Research Paper
Applied Physics
E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016
MHD FLOW IN POROUS MEDIUM WITH INDUCED FIELD AND MASS TRANSFER EFFECT AT CONSTANT TEMPERATURE GRADIENT Shyamanta Chakraborty 1 | Nripen Medhi 2 1
Associate Professor, UGC-HRDC, Gauhati University, Assam, India.
2
Assistant Professor, Dispur College, Guwahati, Assam, India.
ABSTRACT Amagnetohydrodynamic flow of incompressible viscous fluid past a steadily moving infinite vertical porous plate is discussed under the action of transverse magnetic field. Considering a heat flux within the medium due to the constant temperature gradient at the plate, the effects of induced field and fluid concentration on flow and temperature are discussed for smaller values of Eckert number. The expressions and analytical solutions for fluid velocity, temperature, fluid concentration and induced field are obtained following Raptis [1]. The results are shown graphically followed by discussion and conclusion. KEY WORDS: Magnetohydrodynamic, Induced-field, Mass-transfer, Porous-medium, Temperature-gradient. 1. INTRODUCTION Problems of free and forced convective flow under the action of induced magnetic field are frequently encountered in various physical and engineering fields. The phenomenon of natural convective flow is not often caused entirely by the effect of temperature gradients but also due to the applied magnetic field which in turn induces another magnetic field that influence the flow largely. The study of magneto-hydrodynamic flow in the presence of porous walls has many applications in various fields of science and technology. Soundalgekar& Bhatt [1] have studied the oscillatory MHD channel flow and heat transfer. Soundalgekar & Bhatt[2] have studied the oscillatory MHD channel flow and heat transfer. Raptis & Soundalgekar [3] have studied the MHD flow past a steadily moving infinite vertical porous plate with mass transfer and constant heat flux. Drake [4], Soundelgekar [5], Pop [6], Dube [7], Singh [8], Das and Sanyal [9], Mana [10], Seth and Banerjee [11], Mittal et al.[12], and many other authors have studied various problems of MHD flow taking induced magnetic field into account. Singh and Singh [13] has studied on MHD effects on flow of viscous fluid with induced magnetic field. Motivated by the above referenced works and the numerous possible applications in the field Science & Technology specially in Geological Science, Astronomical Science and Biophysical Science, it is our interest to investigate a steady, fully developed MHD convective heat and mass transfer problem of an incompressible fluid flow where an incompressible electrically conducting viscous fluid moves past a steadily moving infinite vertical porous plate under the action of uniform magnetic field that induces another field transverse to it. The expressions and analytical solutions for fluid velocity, temperature, concentration and induced field are obtained following Raptis [1]. The effects of flow parameters like Schmidt Number, Hartmann Number, Prandtl Number, Magnetic Prandtl Number on fluid velocity, temperature, magnetic field, mass transfer, skin-friction and rate of heat transfer, are analyzed graphically and compared to those in absence of induced field. The results observed some significant differences in flow pattern in variation of flow parameters.
2. FORMULATION OF THE PROBLEM The flow of an electrically conducting, incompressible viscous fluid is considered. The plate which is vertical and porous is assumed to be moving steadily in the vertical up- ward direction along which the x-axis is chosen. The y-axis is chosen normal to the plate. Under these conditions, the basic equations of combined free and forced convective flow,under usual Bossinesq’s approximation are derived by Raptis & Soundalgekar [3]. 3. (i)
ASSUMPTIONS The level of concentration of foreign masses are very low such that the Soret and Dufour are negligible.
(ii)
The flow is a free and forced convective flow under Bossinesq’s approximatnion.
(iii)
All the variables (i.e. velocity field, magnetic field, temperature field and mass concentration) are only dependent y-axis.
(iv)
Hall Effect, Polarization effect, Buoyancy effect and the effects of heat radiation are negligible.
4.
MATHEMATICAL FORMULATION We consider a fully developed laminar unidirectional flow of an incompressible, electrically conducting incompressible fluid past a vertical porous plate. The plate is assumed to be moving steadily in the infinite vertical upward direction along which x-axis is chosen, while y-axis is normal to the plate. Let a uniform magnetic field H(y) is applied along the motion of the plate that induced magnetic field B =e H(y) transverse to the motion of the plate.
5.
GOVERNING EQUATIONS Under above assumptions, the velocity and magnetic field magnetic field components are V = u y , vo , 0 ; B = H y , eH y , 0
(1)
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International Educational Scientific Research Journal [IESRJ]
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Research Paper
E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016
H &ď e are being magnetic flied applied and magnetic permeability of the medium. Under above assumptions and the geometry of the problem, the governing equations are as follows. Equation of conservation of momentum đ?œ•đ?‘˘
= ď ľ(ď‚ś2u / ď‚śy2 ) + ď eH y 
V0
đ?œ•đ?‘Ś
đ?œ•đ??ť đ?œ•đ?‘Ś
+ g ď ˘ ( T – T ď‚Ľ) + g ď ď ˘ ( C – Cď‚Ľ) }
(2)
Equation of thermal diffusivity
ď ˛Cp (V0
� � ��
) = k (d2T / dy2) + ď (
đ?œ•đ?‘˘ đ?œ•đ?‘Ś
) 2+
1
ď ł
đ?œ•đ??ť
( 2
(3)
đ?œ•đ?‘Ś
Equation of Mass diffusivity đ?‘‘đ??ś
(�
đ?‘‘đ??ś
) = D ( 2
đ?‘‘đ?‘Ś
(4)
đ?‘‘đ?‘Ś
Equation of Magnetic diffusivity đ?œ•đ??ľ
1
= ďƒ‘ X (v X B) +
đ?œ•đ?‘Ą
ď ł
ďƒ‘2 H
(5)
Introducing non-dimensional quantities as given below u =
�
đ?‘Œ
; y = v0
đ?‘‰đ?‘œ
C= (C – Cď‚Ľ ) / (CW – Cď‚Ľ); T = [(T – Tď‚Ľ )k V0] / (qď ľ) ;
ď ľ
H= (H /ď ľ0) ďƒ–(ď e / ď ˛ ), q is a quantity with dimension M T –3
(6)
Substituting above non-dimensional quantities and removing primes, equations (2) – (5) are as follows. đ?œ•đ?‘˘
− 1 đ?‘ƒđ?‘&#x; đ?œ•đ??ś đ?œ•đ?‘Ś
đ?œ•đ?‘Ś
[ +
đ?œ•2 đ?‘‡ đ?œ•đ?‘Ś 2
đ?œ•2 đ?‘˘
= GmC + GrT+
đ?œ•2 đ??ť
]+
đ?œ•đ?‘Ś 2 1 đ?‘†đ?‘?
+Pr
đ?œ•2 đ??ś
[
đ?œ•đ?‘Ś 2
đ?œ•đ?‘‡ đ?œ•đ?‘Œ
đ?œ•đ??ť đ?œ•đ?‘Œ
đ?œ•đ?‘Ś 2
đ?œ•đ??ť
+M
(7)
đ?œ•đ?‘Ś
đ?œ•đ?‘ˆ
+M =0
(8)
đ?œ•đ?‘Ś
]= 0
(9)
= − PrE [(
đ?œ•đ?‘˘ đ?œ•đ?‘Ś
)2+
1
đ?œ•đ??ť
đ?‘ƒđ?‘š
( )2]
(10)
đ?œ•đ?‘Ś
where, ď ď ľ = , Kinematic viscosity ; ď = Coefficient of viscosity of the medium ; T = Fluid temperature ; ď ˘ = Volumetric Coefficient thermal expansion ;ď ď ˘ = Volumetric ď ˛
Coefficient of mass transfer ; ď Ą = thermal diffusivity ; ď ľm = ď eď ł (đ?‘šđ?‘Žđ?‘”đ?‘›đ?‘’đ?‘Ąđ?‘–đ?‘? đ?‘‘đ?‘–đ?‘“đ?‘“đ?‘˘đ?‘ đ?‘–đ?‘Ąđ?‘–đ?‘Łđ?‘–đ?‘Ąđ?‘Ś) ; ď ł = đ?‘’đ?‘™đ?‘’đ?‘?đ?‘Ąđ?‘&#x;đ?‘–đ?‘?đ?‘Žđ?‘™ đ?‘?đ?‘œđ?‘›đ?‘‘đ?‘˘đ?‘?đ?‘Ąđ?‘–đ?‘Łđ?‘–đ?‘Ąđ?‘Ś, ď e magnetic permeability of the medium ; C = fluid Concentration, ď ˘ = Volumetric Coefficient thermal expansion ; ď ď ˘ = Volumetric Coefficient of mass transfer ; D = Coefficient of mass transfer ; S c đ??ˇ = Schmidt Number ; M = ( H /đ?‘Ł0 ) ďƒ–(ď e /ď ˛), Magnetic Hartmann Number, ď ľ
Pm = ď eď łď ľ Magnetic Prandtl Number; Gr = (g ď ˘ q ď ľ2) / (k v0 4), Grashoff Number ; Gm= [g ď ˘ď ľ (Cw – Cď‚Ľ)]/ (v0 3), Modified Grashoff Number; Pr = (ď Cp / k), Prandtl Number; E = (k v0 3)/(qď ľCp), Eckert Number The corresponding non-dimensional boundary conditions are y= 0, &
yď‚Žď‚Ľ ,
��
u = U,
(
u ď‚Ž0,
T ď‚Ž 0,
đ?‘‘đ?‘Ś
) = -1,
C=1, C ď‚Ž 0,
H=K
H ď‚Ž0
(11)
Where U and K are two assumed values for u and H respectively suitable to the problem
6. SOLUTIONS OF GOVERNING EQUATIONS Solution for Mass Transfer equation under above boundary conditions is đ?‘?(y) = đ?‘’ −
đ?‘†đ?‘? đ?‘Ś
(12)
In order to get solution of the governing equations (7 – 10) we consider following series expansion in terms of E, the Eckert Number. u = u0 + E u1 + 0(E2)
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E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016
2
H= H0 + E H1 + 0(E ) T= T0 + E T1 + 0(E2)
(13)
Using above expansions the governing equations can be divided in increasing order of E (i.e. 1st , 2nd ----) as follows. 1st order of equations đ?œ• 2 đ?‘˘0
đ?œ•đ?‘˘0
-
đ?œ•đ?‘Ś 2
(1 /Pr)
đ?œ•2 đ?‘‡0 đ?œ•đ?‘Ś 2
đ?œ•đ?‘Ś 2
+ GrT0 +Gm C = 0
(14)
=0
(15)
(1+ M) = 0
(16)
đ?œ•đ?‘Ś đ?œ•đ?‘‡0
+
đ?œ•2 đ??ť0
(1 /Pm)
đ?œ•đ??ť0
+M
đ?œ•đ?‘Ś
đ?œ•đ?‘Ś đ?œ•đ??ť0
+
đ?œ•đ?‘Ś
Corresponding non-dimensional boundary conditions are y= 0, y ď‚Žď‚Ľ ,
&
u ď‚Ž0,
��0
u0 = U,
(
T ď‚Ž 0,
C ď‚Ž 0,
đ?‘‘đ?‘Ś
) = -1,
C=1,
H0 = K
H ď‚Ž0
(17)
2nd order of equations đ?œ• 2 đ?‘˘1 đ?œ•đ?‘Ś 2 đ?œ•2 đ?‘‡1 đ?œ•đ?‘Ś 2
-
đ?œ•đ?‘˘1
đ?œ•đ?‘‡1
+
(1 /Pm)
đ?œ•2 đ??ť1 đ?œ•đ?‘Ś 2
đ?œ•đ?‘Ś
+ GrT1 + Gm C = 0 đ?œ•đ?‘˘0
= - Pr [ (
đ?œ•đ?‘Ś
7.
đ?œ•đ??ť1
+M
đ?œ•đ?‘Ś
đ?œ•đ??ť1
+
đ?œ•đ?‘Ś
đ?œ•đ?‘Ś
) 2+
đ?œ•đ?‘˘1
+M
1
(
đ?‘ƒđ?‘š
đ?œ•đ??ť0 đ?œ•đ?‘Ś
(18)
)2]
(19)
=0
đ?œ•đ?‘Ś
(20)
VISCOUS-DRAG AND HEAT-TRANSFER
The physicalquantities of ourinterestat the plate are Skin friction coefficients [ď ´]y= 0 and rate of heattransfer [Q]y= 0 are as follows. đ?‘‘đ?‘˘
ď ´ =ď€ ď€ď ( )y = 0
and
đ?‘‘đ?‘Ś
1
��
k
đ?‘‘đ?‘Ś
Nu = - [
]đ?‘Ś=
0
Using (6), the non-dimensionalviscous drag per unit length i.e. Skin friction and rate of heattransferat the plate y = 0 are given as ��
ď ´p =-( )y = 0
(21)
đ?‘‘đ?‘Ś
Nu = -[
��
]đ?‘Ś=
đ?‘‘đ?‘Ś
(22)
0
Where, ��
��
đ?‘‘đ?‘Ś
đ?‘‘đ?‘Ś
( ) and ( ),are the non-dimensional gradients of velocity and temperature with in the medium.
8.
SOLUTION OF GOVERNING EQUATIONS.
The solutions of governing equations (14)to (20)under their boundary conditions are as follows. đ?‘‡0 = −
1 đ?‘ƒđ?‘&#x;
đ?‘’ đ?‘ƒđ?‘&#x; đ?‘Ś
(21)
đ??ť0 =đ??ś1 đ?‘’ đ?‘š1đ?‘Ś + (đ?‘˜5 —đ?‘?1 )đ?‘’ đ?‘š2 đ?‘Ś − đ?‘˜3 đ?‘’ đ?‘ƒđ?‘šđ?‘Ś + đ?‘˜4 đ?‘’ −đ?‘ đ?‘? đ?‘Ś đ?‘˘0 =đ?‘Ž1 đ?‘’
đ?‘š1 đ?‘Ś
+ đ?‘Ž2 đ?‘’
đ?‘š2 đ?‘Ś
− đ?‘Ž3 đ?‘’
đ?‘†đ?‘? đ?‘Ś
+ đ?‘Ž4
(22)
đ?‘’ đ?‘ƒ đ?‘šđ?‘Ś
(23)
đ?‘˜1 = đ?‘?đ?‘š+1 , đ?‘˜2 = đ?‘ƒđ?‘š (đ?‘€2 − 1), đ?‘€đ?‘ƒđ?‘šđ??şđ?‘&#x; đ?‘?đ?‘š (đ?‘?đ?‘š 3 + đ?‘˜1 đ?‘?đ?‘š − đ?‘˜2 đ?‘?đ?‘š ) đ?‘?đ?‘šđ?‘€đ??şđ?‘š đ?‘˜4 = −đ?‘ đ?‘? 3 − đ?‘˜1đ?‘ đ?‘? + đ?‘˜2 đ?‘ đ?‘?
đ?‘˜3 =
đ?‘˜6 = (
đ?‘š2 đ?‘€đ?‘?đ?‘š
+ 1) − (
đ?‘š1 đ?‘€đ?‘ƒđ?‘š
+ 1),
đ?‘˜5 = K − (đ?‘˜4 −
đ?‘˜3 )
đ?‘š2 đ?‘ đ?‘? đ?‘ƒđ?‘&#x; đ?‘˜7 = đ?‘˜5 ( + 1) + đ?‘˜4 (1 + ) + đ?‘˜3 (1 + ) đ?‘€đ?‘ƒđ?‘š đ?‘€đ?‘ƒđ?‘š đ?‘€đ?‘ƒđ?‘š
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E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016 𝑘1 ± √𝑘1 2 + 4𝑘2 𝑚1,2 =
−𝑘6 ± √𝑘2 2 + 4𝑘7 ,
2
𝑚5,6 =
2
𝐾8=𝑝𝑟 +1 ; 𝑘9 = 𝑝𝑟 (1−𝑀2) 𝑚3,4 =
𝑈+𝑘7
𝑐1 =
𝑎2 = (𝑘5 −
𝑘6
𝑈+𝑘7 𝑘6
𝑐2 =𝑘5 − 𝑐1 , 𝑎1 = −(
)(
𝑚2
𝑚1
𝑈+𝑘7
+1) (
𝑀𝑝𝑚
𝑠𝑐
+1), 𝑎3 = 𝑘4 (1 −
𝑀𝑃𝑚
𝑘6
𝑀𝑝𝑚
−𝑘8 ± √𝑘8 2 − 4𝑘9 2
),
) , 𝑎4 = 𝑘3 (1 +
𝑝𝑟 𝑀𝑃𝑚
).
𝑎5 = 𝑎12 𝑚12 ; 𝑎6 = 𝑎22 𝑚22 ; 𝑎7 = 2𝑎1 𝑎2 𝑚1 𝑚2 ; 𝑎8 = 𝑎3 𝑆𝑐2 𝑎9 = 𝑎42 𝑝12 , 𝑎10 = 2𝑎3 𝑎4 𝑆𝑐 𝑃𝑟 , 𝑎11 = 2𝑎1 𝑎3 𝑚1 𝑆𝑐 , a12 = 2a1 a4 m1 Pr , a13 = 12 a2 a3 m2 , a14 = −2a2 a4 m2 Pr , a15 = a21 = a27 = a31 = a34 = 𝑏7 = 𝑏9 =
−𝑃𝑟 𝑎11 4𝑚12 −𝑃𝑟 2𝑚1
, a22 =
4𝑚12 −𝑃𝑟 2𝑚1
, a28 =
𝑀𝑃𝑚 𝐺𝑟
, a32 =
(𝑚1 −𝑆𝑐 )3 +𝑘6 (𝑚1 −𝑆𝑐 )2 +𝑘7 (𝑚1 −𝑆𝑐 ) −𝑀𝑃𝑚 𝐺𝑟
−𝑃𝑟 𝑎6 4𝑚12 −𝑃𝑟 2𝑚1
−𝑃𝑟 𝑎13
, a23 =
4𝑚12 −𝑃𝑟 2𝑚1
𝑀𝑃𝑚 𝐺𝑟
−𝑃𝑟 𝑎7
, a17 =
, a24 =
4𝑚12 −𝑃𝑟 2𝑚1
−𝑃𝑟 𝑎14 4𝑚12 −𝑃𝑟 2𝑚1
𝑀𝑃𝑚 𝐺𝑟 (−2𝑆𝑐 )3 +𝑘6 (−2𝑆𝑐 )2 +𝑘7 (−2𝑆𝑐 )
, a18 =
, a25 =
, a29 =
𝑀𝑃𝑚 𝐺𝑟 (𝑚1 +𝑃𝑟 )3 +𝑘6 (𝑚1 +𝑃𝑟 )2 +𝑘7 (𝑚1 +𝑃𝑟 )
−𝑃𝑟 𝑎8 4𝑚12 −𝑃𝑟 2𝑚1
, a19 =
−𝑃𝑟 𝑎9 4𝑚12 −𝑃𝑟 2𝑚1
𝑀𝑃𝑚 𝐺𝑟 (2𝑚1 )3 +𝑘6 (2𝑚1 )2 +𝑘7 (2𝑚1 ) 𝑀𝑃𝑚 𝐺𝑟
, a30 =
−𝑃𝑟 𝑎10 4𝑚12 −𝑃𝑟 2𝑚1
,
(2𝑚2 )3 +𝑘6 (2𝑚2 )2 +𝑘7 (2𝑚2 ) 𝑀𝑃𝑚 𝐺𝑟
,
,
(𝑃𝑟 −𝑆𝑐 )3 +𝑘6 (𝑃𝑟 −𝑆𝑐 )2 +𝑘7 (𝑃𝑟 −𝑆𝑐 )
−𝑀𝑃𝑚 𝐺𝑟
, a33 =
, a20 =
𝑀𝑃𝑚 𝐺𝑟
, a26 =
(2𝑃𝑟 )3 +𝑘6 (2𝑃𝑟 )2 +𝑘7 (2𝑃𝑟 )
,
(𝑚2 −𝑆𝑐 )3 +𝑘6 (𝑚2 −𝑆𝑐 )2 +𝑘7 (𝑚2 −𝑆𝑐 )
, 𝑏1 = 𝑐1 𝑚3 , 𝑏2 = (𝑘5− 𝑐1 )𝑚1 , 𝑏3 = −𝑘3 𝑃𝑟 , 𝑏4 = −𝑘4 𝑆𝑐 , 𝑏6 =
(𝑚2 +𝑃𝑟 )2 +𝑘6 (𝑚2 +𝑃𝑟 )+𝑘7 (𝑚2 +𝑃𝑟 ) −𝑃𝑟 𝑏22
−𝑃𝑟 𝑏12 𝑃𝑚 (4𝑚32 −𝑃𝑟 2𝑚3 )
,
−𝑃 2𝑏 𝑏
𝑟 1 2 𝑏 8 = 𝑃𝑚{(𝑚3+𝑚4)2−𝑃𝑟(𝑚3+𝑚4)},
−𝑃𝑟 𝑏32 𝑃𝑚 {(2𝑃𝑟 )2 −𝑃𝑟 (2𝑃𝑟 )}
𝑏16 =
−𝑃𝑟 𝑎12
,a16 =
(𝑚1+𝑚2 )3 +𝑘6 (𝑚1 +𝑚2 )2 +𝑘7 (𝑚1 +𝑚2 )
𝑃𝑚 (4𝑚42 −𝑃𝑟 2𝑚4 )
𝑏11 =
−𝑃𝑟 𝑎5 4𝑚12 −𝑃𝑟 2𝑚1
−𝑃𝑟 𝑏42
, 𝑏10 =
−𝑃𝑟 2𝑏3 𝑏4 𝑃𝑚 [(𝑃𝑟 −𝑆𝑐 )2 −(𝑃𝑟 −𝑆𝑐 )𝑃𝑟 ]
𝑃𝑚 [(−2𝑆𝑐 )2 −𝑃𝑟 (−2𝑆𝑐 )]
, 𝑏12 =
𝑀𝑃𝑚 𝐺𝑟 (2𝑚4 )2 +𝑘6 (2𝑚4 )2 +𝑘7 (2𝑚4 ) 𝑀𝑃𝑚 𝐺𝑟
(𝑃𝑟 −𝑆𝑐 )3 +𝑘6 (𝑃𝑟 −𝑆𝑐 )+𝑘7 (𝑃𝑟 −𝑆𝑐 )
−𝑃𝑟 2𝑏1 𝑏3 𝑃𝑚 [(𝑚3 +𝑃𝑟 )2 −𝑃𝑟 (𝑚3 +𝑃𝑟 )]
, 𝑏17 =
, 𝑏21 =
, −𝑃𝑟 2𝑏1 𝑏4
, 𝑏13 =
𝑀𝑃𝑚 𝐺𝑟
, 𝑏18 =
(𝑚3 +𝑚4 )2 +𝑘6 (𝑚3 +𝑚4 )+𝑘7 (𝑚3 +𝑚4 ) 𝑀𝑃𝑚 𝐺𝑟
(𝑚3 +𝑃𝑟 )3 +𝑘6 (𝑚3 +𝑃𝑟 )+𝑘7 (𝑚3 +𝑃𝑟 )
, 𝑏14 =
𝑃𝑚 [(𝑚3 −𝑆𝑐 )2 −𝑃𝑟 (𝑚3 −𝑆𝑐 )]
, 𝑏22 =
−𝑃𝑟 2𝑏2 𝑏3 𝑃𝑚 [(𝑚4 +𝑃𝑟 )2 −𝑃𝑟 (𝑚4 +𝑃𝑟 )]
𝑀𝑃𝑚 𝐺𝑟 (2𝑃𝑟 )3 +𝑘6 (2𝑃𝑟 )+𝑘7 (2𝑃𝑟 ) 𝑀𝑃𝑚 𝐺𝑟
(𝑚3 −𝑆𝑐 )3 +𝑘6 (𝑚3 −𝑆𝑐 )+𝑘7 (𝑚3 −𝑆𝑐 )
, 𝑏19 =
, 𝑏23 =
, 𝑏15 =
−𝑃𝑟 2𝑏2 𝑏4 𝑃𝑚 [(𝑚4 −𝑆𝑐 )2 −𝑃𝑟 (𝑚4 −𝑆𝑐 )]
𝑀𝑃𝑚 𝐺𝑟 (−2𝑆𝑐 )3 +𝑘6 (−2𝑆𝑐 )+𝑘7 (−2𝑆𝑐 )
,
, 𝑏20 =
𝑀𝑃𝑚 𝐺𝑟 (𝑚3 +𝑃𝑟 )3 +𝑘6 (𝑚3 +𝑃𝑟 )+𝑘7 (𝑚3 +𝑃𝑟 )
, 𝑏24 =
𝑀𝑃𝑚 𝐺𝑟 (𝑚3 −𝑆𝑐)3 +𝑘6 (𝑚3 −𝑆𝑐 )+𝑘7 (𝑚3 −𝑆𝑐 )
𝑇1 = 𝑐4 𝑒 𝑝𝑟𝑦
𝑎15 𝑒 𝑚1𝑦
+
+
𝑎16 𝑒 𝑚2𝑦
+
𝑎17 𝑒(𝑚1+𝑚2)𝑦 + 𝑎18 𝑒 −2𝑠𝑐 𝑦
+
𝑎19 𝑒 𝑝𝑟𝑦 + 𝑎
20
𝑒
(𝑝𝑟−𝑠
𝑎21 𝑒(𝑚1−𝑠𝑐 )𝑦
𝑐)𝑦 +
+
𝑎22 𝑒 (𝑚1+𝑝𝑟)𝑦
𝑎23 𝑒 (𝑚2−𝑠𝑐 )𝑦
--
--
𝑎24 𝑒 (𝑚2+pr )𝑦 +𝑏5 𝑒 2𝑚3𝑦 +𝑏6 𝑒 2𝑚4𝑦 +𝑏7 𝑒 (𝑚3 +𝑚4)𝑦 +𝑏8 𝑒 2𝑝𝑟 𝑦 +𝑏9 𝑒 −2𝑠𝑐 𝑦 +b1𝑏10 𝑒 (𝑝𝑟 −𝑠𝑐 )𝑦 +𝑏11 𝑒 (𝑚3+𝑝𝑟)𝑦 +𝑏12 𝑒 (𝑚3−𝑠𝑐 )𝑦 +𝑏13 𝑒 (𝑚3+pr )y +𝑏14 𝑒 (𝑚3−𝑆𝑐)𝑦 . 𝐻1 =𝑐6 𝑒 𝑚5𝑦 + 𝑐7 𝑒 𝑚6𝑦 +𝑐8 𝑒 𝑝𝑟𝑦 +𝑎25 𝑒 2𝑚1𝑦 + 𝑎26 𝑒 2𝑚2𝑦 +𝑎27𝑒(𝑚1+𝑚2)𝑦 +a28 𝑎28 𝑒 −2𝑠𝑐 𝑦 + 𝑎29 𝑒 2𝑝𝑟 𝑦 +𝑎30 𝑒 (𝑝𝑟 −𝑠𝑐 )𝑦 +𝑎31 𝑒 (𝑚1−𝑠𝑐 )𝑦 + 𝑎32 𝑒 (𝑚1+𝑚2)𝑦 -𝑎33 𝑒 (𝑚2−𝑠𝑐 )𝑦 − 𝑎34 𝑒 (𝑚2+pr )𝑦 +𝑏15 𝑒 2𝑚3𝑦 +𝑏16 𝑒 𝑚4𝑦 +b17𝑒 (𝑚3+𝑚4)𝑦 +𝑏18 𝑒 2𝑝𝑟𝑦 +b19𝑏24 𝑒 −2𝑠𝑐 𝑦 +𝑏20 𝑒 (𝑝𝑟−𝑠𝑐 )𝑦 +𝑏21 𝑒 (𝑚3+pr )𝑦 +𝑏22 𝑒 (𝑚3−sc )𝑦 +b23𝑏23 𝑒 (𝑚3+pr )𝑦 +𝑏24 𝑒 (𝑚3−𝑠𝑐 )𝑦 . 𝑢1 = − 𝑠𝑐
1 𝑀𝑝𝑚
(𝑐6 𝑚5 𝑒 𝑚5𝑦 + 𝑐7 𝑚6 𝑒 𝑚6𝑦 + 𝑐8 𝑝𝑟 𝑒 𝑝𝑟𝑦 + 𝑎25 2𝑚1 𝑒 2𝑚1𝑦 + 𝑎26 2𝑚2 𝑒 2𝑚2𝑦 + 𝑎27 ( 𝑚1 + 𝑚2 ) 𝑒 (𝑚1+𝑚2)𝑦 + 𝑎28 (-2Sc) 𝑒 −2𝑠𝑐 𝑦 + 𝑎29 2𝑒 2𝑝𝑟𝑦 + 𝑎30 ( 𝑝𝑟 -
𝑒 (𝑝𝑟−𝑠𝑐 )𝑦
)
+
𝑒 (𝑚2+pr)𝑦
𝑎34 ( 𝑚2 + 𝑝𝑟
)
𝑏21 ( 𝑚3 + 𝑝𝑟 )
𝑒 (𝑚3+pr )𝑦
+
𝑎26 𝑒 2𝑚2𝑦
+𝑏15 𝑒
2𝑚3 𝑦
𝑎31
2𝑚4 𝑦
𝑚1
𝑏15 2𝑚3
+𝑏17 𝑒
+
-
𝑒 2𝑚3 𝑦
𝑏22 ( 𝑚3 - 𝑠𝑐 )
𝑎27 𝑒 (𝑚1+𝑚2)𝑦
+
+𝑏16 𝑒
+
+
(
𝑠𝑐 +
𝑏16 𝑚4
𝑒 (𝑚3−𝑠𝑐 )𝑦
𝑎28 𝑒 −2𝑠𝑐 𝑦
(𝑚3 +𝑚4 )𝑦
+𝑏18 𝑒
𝑒 (𝑚1−𝑠𝑐 )𝑦
)
2𝑝𝑟 𝑦
+
𝑒 2𝑚4𝑦
+
+
𝑎32
+
𝑏17 ( 𝑚3 + 𝑚4
𝑏23 ( 𝑚3 + 𝑝𝑟 )
𝑎29 𝑒 2𝑝𝑟𝑦
+𝑏19 𝑒
−2𝑠𝑐 𝑦
+
+𝑏20 𝑒
𝑚1
( )
𝑒 (𝑚3+𝑚4)𝑦
𝑒 (𝑚3+pr )𝑦
+
𝑎30 𝑒 (𝑝𝑟 −𝑠𝑐 )𝑦 (𝑝𝑟 −𝑠𝑐 )𝑦
𝑝𝑟
+
+𝑏21 𝑒
+
)
𝑏18 2𝑝𝑟
𝑏24 ( 𝑚3 - 𝑠𝑐 ) +
𝑒 (𝑚1+𝑝𝑟)𝑦
𝑒 (𝑚3−𝑠𝑐 )𝑦
𝑎31 𝑒 (𝑚1−𝑠𝑐 )𝑦
(𝑚3 +pr )𝑦
+𝑏22 𝑒
𝑒 2𝑝𝑟𝑦
+
(𝑚3 −𝑠𝑐 )𝑦
+
𝑏19
]-
𝑎33
-
−
(-
2𝑠𝑐
1 𝑀
[
( )
𝑐6
𝑚2
𝑒 (−2𝑠𝑐 𝑦) 𝑒 𝑚5 𝑦
+
+
𝑐7
𝑠𝑐
)
𝑒 (𝑚2−𝑠𝑐 )𝑦
𝑏20 ( 𝑝𝑟 - 𝑠𝑐 𝑒 𝑚6 𝑦
+ 𝑐8
)
𝑒 𝑝𝑟𝑦
𝑒 (𝑝𝑟−𝑠𝑐 )𝑦 + 𝑎25
+𝑏23 𝑒
(𝑚3 +𝑝𝑟 )𝑦
+𝑏24 𝑒
(𝑚3 −𝑠𝑐 )𝑦
) + 𝑏21 +
𝑏22 + 𝑏23 + 𝑏24 )) 𝑚5 𝑐72 = 𝑐8 𝑝𝑟 + 𝑎25 2 𝑚1 + 𝑎26 2 𝑚2 + 𝑎27 (𝑚1 + 𝑚2 ) + 𝑎28 (-2 𝑠𝑐 ): 𝑝𝑟 𝑎29
+ 𝑎30 (𝑝𝑟 - 𝑠𝑐 ) + 𝑎31 (𝑚1 - 𝑠𝑐 ) + 𝑎32 (𝑚1 + 𝑝𝑟 ) - 𝑎33 (𝑚2 - 𝑠𝑐 ) - 𝑎34 (𝑚2 + Pr) + 𝑏15 2
𝑚3 + 𝑏16 2 𝑚4 + 𝑏17
(𝑚3 + 𝑚4 ) + 𝑏18 2 𝑝𝑟 :
International Educational Scientific Research Journal [IESRJ]
+
𝑒 𝑚1 𝑦
𝑎32 𝑒 (𝑚1+pr )𝑦 - 𝑎33 𝑒 (𝑚2−𝑠𝑐 )𝑦 - 𝑎34 𝑒 (𝑚2+pr )𝑦
𝑐71 = ((𝑐8 + 𝑎25 + 𝑎26 + 𝑎27 + 𝑎28 + 𝑎29 + 𝑎30 + 𝑎31 + 𝑎32 - 𝑎33 - 𝑎34 + 𝑏15 + 𝑏16 + 𝑏17 + 𝑏18 + 𝑏19 + 𝑏20
𝑐73 = 2
-
101
Research Paper
E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016
đ?‘?74 = đ?‘?19 (-2 đ?‘ đ?‘? ) + đ?‘?20 (đ?‘?đ?‘&#x; - đ?‘ đ?‘? ) + đ?‘?21 (đ?‘š3 + đ?‘?đ?‘&#x; ) + đ?‘?22 (đ?‘š3 - đ?‘ đ?‘? ) + đ?‘?23 (đ?‘š3 + đ?‘?đ?‘&#x; ) + b24 đ?‘?24 (đ?‘š3 - đ?‘ đ?‘? ): đ?‘?7 = (1 / (đ?‘š5 - đ?‘š6 )) (đ?‘?72 + đ?‘?73 + đ?‘?74 - đ?‘?71 ): đ?‘?6 = -(đ?‘?7 + c8 + đ?‘Ž25 + đ?‘Ž26 + đ?‘Ž27 + đ?‘Ž28 + đ?‘Ž29 + đ?‘Ž30 + đ?‘Ž31 + đ?‘Ž32 - đ?‘Ž33 - đ?‘Ž34 + đ?‘?15 + đ?‘?16 + đ?‘?17 + đ?‘?18 + đ?‘?19 + đ?‘?20 + đ?‘?21 + đ?‘?22 + đ?‘?23 + đ?‘?24 )) 9.
RESULT AND DISCUSSION
The purpose of study is to discuss the effects of induced field (pm) and mass transfer (Sc) on velocity field(u), temperature field (T), magnetic field (H), skin friction at the đ?‘‘đ?‘˘ đ?‘‘đ?‘‡ plate [ď ´ ]y= 0 , and the rate of heattransferat the plate [Q] y= 0 . In calculating numerical values of the physical quantities u, H, T, , , ď ´ and Q, we have considered đ?‘‘đ?‘Ś
đ?‘‘đ?‘Ś
Gm = 4 (arbitarily); E=0.001 so to satisfy the series expansion [13]; Gr = 10 because it relates to the various problems in engineering and technologycal activities; Mass transfer parameter Sc varied as 0.22 (for Hydrogen), 0.6 (for Water vapour ) & 0.75 (for Oxygen). The physical parameters whose effects on flow are the objectives of this study i.e the induced field parameter Pm (Magnetic Prandtl Number )is varied as 0.5 to 4.5 while applied field parameter M (Hartmann Number) is varied from 5.0 to 20.0; We consider that Pm  0.5 means slow affect of induced field while Pm  2.5 moderate affect of the induced field. Further the parameter relates viscous with thermal diffussion i.e. Prandtl Number (Pr) is varied as 0.015 ( for mercury) , 0.16 ( for noble gase ), 0.7 ( for water at 20 0 c ).Following are the results from the figures (1-5) drawn using numerical values of the physical parameters given above. Fig. 1(i – ix): Velocity Profile, variation of u with Pm for Sc = 0.22; Gr=10; Gm =4; U=1; K=.5; E=0.001;
3500
Pr=0.015: M=5.0
Pr=0.015: M=10.0
1E+09
250000
3000
Pr=0.015: M=15:.0
0
u
Pm= 2.5
2500
200000
-1E+09
Pm=2.5 2000 1500
150000
-2E+09
u
-3E+09
500
Pm=0.5
0 0 3500
0.5
y
0.5
-7E+09
1
-50000
Pm=2.5
-8E+09
250000
Pr=0.16: M=5.0
Pm= 1.5
-6E+09
Pm=0.5
0 1.5
1
-5E+09
Pm=1.5 0
y 1
0.5
-4E+09
50000
Pm=1.5
y
u
100000
1000
0
450000
Pr=0.16: M=10.0
Pr=0.16: M=10.0
400000
3000
200000
Pm=2.5
2500
350000
u
u
150000
1500
100000
Pm=2.5
300000
2000
1000
Pm=1.5
50000
Pm=0.5
0
500 0 0
0.5
1
y 1.5
Pm=2.5
u
250000 200000 150000
0 -50000
International Educational Scientific Research Journal [IESRJ]
0.5 Pm=0.5
y
Pm=1.5
100000
Pm=1.5
50000 0 1
-50000 0
0.5
Pm=0.5 y
1
102
Research Paper 16000
E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016 250000
Pr=0.7: M=5.0
14000
450000
Pr=0.7: M=10.0
200000
u
12000
Pm=2.5
10000
350000
Pm=2.5
300000
u
150000
Pm=2.5
250000
u
8000
Pr=0.7: M=15.0
400000
200000
100000
150000
6000
Pm=1.5
50000
4000 2000 0 0
Pm=1.5 Pm= 0.5
y
0.5
1
100000
Pm=1.5
50000 0
y 0.5
0
Pm=0.5
1
-50000
0
Pm=0.5
y
-50000 0
0.5
1
Fig 1. Shows variation of fluid velocity with Pm, M & Pr. It is clear from the figures that when M remains unchanged, u increases with Pm; the rate of increase is higher for higher values of Pm. keeping Pm constant, and M rises up, u increases with largely; e.g. at y = 1.0, & Pm = 1.5, u = 815.7 for M=5.0 while u = 22581.4 for M = 10.0; similarly for Pm = 0.5 and Pm = 2.5. For higher values of M (> 10) and lower value of Pr, velocityu reverses its direction. The variation of u with Pm almost same in nature when Pr increases, keeping M unchanged. Fig 2 (i – ix): Temperature Profile, variation of T with Pm for Sc = 0.22; Gr =10; Gm =4; U=1; K=.5; E=0.001;
Pr=0.015 M=5.0
0 0
-50
0.5
1
Pr=0.015 M=10
20000
y 1.5
y
0
T
0
-100
0.2
0.4
-20000
0.6
Pm=0.5
-250
Pm= 1.5
-80000
Pm= 2.5
-100000
Pr=0.16 M=5.0
0
0.5
1
T
-1000
Pm= 0.5
-1500
-2500 -3000 -3500
y
0 -200000
0
0.2
-400000
0.4
0.6
Pm= 1.5
-800000
Pm= 2.5
-800000 -1000000
Pm= 2.5
y 0
0.8
0.2
0.4
0.6
Pm= 0.5
-4000000
T
Pm= 1.5
-6000000
Pm= 1.5
-8000000
Pm= 2.5
Pm= 2.5
-4000
-1200000
-10000000
-4500
-1400000
-12000000
International Educational Scientific Research Journal [IESRJ]
Pr=0.16 M=15.0
0 -2000000
-600000
Pm= 1.5
-600000
Pm= 0.5
T
-2000
Pm= 0.5
2000000
Pr=0.16 M=10.0
200000
y 1.5
0.4 y 0.6
-1000000
0 -500
-200000
Pm= 2.5
-120000
-450 500
0.2
Pm= 1.5
-350 -400
0
-400000
-60000
-300
0
T
-40000
-200
Pr=0.015: M=15:
0.8
T
Pm= 0.5
-150
200000
103
Research Paper
E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016
Pr=0.7 M=5.0
0 0
0.5
1
1.5
-1000000 0
0.2
0.4
-2000000
T -15000
Pm= 1.5
Pr=0.7 M=15.0
-10000000
0
0.2
T
-3000000 -4000000
-30000000
Pm= 1.5
y
0
0.8
-20000000
-5000000
-20000
0.6
Pm= 0.5
T
-10000
10000000
y
0
Pm= 0.5
-5000
Pr=0.7 M=10.0
1000000
y
0.4
0.6
Pm=0.5
Pm=1.5
-40000000
-6000000
Pm= 2.5
-50000000
-7000000
-25000
Pm= 2.5
-8000000 -9000000
-30000
Pm=2.5
-60000000 -70000000
Fig 2. Show variation of fluid Temperature (T) with Pm, M & Pr. The figures show that within the medium, T decreases with the rise of Pm for all values of M (= 5.0, 10.0, 15.0) & Pr (= 0.015, 0.16, 0.7). Keeping Pm constant, T decreases with increase of M largely ( e.g. at y = 0.8 & Pm = 1.5, T = -100.26 for M=5.0 while T = �� 3968069.0 for M = 10.0; similarly for Pm = 0.5 and Pm = 2.5 ); this implies higher rate of heat transfer -[ ] at higher values of M. When M & Pm are at constant, T ��
decreases with increase of Pr ; this implies lower the thermal diffusion, lower fluid temperature. The nature of variation of T with Pm at constant M is almost same for different higher values of Pr (= 0.015, 0.16, 0.7). Fig 3 (i – iii): Magnetic field Profile variation of H with Pm, for Sc = 0.22; Gr=10; Gm =4; U=1; K=.5; E=0.001;
500
y y
0
H
0.5
Pm= 0.5 1.5 1
y
0.5
1
-40000
Pm= 1.5
Pm= 1.5
0
-1500
-20000
-80000
-30000
-140000 -2500
-160000
0.6
0.8
Pm= 1.5
Pm= 2.5 -40000
-120000 -2000
0.4
H
-60000
-100000
0.2
-10000
-1000
Pm= 2.5
Pm= 0.5
0
-20000
H
Pr=0.015: M=15.0
10000
Pm= 0.5
0 0
0 -500
Pr=0.015: M=10.0
20000
Pr=0.015: M=5.0
-50000
Pm= 2.5
-60000
Fig 3. Show variation of magnetic field with Pm, M & Pr. Away from the plate, H decreases for all values of M,Pm & Pr ; the rate of decrease is more when Pm is higher; similarly for M when Pm is constnt. Field a very near to the plate is less variant of Pm depending upon applied field ( e.g. field is less variant within y ≤ 2.0 when M = 5.0 while it is y ≤ 5.0 when M = 10). When M increases at constant Pm, H decreases largely (e.g. at y = 0.4, & Pm = 1.5; H = -21.022 for M=5.0 while H = -162.344 for M = 10.0 and H = --1437.68 for M = 15.0); similarly for Pm = 0.5 and Pm = 2.5.
International Educational Scientific Research Journal [IESRJ]
104
Research Paper
E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016
Fig 4 (i – ix): Shear Stress ( ), variation with Pm & M
2000 1800 1600 1400 1200 1000 800 600 400 200 0 -200 0 60
60
y=0, Pr=0.016
for Sc = 0.22; Gr =10; Gm =4; U=1; K=.5; E=0.001;
70
Pr=0.015, M=5.0
50
M= 5.0
Sc=0.22
40
M= 10.0
Sc=0.6
M= 20.0
20
Sc=0.75
40
Sc=0.6
30
Sc=0.75
20
10
2
6 Pm 8
4
10
Pm
0 0
1
2
3
40 30
Sc =0.22
50 40
Sc = 0.6
30
Sc =0.75
Sc=0.6 Sc=0.75
20 10
Pm 0 2
10
Pm
70
Pr=0.7, M=5.0
60
1
Sc =0.22 Sc=0. 6 Sc=0.75
20
30 20
10
10
Pm
0 1
2
Pm
0 3
3
0
1
100 90 80 Sc=0.22 70 60 Sc=0.6 50 40 Sc=0.75 30 20 10 0
40
30
Pm
Pr=0.7, M=10.0
50
40
Sc = 0.6 & 0.75
0
2
60 50
Sc = 0.22
30
10 0
0
1
3
40
20
3
2
Pr=0.16, M=20.0
50
20
0 1
1
60
60
Sc=0.22
0 70
70
50
Pm
0
Pr=0.16, M=10.0
0
Sc=0.22
50
30
Pr=0.16, M=5.0
0
Pr=0.015, M=10.0
60
2
2
3
Pr=0.7, M=20.0
Sc=0.22 Sc=0.6 Sc=0.75
Pm 0
1
2
3
Fig 4. Show variation of Shear stress () experienced by the fluid motion at the plate ( y=0 ) with Pm, M & Pr. at the plate y = 0, varies inversely with Sc when M, Pm &Pr are fixed. For all values of M, decreases rapidly with the increase of Pm within lower values of Pm 2.0 (approx.), while increases slowly beyond it. When Pm, M &Pr are constant, increases with the rise of M ; decreases with rise of Sc at constant Pm, Pr& M. When field is high M 20 , the variation of with Sc becomes less significant but becomes prominent when both M &Pr are high. This implies the strong effect of Pron .
International Educational Scientific Research Journal [IESRJ]
105
Research Paper
E-ISSN No : 2454-9916 | Volume : 2 | Issue : 11 | Nov 2016 đ?’…đ?‘ť
Fig 5: Rate of heat transfer, proportional to (
đ?’…đ?’š
) Variation with Pm & M for Gr=10; Gm =4; U=1; K=.5; E=0.001;
500
2500000
Pm=2.5
400 350
Pr=0.015, M=10.0
Pr=0.015, M=5.0
450
Q
2000000
Pm=2.5
1500000
300 250
Pm=1.5
200
1000000
Q
Pm= 0.5
500000
150
Pm=1.5
100
0
Pm=0.5
50
0
0 0
0.2
6000
0.4
0.6
0.8
y 1
60000000
y 0.6
0.8
Pr=0.16, M=5.0
Pm=0.5
50000000
Pm=2.5
Q
0.4
-500000
Pr=0.16, M=5.0
5000
0.2
40000000
4000
Q
Pm=1.5
3000
30000000
Pm=2.5
20000000 2000
Pm=0.5
1000
Pm=1.5
10000000 0 0
0 0
0.2
0.4
0.6
0.8
y 1
-10000000
0.2
0.4
y
0.6
0.8
Fig 5. Show variation of Q away from the plate. A little away from the plate y >0 (depending upon M), Q is less variant for the variation of Pm and Pr ( e.g. for M= 5.0, y ≤ 4.0 aprox. while M= 10.0, y ≤ 2.0 aprox.); beyond this, Q increases sharply with the increase of Pm. The limiting value of y beyond which Q varies linearly with Pm depends upon M. The rate of increase of Q within the medium is more when Pm is higher. The variation of Q within the medium is higher when either of M &Pr is higher and when both M &Pr are higher Q is much higher within the medium. 10.
CONCLUSION  
 

11.
Applied field(H), Induced field (Pm) and thermal diffusivity factor ( Pr) have influence over fluid velocity. Fluid velocity increases with the increase of applied field and induced field. The direction of fluid velocity reverses when M & Pm is higher but Pr is lower. The effect of H, Pm ,Pr on Fluid temperature (T) is opposite in nature to the variation of velocity as above. Temperature is much low when all the three parameters Pm, M &Pr are higher. This implies strong thermal diffusivity effect with magnetic interaction and also higher rate of heat transfer within the medium. Away from the plate, Magnetic field within the medium decreases with the rise of induced field; this implies strong magnetic diffusivity within the medium away from the plate. At very close to the plate, magneric field is almost uniform ( i.e. less variant with induced field) depending upon applied field. The effect of applied field on shear stress significantly depends upon induced field. The nature of variation of shear stress is almost opposite within lower level of induced field and beyond it; the level depends upon applied field. Bothinduced field and mass transfer parameter have significant role over the shear stress. Applied field, Induced field and thermal diffusivity factor have distinctive influence over the heat transfer. Heat transfer is more at higher induced field. It is further higherwhen thermal diffusivity factor ( Pr) is higher along with higher applied field (H)but less induced field(Pm); this implies higher rate thermal and magnetic interaction. REFERENCES
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