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Magnetic effects on free convection flow of Nano fluids through porous
medium past an infinite vertical plate in slip flow regime in the presence of
variable heat source and variable suction
Dr Rajendra Kumar Dhal
*Dr Banamali Jena
**Mr P M Sreekumar
J.N.V. Hadgarh, Keonjhar, Odisha- 758023
Email: [email protected]
*J.N.V. Goshala, Sambalpur, Odisha- 768024
Email:[email protected]
**J.N.V. Joura, Morena, MP-476221
Email:[email protected]
ABSTRACT: Magnetic effects on free convection flow of nano fluid through porous medium
past an infinite vertical plate in slip flow regime in the presence of both variable heat source
and variable suction for water- Cu and water - Al2O3 has been studied. The influences of the
various parameters on the flow field, Heat field, mean value of skin friction and tangential
value of phase angle are extensively discussed from graphs and tables.
KEY WORDS: MHD, Free convection, Heat Transfer, variable Heat Source, slip flow regime,
variable Suction, nano fluid, porous medium.
1.INTRODUCTION: Free convection fluids like oil, water and ethylene glycol have poor heat
transfer capacity due to their poor thermal conductivity. To enhance the thermal conductivity of
these fluids, investigatorshave added nano particles (nano size) of base metals (Al, Cu), Oxides
(Al2O3, TiO2), Nitriles (AlN, SiN) and Carbides (SiC), etc. to base fluids. Thus the convective
heat transfer of the base liquids will increase. Nano fluids contribute to lower heat exchanger
size because of their thermal and flow characteristics. Hence, nano fluids are used in
microelectronics and in chips of computer devices.Nano fluids in the presence or absence of
magnetic field have enumerable applications in the industries due to unique chemical and
physical properties of nanometer sized materials. Sundry utilizations of nano fluids, the cooling
applications of nano fluids incorporate silcon mirror cooling, electronics cooling, vehicle
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cooling, transformer cooling and so on, are some bright instances. The hypothesis of nano fluid
was first presented by S. U. S. Choi and has been a field of dynamic research area for two
decades. Choi et. al. [1] has demonstrated experimentally that the injection of nano particles
enhances the themal conductivity of the fluid. Buongiorno [2] developed a mathematical model
for nanofluid and explored its various transport mechanisms of nano fluids. Mahian et. al. [3]
have discussed the irreversible analysis of the vertical annulus using water-TiO2 fluid with
MHD flow effects. Shehzad et. al. [4] have studied the MHD mixed convective peristaltic
motion of nano fluid with joule heating and thermophoresis effects. Venkataramanaiah et. al.
[5] have clarified about the nano particles effect on MHD boundary layer flow of Williamson
fluid over a stretching sheet. Zubair et. al. [6] have discussed the Heat and Mass Transfer
analysis of MHD Nano fluid flow with Radiative Heat Effects in the presence of Spherical Au-
Metallic Nano particles.Khan et. al. [7] have discoursed about the MHD Williamson nano fluid
with chemical reaction. Thumma et. al. [8] have investigated about the numerical study of heat
source on dissipative magnetic nano fluid flow from a non linear inclined stretching surface.
Vedavathi et. al. [9] have discussed on heat transfer on MHD nano fluid flow over a semi
infinite flat plate embedded in a porous medium with radiation absorption, heat source and
diffusion thermo effect. Mohyud-Din et. al. [10] have presented a study of heat and mass
transfer on magnetohydrodynamic (MHD) flow of nano particles. D Vidyanandha Babu [11]
has studied the effect of Dufour and thermal radiation on convection radiative nano fluid flow
with suction and heat source. Govardhan et. al. [12] have studied on Heat and Mass transfer in
MHD Nano fluid over a STretching Surface along with Viscous Dissipation effect.
In this problem, we try to investigate the Magnetic effects on free convection flow of nano fluid
through porous medium past an infinite vertical plate in slip flow regime in the presence of
variable heat source and variable suction both for water-Cu and Water-Al2O3.
2. FORMULATION OF PROBLEM: An unsteady free convection two dimensional flow of an
incompressible, electrically conducting viscous nano fluid through porous medium past an
infinite vertical plate in slip flow regime in the presence of chemical reaction, variable heat
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source and variable suction is considered. Let πβ² axis is taken along the plate in vertical upward
direction and πβ² axis is normal to the plate. A uniform magnetic field strength H0 is applied
normal to the plate. Initially, sorrounding temperature is πββ² . Also, the temperature at the plate
is ππ€β² . For π‘ β² β₯ 0, the temperature and mass concentration changes periodically. As the plate is
infinite along πβ² axis, so all the physical quantities are in dependentof π₯β² and are the functions
of π¦β² and π‘ β²only. It is assumed that viscous dissipation and joulean dissipation are neglected,
density varies in the body force term and variable heat source is taken as π£0
2
ππ 1 + ππ΅πππ β² π‘ β²
.
Then by usual Boussinesq's approximation the unsteady flow is governed by the following
equations.
Equation of Continuity:
ππ£ β²
ππ¦ β²= 0 βΉ π£ β² = βπ0 1 + ππ΄πππ β²π‘ β² (1)
The Equation of Motion:
ππ’ β²
ππ‘ β²β π0 1 + ππ΄πππ β²π‘ β²
ππ’ β²
ππ¦ β²=
πππ
πππ
π2π’ β²
ππ¦ β²2 + π π½ ππ π β² β πββ² + π π½π ππ πΆ β² β πΆβ
β² βππ΅0
2π’ β²
πππβ
π’ β²
πΎβ²(2)
The Energy Equation:
ππ β²
ππ‘ β²β π0 1 + ππ΄πππ β²π‘ β²
ππ β²
ππ¦ β²=
πππ
ππΆπ ππ
π2πβ²
ππ¦ β²2β
π02
ππ 1 + ππ΅πππ β²π‘ β² π β² β πβ
β² (3)
with the following boundary conditions
π‘ > 0: π’β² = πΏβ²
ππ’ β²
ππ¦ β², π β² = ππ€
β² + π ππ€β² β πβ
β² πππ β²π‘β²ππ‘π¦ = 0,
π’β² = 0 , π β² = πββ² ππ‘π¦ β β
(4)
where πππ is the dynamic viscosity,πππ is the thermal diffusivity,πππ is the effective density,
ππΆπ ππ
is the heat capacity and π½ ππ is the coefficient of volumetric expansion of heat of nano
fluid. They are defind as
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πππ =ππ
1βπ 2.5 ,πππ = 1 β π ππ + πππ , ππΆπ ππ
= 1 β π ππΆπ π
+ π ππΆπ π ,
πππ = ππ + 2ππ β 2π ππ β ππ
ππ + 2ππ + 2π ππ β ππ ππ , π½ ππ = 1 β π π½ π + π π½ π
π is the electrical conductivity of the fluid, g is the acceleration due to gravity and πis the
volume of nano fluid.
Let us introduce the dimensionless quantities
π’ =
π’ β²
π0 , π‘ =
π02π‘ β²
ππ , π¦ =
π0π¦ β²
ππ , π =
π β²βπββ²
ππ€β² βπβ
β² , πΊπ =π π½ πππ ππ€
β² βπββ²
π03 ,
ππ =ππ ππΆπ
π
ππ , π =
ππππ΅02
πππ02 , π =
π β²
π03 ππ , πΎ =
πΎβ²π02
ππ , π =
π0πΏβ²
ππ
(5)
where Gr is the Grashof number, K is permeability of porous medium, M is magnetic
parameter, Pr is Prandtl number, h is rarefraction parameter, A is suction parameter and B is
heat source parameter.
Substituting equation (5) in equations (2) to (3) with boundary condition (4), we have
ππ’
ππ‘β 1 + ππ΄ππππ‘
ππ’
ππ¦= π0
π2π’
ππ¦ 2 β π1ππ’ β1
πΎπ’ + π2πΊππ(6)
ππ
ππ‘β 1 + ππ΄ππππ‘
ππ
ππ¦=
1
πππ3
π2π
ππ¦ 2 β 1 + ππ΅ππππ‘ π (7)
with boundary conditions
π’ = hβu
βy, π = 1 + πππππ‘ at π¦ = 0
π’ = 0, π β 0, asπ¦ β β (8)
3. METHOD OF SOLUTION:
Assuming small amplitude oscillationπ βͺ 1, we can represent the velocity u and temperature
πnear the plate as follows
π’ = π’0 π¦ + ππ’1 π¦ ππππ‘
π½ = π½π π + ππ½π π ππππ (9)
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Substituting (9) in (6) to (8) and equating the coefficient of harmonic and non harmonic terms
with neglecting the coefficient of π2, we get
π0 π’0β²β² + π’0
β² β π1π + 1
πΎ π’0 = βπΊππ2π0 (10)
π0π’1β²β² + π’1
β² β ππ + π1π + 1
πΎ π’1 = βπ’0
β² π΄ β πΊππ2π1 (11)
π3π0β²β² + πππ0
β² β πππ0 = 0 (12)
π3π1β²β² + πππ1
β² β ππ 1 + ππ π1 = πππ΅π0 β πππ΄π0β² (13)
with the following boundary conditions
π’0 = hβu0
βy, π’1 = h
βu1
βy, π0 = 1, π1 = 1 at π¦ = 0
π’0 = 0, π’1 = 0, π0 = 0, π1 = 0 atπ¦ β β (14)
Solving the equations (10) to (13) using boundary condition (14) and (9), we get
u = b2 eβa2y β b1 e
βa1y + b8eβa4y + b5eβa2y + b6eβa1y + b7 eβa3y Ο΅eiΟt (15)
ΞΈ = eβa1y + b4eβa3y + b3eβa1y Ο΅eiΟt (16) The non-
dimensional skin friction
π0 = ππ’
ππ¦
π¦=0
= βπ2π2 + π1π1 + βπ4π8 β π2π5 β π1π6 β π3π7 πππππ‘
= βπ2π2 + π1π1 + πππππ‘ πΉπ + ππΉπ = ππ + π πΉ ππ ππ‘ +πΌ (17)
where πΉ = πΉπ2 + πΉπ
2 , π‘πππΌ =πΉπ
πΉπ , ππ = βπ2π2 + π1π1 Is the skin friction of mean velocity
and F= βπ4π8 β π2π5 β π1π6 β π3π7
The non-dimensional Nusselt number
ππ’ = β ππ
ππ¦
π¦=0
= π1 + π3π4 + π1π3 πππππ‘ = π1 + πππππ‘ π1 + ππ2
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=π1 + π πππ ππ‘ +π½ (18)
where π = π12 + π2
2π‘πππ½ =π2
π1, Ο
0=
1
1βΟ 2.5 1βΟ +ΟΟsΟf
, Ο1
=1
1βΟ +ΟΟsΟf
,
π2 = 1 β Ο + ΟΞ²s
Ξ²f
, π3 = ππ +2ππβ2π ππβππ
ππ +2ππ+2π ππβππ
1βπ +π ππΆπ
π ππΆπ
π
,
π1 =ππ+ ππ 2+4π3ππ
2π3, π2 =
1+ 1+4π0 π1π+ 1
πΎ
2π0 , π3 =
ππ+ ππ2+4πππ3 1+ππ
2π3,
π4 =1 + 1 + 4 π0 ππ + π1π +
1
πΎ
2π0 , π1 =
π1πΊπ
π0π12 β π1 β ππ1π +
1
πΎ
, π2
=π1 1 + ππ1
1 + ππ2, π3 =
πππ΅ + πππ΄π1
π3π12 β πππ1 β ππ 1 + ππ
,
π4 = βπ3, π5 =π΄π2
π0π22 β π2 β ππ + π1π +
1
πΎ
, π6 = β π΄π1π2 + πΊππ2
π0π12 β π1 β ππ + π1π +
1
πΎ
,
π7 = βπΊππ2π4
π0π32 β π3 β ππ + π1π +
1
πΎ
, π8 = β ππ3π5 + ππ1π6 + ππ3π7 + π5 + π6 + π7
1 + ππ4
4. RESULT AND DISCUSSION:
In this paper, Magnetic effects on free convection flow of nano fluid through porous medium
past an infinite vertical plate in slip flow regime in the presence of variable heat source and
variable suction has been compared between water-Cu and water-Al2O3. The effect of the
parameters Gr, M, K, Pr, A, B, π and h on flow characteristics have been studied and shown by
means of graphs and tables. In order to have physical correlation, we choose suitable values of
flow parameters and values of parameters for two different nano fluids in table-1. The graph of
heat and velocities are taken w.r.t. to distance y. The mean value of shearing stress and tangent
value of phase angle are shown in table.
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Velocity Profile: The velocity profiles are depicted in Fig 1-7. Figure-1 shows the effects of the
parameter M on velocity profile at any point of the fluid when Pr = 6.86, K = 2, A = 2, B = 2,
Gr = 2, t = 0.5, Ο = 0.05and h = 0.2. It is noticed that the velocity decreases with the increase
of Magnetic parameter (M). Initially, the velocity of water-Cu nano fluid is more than the
water-Al2O3 nano fluid.
Figure-(2) shows the effects of the parameter K on velocity profile at any point of the fluid
when Pr = 6.86, M = 2, A = 2, B = 2, Gr = 2, t = 0.5, Ο = 0.05and h = 0.2. It is noticed that the
velocity increases with the increase of permeability of porous medium (K). Initially the velocity
of water-Cu nano fluid is more than water-Al2O3 nano fluid.
Figure-(3) shows the effect of the parameter Gr on velocity profile at any point of the fluid
when Pr = 6.68, M = 2, A = 2, B = 2, K = 2, t = 0.5,Ο = 0.05and h = 0.2. It is noticed that the
velocity increases with the increase of Grashof number (Gr).
Figure-(4) shows the effect of the parameter A on velocity profile at any point of the fluid,
when Pr = 6.86, M = 2, K = 2, B = 2, Gr = 2, t = 0.5,Ο = 0.05and h = 0.2. It is noticed that the
velocity increases with the increase of Suction parameter (A). Initially, the velocity of water-Cu
nano fluid is more than water-Al2O3 nano fluid.
Figure-(5) shows the effect of the parameterπ on velocity profile at any point of the fluid, when
Pr = 6.86, M = 2, K = 2, A = 2, B = 2, Gr = 2, t = 0.5 and h = 0.2. It is noticed that the velocity
increases with the increase of volume fraction of nano fluid π . The velocity of water-Cu nano
fluid is more than water-Al2O3 nano fluid near the plate.
Figure-(6) shows the effect of the parameter h on velocity profile at any point of the fluid, when
Pr = 6.86, M = 2, K = 2, A = 2, B = 2, Gr = 2, t = 0.5 and Ο = 0.05.It is noticed that the
velocity increases with the increase of rarefaction parameter (h). The velocity of water-Cu nano
fliuid is more than water-Al2O3 nano fluid near the plate.
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Figure-(7) shows the effect of the parameter B on velocity profile at any point of the fluid,
when Pr = 6.86, M = 2, K = 2, A = 2, Gr = 2, t = 0.5, π = 0.05, and h = 0.2. It is noticed that
the velocity increases slowly with the increase of Source parameter (B). The velocity of water-
Cu nano fluid is more than water-Al2O3 nano fluid near the plate.
Heat Profile: The heat profiles are depicted in Fig 8-10. Figure-(8) shows the effect of the
parameter B on heat profile at any point of the fluid, when Pr = 6.86, A = 2, t = 0.5 and π =
0.05. It is noticed that the temperature slowly rises in the increase of heat source (B).The heat
of water-Cu nano fluid is less than the water- Al2O3 at every point of the flow field.
Figure-(9) shows the effect of the parameter A on heat profile at any point of the fluid, when Pr
= 6.86, B = 2, t = 0.5 and π = 0.05. It is noticed that the temperature slowly falls with the
increase of Suction Parameter (A). The heat of water-Cu nano fluid is less than water-Al2O3
nano fluid at every point of the flow field.
Figure-(10) shows the effect of the parameter π on heat profile at any point of the fluid, when
Pr = 6.86, A = 2, B = 2. It is noticed that the temperature rises with the increase of heat source
(B). The heat of water-Cu nano fluid is less than water-Al2O3 nano fluid at every point of the
flow field, but the rate of rise of heat of water-Cu nano fluid is more than water-Al2O3nano
fluid.
Skin Friction: The mean value of shearing stress and Tangent value of phase angle is depicted
in Table-(2), which illustrates the effects of the parameters M, K, Gr, A, h and π on mean value
of shearing stress and Tangent value of phase angle at the plate. It is noticed that Mean value of
skin friction at plate decreases with the increase of magnetic parameter (M), Suction parameter
(A) and rarefaction parameter (h), where as decreases with the increase of permeability of
porous medium (K), Grashof number (Gr) and volume fraction of nano fluid π . But the
Tangent value of phase angle increases with the increase of volume fraction of nano fluid π ,
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magnetic parameter (M) and Suction parameter (A), where as decreases with the increase of
permeability of porous medium (K) and rarefaction parameter (h).
Table-1
Sl.no parameters Base fluid
(water)
Nano
particles
( Cu)
Nano
particles
(Al2O3
1 Density 997.1 8933 3970
2 Specific heat at
constant
pressure
4179 386 765
3 coefficient of
volumetric
expansion
2.064X10β4 9.8X10β6 8.5X10β6
4 Thermal
diffusivity
0.613 401 40
Table-2
When B=2 and Sc=6.8 Mean value of
shearing
stress ππ
For
Al2 O3 -water
Mean value
of shearing
stress ππ
Cu-Water
Tangent
value of
phase
angle
π‘πππΌ
Cu-Water
Tangent
value of
phase
angle
π‘πππΌ
Al2 O3 -
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water
Initially
h=0.2,M=5,K=2,A
=2, Gr=2,π =
0.05
M=5 1.1026 1.3481 0.1581 0.1402
M=7 1.0188 1.2384 0.1629 0.1414
M=8 0.9530 1.1920 0.1653 0.1428
K=3 1.1433 1.3946 0.1544 0.1365
K=5 1.2124 1.4658 0.1482 0.1296
A=4 1.1026 1.3456 0.3022 0.2645
A=6 1.1026 1.3419 0.4469 0.3893
Gr=4 2.2053 2.6965 0.1581 0.1402
Gr=6 3.3079 4.0451 0.1581 0.1402
h=0.4 0.8162 0.9631 0.1616 0.1467
H=0.6 0.6479 0.7474 0.1641 0.1411
π
= 0.09
1.1059 1.3759 0.1636 0.1439
π
= 0.1
1.1084 1.4616 0.1691 0.1476
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Fig-(1): Effect of M on Velocity profile for water-Cu and water-Al2O3nano fluid, when Pr
= 6.86, K = 2, A = 2, B = 2, Gr = 2, t = 0.5, π = 0.05 and h = 0.2.
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Fig-(2): Effect of K on Velocity profile for water-Cu and water-Al2O3 nano fluid, when Pr
= 6.86, M= 2, A = 2, B = 2, Gr = 2, t = 0.5, π = 0.05 and h = 0.2.
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Fig-(3): Effect of Gr on Velocity profile of water-Cu and water-Al2O3 nano fluid, when Pr
= 6.86, K = 2, M = 2, A = 2, B = 2, t = 0.5, π = 0.05 and h = 0.2.
Fig-(4): Effect of A on velocity profile of water-Cu and water-Al2O3 nano fluid, when Pr =
6.86, M = 2, K = 2, B = 2, Gr = 2, t = 0.5, π = 0.05 and h = 0.2.
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Fig-(5): Effect of π on Velocity profile of water-Cu and water-Al2O3 nano fluid, when Pr
= 6.86, M = 2, K = 2, A = 2, B = 2, Gr = 2, t = 0.5 and h = 0.2.
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Fig-(6): Effect of h on Velocity profile of water-Cu and water-Al2O3 nano fluid, Pr =
6.86, M = 2, K = 2, A = 2, B = 2, Gr = 2, t = 0.5 and π = 0.05.
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Fig-(7): Effect of B on Velocity profile of water-Cu and water-Al2O3 nano fluid, when Pr =
6.86, M = 2, K = 2, A = 2, Gr = 2, t = 0.5, π = 0.05 and h = 0.2.
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Fig-(8): Effect B on Heat profile of water-Cu and water-Al2O3 nano fluid, when Pr = 6.86,
A = 2, t = 0.5 and π =0.05.
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Fig-(9): Effect of A on Heat profile for water-Cu and water-Al2O3 nano fluid, when Pr =
6.86, B = 2, t = 0.5 and π = 0.05.
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Fig-(10): Effect of π on Heat profile of water-Cu and water-Al2O3 nano fluid, when Pr =
6.86, B = 2, t = 0.5 and A = 2.
6. CONCLUSION:
The following results are obtained due to the magnetic effects on free convection flow of nano
fluid through porous medium past an infinite vertical plate in slip flow regime in the presence
of variable heat source and variable suction:
i. The velocities of both water-Cu and water-Al2O3 nano fluids decrease with the increase
in M, but increases with the increase of other parameters. Also velocity of water-Cu nano
fluid is more than the water-Al2O3 nano fluid near the plate.
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ii. Heat diminishes with the enhancement of the value of suction parameter (A), but rises
with the increase of B and π.
iii. Mean value of Skin friction at the plate decreases with the increase of M, A and h, where
as increases with the increase of K, Gr and π. But the tangent value of phase angle
increases with the increase of π, M and A, where as decreases with the increase of K and
h. No change is marked in case of increase in Gr.
iv. Both mean value of Skin friction and tangent value of phase angle is more in case of
water-Cu nano fluid than water-Al2O3 nano fluid for all parameters.
Reference
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Available online at http://euroasiapub.org
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Email:- [email protected], http://www.euroasiapub.org
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