Study the Micropolar Fluid Flow near the Stagnation on a ... · (1984) considered the laminar mixed...

Study the Micropolar Fluid Flow near the Stagnation on a Vertical Plate

with Prescribed Wall Heat Flux

Gitima Patowary1, Dusmanta Kumar Sut

2

1Assistant Professor, Dept. of Mathematics, Barama College, Barama, Assam, email:

[email protected] 2Assistant Professor, Dept. of Mathematics, Jorhat Institute of

Science and Technology, Jorhat, Assam, email: [email protected]

Abstract

This paper investigates the

influence of the material parameter,

buoyancy parameter and Prandtl number

on the skin friction coefficient and the heat

transfer rate at the surface on a steady,

two dimensional flow of an incompressible

micropolar fluid near the stagnation point

on a vertical plate with prescribed surface

heat flux in presence of a magnetic filed.

The free stream velocity and the surface

heat flux are assumed to be proportional to

the distance from the stagnation point.

Similarly transformation is employed to

transform the governing partial differential

equations to a set of ordinary differential

equations. The effects of the material

parameter, buoyancy parameter and

Prandtl number on the skin friction

coefficient and the heat transfer rate at the

surface are discussed and the

corresponding velocity, temperature and

microrotation profiles are shown

graphically. Both assisting and opposing

flows are considered and it is found that

dual solutions exist for both cases.

Key words: micropolar fluid, vertical

plate, mixed convection, stagnation point

flow, magnetic parameter, boundary layer,

fluid mechanics, shooting method.

1. Introduction The theory of micropolar fluid, first

introduced and formulated by Eringen

(1966, 1972) and derived the constitutive

laws of fluid with micro-structure has been

a field of very active research for the last

decades. This theory is capable to explain

the complex fluids behaviour such as

liquid crystals, polymeric suspensions,

animal blood etc. by taking into account

the effect arising from local structure and

micro-motions of the fluid elements. An

extensive review of micropolar fluids and

their applications has been done by Ariman

et al. (1973).

Many researchers have been

studied the free or mixed convection flow

of a micropolar fluid towards a vertical

surface. Bhargava et al. (2003) have

analyzed the effect of temperature

dependent heat sources on the fully

developed free convection electrically

conducting micropolar fluid between two

parallel porous vertical plates in a strong

magnetic field by solving the governing

differential equations using the quasi

linearization method. Jena and Mathur

(1984) considered the laminar mixed

convection boundary layer flow of a

micropolar fluid from an isothermal

vertical flat plate. Gorla (1995) studied the

unsteady laminar mixed convection

boundary layer flow of a micropolar fluid

over a vertical flat plate by assuming the

free stream velocity undergoes arbitrary

Dusmanta Kumar Sut et al, Int. J. Comp. Tech. Appl., Vol 2 (6), 1768-1778

IJCTA | NOV-DEC 2011 Available [email protected]

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variation with time. Lok et al. (2006)

studied the unsteady mixed convection

boundary layer flow of a micropolar fluid

near the stagnation point flow towards a

horizontal permeable plate immersed in a

micropolar fluid. They obtained the

solutions using homotopy analysis method

and found that their solutions are in

favourable agreement with the numerical

solutions previously reported by Attia

(2006). Azizah et al. (2010) studied

considering a steady mixed convection

flow of micropolar fluid near the

stagnation point on a vertical plate with

prescribed surface heat flux. The similar

problem in a viscous fluid was studied by

Ramachandran et al. (1988) by considering

both cases of an arbitrary surface

temperature and arbitrary surface heat flux.

They found that dual solutions exist for

certain range of the buoyancy parameter,

for opposing flow case only. Devi et al.

(1991) extended this problem to the

unsteady case, while Lok et al (2005)

investigated the case when the plate

immersed in a micropolar fluid. They also

fund that dual solution exist for the

opposing flow only. Azizah et al. (2010)

have showed that dual solutions also exist

for the prescribed surface heat flux case,

for both assisting and opposing flows.

Motivated by the above

investigations, the present paper considers

a steady mixed convection flow of a


on a vertical plate with prescribed heat flux

in presence of steady magnetic field. The

objective of the present study is to show

that dual solutions also exist for the

prescribed surface heat flux case, for both

assisting and opposing flows in presence of

steady magnetic field.

2. Mathematical formulation

A steady, two dimensional flow of

an incompressible electrically conducting


on a vertical flat plat place in the plane

0y of a Cartesian system of coordinates

Oxy in presence of a magnetic field 0B

applied in the normal direction to the walls

is considered. The fluid occupies the half

plane 0y . It is assumed that the free

stream velocity )(xU and the surface heat

flux )(xqw are proportional to the distance

x from the stagnation point, i.e.,

axxU )( and bxxqw )( , where a and

b are the constants. Under these

assumptions, the simplified governing

equation equations are given by

Equation of continuity is

0

y

v

x

u (1)

Equation of momentum is

uBTTgy

N

y

u

dx

dUU

y

uv

x

uu

2

02

2

)(

(2)

Equation of angular momentum is

y

uN

y

N

y

Nv

x

Nuj 2

2

2

(3)

Equation of energy is

2

2

y

T

y

Tv

x

Tu

(4)



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The appropriate physical the boundary conditions of equations are

wq

y

T

y

unNvu

,,0,0 at 0y

TTNxUu ,0),( as y (5)

where u and v are the components of

velocity in x and y direction respectively,

is the dynamic viscosity, is the

vortex viscosity, is the fluid density,

is the spin gradient viscosity, is the thermal diffusivity, is the thermal

expansion coefficient, g acceleration due

to gravity, T is the fluid temperature, j is

the micro inertia density, N is the micro

rotation vector, 0B is the external magnetic

field and wq is the surface heat flux. It

should be noted that n is a constant such

that 10 n (Azizah et al., 2010). The

case n=0 is called strong concentration by

Guram and Smith (1980), which indicates

N=0 near the wall, represents concentrated

particle flows in which the microelements

close to the wall surface are unable to

rotate (Jena and Mathur, 1981). The case

n=1/2 indicates the vanishing of anti

symmetric part of the stress tensor and

denotes weak concentrations (Ahmadi,

1976). The case n=1, as suggested by

Peddieson (1972), is used for the modeling

of turbulent boundary layer flows. The

case n=1/2 for weak concentration

(Azuzah et al., 2010). Here we follow by

assuming jKj )2/1()2/(

(Azuzah et al., 2010). This assumption is

invoked to allow the field of equations

predicts the correct behaviour in the

limiting case when the microstructure

effects become negligible and the total spin

N reduces to the angular velocity (Ahmadi,

1976, Yucel, 1989).

The governing equations subject to

the boundary conditions can be expressed

in a simpler form by introducing the

following transformations:

x

vy

u

, (6)

2/12/13

2

12

1

)()(),(),()(,

v

a

q

TTkxg

v

aNxfavy

v

a

w

(7)

Where is the thermal conductivity, is the stream function.

The transformed ordinary differential equations are:

01)1( 2 hKfRffffK (8)

0)2(2

1

fgKgfgfg

K (9)

01

ffPr

(10)

subject to the boundary conditions (5) which become

1)0(),0()0(,0)0(,0)0( fngff

0)(,0)(,1)( gf as (11)

where we have taken avj / as a reference length (Nazar et al 2004). Here prime denote

differentiation with respect to , /K is the material parameter, /vPr is Prandtl

number and 2/5

Re/ xxGr , a

BR

2

00 is the buoyancy parameter. Further



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)/( 24 kvxqgGr wx is the local Grashof number and vUxx /Re is the local Reynolds

number. Here is a constant with 0 and 0 correspond to the opposing and assisting flows respectively, while 0 is for pure forced convection flow. When 0K (viscous

fluid), the present problem reduces to those of Ramachandran et al (1988) and when 00 B ,

the present problem reduces to those of Azizah et al. (2010).

The physical quantities of interest are the skin friction coefficient fC and the local

Nusselt number xNu , which are defined as

)(,

22

TTk

xqNu

UC

w

w

x

w

f

(12)

Where the wall shear stress w and the wall heat flux wg are given by

00

,

y

w

y

wy

TkqkN

y

uk (13)

Using (6), we obtain

)0(/1Re/),0(])1(1[Re2

1 2/12/1 xxx NufKnC f (14)

3. Results and Discussion

In this paper, we have attempted to

develop a mathematical analysis for

investigating a steady, two dimensional

flow of an incompressible micropolar fluid

near the stagnation point on a vertical plate

with prescribed surface heat flux in

presence of magnetic field. The systems of

non linear equations (8-10) subject to the

boundary conditions (11) have been solved

numerically by applying Runge-Kutta

Shooting techniques for some values of the

governing parameters, namely the material

parameter K, buoyancy parameter , Prandtl number Pr and Magnetic

parameter R, while we consider n=1/2

throughout the paper, which represents

weak concentration of fluids particles at

the surface. The dual solutions were

obtained by setting two different values of

boundary layer thickness which

satisfies the boundary condition (11) and

produce two different velocity,

temperature and microrotation profiles as

shown in Figs 1-3. The variations of the

skin friction coefficient )0(f and the

surface temperature )0( with the mixed

convection or buoyancy parameter for different values of Pr and K=1 are

presented in Figs. 4 and 5, respectively.

These figures show that dual solutions

exist for both assisting )0( and

opposing )0( flows, For the assisting

flows, the solution exists for all values of

, while for the opposing flow it exists up

to a critical value, say c . Between the two

solutions as presented in Figs. 4 and 5, we

expect that the upper branch solution is

stable and most physically relevance, while

the lower branch solution is not, since it is

the only solution for the forced convection

case )0( .

4. Conclusion

The problem of mixed convection flow towards a vertical plate with a

prescribed surface heat flux immersed in

an incompressible micropolar fluid in

presence of a magnetic field has been

studied numerically. The governing partial

differential equations were transformed

into ordinary differential equations using

similarity transformation, before being

solved numerically by shooting method.

We discussed the effects of the material

parameter, buoyancy parameter, the

Prandtl number, Magnetic parameter on



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the fluid flow and heat transfer

characteristics. We found that dual

solutions exist for both assisting and

opposing flows. The solutions for the

assisting flow )0( could be obtained

for all values of the buoyancy parameter ,

while for the opposing flow )0( , the

solutions were obtained only in the range

of )0( c where c is the minimum

value of for which the solution exists. It is also found that micropolar fluids as well

as fluids with larger Prandtl number

increase the range of for which the solution exists.

REFERENCES

1. Ahmed, G., Self-Similar Solution of Incompressible Micropolar Boundary

Layer Flow over a Semi Infinite Plate,

Int. J. Eng. Sci., Vol. 14, pp. 639-646

(1976)

2. Ariman, T., M. A. Turk and N. D. Sylvester, Microcontinum Fluid

Mechanics- A Review, Int. J. Eng. Sci.,

Vol. 11, pp. 905-930 (1973)

3. Attia, H. A., Investigation of Non-Newtonian Micropolar Fluid Flow with

Uniform Suction/Blowing and Heat

Generation, Turkish J. Eng. Environ.

Sci., Vol. 30, pp. 359-365 (2006)

4. Bhargave, R., L. Kumar and H. S. Takhar, 2003, Numerical Solution of

Free Convection MHD Micropolar

Fluid Flow between Two Parallel

Porous Vertical Plates, Int. J. Eng. Sci.,

Vol. 41, pp. 123-136 (2003)

5. Cebeci, T. and P. Bradshaw, Physical and Computational Aspects of

Convective Heat Transfer, Springer,

New York (1988)

6. Chen, C. H., Laminar Mixed Convection Adjacent to Vertical,

Continuously Stretching Sheets, Heat

Mass Transfer, Vol. 33, pp. 471-476

(1998)

7. Devi, C.D.S., H. S. Takhar and G. Nath, Unsteady Mixed Convection Flow in

Stagnation Region Adjacent to a

Vertical Surface, Heat Mass Transfer,

Vol. 26, pp. 71-79 (1991)

8. Erignen, A. C., Theory of Micropolar Fluids, J. Math. Mech., Vol. 16, pp. 1-

18 (1966)

9. Erignen, A.C., Theory of Thermo micropolar fluids, J. Math. Anal. Appli.,

Vol. 38, pp. 480-495 (1972)

10. Gorla, R. S. R., Unsteady Mixed Convection in Micropolar Boundary

Layer Flow on a Vertical Plat, Fluid

Dyn. Res., Vol. 15, pp. 237-250

(1995)

11. Guram, G.S. and A.C. Smith, Stagnation Flows of Micropolar

Fluids with Strong and Weak

Interactions, Comp. Math. Appl.,

Vol. 6, pp. 213-233 (1980)

12. Ishak, A., R. Nazar, N.M. Arifin and I. Pop, Dual Solution in Mixed

Convection Flow near a Stagnation

Point on a Vertical Porous Plate, Int.

J. Ther. Sci., Vol. 47, pp. 417-422

(2008)

13. Jena, S. K. and M. N. Mathur, Similarity Solutions of Laminar Free

Convection Flow of a

Thermomicropolar Fluid Past a

Non-Isothermal Vertical Flat Plat,

Int. J. Eng. Sci., Vol. 19, pp. 1431-

1439 (1981)

14. Jena, S. K. and M. N. Mathur, Mixed Convection Flow of a Micropolar

Fluid from an Isothermal Vertical

Plate, Comp. Math. Appl., Vol. 10,

pp. 291-304 (1984)

15. Lok, Y. Y., N. Amin, D. Campean and I. Pop, Steady Mixed Convection

Flow of a Micropolar Fluid near the

stagnation point on a Vertical

Surface, Int. J. Numer. Methods Heat

Fluid Flow, Vol. 15, pp. 654-670

(2005)

16. Lok, Y. Y., N. Amin and I. Pop, Unsteady Mixed Convection Flow of

a Micropolar Fluid Near the

Stagnation Point on a Vertical

Surface, Int. J. Ther. Sci., Vol. 45,

pp. 1149-1157 (2006)



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17. Nazar, R., N. Amin, D. Philip and I. Pop, Stagnation Point Flow of a

Micropolar Fluid towards a

Stretching Sheet, Int. J. Non-Linear

Mech., Vol. 39, 1227-1235 (2004)

18. Peddieson, J., An Application of the Micropolar Fluid Model to the

Calculation of Turbulen Shear Flow,

Int. J. Eng. Sci., Vol. 10, pp. 23-32

(1972)

19. Ramachandran, N., T. S. Chen and B. F. Armaly, Mixed Connection in

Stagnation Flows Adjacent to

Vertical Surfaces, J. Heat Transfer,

Vol. 110, pp. 373-377 (1988)

20. Ridha, A., Three-Dimensional Mixed Convection Laminar Boundary-

Layer near a Plane of Symmetry, Int.

J. Eng. Sci., Vol. 34, pp. 659-675

(1996)

21. Yacob, Nor Aziah, Ishak Anuar, I. Pop., Non-Newtonian Micropolar

Fluid Flow Towards a Vertical Plate

with Prescribed Wall Heat Flux,

Australian Journal of Basic and

Applied Sciences, Vol. 4(8), pp.

2267-2273 (2010)

22. Yucel, A., Mixed Convection in Micropolar Fluid Flow over a

Horizontal Plate with Surface Mass

Transfer, Int. J. Eng. Sci., Vol. 27,

pp. 1593-1602 (1989)

23. Ziabakhsh, Z., G. Domairry and H. Bararnia, Analytical Solution of Non-

Newtonian Micropolar Fluid Flow

with Suction/Blowing and Heat

Generation, J. Taiwan Inst. Chem.

Eng., Vol. 40, pp. 443-451 (2009)



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-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

f'

Fig. 1. Variation velocity profiels )(f against for various values of K taking Pr=0.7,

1 , R=0.1



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-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fig. 2. Variation temperature profiles )( against for various values of K taking

Pr=0.7, 1 , R=0.1



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0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

->

Fig. 3. Variation of microrotation profiles )(g against for various values of K taking

Pr=0.7, 1 , R=0.1



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0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fig. 4. Variation velocity profiles )(f against for various values of Pr taking K=1,

1 , R=0.1



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0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fig. 5. Variation temperature profiles )( against for various values of Pr taking

K=1, 1 , R=0.1



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Study the Micropolar Fluid Flow near the Stagnation on a ... · (1984) considered the laminar mixed...

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