtl number

M : Hartmann number

: Angular velocity

: Fluid viscosity

A : Pre-exponential factor D :Molar diffusivity

2.0 Mathematical Formulation

A magnetohydrodynamic flow of viscous, incompressible, electrically conducting fluid past an infinite plate in a porous medium under suction velocity is considered. The x- axis is taken along the plate in the direction of the flow and y- axis normal to it. A uniform magnetic field is applied normal to the direction of the flow. It is assumed that the magnetic Reynolds number is less than unity so that the induced magnetic field is neglected in comparison to the applied magnetic field. We further assumed that all the fluid properties are constant except that of the influence of density variation with temperature. Thus, the basic flow in the medium is entirely due to buoyancy force caused by temperature difference between the wall and the medium. Initially at t 0 , the plate as well as fluid is assumed to be at the same temperature and the concentration of species is very low so that the Soret and Dofour effect are neglected [9]. When t 0 , the

T

w

temperature of the plate is instantaneously raised (or lowered) to ' and the concentration of

w

species is raised (or lowered) to C ' .

Under the above assumptions and taking the usual Boussinesqs approximation into account, the governing equations for continuity, momentum, concentration and energy are presented below:

dv'

0

dy '

(2.1)

dU '

1 d dU '

B2U '

v'

dy'

dy '

dy '

g * T ' T

g C ' C 0

(2.2)

'

dC '

v

dy '

d 2C '

D

dy '2

AC ' C

(2.3)

' dT ' k d 2T ' 1 dq

v t r

(2,4)

dy '

c p

dy '2

c p

dy '

The boundary conditions:

0

U ' v

T '

y '

q

k

C ' C

at y ' 0

(2.5)

w

U ' 0 ,

T ' T

C ' C

as y'

From equation (2.1), we take

v' y' v

0

The minus sign indicates that the suction is towards the plate.

By using Rosseland approximation qr

takes the form [10]

4 dT 4

qr 3k dy

(2.6)

The temperature difference within the fluid assumed sufficiently small such that T 4 may be expressed as a linear function of the temperature. Expanding T 4 in a Taylor series about T and neglecting higher order terms, we have

T 4 4T

3T 3T 4

(2.7)

Substituting equation (2.7) into (2.6), we obtain

16 T 3 d 2T q r 3k dy 2

(2.8)

Using the following non dimensional quantities:

v y ' U '

T ' T

C ' C

g * q

kv

y 0 U

C

G 0

v q

C C

r v 3

0 w 0

kv0

g C

• C

B

c p

16T 3

G w

M 0 Pr

R

Sc

A

v

0

rc 3

0

v0 kt

3k *k D

v 2

on equations (2.2)-(2.5), the dimensionless governing equations for momentum, energy and concentration and their boundary conditions are:

d 2C dy2

• Sc dC

  dy

  • Sc 0

    (2.9)

    d 2 Pr d

    dy2

    0

    1 R dy

    (2.10)

    d

    dU dU M 2U G

    G C

    (2.11)

    dy

    dy dy

    r rc

    With the boundary conditions

    U y1

    Cy 1,

    d

    dy y0

    1

    on y 0

    (2.12)

    U y 0

    Cy 0

    y 0

    as y

    The fluid viscosity was assumed to obey the Reynold model [11].

    e

    (2.13)

    Using equation (2.13) in equation (2.11), we obtain

    d dU dU 2

    dy e

    dy dy M

    U Gr GrcC

    (2.14)

    U y 1, on

    y 0

    U y 0

    as y

    3.0 Method of Solution

    Solving equations (2.9) and (2.10), we have

    Cy emy

    and

    y 1 eny

    n

    (3.1)

    To solve equation (2.14), we employ asymptotic technique as follows: Let O where 1

    U U0

    U1

    2U

    … h..o.t

    (3.2)

    2

    Putting equation (3.2) into equation (2.14) and collecting the order of , we have Order 0

    d 2U dU 2

    0 0 M

    U0 Gr GrcC

    (3.3)

    dy2 dy

    U0 y 1

    on y 0

    U0 y 0 as

    y

    Order

    d dU d 2U dU 2

    0 1 1 M

    U1 0

    (3.4)

    dy dy dy2 dy

    U1 (0) 0 U1 () 0

    Order 2

    1. d 2 dU

       d dU d 2U dU 2

       0

       1 2 2 M

       U2 0

    2. dy dy

    dy dy dy2 dy2

    (3.5)

    U2 0 0

    U2 () 0

    Solving equations (3.3) (3.5), we have

    U y a ery a eny a emy

    (3.6)

    0 5 7 8

    1

    9

    11

    U y a ey a

    enr y a

    e2ny a

    emny

    (3.7)

    12

    13

    U y a

    ey a

    e2nr y a

    e3ny a

    e2nmy a

    e ny

    (3.8)

    2 14 16 17 18 19

    Recall

    0 1 2

    U y U U 2U

    (3.2)

    Based on the solution obtained in equations (3.6)-(3.8), we finally calculate equation (3.2) to be

    5

    7

    8

    9

    U y a ery a eny a emy a ey a

    en r y a

    e2ny a

    em n y

    11 12 13

    2 a

    ey a

    e2n r y a

    e3ny a

    e2n my a

    e n y

    (3.9)

    14 16 17 18 19

    n

    Pr

    1 R

    m 1 Sc

    2

    Sc2 4Sc

    r 1 1

    2

    1 4M 2 1 1

    1 4M 2

    2

    1 1

    1 4M 2 a

    1

    Gr

    a

    Grc

    a 1 a a

    a a5r r a

    7

    n

    m2 m M 2

    2a7n a

    n2 n M 2

    8

    5

    7

    8

    a8m m

    2

    11 n r2 n r M 2 1

    n 12

    4n2 2n M 2 13

    m n2 m n M 2 1

    n

    a r2 2na r 2na n r2 2n2a

    n r 3a 12na

    a a a a a

    5 5 11

    11 a17 7 12

    9 11 12

    13 16

    2n2 2n r2 2n r

  • M 2

  2 9n2 3n M 2

  a m2 2na m 2na

  m n2 2n2a

  m n

  2a

  • na

  a 8

  8 13

  13 a19

  9 9

  18 2n2 2n m2 2n m M 2

  a14 a16 a17 a18 a19

  n n2 n M 2

  The physical quantity of most interest in such problem is the skin-friction coefficient which is

  U

  defined by y

  y0

  From equation (3.9) we calculate as follows:

  ra5 na7 ma8 a9 n ra11 2na12 m na13

  14

  2 a 2n ra 3na 2n ma na

  16 17 18 19

  (3.10)

  4.0 Discussion of Results and Conclusion

  For the purpose of discussing the effect of various parameters on the flow profiles and the temperature distribution within the boundary layer, analysis had been carried out for various

  values of

  M ,Grt ,Grc ,, and R with fixed values of Pr, and

  Sc.The values of Pr, and Sc were

  taken to be 0.71 and 0.6 respectively for plasma. These parameters were assigned the values

  M 1,Gr

  5.0,Grc 1.0,

  0.1, R 0.5, 0.1except where stated otherwise. It should be

  noted that increase in viscosity parameter leads to decrease in viscosity as given by the

  relation in equation (2.13). Also,

  0, 0

  and

  0 indicate generative, no reaction and

  destructive chemical reaction respectively. The variation of the skin-friction coefficient for

  various values of , M , R and

  Grt

  with

  Pr 0.71is shown in table 4.1. It can be seen from this

  table that the skin-friction coefficient increases as the viscosity parameter, mass Grashof number or the thermal Grashof number increases. Increasing of the magnetic parameter, the radiation parameter or reaction parameter leads to a decrease in the skin-friction.

  The effect of on the dimensionless velocity U is illustrated in table 4.2. From the table, we can see that increase in viscosity parameter increased the fluid velocity near the plate only while reverse is the case as we move away from the plate.

  In figure 4.1, we show the distribution of velocity for various values of delta. It could be seen that increase in delta increases velocity. In figure 4.2, we show that generative chemical reaction leads to increase in fluid velocity while increase in destructive chemical reaction lowers the velocity.

  Also, in figure 4.3, we show the distribution of velocity for various values of radiation parameter. It could be seen that increase in radiation parameter increases the velocity. We displayed in figure 4.4 and 4.5 the effect of thermal and mass Grashof number respectively. It is observed that an increase in the values of both parameters leads to increase in the velocity and vice versa. The velocity U of the fluid decreases as the magnetic parameter M increases as shown in figure 4.6. It is observed that the velocity increased to its maximum value near the plate and then decreased to zero. Figure 4.7 shows the temperature profile for various values of

  radiation parameter. It could be seen that increase in radiation parameter R reduces temperature of the fluid. Also it is noticed that a decrease in the fluid temperature with maximum value at the plate and minimum at a distance away from the plate. The effect of reaction parameter on the concentration of chemical species is shown in figure 4.8. We noticed that increase in reaction parameter reduces the concentration of the chemical species.

  Table 4.1: The values of skin-friction when 0

  0.0752 0.0752 0.0751

  10 0.0754 0.0753

  0.1211 0.1209 0.1208

  9 0.1214 0.1213

  0.195 0.1947 0.1943

  8 0.1959 0.1954

  0.3143 0.313 0.3124

  7 0.3164 0.3153

  0.5068 0.5044 0.5021

  6 0.5119 0.5093

  4 1.3358 1.3244 1.313 1.3016 1.2902

  5 0.8287 0.8228 0.817 0.8115 0.8061

  2.084 2.0676 2.0491

  3 2.1099 2.0981

  3.1637 3.1743 3.1737

  2 3.1087 3.1418

  4.0395 4.2136 4.3742

  1 3.6507 3.8519

  1

  1

  1

  1

  0 1

  y U 0.2 U 0.1 U 0.0 U 0.1 U 0.2

  M R Grt Grc

  Table 4.2: Velocity U distribution for various values of

  -0.2	1.0	0.5	5	1	0.1	6.229
  -0.1	1.0	0.5	5	1	0.1	7.3786
  0.0	1.0	0.5	5	1	0.1	8.8271
  0.1	1.0	0.5	5	1	0.1	10.5683
  0.2	1.0	0.5	5	1	0.1	12.6023
  0.1	0.0	0.5	5	1	0.1	28.8078
  0.1	0.5	0.5	5	1	0.1	18.8078
  0.1	1.0	0.1	5	1	0.1	6.0487
  0.1	1.0	0.5	5	0	0.1	9.6502
  0.1	1.0	0.5	5	2	0.1	11.4864
  0.1	1.0	0.0	5	4	0.1	7.6376
  0.1	1.0	0.1	5	1	-0.12	6.2558
  0.1	1.0	0.5	5	1	0.0	10.6327
  0.1	1.0	0.5	5	1	0.1	10.5683
  0.1	0.5	0.5	5	1	0.2	18.7082
  0.1	1.0	0.5	8	1	0.1	17.5009
  0.1	1.0	0.5	10	1	0.1	22.1227
  0.1	1.0	0.0	10	1	0.1	11.0303

  4.5

  4

  3.5

  3

  delta =-0.2

  delta =-0.1

  delta =0.0

  delta =0.1

  delta =0.2

  4.5

  4

  3.5

  3

  omega =-0.12

  omega =0.0

  omega =0.1

  omega =0.2

  velocity

  velocity

  2.5 2.5

  2 2

  1.5 1.5

  1 1

  0.5 0.5

  0

  0 2 4 6 8 10 12 14 16 18 20

  y

  0

  0 2 4 6 8 10 12 14 16 18 20

  y

  Fig 3.1: velocity U y distribution for various values of Fig 3.2: velocity U y distribution for various values of

  4.5

  4

  3.5

  3

  velocity

  2.5

  2

  1.5

  1

  0.5

  0

  R=0.0 9

  R=0.2

  R=0.5 8

  7

  6

  velocity

  5

  4

  3

  2

  1

  0

  0 2 4 6 8 10 12 14 16 18 20

  Grt =5.0

  Grt =8.0

  Grt =10.0

  y 0 2 4 6 8 10 12 14 16 18 20

  y

  Fig 3.3: velocity U y distribution for various values of R Fig 3.4: velocity U y distribution for various values of Gr

  14

  6

  Grc=0.0 12

  Grc=1.0

  5 Grc=2.0

  Grc=4.0 10

  M =0.0

  M =0.5

  M =1.0

  4

  velocity

  8

  velocity

  3 6

  2 4

  1 2

  0

  0 2 4 6 8 10 12 14 16 18 20

  y

  0

  0 2 4 6 8 10 12 14 16 18 20

  y

  Fig 3.5: velocity U y distribution for various values of Grc Fig 3.6: velocity U y distribution for various values of M

  1

  0.9

  0.8

  6 0.7

  concentration

  4 0.6

  omega =0.0

  omega =0.2

  omega =0.4

  omega =0.6

  omega =0.8

  omega =1.0

  2

  0

  0

  0.5

  1

  1.5

  R

  0

  2

  4

  6

  y

  2 10 8

  0.5

  0.4

  0.3

  0.2

  0.1

  0

  0 2 4 6 8 10 12 14 16 18 20

  y

  Fig 3.7: temperature y distribution for various values of R Fig 3.8: Conc. Cydistribution for various values of

  5.0 Numerical Approach

  Applying Crank-Nicolson formula to equations (2.9), (2.10) and (2.14)

  P3Ci1 Q3Ci, R3Ci1 0

  P2i1 Q2i, R2i1 0

  (5.1)

  (5.2)

  P1Ui1 Q1Ui, R1Ui1 D1i

  (5.3)

  With corresponding boundary conditions (2.12) becoming

  Ui 0 1

  d

  dy

  1

  i y0

  Ci 0 1

  Ui 0

  i 0

  Ci 0

  (5.4)

  Where

  P

  ySc

  Q 2 y2 S

  R

  ySc

  1. yPr

  2. 2

     3 1 2 3

     c 3 1

     2

     2 1

     21 R 2

  yPr

  P

  a6 y

  Q 2 a y2 M 2

  R

  a6 y

  2 1

  1

  5

  21 R

  1 1 2 1 1

  1 1 2

  D1i

  y2 a a

  a e

  a d

  2 dy

  a3 Gr

  a4 GrcC

  a5 a3

• a4

a6 a1

a2

1

Thus, y is a constant mesh size along y direction respectively. We need a scheme to find single values at next level time in terms of known values at an earlier time level. A central difference

approximation for the first order derivatives of C,

and U with respect to y and a central

difference approximation for the second order derivative of C,

and U with respect to y are

used.

5.1 Tables and Graphical Presentations: Numerical Solution

To ensure the validity of our analytical solutions, we have compared our numerical solutions with the exact solutions obtained for concentration, temperature and velocity for some varying fluid parameters.

Table 5.1 to 5. 3 show comparison between analytical and numerical values for concentration,

temperature and velocity respectively M 1.0,Gr 5.0,Grc 1.0, 0.1, R 0.5, 0.1 . It

was clearly seen from these tables that the error decreases as the value of y approaches 10. Hence the results are in good agreement. Also, figure 5.1 to 5.3 for concentration, temperature and velocity below show that the curves corresponding to exact and numerical solutions are lying very close to each other which further confirm the accuracy of our model.

Table 5.1: Comparison between exact and Table 5.2: Comparison between exact and numerical values for concentration numerical values for temperature

0.0475

y	Exact	Numerical	Error		y	Exact	Numerical	Error
0	1	1	0		0	2.1127	2.1127	0
2	0.2529	0.2524	0.0005		2	0.8198	0.8188	0.001
4	0.064	0.0637	0.0003		4	0.3181	0.3173	0.0008
6	0.0162	0.0161	0.0001		6	0.1234	0.1229	0.0005
8	0.0041	0.0041	0		8	0.0479	0.0004
10	0.001	0.001	0		10	0.0186	0.0183	0.0003

Table 5.3: Comparison between exact and numerical values for velocity

y	Exact	Numerical	Error
0	1	1	0
2	3.1743	3.5772	0.4029
4	1.3016	1.4618	0.1602
6	0.5044	0.5654	0.0610
8	0.1947	0.2174	0.0227
10	0.0752	0.0834	0.0082

1

numerical

0.9 analytical

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 2 4 6 8 10 12 14 16 18 20

Figure 5.1: Concentration profiles

2.5

5

numerical numerical

analytical

2

4.5 analytical

4

1.5

3.5

3

2.5

1 2

1.5

0.5 1

0.5

0

0 2 4 6 8 10 12 14 16 18 20

0

0 2 4 6 8 10 12 14 16 18 20

Figure 5.2: Temperature profiles Figure 5.3: Velocity profiles

Concluding remarks

In this work, the problem of boundary layer flow of a steady viscous, incomprehensible electrically conducting fluid with variable viscosity over a continuously moving vertical porous plate in the presence of magnetic field and radiation had been investigated. An asymptotic method was employed to solve the resulting coupled differential equations. The obtained results were compared with Crank-Nicolson finite difference method of implicit scheme and found to be in good agreement. Also, graphical illustrations presented further revealed the relationship between varying parameter affecting fluid behaviour. Above all, the result is an improvement over the previous work by other authors in that it caters for temperature dependent viscosity rather than constant viscosity which is simple to comprehend.

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