Pergamon
Int. Comm. Heat Mass Transfer, Vol. 27, No. 4, pp. 581590, 2000 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 07351933/00/Ssee front matter
PII S07351933(00)001408
VISCOUS DISSIPATION EFFECTS ON MHD FREE CONVECTION FLOW OVER A NONISOTHERMAL SURFACE IN A MICROPOLAR FLUID
M.A. E1Hakiem Department of Mathematics Faculty of Science South Valley University Aswan, Egypt
(Communicated by J.P. Hartnett and W.J. Minkowycz)
ABSTRACT An analysis is presented to study the effect of viscous dissipation, the thermal dispersion and Joule heating on MHDfree convection flow with a variable plate temperature in a micropolar fluid in the presence of ut~iformtransverse magnetic field. Results for velocity, angular velocity and thermal functions are displayed for a range of values of the material parameters of micropolar fluid, thermal dispersion and magnetic field parameters. The presence of dissipation increases both the skin friction and the rate of heat transfer at the surface. The friction factor and heat transfer rate increase with an increase in the thermal dispersion parameter and it decrease with an increase in the magnetic field parameter Mn and micropolar parameter A. © 2000 ElsevierScienceLtd
Introduction Magnetohydrodynamic free convection flow of an electrically conducting fluid over a hot vertical wall in the presence of a strong magnetic field has been studied by Sparrow and Cess [1] and Riley [2]. Hossain and Ahmed [3] have studied the combined effect of forced and free convection with uniform heat flux in the presence of a strong magnetic field. Gebhart [4] has shown that the viscous dissipation effect plays an important role in natural convection in various devices which are subjected to large variations of gravitational force or which operate at high rotational speeds. Takhar and Soundalgekar [5] have studied the effects of viscous and Joule heating on the problem posed by Sparrow and Cess [1], using the series expansion method of Gebhart [4]. Hossain [6] studied the effect of viscous and Joule heating on the flow of an electrically conducting, viscous, incompressible fluid past a semiinfinite plate with surface temperature varying linearly with the distance from the leading edge in the presence of uniform transverse magnetic field. 581
582
M.A. E1Hakiem
Vol. 27, No. 4
Eringen [7] has proposed the theory of micropolar fluids which takes account of the inertial characteristics
of the substructure particles, which are allowed to undergo rotation. The theoD, of
thermomicropolar fluids has been developed by Eringen [8]. Mohammadien and Gorla [9] studied the effects of transverse magnetic field on mixed convection in a micropolar fluid on a horizontal plate with vectored mass transfer. The effect of a transverse magnetic field on natural convection in micropolar fluids was studied by Mohammadien et al [10]. EIFlakiem [11] studied the effects of a transverse magnetic field on natural convection with temperature dependent viscosity in micropolar fluids. Thermal dispersion effects on nonDarcy natural convection with lateral mass flux was studied by Murthy and Singh [12].
In the present paper, we propose to study the effect of thermal dispersion, viscous and Joule heating on the flow' of
an incompressible, electrically conducting micropolar fluid past a semiinfinite plate
whose surface temperature varies linearly with the distance from the leading edge. The plate is subjected to a uinform transverse magnetic field. Numerical solutions are obtained for the flow and temperature fields for several values of the material properties of the micropolar fluid, the values of thermal dispersion parameter and the magnetic field strength parameter.
.Basic Equations Let us consider a steady, twodimensional, laminar free convection boundary layer flow of an incompressible and electrically conducting micropolar fluid through a uniformly distributed transverse magnetic field of strength B0 in presence of thermal dispersion. The governing equations of motion can be written in the following form: Mass equation: ~u
+ ~ 0x
0y
: o
(l)
Momentum equation: U
Ou 
Ou
+ ~ = g*4(TT~,)+(~ & Oy 
K 2 +) a u p ey ~
+
K ON ~oB~
[email protected]
p
u
(2)
Angular momentum equation:
ON c3N_ y 82N u c3x +v @ PJ @2
K ~u_ pj(2N+ @ )
(3)
Energy equation:
OT u+~
~3T 0 (a~. OT ) + . . . .
v ( Ou ) 2 + c~oB20u2
{4)
Vol. 27, No. 4
VISCOUS DISSIPATION EFFECTS ON MHD FLOW
583
Here u and v are velocity components associated with x and y the directions measured along and normal to the vertical plate, respectively; T the temperature of the fluid in the boundary layer; N the angular velocity; g'the acceleration due to gravity; fl the coefficient of thermal expansion; K the vortex viscosity; y the spin gradient viscosity; j the microrinertia per unit mass; k the thermal conductivity; p the density of the fluid; G 0 the electrical conductivity; v the kinematic coefficient of viscosity; Cpthe specific heat at constant pressure; T the temperature of the ambient fluid and a y the effective thermal diffusivity and defined as: aft or +~za,
a d = y:ud
The thermal dispersion is introduce by assuming the effective thermal diffusivity a # components: a
to have two
the molecular diffusivity and ~ze the diffusivity due to thermal dispersion. Where Y: the
dispersion coefficient and d the pore diameter.
The boundary conditions are u=v=N=O,
at y = 0
T=Tw(x )
u>O,
N>O,
(5)
as y >oo
T+T~
To reduce equations (14) to ordinary differential equations, the stream function
and v =   &
c~ defined by u =  
~v
is introduced with the following set of transformations: I
1
:
~(rl,~) = ( g * f l S ) ~ v 2 x f ( r l , 4 ) , 3
1
N(rl,~ ) = ( g , f l S ) 4 v 2g(r/,¢) '
~ _ g'flx CP
(6)
(z ~)
OoBo~

O(rl,~)  (Tw  T°~) ,
1
rI = ( g * f l S ) : v ~y
(7~,  T®) = Sx,
Mn=
,
p(g'~S): Upon substituting expressions in equation (6) into equations (24) and using the boundary condition (5) one finds: (l+A)f'+Ag'+ff"(M+f')f'+O=~[f'
. . . . ~g
5
_ [ , O fc~]
,~f.
2cg"  A B ( Z g + f " )  f ~ + g ' f = g [ J ~   g ~  1
(Pr)' [(1 + s f 50" + soy"] + f O '  f ' O = ~ [ f ' c?O _ O' O f _ M n ( f ' ) 2 _ ( f , ) 2 ]
(7) (8) (9)
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M.A. E1Hakiem
Vol. 27, No. 4
The transformed boundary conditions are given by
0(0, 4) : 1
f ( 0 , 4 ) = f ' ( o , ¢ ) = g(0,g)= 0,
(lO)
f'(m,g) = g(m,~)= 0(m,g)= 0
In the previous equations, primes denote differentiations with respect to 7/ only, the Prandtl number
Pr=pvCp/k,
A=K/pv,
2 = y / p v j , B=v(g*flS)
and s = (dy~x/ a)(g*Sfl) 112.
The physical quantities of interest are the local friction factor v w and the local heat transfer qw. These are defined by 1 rw
3
= [(# + K)o~y + KN]y: o = p v 2 ( g ' ~ S ) ~ xO + A ) f " ( o , ¢ ) ,
(i 1)
and 1

1
q~ = kv~(g*fls)a(T~ T~)O '0 (,4)
(12)
The wall couple stress is defined by
M
 Z(O~N)y=o= ~g*~Svlxg'(O)
(13)
c9/ Results and Discussion The governing boundary layer equations (710) for the velocity, temperature and microrotation have been solved numerically using Kellerbox implicit scheme [13]. The resulting solutions for velocity, angular velocity and temperature functions are shown graphically in Figures. (16)andthe numerical values for functions proportional to shearstress, wall couple stress and heat transfer coefficients are presented in Table I. Here, we have chosen Pr=0.72, B=0.1, ~. =0.5 while, A, Mn, s and ~were varied over a range.
We now discuss the effect of thermal dispersion, viscous and Joule heating on the shearstress and the rate heat transfer at the surface of the wall. From Table I it may be concluded that as the buoyancy and thermal dispersion parameters increase, both the skin friction and the surface heat transfer rate increase. The friction factor and the rate of heat transfer due to the presence of thermal dispersion, viscous and Joule parameter A.
heating decrease with an increase in the magnetic field parameter Mn and micropolar
Vol. 27, No. 4
VISCOUS DISSIPATION EFFECTS ON MHD FLOW
"FABLE 1 for various values o f
Variation o f s
0.0
! A
"
'rd.
1
0.0
4
0,5
1
4
5
1
4
0.2
0.0
1
f"(0) 0.0 0.5 1.0 0,0 0.5 1.0 0.0 0.5 1.0 0.0 [1.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5
1.0 4
0.5
1
4
5
1
4
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5
1.0 16
oo
1
0.0 0.5
4
0.0 0.5 1,0 0.0 I).5 1.0 0.0 0.5 1.0 0.0 05 10 00 0.5 1.0
1.0
0.5
I
4
5
1
4
0.61221 0.62226 0.63272 0,41708 0.42271 0.42881 0.47436 0.48185 0.48980 0.33024 0.33482 0.33979 0.18611 0.18854 0.19118 0.14025 0.14277 0.14570 0.61377 0.62385 0.63436 0.41733 0.42290 0.42894 0.47553 0.48296 0.49071 0.33045 0.33504 0. 34002 0.18385 0.18599 0.18822 0.13856 0.14045 0.14249 0.61944 0,62974 0.64047 0.41829 0.42387 0.42994 0.47982 0.48743 0.49537 0.33125 0.33585 0.34086 0.18498 0.18717 0 18945 0 13882 0.14[)72 0.14276
A, Mn and s.
 0'(0) 0.45359 0.40463 0.35375 0.34402 0.31)605 026540 0.42829 0.39206 0.35429 0.33230 0.30076 0.26691 0.34533 0.33144 0.31690 0.28108 0.25909 0.23490 0. 46411 0.41458 0.36308 0.34819 0.31018 0.26950 0.43727 0,40067 0.36252 0.33609 0.30432 0.27019 0.34955 0.33662 0.32315 0.29191 0.27546 0.25780 0.50545 0.45367 0.39969 0.36465 0. 32549 0.28341 0.47270 0.43457 0.39473 0.351()7 0.31840 [).28325 0.36868 0 35545 0.34164 0.30201 0.28529 0.26732
± g'(0) 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0,01475 0,01513 0,01640 0.00822 0.00848 0.00876 0.07946 0.08085 0.0231 0.05280 0.05458 0.05654 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.01488 0.01525 0.01562 0.00826 0.00852 0.00881 0.07723 0.07846 0.07972 0.05129 0.05242 0.05363 0,00000 0.00000 0.00000 0,00000 0.00000 0,00000 /).1)1536 0.01574 0,01612 0,00840 0.00866 0.00895 0.07829 0.07954 0.08083 0.05163 0.05276 0.05398
F
585
586
M.A. EIHakiem
Vol. 27, No. 4
Figures (16) represent, velocity, temperature and microrotation component for different values of Table 1 and functions proportional to friction factor, heat transfer rate and wall couple stress, the micropolar parameter A, the thermal dispersion parameter s and magnetic field parameter Mn in the absence, as well as presence, of viscous and Joule heating. We may conclude that dissipative heat increases the velocity field and temperature field while increasing the values of Mn and A will reduce the velocity field and increase temperature field. We observe that the microrotation changes sign from negative to positive values within the boundary layer. The magnitude ofmierorotation increases with an increase m A, while it decrease with increasing values of Mn. The velocity, magnitude ofmicrorotation and temperature increase with increase of thermal dispersion parameter s in the presence of viscous, Joule heating, magnetic field parameter Mn and the micropolar parameter A.
.." . " " ~ , ~ ,:'Z"~" . . . . . \ ' x . \ _~~'  ~  ~
0 o
',=00 ~=05 \=15
[
............................. 2
i=00 1= I 0 105
4 q
FIG I
Velocity profiles for variousvalues of A and ~ with Mn~I.O and s=l.O.
",

~=o o
04
,.
2
4 q
FIG 2
Temperature profiles for various values of A and ~ with Mn = 1.0
a n d s = 1.0.
Vol. 27, No. 4
VISCOUS DISSIPATION EFFECTS ON MHD FLOW
587
0.02
001
£ (:7) 001
0 02
~
A=50
I ......... ~=I0 ~.=0.5 ~0.0
0 03 2
4
q
FIG. 3 Microrotation profiles for various values of A a,ld ~: with Mn=1.0 and s = 1.0.
03
I ....~_1.0 = ~=05
02
~
,
~s=O0
0.1
0
2
4 q
FIG. 4 Velocity profiles for various values of ~, s and Mn with A =0.5.
588
M.A. ElHakiem
Vol. 27, No. 4
1Or
Mn=l 0
O4
0.2
0 0
2
4
q FIG. 5 Temperature profiles for various values of and ~, s and Mn with A =0.5.
0 020 . _ .  .......... ..... 0.015
0"010 ,'" t:33
"Mn=lO Mn=40
,'" ,'
/
/
/
/
~ ~
~
~
0005
0.005
2
4
q FIG. 6 Microrotation profiles for various values of ~, s and Mn with A =0.5.
,, ", ',
Vol. 27, No. 4
VISCOUS DISSIPATION EFFECTS ON MHD FLOW
589
Concluding Remarks In this paper, we presented an analysis for the effect of thermal dispersion, viscous and Joule heating on the flow of an electrically conducting and viscous incompressible micropolar fluid past a semiinfinite plate whose temperature varies linearly with the distance from the leading edge in the presence of uniform transverse magnetic field. The governing boundary layer equations for the velocity, temperature and microrotation have been solved numerically. All the results are presented for various values of the thermal dispersion, micropolar and magnetic parameters. This friction factor and the rate of heat transfer decrease with the magnetic parameter Mn and the micropolar parameter A and it increase with the thermal dispersion parameter s increase. Using the magnetic field in micropolar fluids could serve as an effective drag reducing mechanism.
Nomenclature B0
Magnetic field intensity
Cp
Specific heat of fluid
f
Nondimensional stream velocity
g
Nondimensional angular velocity
g*
Acceleration due to gravity
j
Microinertia per unit mass
k
Thermal conductivity of fluid
K
Vortex viscosity
Mn
Magnetic parameter
N
Angular velocity
Pr
Prandtl number
q
Local heat transfer
s
Thermal dispersion parameter
S
Constant in equation (6)
T
Temperature
u
Vertical velocity component
v
Horizontal velocity component
x
Vertical coordinate
y
Horizontal coordinate
o'
Thermal diffusivity
7
Spin gradient viscosity
A, 2. Dimensionless material properties t~
Dimensionless temperature
r/, ~
Dimensionless coordinates
590
M.A. E1Hakiem
H
Dynamic viscosity
v
Kinematic viscosity
13
Coefficient of thermal expansion
/9
Density
cr0
Electrical conductivity
rw
Local friction factor
~'
Stream function
Vol. 27, No. 4
Subscripts w
Refers to conditions at the wall
ze
Refers to conditions far away from the wall
Superscripts '
Differentiation with respect to r/ References
1. E. M Sparrow and R. D. Cess, Int. J HeatMass Tran.~er 3,267 (196l). 2. N. Riley, J FluidMech. 18, 577 (1964). 3. M. A. Hossain and M. Ahmed, Int. J HeatMass Transjer 33, 571 (1990). 4. B. Gebhart, J lZluidMech 14, 225 (1962). 5. H S. Takhar and V. M Soundalgekar, Appl. Sci. Res. 36, 163 (1980). 6. M. A. Hossain, Int. J HeatMass Transfer, vol. 35, No. 12, 3485 (1992). 7. A. C. Eringen, J Mathematics and Mechanics, 16, 1 (1966). 8. A. C. Eringen, d~ MathematicalAnalysis and Applications, 38, 488 (1972). 9. A. A. Mohammadein and R. S. R. Gorla, Int. ~ ActaMechanlca, 118, 337 (1996). 10. A. A. Mohammadein, M. A E1Hakiem and R S. R. Gorla, J Appl. Mech. and Eng. vol. 1, No. 3, 337 (1996). ll. M. A. EIHakiem, o( AppL Mech. andEng vol. 3, No. 2, 287 (1998). 12. P. V. S. N. Murthy and P. Singh, J Heat andMass Transifer 33, 1 (1997). 13. T. Cebeci and P. Bradshaw, Physical and Computational Aspects" ~?fConvective Heat Transfer, SpringerVerlag, Berlin, (1984).
Received April 7, 2000