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Vol. 1 Issue V, December 2013 ISSN: 2321-9653
INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T)
Mixed Convection in MHD Slip Flow of Alumina Water Nanofluid Over a Flat Plate Padam Singh#1, Manoj Kumar*2 #
Department of mathematics Statistics and Computer Science,
G.B. Pant University of Agriculture and Technology, Pantnagar, Uttarakhand, India - 263145
Abstract— Heat transfer in magnetohydrodynamic (MHD) slip flow of an incompressible, viscous, electrically conducting, mixed convective and steady alumina-water nanofluid over a flat plate has been analyzed. The governing equations are transformed into a set of simultaneous ordinary differential equations by using similarity transformation. The set of equations thus obtained has been solved using Adaptive Runge-Kutta method with shooting technique. The effects of magnetic parameter and heat source parameter on velocity and temperature distribution, shear stress and temperature gradient were depicted graphically and analyzed. Significant changes were observed in the heat transfer rate. Keywords— Magnetohydrodynamic, Heat source, Boundary layer slip, Volume fraction and Mixed convection. 2010 Mathematics Subject Classification: 74F10, 76W05, 76N20, 65M06, 76R99.
I. INTRODUCTION Wang et al. [4] were studied the mixed convective boundary layer flow of non- Newtonian fluids along vertical wavy plates. The authors found that Prandtl number and buoyancy parameters were seen to enhance the influence of plate surfaces on the local Nusselt number in Newtonian fluids or non-Newtonian fluids. Moreover, the irregular surfaces have higher total heat flux than that of corresponding flat plate for any fluid. Vadasz et al. [5] investigated the heat transfer enhancement in nanofluid suspensions. The results were shown excessive improvement in the thermal conductivity of the suspension. Molla and Yao [7] investigated mixed convective heat transfer of non-Newtonian fluid over a flat plate using a modified power law viscosity model. The results were obtained for a shear thinning fluid in terms of the velocity and temperature distribution, and for wall shear stress and heat transfer rates. Ahmad and Pop [8] studied the steady mixed convection boundary layer flow past a vertical flat plate embedded in a porous medium filled with nanofluids. The effects of various parameters on velocity
distribution were analyzed. Bachok et al. [9] were analyzed heat transfer characteristics of mixed convective flow over a permeable vertical flat plate embedded in an anisotropic fluid. They were found that dual solutions exist for both assisting and opposing flows. Mohammad et al. [10] studied the heat transfer of an alumina-water nanofluid flow inside a wide rectangular micro channel. Results show that the velocity and temperature difference between the phases is very small and negligible. The average Nusselt number increases as the Reynolds number and volume concentration increase and also with the decay in the nanoparticles size. Aladag et al. [11] made experimental investigation of the viscosity of nanofluids at low temperature. It has been found that carbon nano tube water based nanofluid behaves as Newtonian fluid at high shear rate whereas Alumina water based nanofluid is nonNewtonian. Hamed and Kasera [12] studied the variation iteration method solution for mixed convection over horizontal flat plate and analyzed various parameters numerically and graphically.
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Vol. 1 Issue V, December 2013 ISSN: 2321-9653
INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T)
The objective of present paper is to study the mixed convective heat transfer in MHD slip flow of alumina water nanofluid over a flat plate.
Here
,
are velocity slip factor and
thermal slip factor with initial values
,
respectively.
TABLE 1 NANOFLUID PROPERTIES
II. MATHEMATICAL DESCRIPTION The physical model of the problem is given here along with flow configuration and coordinate system. Present problem deals with analysis of two dimensional MHD boundary layer slip flow of alumina-water nanofluid. The magnetic field B is imposed in transverse direction to the flow. The plate length is considered infinite and the uniform velocity at infinity is u∞. The temperature on the surface of the plate is Tw, and far from the surface it is T∞. The continuity, momentum and energy equations representing flow are as following:
y
Dynamic Viscosity [1] Density [2] Specific heat [3] Thermal conductivity [4]
Kinematic Viscosity Thermal Diffusivity
T
v
u
TABLE 2 THE PHYSICAL PROPERTIES OF ALUMINA AND WATER AT ROOM TEMPERATURE ARE
x
u
B
Tw
Density (Kg/m3)
Thermal Conductivity (W/m.K)
Specific Heat (J/Kg.K)
Alumina
3970
36
769
Water
1000.52
0.597
4181.8
Figure 1: Flow configuration and Coordinate system (1) III. METHOD OF SOLUTION (2)
To solve the governing equations (2) and (3) with the boundary conditions (4) following similarity transformation has been introduced:
(3) ,
,
With the following boundary conditions: , , (4)
, and and
,
and ,
at y = 0 when
Here
, ,
,
,
.
are constants and
which satisfies equation (1) with
Page 23
where
is the stream function and
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Vol. 1 Issue V, December 2013 ISSN: 2321-9653
INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T) VALUES OF
After using above transformation, the equations (2) and (3) reduce to the nonlinear differential equations as follows:
AND FOR OBTAINED BY SHOOTING METHOD
Parameters (5) 0
(6) Magnetic Parameter (M) and the boundary conditions (4) reduce as follows: ,
,
,
] (7)
where
is the velocity slip parameter and
is the thermal slip parameter with initial values
and
.
To solve the set of non-linear differential equations (5) and (6) subject to the boundary conditions (7) Adaptive Runge-Kutta method with shooting technique has been applied. This method is based on the discretization of the problem domain and the calculation of unknown boundary conditions. The domain of the problem is discretized and the boundary conditions for are replaced by , and where ; is sufficiently large value of corresponding to step size at which the boundary conditions (7) for is satisfied. Onto account of the consistency and to fulfill stability criteria and step size have been taken. To solve the problem the nonlinear equations (5) and (6) are first converted into first order ordinary linear differential equations as follows: ; ; ;
Heat Source Parameter (S)
]
0.1
0.4161 0.5121
-0.2803 -0.3483
0.3
0.6581
-0.4363
0.5
0.7731
-0.4865
0.7
0.8693
-0.5287
1.0
0.9923
-0.5689
0
0.7658
-0.7868
0.1
0.7728
-0.4907
0.3
0.8054
-0.6509
0.7
0.6937
-1.7999
1.0
0.7189
-0.7799
1.5
0.7203
-0.7745
IV. RESULT AND DISCUSSION The computations have been made for velocity, temperature, temperature gradient, Shear stress profile and other physical parameters involved in the flow. The results were depicted graphically to analyze them and their physical explanation is also given corresponding to different values of magnetic parameter and heat source parameter. The physical and thermal properties of alumina water nanofluid corresponding to different volume fraction are also tabulated below in table 4.
;
There are three conditions on the boundary and two conditions at as given in equation (7). Shooting technique has been used to find required missing initial conditions. The value of unknowns and has been tabulated in table 3 TABLE 3
Figures 2-3 exhibit the velocity and shear stress profiles obtained by the numerical simulations for various values of magnetic parameter M. It is noticed that the velocity increases with an increase in the magnetic parameter while shear stress profile increases upto and then decreases asymptotically. The temperature and temperature gradient profile decrease with an increase in the magnetic parameter M as shown in figures 4 - 5. Figures 6-7 show the effect of heat source parameter on velocity and shear stress profile respectively. It has been observed that the velocity decreases
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INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T)
and shear stress increases with an increase in the heat source parameter. The effects of heat source parameter on temperature and temperature gradient have also been studied through figures 8 and 9. It is noticed that the temperature decreases upto and then increases. Similarly temperature gradient decreases upto and then increases. 1.0
Fig. 4 Effect of magnetic parameter on temperature 0.8 0.0
0.6
f '( )
M = 0, 0.1, 0.3, 0.5, 0.7, 1.0
-0.2
M = 0, 0.1, 0.3, 0.5, 0.7, 1.0
0.4 -0.4
'( )
0.1, 0.1, S 0.1, Gr 0.1, Pr 6.65564065
0.2
0.0 0
2
4
6
-0.6
8
0.1, 0.1, S 0.1, Gr 0.1, Pr 6.65564065
10 -0.8
Fig.2 Effect of magnetic parameter on velocity
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Fig.5 Effect of magnetic parameter on temperature gradient
1.0
0.1, 0.1, S 0.1, Gr 0.1, Pr 6.65564065
0.8
1.0
M = 0, 0.1, 0.3, 0.5, 0.7, 1.0 0.8
0.6
f ''( ) 0.6
0.4
f '( )
S = 0, 0.1, 0.3, 0.7, 1.0, 1.5
0.4
0.2
0.1, 0.1, Gr 0.1,
0.2
Pr=6.65564065, M=0.5
0.0 0
2
4
6 0.0 0
Fig. 3 Effect of magnetic parameter on Shear stress
2
4
6
8
Fig.6 Effect of heat source on velocity.
1.0
0.1, 0.1, S 0.1, Gr 0.1, Pr 6.65564065
0.8
1.0
0.1, 0.1, Gr 0.1, 0.8
M = 0, 0.1, 0.3, 0.5, 0.7, 1.0
0.6
( )
0.6
0.4
f ''( ) 0.4
0.2
0.0 0.0
0.5
1.0
1.5
Page 25 2.0
2.5
3.0
3.5
0.2
4.0
0.0
Pr=6.65564065, M=0.5 S = 0, 0.1, 0.3, 0.7, 1.0, 1.5
10
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Vol. 1 Issue V, December 2013 ISSN: 2321-9653
INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T) : Specific heat capacity
: Thermal Grashof number g
: Gravitational acceleration
H
: Heat source :Thermal conductivity of water : Thermal conductivity of nanofluid : Thermal conductivity of alumina particles
Fig.7 Effect of heat source on shear stress
1.5
0.1, 0.1, Gr 0.1, Pr=6.65564065, M=0.5
1.0
L
: Reference length
M
: Magnetic parameter
n
: Empirical shape factor
Pr
: Prandtal number
S
: Heat source parameter
0.5
Tw : Temperature at the wall
( )
: Thermal slip factor
0.0
-0.5
S = 0, 0.1, 0.3, 0.7, 1.0, 1.5
T∞
:Temperature at infinity
u∞
: Uniform velocity at infinity
-1.0
0
2
u, v : Velocity components in x and y direction 4
6
8
: Thermal slip parameter
Fig.8 Effect of heat source on temperature
:Thermal expansion coefficient : Velocity slip parameter
2
: Electrical conductivity of nanofluid 1
: Spherecity : Volume fraction of nanoparticles
0
'( )
: Density of water S = 0, 0.1, 0.3, 0.7, 1.0, 1.5
-1
: Density of alumina particles : Nanofluid density
-2
0.1, 0.1, Gr 0.1, Pr=6.65564065, M=0.5
-3 0
2
4
6
: Dynamic viscosity of nanofluid : Kinematic viscosity of nanofluid
8
Fig.9 Effect of heat source on temperature gradient
: Velocity slip factor.
V. NOMENCLATURE : Magnetic field
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Vol. 1 Issue V, December 2013 ISSN: 2321-9653
INTERNATIONAL JOURNAL FOR RESEARCH IN AP PLIED SCIENCE AN D E N G I N E E R I N G T E C H N O L O G Y (I J R A S E T) TABLE 4 PHYSICAL PARAMETERS FOR AL2O3 NANOFLUID AT 200C
Vol. Fr. Ď•
Thermal Conductivity Knf
Dynamic Viscosity Âľ nf
Prandtal Number Pr
Thermal Diffusivity
Kinematic Viscosity
Density
Heat Capacity
0.00
0.5970000
1.0020000
7.0186994
0.1426872
1.0014792
1000.5200
4181.800
0.01
0.6142114
1.0274950
6.9385106
0.1437426
0.9973600
1030.2148
4147.672
0.02
0.6317568
1.0539075
6.8622847
0.1448988
0.9943372
1059.9096
4113.544
0.03
0.6496460
1.0812805
6.7898403
0.1461537
0.9923606
1089.6044
4079.416
0.04
0.6678893
1.1096592
6.7210103
0.1475057
0.9913875
1119.2992
4045.288
0.05
0.6864972
1.1390917
6.6556406
0.1489536
0.9913818
1148.9940
4011.160
0.06
0.7054809
1.1696288
6.5935894
0.1504967
0.9923135
1178.6888
3977.032
0.07
0.7248517
1.2013245
6.5347257
0.1521346
0.9941582
1208.3836
3942.904
0.08
0.7446217
1.2342358
6.4789292
0.1538674
0.9968963
1238.0784
3908.776
0.09
0.7648033
1.2684233
6.4260890
0.1556954
1.0014792
1000.5200
4181.800
0.10
0.7854095
1.3039515
6.3761029
0.1576193
0.9973600
1030.2148
4147.672
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Bachok, N., Ishak, A., Pop, I. Mixed convection boundary layer flow over a permeable vertical flat plate embedded in a anisotropic porous medium, mathematical problems in engg., ID 659023, pp. 1-12, 2010. K., Mohammad, Abbassi, A., Avval, M.S., Frijns, A., Darhuber, A., Harting, J. Experimental and numerical investigation of nanofluid forced convection inside a wide microchannel heat sink, App. Thermal Engg., vol.36, pp.260 -268, 2012. Aladag, B., Halelfadl, S., Doner, N., Mare, T., Duret, S. and Estelle, P. Experimental investigations of the viscosity of nanofluids at low temperatures, App. Energy, vol.97, pp.876-880, 2012. Hamed, S., Kasra, A. VIM Solution for mixed convection over horizontal moving porous flat plate, progress in applied mathematics, vol.6(1), pp.12-29, 2013.