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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 98
I J E N S IJENS © February 2020 IJENS -IJMME-4334-201601
MHD Mixed Convection in Square Enclosure Filled with non-Newtonian Nanofluid with Semicircular-
Corrugated Bottom Wall 1*Farooq H. Ali ,1Hameed K. Hamzah, 1Ahmed, Saba Y. 3, Emad D. Aboud2, Hayder K. Rashid1Hussein M. Jassim
Iraq. –Hilla–Babylon City-University of Babylon -Mechanical Engineering Department -College of Engineering1
Iraq. –llaHi–Babylon City-University of Babylon -Ceramic Engineering Department -College of Material Engineering2
Iraq. –Hilla–Babylon City-University of Babylon -Automobile Engineering Department -Musayab-College of Engineering/ Al3
* Corresponding author: Farooq Hassan Ali
[E-mail addresses:farooq_hassan77@yahoo.com]
Abstract-- Many researchers have been tried to discuss the
parameters that used to enhance the heat transfer rate
within different shapes of enclosure that exerted by an
external magnetic field with a lid driven. So, in this research,
combined effect of non-Newtonian fluid that contains Nano-
particles Al2O3 within new square enclosure heat geometry
shapes (four cases that will be elaborated in this research:
case I :small wavy notched shape, case II: small semicircle
notched, case III: bigger semicircle notched and the last one
case IV has a large wavy notched) have been solved
numerically by finite element method that base on the
Galerkin weighted residual formulation where used in
COMSOL Multiphasic under a relevant dimensionless
parameters: 0.001 Ri 1, 0 Ha 60, solid volume
fraction 0 0.1, power law index 0.2 n 1.4, Grashof
number Gr=100.
However, the innovation of the boundary
conditions are a new heating geometry shape wall has an
alternating effect with the variation of the power index
parameters and Richardson number. It can be shown that
case II of small circular notched is the best cases that
justified of improving forced and mixed convection heat
transfer. The magnetic field strength interplay negative
effect on improving the forced convection flow within the
enclosure. The average Nusselt number increases by
evolution of the power law index value while independent on
the shape geometry of the heating cavity wall.
Index Term-- MHD, Square enclosure, Non-Newtonian,
nano-fluid, semicircular corrugated hot wal
1. INTRODUCTION
There are many engineering applications of enclosure
with lid driven at different wall heating geometrical
shapes with presence of external magnetic field. The
effect of magnetic field is to produce a force that will
counteract the fluid flow direction which induced by
buoyancy force. The mechanistic of this type of forces
due to the applied magnetic field reported in details by
many previous researchers. However, the magnetic field
strength plays an important role in many engineering
requests. The mechanism of external magnetic field
applied on electrically conducting fluid called Magneto-
hydrodynamics (MHD). Besides, the working fluid and
nanofluid particle properties are very important to restrict
the convection flow. The mutually effect between the
power index for dilatant fluid and volume fraction of
nanoparticles on convective heat transport with presence
magnetic field can be detected in many applications.
These applications associated with the alternating effect of
different cavity conditions, have typical in many thermal
systems, electronic devices, MHD flow meter, plasma
physics and geophysics. On the other hand, fluid type and
geometry shape have a relevant interplay effect on flow
pattern. These modes of flow exerted on the effect of heat
transfer rate that occur in many actual applications like
chemical processing. However, all applications need
effective cooling and heat dissipation systems for all its
work on time. Moreover, the turbulent flow modes will
meliorates heat transfer rate. For this reason, one or more
wall will move at suitable velocity to rising turbulent
mode influence on the effectiveness of heat transfer rate.
Hence, enormous papers illustrated and propose new
enclosure conditions with lid driven dominant flush
situation to get more superiority of convection. Marchi et
al., 2009 [1] used the finite volume method to solve flow
problems in square enclosure. Where, all parameters
changed by keeping the lid velocity constant. Which
presented second-order accuracy in the approximation
solution to solve the governing equations with 42
variables numerically. However, the numerical solution
was used multiple Richardson extrapolations to degrading
the discretization error until the achieved machine round-
off error. On the other hand, Al-Salem et. al , 2011 [2]
investigated a numerical technique depends on the finite
volume method to study the MHD mixed convection
cavity. The model was adiabatic in vertical walls as well
as fixed temperature at the sliding top. Moreover, the heat
source of bottom wall modeled as linearly source. This
study used to reveals the effect of different lid movement
directions on the heat transfer enhancement. However,
many parameters were merged together to get new evolve
in mixed convection behavior such as: Reynolds number
and Grashof number that illustrated by stream function
and isotherm contour. In addition, Billah et. Al, 2011 [3]
solved the lid driven square cavity problem numerically
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I J E N S IJENS © February 2020 IJENS -IJMME-4334-201601
after heated by hollow circular cylinder. The position of
hollow cylinder is in the middle of the cavity. In Billah
work, the moving wall was vertical left one. In contrast,
the right vertical wall fixed. In addition, other walls,
which thermally insulated. The same convective flow
characteristic especially Richardson number, thermal
conductivity for solid particles that help to produce
nanofluid as well as Nusselt number which used to
explore the cylinder diameter values that effect on heat
transfer performs. The extrapolation of the mutually effect
between these parameters that equivalent to streamlines
and isotherm contours for multiples situations.
Sheikholeslami, et.al, 2012 [4] investigated numerically
by using the finite volume method the natural convection
heat transfer of Cu-water nanofluid in a circular enclosure
containing hot sinusoidal circular wall with the presence
of horizontal magnetic field. It is examined the effect of
different parameters on the fluid flow and heat transfer
rate such as: Hartmann number range (0-100), Rayleigh
number range (103-105), nanoparticle volume fraction (0-
0.06) and the number of undulations (3and 6) at constant
amplitude (A=0.5) and Prandtl number (Pr=6.2). It is
shown that increasing the Hartmann number makes the
isotherms parallel to each other i.e. all of the heat transfer
process is carried out by conduction. As the Rayleigh
number, number of undulations and nanoparticle volume
fraction increase, the average Nusselt number increases
and opposite effect with increasing Hartmann number.
The enhancement of heat transfer remains constant with
the increasing of Hartmann number on Ra=103 while, the
maximum value of the enhancement occurs at Ha=20 and
60 with Ra=104 and 106.
Moreover, Bhattacharya, et. al , 2013 [5] investigated the
trapezoidal cavity with constant top lid velocity. The
temperature of the bottom is greater than the top wall.
While, the sides are adiabatic with an inclination angle of
45o.The steady flow features affected by lid moving as
well as temperature distribution were tested. Which
investigated the combined effects of lid velocity and
buoyancy force on the altered convection heat transfer
with a trapezoidal enclosure and found that the lid
velocity plays an important role for heat transfer rate as
well as the features of hot temperature at the bottom
trapezoidal wall (isothermally or sinusoidal). Also, it is
found that the heat transfer rate equivalent to the isotherm
bottom wall condition will increase compared with non-
isotherm. Interestingly, many researchers used non-
Newtonian fluid to occupy the cavity regime to benefit
from its properties to improve the predicted heat
convection rate, for example Matin and Khan, 2013 [6]
numerically investigated the free convection due to
temperature difference between two concentrated
cylinders depends on the finite volume method. Besides,
the impacts of power-law index for non-Newton fluid
with nanoparticles is presented and treatment in details.
Heat and fluid flow for sinusoidal left wall of enclosure
with Cu-water nanofluid have been studied numerically
by Finite volume method under constant heat flux in
Sheikholeslami, et.al, 2013 [7] Effect of governing
parameters such as nanoparticles volume fraction (0-
0.04), Rayleigh number (103-105), Hartmann number (0-
100) and dimension- less amplitude of the sinusoidal wall
(0.1-0.3) are investigated at constant Pr=6.2 through the
isotherms, streamlines, average Nusselt numbers on the
left wall of the enclosure. It can be shown that the
increase of the nanoparticles volume fraction,
dimensionless amplitude of the sinusoidal wall and
Rayleigh number increases the average Nusselt number.
While Nusselt number decreases with the increase of the
Hartmann number.
Also, Kefayati, 2014 [8] presented a numerical difference
method depends on Lattice Boltsman Method to simulate
the enclosure with non_Newtonian molten polymer. The
model under consideration contains two adiabatic
horizontal walls. While, the left wall is hot and the right
one is heated by sinusoidal for temperature behavior.
Moreover, it is illustrated that the power law index has
been used to model the behavior of natural convection
within enclosure has a wide effect on the rate of heat
transfer. It is concluded that, the decreasing in power
index at range (1≤ n ≤ 1.5) will enhanced the heat transfer
rate. There are several paper deals with cavity wall
moving to provides turbulent modes throughout the
cavity. Another study for Kefayati, et.al, 2014 [9], deals
with non-Newtonian power-law fluid flow in a sinusoidal
heated cavity have been analyzed by finite difference
Lattice Boltzmann method. A different parameters have
been studied their effectssuch as: the Rayleigh number, Ra
(104 and 105), Hartmann number range (0- 60) and the
power-law index (0.5- 1.5) at constant Prandtl number Pr
= 10. It is concluded that the enhancement of heat transfer
increases with the increase of Rayleigh number for both
Newtonian and Non-Newtonian while decreases with the
increase of Hartmann number, without the magnetic field,
heat transfer in pseudoplastic fluids is enhanced by
decreasing the power-law index from n = 1 to 0.5 while,
the heat transfer reduced as the power-law index increases
from n = 1 to 1.5 in dilatant fluids. MHD free convection
of two dimensional trapezoidal cavity with non-
uniformaly heated bottom wall have been studied
numerically by Hossain and Abdul Alim, 2014 [10]. The
fluid inside the cavity is concerned with Rayleigh range
(103 -107), Pr (0.026, 0.7, 1000), Ha number of 50 and the
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I J E N S IJENS © February 2020 IJENS -IJMME-4334-201601
cavity tilt angle range (0o, 30o, 45o). It is found that the
average Nusselt number for the non-uniform heating
bottom wall increases with the increase of Rayleigh
number and the maximum heat transfer rate near the edge
of the wall while the minimum rate at the center of the
wall with irrespective all of the tilt angles, all range of
Rayleigh and different Prandtl number.
Hussein and Hussain, 2015 [11] discussed the influence
of MHD within the parallelogram cavity with declination
angles. The problem investigated numerically by finite
volume method and enclosure has two vertical sliding
walls positioned at left and right sides. These walls move
in a parallel direction with constant velocity. While, the
horizontal sides of the parallelogram inclination cavity are
thermally insulated. The cavity affected by the horizontal
magnetic field. It is found that the rising in strength of
Hartman number which induced by increasing the
magnetic field strength will decrease the rate of
convection rapidly. The heating geometry form have been
used as a severe factor for improved or enhance enclosure
convection flow, where, Rabbi, et. al, 2016 [12]
elucidated and presented a suitable numerical
investigation for comparison between two different
bottom shapes cavity geometries, (a) triangular and (b)
semi-circular notched cavity. The focal point was
studying the effect of external MHD on the performance
of pure mixed convection heat transfer for both geometry
configurations. Moreover, these cavities filled with Ferro
fluid, as a working fluid. In addition, it is illustrated many
relevant factors like Richardson number, Hartman number
and Grashof number and its contributed for flow
convection that can be realized by average Nusselt
number values. It is found that, semi-circular heater
configuration cavity enhances convection heat transfer
rate compared to triangular cavity. Moreover, the strength
of the magnetic field will be reduced or downplays of
convection rate. Besides, forced convection is dominant
compared with natural convection for Ri < 0.1. Kefayati,
2016 [13] analyzed the natural convection by using finite
difference within square cavity affected by external
magnetic field that exerted. The working fluid is non-
Newtonian liquid that used to study the power law effect
on improving the heat transfer rate. However,
multifarious parameters like: flow type, the entropy that
induced by heat transfer and Rayleigh number introduced
with good analyzing. Evidently, that many exciting
parameters and boundary conditions play as crucial effect
for activating and improve the convective heat transfer
within the enclosure. These multifarious factors and
boundary conditions will be produced a complex problem
and thus make the solution is so difficult. MHD mixed
convection in a lid-driven cavity having a triangular
corrugated bottom wall and filled with a non-Newtonian
power-law fluid under the influence of an inclined
magnetic field have been investigated by Selimefendigil
and Chamkha, 2016 [14] under Richardson number range
(0.01-100), Hartmann number range (0-50) and angle
inclination of the magnetic field range (0o-90o). it is
concluded that the local heat transfer is reduced as the
Hartmann number increases for all power index. The
effect of natural convection is more virtual for thining
shear fluid while the buoyancy force is poor for dilatant
fluid than Newtonian fluid. The local heat transfer
enhanced as the Richardson number increased while the
average Nusselt number decreased. Finally, the average
Nusselt number enhanced as the inclination angle of the
magnetic field increased.
Bondarenko, et al, 2018 [15] illustrated the evolution in
heat transfer rate due to mixed convection in cavity filled
by nanofluid augment the porous media effect and moving
the wall at the enclosure top. The impact of the
Richardson number treated as control parameter, which is
merging the natural and forced convection. Moreover, it is
discussed the corresponding Nusselt number. While,
Arpasanda and Fazeli, 2018 [16] numerically investigated
the mixed convection with parallelogram cavity exerted
by magnetic field MHD. Moreover, the analysis involved
the effect of porous media and nanoliquid that filled the
enclosure with convective effective. In addition, the flow
parameters explored or depicted by isotherm and
streamlines depends on the permeability and Richardson
number. Also, Pal, et. al, 2018 [17] numerically studied
the square enclosure that filled with Cu-water nanofluid
and absence the magnetic field. Besides, the upper wall is
adiabatic and having a constant horizontal velocity vector.
In addition, the bottom wall is thick and has wavy shape.
Many parameters discussed such as: Richardson number
and nanoparticle volume fraction but the focal point in
this manuscript is the effect of bottom wavy wall heating.
Where, the numerical analyzed of lower thick and wavy
wall heated effect has drawn attention for equivalent with
the present research. During the last year, Bakar, et. al,
2019 [18] studied numerically the impact of the
inclination of magnetic field on heat transfer and flow
pattern within the square enclosure. The numerical
solution that done by Bakar depending on finite volume
method. However, the square cavity consists of cold wall
at the bottom and moving hot wall at the top in addition
the vertical walls are thermally insulated. It is observed
that the Nusselt number can be presented as a function of
magnetic field angle. Where, the heat transfers rate
increase as the angle of magnetic field increase. Also, it is
noticed that, the increasing of Richardson number has a
negative effect on the heat transfer rate. As mentioned
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previously by other works, for a mixed convection, the
effect of the magnetic field has reduced the heat transfer
rate.
The intrinsic evolution of this study is the inclusive,
combined effect of non-Newtonian fluid that contains
nanoparticles Al2O3 within new enclosure heat geometry
shapes (four cases that will be elaborated in subsequent
articles) that motivating by external magnetic field at
constant horizontal lid_ driven velocity. Several numerical
methods may be used to satisfy the high accuracy of
results for own complex problem. However, from above
previous works and our experience found that, the finite
element method that base on Galerkin weighted residual
formulation where used in COMSOL Multiphysics is a
confidential method. Besides, the bulk of researcher
performed that the increasing of magnetic field will be
reduced the heat transfer rate with independent of any
other parameters. Also, it is reported that the heating
geometry, wall shapes have the capability to increase the
effects of convection heat flow within the enclosure.
Moreover, the motivating contribution of power law index
reduces the effect of the average volume fraction of
nanofluid.
The new shapes of heating configuration for lid driven
enclosure are the basic reason of attraction for this distinct
problem situation, as well as observed by tracking papers
that survey, which cited above. However, the modern
engineering, theoretical works of cavity that analyzed
numerically with non-Newtonian fluid having
nanoparticles that is relevant to mixed convection affected
by external magnetic field with moving one wall at
different shape of the cavity heat lower wall, depends on
the papers above and our knowledge is not enough to
discuss.
2. PHYSICAL MODEL DESCRIPTION
The problem description of the present work is displayed
in Figure (1). It involved of the square enclosure of two-
dimensional with width and height equal to (L). The two
vertical sided left and right walls are thermally insulated
with no-slip condition. Top horizontal wall kept at low
temperature (Tc) with movement lid-driven action of
velocity (U) from left to right direction. Bottom wall was
kept at high temperature (Th) with no-slip condition, the
configuration of this wall varied from sinusoidal
semicircular to semicircular shapes with different radius.
The enclosure occupied by a mixture of non-Newtonian
fluid and Al2O3-water nanofluid. The thermo-physical
properties of the nanofluid were listed in the table (1).
Magnetic fields with strength Bo was applied in horizontal
direction effect was considered. Many assumptions were
considered to simplify the study such as laminar flow,
steady state, two-dimensional, incompressible, neglect
viscous dissipation and heat generation. Only density
properties were considered to be variation with the
Boussinesq approximation model.
3. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
Following non-dimensional parameters and numbers are existed to transform the dimensional equations (12-15) to non-
dimensional form (16-19) [20].
𝑋, 𝑌 =𝑥, 𝑦
𝐿, 𝑈, 𝑉 =
𝑢, 𝑣
𝑢𝑜, 𝑃 =
𝑝
𝜌𝑛𝑎𝑢𝑜, 𝜃 =
𝑇 − 𝑇𝑐𝑇ℎ − 𝑇𝑐
(1)
𝐺𝑟𝑎𝑠ℎ𝑜𝑓 𝑛𝑢𝑚𝑏𝑒𝑟 (𝐺𝑟) =𝛽𝑏𝑓𝑔𝑦𝐿(𝑇ℎ − 𝑇𝑐)
𝜈𝑏𝑓2 ,
𝑃𝑟𝑎𝑛𝑑𝑡𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 (𝑃𝑟) =𝜈𝑏𝑓𝛼
, 𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 (𝑅𝑒) =𝑢𝑜𝐿
𝜈𝑏𝑓, 𝑅𝑖𝑐ℎ𝑟𝑑𝑠𝑜𝑛 𝑛𝑢𝑚𝑏𝑒𝑟
=𝐺𝑟
𝑅𝑒2, 𝐻𝑎𝑟𝑡𝑚𝑎𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 (𝐻𝑎) = 𝐿Β𝑜√
𝜎𝑏𝑓𝜇𝑏𝑓
(2)
3-1 The properties of nanofluid are described by the following equations [20].
Density (𝜌𝑛𝑎)
𝜌𝑛𝑎 = (1 − 𝜑)𝜌𝑏𝑓 + 𝜑𝜌𝑠𝑜 (3)
Heat capacity (𝜌𝑐𝑝)𝑛𝑎
(𝜌𝑐𝑝)𝑛𝑎 = (1 − 𝜑)(𝜌𝑐𝑝)𝑏𝑓 + 𝜑(𝜌𝑐𝑝)𝑠𝑝 (4)
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The coefficient of thermal expansion (𝜌𝛽)𝑛𝑎
(𝜌𝛽)𝑛𝑎 = (1 − 𝜑)(𝜌𝛽)𝑏𝑓 + 𝜑(𝜌𝛽)𝑠𝑜 (5)
Thermal conductivity (𝑘𝑛𝑎)
𝑘𝑛𝑎 = 𝑘𝑏𝑓(𝑘𝑠𝑝 + 2𝑘𝑏𝑓 + 2𝜑(𝑘𝑏𝑓 − 𝑘𝑠𝑝)
𝑘𝑠𝑝 + 2𝑘𝑏𝑓 − 𝜑(𝑘𝑏𝑓 − 𝑘𝑠𝑝)) (6)
Thermal diffusivity (𝛼𝑛𝑎)
𝛼𝑛𝑎 =𝑘𝑛𝑎
(𝜌𝑐𝑝)𝑛𝑎 (7)
Electric conductivity (𝜎𝑛𝑎)
𝜎𝑛𝑎 = 𝜎𝑏𝑓(1 +3(𝛾 − 1)𝜑
(𝛾 + 2) − (𝛾 − 1)𝜑) (8)
Where 𝛾 represent the ratio (𝜎𝑠𝑝
𝜎𝑏𝑓)
In the above equations (𝜑) denotes the volume fraction of the solid nanoparticle (Al2O3) that added to the pure liquid (water).
Thermal and electrical conductivity denoted by equations (6 and 8) is expressed by using Maxwell-Granetts model [19].
In this study, the shape of nanoparticles was not considered, therefore Maxwell-Granetts model is chosen for it. This model is
suitable for spherical nanoparticles and small temperature gradients [20].
3-2 Non-Newtonian fluid equations
The equations for the shear thinning non-Newtonian fluid suspended by nanofluid can described by Ostwald-DeWaele model or
power-law model. The shear stress tensor equation is[20]:
𝜏𝑖𝑗 = 2𝜇𝑛𝑎𝐷𝑖𝑗 = 𝜇𝑛𝑎 (𝜕𝑢𝑖
𝜕𝑥𝑗+
𝜕𝑢𝑗
𝜕𝑢𝑖) (9)
𝜇𝑛𝑎 represented dynamic viscosity coefficient, 𝐷𝑖𝑗 represented the deformation rate of fluid in two-dimensional Cartesian
coordinates.
𝜇𝑛𝑎 =𝜇𝑏𝑓
(1−𝜑)2.5 (10)
𝜇𝑏𝑓 denotes the dynamic viscosity of base fluid (water) which expressed in the following equation [20].
𝜇𝑏𝑓 = 𝑀{2 [(𝜕𝑢
𝜕𝑥)
2
+ (𝜕𝑣
𝜕𝑦)
2
] + (𝜕𝑣
𝜕𝑥+
𝜕𝑢
𝜕𝑦)
2
} 𝑛−1
2 (11)
M is the coefficient of consistency, n is the power law index, the variation of (n) from value of
Unity refers the variation from pure Newtonian fluid, when (n) 1
the non-Newtonian fluid is a dilatant fluid.
3-3 Basic Equations
The equations describe the fluid flow and heat transfer for non-Newtonian nanofluid inside square lid-driven enclosure are
continuity, momentum and energy in dimensional form are [20]:
𝜕𝑢
𝜕𝑥+
𝜕𝑣
𝜕𝑦= 0 (12)
𝜌𝑛𝑎 (𝑢𝜕𝑢
𝜕𝑥+ 𝑣
𝜕𝑢
𝜕𝑦) = −
𝜕𝑝
𝜕𝑥+ (
∂𝜏𝑥𝑥𝜕𝑥
+∂𝜏𝑦𝑦𝜕𝑦
) + 𝜎𝑛𝑎𝐵𝑜2(𝑣𝑠𝑖𝑛𝛾𝑐𝑜𝑠𝛾 − 𝑢𝑠𝑖𝑛2𝛾) (13)
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𝜌𝑛𝑎(𝑢𝜕𝑣
𝜕𝑥+ 𝑣
𝜕𝑣
𝜕𝑦= −
𝜕𝑝
𝜕𝑦+ (
∂𝜏𝑥𝑦𝜕𝑥
+∂𝜏𝑦𝑦𝜕𝑦
) + 𝑔(𝜌𝛽)𝑛𝑓(𝑇 − 𝑇𝑐) + 𝜎𝑛𝑎𝐵𝑜2(𝑢𝑠𝑖𝑛𝛾𝑐𝑜𝑠𝛾 − 𝑣𝑐𝑜𝑠2𝛾)(14)
(𝜌𝑐𝑝)𝑛𝑎(𝑢
𝜕𝑇
𝜕𝑥+ 𝑣
𝜕𝑇
𝜕𝑦) = 𝑘𝑛𝑎 (
𝜕2𝑇
𝜕𝑥2+
𝜕2𝑇
𝜕𝑦2) (15)
After using the parameters in the equation (1) and non-dimensional number in the equation (2) the basic governing equation
transforms to the non-dimensional form as follows:
𝜕𝑈
𝜕𝑋+
𝜕𝑉
𝜕𝑌= 0 (16)
𝑈𝜕𝑈
𝜕𝑋+ 𝑉
𝜕𝑈
𝜕𝑌= −
𝜕𝑃
𝜕𝑋+
1
𝑅𝑒
𝜌𝑏𝑓𝜌𝑛𝑎
1
(1 − 𝜑)2.5[2
𝜕
𝜕𝑋(
𝜇𝑏𝑓𝑀
𝜕𝑈
𝜕𝑋) +
𝜕
𝜕𝑌(
𝜇𝑏𝑓𝑀
(𝜕𝑈
𝜕𝑌+
𝜕𝑉
𝜕𝑋))]
+𝐻𝑎2
𝑅𝑒
𝜎𝑛𝑎𝜎𝑏𝑓
(𝑉𝑠𝑖𝑛𝛾𝑐𝑜𝑠𝛾 − 𝑈𝑠𝑖𝑛2𝛾) (17)
(𝑈𝜕𝑉
𝜕𝑋+ 𝑉
𝜕𝑉
𝜕𝑌) = −
𝜕𝑃
𝜕𝑌+
1
𝑅𝑒
𝜌𝑏𝑓𝜌𝑛𝑎
1
(1 − 𝜑)2.5[2
𝜕
𝜕𝑌(
𝜇𝑏𝑓𝑀
𝜕𝑉
𝜕𝑌) +
𝜕
𝜕𝑋(
𝜇𝑏𝑓𝑀
(𝜕𝑈
𝜕𝑌+
𝜕𝑉
𝜕𝑋))] +
(𝜌𝛽)𝑛𝑎𝜌𝑛𝑎𝛽𝑏𝑓
𝑅𝑖𝜃
+𝐻𝑎2
𝑅𝑒
𝜎𝑛𝑎𝜎𝑏𝑓
𝜌𝑏𝑓𝜌𝑛𝑎
(𝑈𝑠𝑖𝑛𝛾𝑐𝑜𝑠𝛾 − 𝑉𝑐𝑜𝑠2𝛾) (18)
(𝑈𝜕𝜃
𝜕𝑋+ 𝑉
𝜕𝜃
𝜕𝑌) =
𝛼𝑛𝑎𝛼𝑏𝑓
1
𝑅𝑒(
𝜕2𝜃
𝜕𝑋2+
𝜕2𝜃
𝜕𝑌2) (19)
The boundary conditions related to the present work are described by the following equations:
𝑇𝑜𝑝 𝑤𝑎𝑙𝑙: 𝑌 = 𝐿, 0 ≤ 𝑋 ≤ 𝐿, 𝑈 = 𝑈𝑜 , 𝑉 = 0, 𝜃 = 0 (20𝑎)
𝐵𝑜𝑡𝑡𝑜𝑚 𝑤𝑎𝑙𝑙: 𝑌 = 0,0 ≤ 𝑋 ≤ 𝐿, 𝑈 = 𝑉 = 0, 𝜃 = 1 (20𝑏)
𝑅𝑖𝑔ℎ𝑡 𝑤𝑎𝑙𝑙: 𝑋 = 𝐿, 0 ≤ 𝑌 ≤ 𝐿, 𝑈 = 𝑉 = 0,𝜕𝜃
𝜕𝑋= 0 (20𝑐)
𝐿𝑒𝑓𝑡 𝑤𝑎𝑙𝑙: 𝑋 = 0,0 ≤ 𝑌 ≤ 𝐿, 𝑈 = 𝑉 = 0,𝜕𝜃
𝜕𝑋= 0 (20𝑑)
The heat transfer expressed by local and average Nusselt number in the next two equations
𝑁𝑢𝐿𝑜𝑐 = −𝜕𝜃
𝜕𝑌𝑌=0 (21)
𝑁𝑢𝑎𝑣𝑒 = ∫ 𝑁𝑢𝐿𝑜𝑐
1
0
𝑑𝑋 (22)
4. MESH GENERATION, GRID INDEPENDENCY, NUMERICAL PROCEDURE AND COMPARISON
The mesh generation of the numerical domain is shown in Figure (2). The mesh was cluster near the boundary walls,
while it disperse entire the domain since the change in fluid flow and heat transfer occurs strongly near the boundary walls. Table
(2) shows the grid independency of the present study at various power law index (n), case I. Ri=1, Ra=105, ϕ=0.06 and Ha=30.
The average Nusselt number was chosen for checked the grid independency. In this check, the solution was attained to a certain
number of elements and begins to increase this number until obtained the solution with high accuracy to one or more independent
variable (average Nusselt number).
The Galerkin finite element method with the weighted residual approach was utilized to solve the non-dimensional
governing equations. Non-extending zone in the computational domain are built up, and every independent variables are closed by
utilizing interruption function in those zones. Finite elements with the Lagrangian scheme of various orders are used to interrupt
the independent like as
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𝜂 = ∑ 𝜂𝐾
𝑁
Κ=1
Ψ𝐾(𝑥, 𝑦) (23)
Where 𝜂 refer to any independent variable.
Interruptions were replaced into the non-dimensional governing equations to obtain in residuals for every control equation. The
solution convergence is achieved when the errors of each independent variable is not greater than 10-6 with the following
convergence criteria.
|η𝑛+1−η𝑛
η𝑛+1| = 10−6 (24)
The validation of the numerical code is obtained by comparing the results of the present code with a paper published by
Kefayati, 2015 [20]. Kefayati investigated the numerical solution of mixed convection of two sided lid-driven in square enclosure.
Figure (2) represents the comparison of streamlines and isotherms contour for different power law index and Table (3A and 3B)
show the comparison of stream function and average Nusselt numbers. The comparison obtained excellent similarity in the fluid
flow and heat transfer results.
5. RESULTS AND DISCUSSION In this work, numerically analyzed four cases of
enclosure that contain a non-Newtonian fluid as the
working fluid of nanoparticle with different solid volume
fraction adopted by external magnetic field. Each
enclosed has a square structure with the vertical sides are
thermally insulated while the top wall is cold and moving
to right with constant velocity. The distinguishing
parameters are the shape of bottom hot wall. Where, the
first case has small wavy notched shape, the second case
has small semicircle notched, while the bigger semicircle
notched can be shown in the third case until the last case has a large wavy notched. The cavity geometry shape for
all cases that presented in this manuscript can be shown in
Figure (1). The geometry of hot wall shapes of the
enclosure under analyzing is a distinct parameter that’s
used to improve the convection flow within the enclosure.
Where, this condition plays an important role to enhance
many encountered engineering applications. Moreover,
the relevant dimensionless parameters that used to
investigate these cases are: Richardson numbers, Hartman
numbers, index power law and solid volume fraction.
Where, the range of Hartman number is 0 ≤ Ha ≤ 60, solid
volume fraction 0 ≤ φ ≤ 0.1 and Richardson number values are 0.001 ≤ Ri ≤ 1. The range of the Richardson
number restricted to convection type that dominant where,
the buoyancy force effect is negligible, as well as the
natural convection. However, When Ri=Gr/Re2
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Also, the isotherm line will declination to right at and
distorted at (Ha=0) then try be parallel and quasi uniform
as Harman number increase. Where, the isotherm lines
cluster positioned adhere and covered the surface of hot
wall for wavy cylinder in all situations. In a brief manner,
the investigation above reveals that the power index will provoke and modified the convection heat transfer rate in
contrast, the heat transfer is weaker, due to the
enhancement of the intensity of external magnetic field.
The results of Figures (5-7) are similar that to the results
of figure (4). Nevertheless, for all cases under analyzed,
the convection flow will slow down as magnetic field
strength intensity increase in low Richardson number
(Ri=0.001). Interestingly, case II has the best results
compare with other cases, since the streamlines, function
values are greater than the other cases. Also, it is observed
from the corresponding figures that the stoop of isotherm
lines is more distorted and turned to left positioned with the absence of magnetic field. The mixed convection can
be seen when Ri=1. In this case the buoyancy force will
be more strong compare with the low Richardson number
(Ri=0.001) that explained before. Figures (8-11) represent
the mixed convection effect on the behavior of
streamlines function and isotherm contour. It is noticed
that, the effect of the new value of Richardson number is
to increase the size of rotating eddy in all situations for all
cases for streamlines function. Where, the increasing
value for Richardson number make the vortex of
streamlines function absorb energy, due to increasing the buoyancy force. It is very evident that the behavior of
streamlines and isotherm counter still the same for all
cases. In fact, the streamlines function value for new value
of Richardson number is less that of the previous figure
with (Ri=0.001). However, the case II is still to be the best
situation in order to improve the convective heat transfer
rate.
Moreover, the pertinent parameter can be used to
investigate the modified convection heat flow is the
average Nusselt number. However, the Figure (12)
represents the variation of the Nusselt number at constant
volume solid fraction (0.06) with different values of law-index number of all cavity cases by taking in account the
magnetic field strength that observed by Hartman number
contrast with different values for Ri number. The
Richardson number is very benefit to realize the
contribution of lid _driven velocity on flow behavior.
Moreover, this figure emerges the mutual effect of
Richardson number and Hartman numbers (0-60) for all
cases under research. However, the Nusselt number
reveals the effect cavity wall shape and flow features. The
shape effect on a Nusselt number is extremely significant.
Where, the second case has greater values for Nusselt number as comparing with the first and fourth cases
independently on the value of power index. These results
emphasized the effect of geometry shape of heating wall
on heat transfer rate and Nusselt number. Moreover, the
power index values will be enhanced the heat transfer rate
for all cases that possess the same behavior. However, the
maximum values of the Nusselt number adopted in the
situation where no magnetic forces are present (Ha=0).
On the other hand, the increase in re number will decrease
the Nusselt number. The corresponding curves of Ha
equal to 30 Tesla for cases III and IV imposed the unique
response to increasing in power law index in contrast with
cases I and II. In addition, this figure noticed that the
cavity shape of cases III and IV would help magnetic field to enhance heat transfer rate for small range of Nusselt
number or marginally as power index increase. Figure
(13) shows the variation of average Nusselt number with
the changing of the solid volume fraction. This figure
illustrated that the average Nusselt number, hence heat
transfer rate modified as the power index increase for all
values of volume fraction and in all cases. It is concerned
that, case II has the maximum value for the average of
Nusselt number, but for a restricted volume fraction range
(0.04 ≤ φ ≤ 0.08). This refers to that, the volume fraction
contributing is clear and the combined effect with a
heating geometry shape wall helped to make case II distinct. On the other hand, Figure (14) represents the
variation of average Nusselt number with strength of
external magnetic field that represented by Hartman
number. This figure, highlighted the negative effect of
Hartman number on the intensity of convection heat
transfer rate for cases regardless other restricted values.
6. CONCLUSION
Many researchers have tried to discuss the parameters
that can be used to enhance the heat transfer rate within an
enclosure that exerted by an external magnetic field with lid driven. Several dimensionless parameters included
through the numerical investigation. However, the
innovation of the boundary conditions are a new heating
geometry shape wall alternating effect with variation of
power index parameters at different values of Richardson
number and the solid volume fraction. The conclusion can
be summarized in a brief manner as below:
1. Case II of small circular notched is the best case that justified of improving forced and mixed
convection heat transfer rate. Besides, the
geometry shape is an important to improve or
modified the cavity in terms of heat dissipation. Also, the solid volume fraction has a combined
effect with the closure shape geometry for
enhancement recirculation streamlines function
eddy.
2. The magnetic field strength interplay negative effect on improving the forced convection flow
within the enclosure.
3. The average Nusselt number increases by evolution of power index value and independent
of shape geometry of the heating cavity wall.
4. The isotherm contour behavior is pertinent with the Hartman number effect. Where, the strong
magnetic field (high Hartman number)
declination the isotherm to be uniform similar to
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that of conduction fluid and diminish the
convection flow effect.
5. The flow pattern can be controlled by the strength of magnetic fields and hot wall shape
geometry as well as the nanofluid type that can
be clarified by power index and solid volume fraction.
There are mutually effects between power law index
and solid volume fraction can be used to improve the flash
situation of cavity in term of heat transfer rate with
absence magnetic field effect depending on heating wall
shape geometry.
REFERENCES [1] Marchi, Carlos Henrique, Roberta Suero, and Luciano Kiyoshi
Araki. "The lid-driven square cavity flow: numerical solution with
a 1024 x 1024 grid." Journal of the Brazilian Society of
Mechanical Sciences and Engineering 31.3 (2009): 186-198.
[2] Al-Salem, Khaled, et al. "Effects of moving lid direction on MHD
mixed convection in a linearly heated cavity." International
Journal of Heat and Mass Transfer 55.4 (2012): 1103-1112.
[3] Billah, M. M., et al. "Numerical analysis of fluid flow due to
mixed convection in a lid-driven cavity having a heated circular
hollow cylinder." International Communications in Heat and
Mass Transfer 38.8 (2011): 1093-1103.
[4] Sheikholeslami, M., et al. "Natural convection of nanofluids in an
enclosure between a circular and a sinusoidal cylinder in the
presence of magnetic field." International Communications in
Heat and Mass Transfer 39.9 (2012): 1435-1443.
[5] Bhattacharya, Madhuchhanda, et al. "Mixed convection and role
of multiple solutions in lid-driven trapezoidal
enclosures." International Journal of Heat and Mass Transfer 63
(2013): 366-388.
[6] Matin, Meisam Habibi, and Waqar Ahmed Khan. "Laminar
natural convection of non-Newtonian power-law fluids between
concentric circular cylinders." International Communications in
Heat and Mass Transfer 43 (2013): 112-121.
[7] Sheikholeslami, M., et al. "Natural convection heat transfer in a
cavity with sinusoidal wall filled with CuO–water nanofluid in
presence of magnetic field." Journal of the Taiwan Institute of
Chemical Engineers 45.1 (2014): 40-49.
[8] Kefayati, GH R. "Simulation of non-Newtonian molten polymer
on natural convection in a sinusoidal heated cavity using
FDLBM." Journal of Molecular Liquids 195 (2014): 165-174.
[9] Kefayati, GH R. "Simulation of magnetic field effect on natural
convection of non-Newtonian power-law fluids in a sinusoidal
heated cavity using FDLBM." International Communications in
Heat and Mass Transfer 53 (2014): 139-153.
[10] Hossain, Muhammad Sajjad, and Mohammad Abdul Alim.
"MHD free convection within trapezoidal cavity with non-
uniformly heated bottom wall." International Journal of Heat and
Mass Transfer 69 (2014): 327-336.
[11] Hussein, Ahmed Kadhim, and Salam Hadi Hussain.
"Characteristics of magnetohydrodynamic mixed convection in a
parallel motion two-sided lid-driven differentially heated
parallelogrammic cavity with various skew angles." Journal of
Thermal Engineering 1.3 (2015): 221-235.
[12] Kefayati, GH R. "Simulation of heat transfer and entropy
generation of MHD natural convection of non-Newtonian
nanofluid in an enclosure." International Journal of Heat and
Mass Transfer 92 (2016): 1066-1089.
[13] Rabbi, Khan Md, et al. "Numerical investigation of pure mixed
convection in a ferrofluid-filled lid-driven cavity for different
heater configurations." Alexandria Engineering Journal 55.1
(2016): 127-139.
[14] Selimefendigil, Fatih, and Ali J. Chamkha.
"Magnetohydrodynamics mixed convection in a lid-driven cavity
having a corrugated bottom wall and filled with a non-Newtonian
power-law fluid under the influence of an inclined magnetic
field." Journal of Thermal Science and Engineering
Applications 8.2 (2016): 021023.
[15] Bondarenko, Darya S., et al. "Mixed convection heat transfer of a
nanofluid in a lid-driven enclosure with two adherent porous
blocks." Journal of Thermal Analysis and Calorimetry 135.2
(2019): 1095-1105.
[16] Ghaffarpasand, Omid, and Dariush Fazeli. "Numerical analysis of
MHD mixed convection flow in a parallelogramic porous
enclosure filled with nanofluid and in the presence of magnetic
field induction." Scientia Iranica. Transaction F,
Nanotechnology 25.3 (2018): 1789-1807.
[17] Pal, S. K., S. Bhattacharyya, and I. Pop. "Effect of solid-to-fluid
conductivity ratio on mixed convection and entropy generation of
a nanofluid in a lid-driven enclosure with a thick wavy
wall." International Journal of Heat and Mass Transfer 127
(2018): 885-900.
[18] Bakar, N. A., R. Roslan, and I. Hashim. "Mixed convection in lid-
driven cavity with inclined magnetic field." Sains
Malaysiana 48.2 (2019): 451-471.
[19] Levin, M. L., and M. A. Miller. "Maxwell a treatise on electricity
and magnetism." Uspekhi Fizicheskikh Nauk 135.3 (1981): 425-
440.
[20] Kefayati, GH R. "FDLBM simulation of magnetic field effect on
mixed convection in a two sided lid-driven cavity filled with non-
Newtonian nanofluid." Powder technology 280 (2015): 135-153.
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Nomenclature
Cp Specific heat at constant pressure (KJ/kg.K) Re Reynolds number
g Gravitational acceleration (m/s2) NuLoc Local Nusselt number on the hot wall
k Thermal conductivity (W/m.K) 𝑁𝑢𝑎𝑣𝑒 Average Nusselt number on the hot wall
Bo Magnetic field U Dimensionless velocity component in x-direction
Ha Hartman number u Velocity component in x-direction (m/s)
P Dimensionless pressure V Dimensionless velocity component in y-direction
p Pressure (Pa) v Velocity component in y-direction (m/s)
Pr Prandtl number (νf/αf) X Dimensionless coordinate in horizontal direction
Ra Rayleigh number (𝑔𝛽𝑓𝐿3 𝛥𝑇 𝜈𝑓𝛼𝑓)⁄ x Cartesian coordinates in horizontal direction (m)
T Temperature (K) Y Dimensionless coordinate in vertical direction
Tc Temperature of the cold surface (K) y Cartesian coordinate in vertical direction (m)
Th Temperature of the hot surface (K) Gr Grashof number
Ri Richardson number n Power law-index
D Deformation rate M Coffecient of consistency
L Length and height of enclosure (m)
Greek symbols
α Thermal diffusivity (m2/s) μ Dynamic viscosity (kg.s/m)
θ Dimensionless temperature (T-Tc/Th-Tc) ν Kinematic viscosity (μ /ρ)(Pa. s)
Ψ Dimensional stream function (m2/s) β Volumetric coefficient of thermal expansion (K-1)
ψ Dimensionless stream function ρ Density (kg/m3)
φ Volume fraction 𝜂 Independent variable
𝜎 Electrical conductivity 𝜏 Shear stress
Subscripts
c Cold h hot
bf Base fluid (pure) na Nanofluid
so solid
Abbreviations
Amp. Amplitude (m) N Number of corrugations
Loc Local ave average
Fig. 1, Simplified Diagram of the Present Study.
Fig. 2. Mesh of the Present Work.
Case I
oB
Case II
oB
Case III
oB
Case IV
oB
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].20310 K [= T nanoparticles at 3O2Alphysical properties of water with -Thermo. 1Table Physical Properties Fluid Phase (Water) Al2O3
(J/kg k)p C 4178 765
)3ρ (kg/m 993 3970
k (W/m k) 0.628 40
(1/k)5 βx10 32.2 0.85
(kg/ms)6 μx10
(nm) pd
695
0.385
-
33
Grid size Number of
elements
Average Nusselt number, Nu
n=0.2 n=0.6 n=1 n=1.4
G1 4317 0.67988 0.69817 0.75304 0.85146
G2 5943 0.67978 0.69912 0.75557 0.85548
G3 9999 0.67961 0.69958 0.75712 0.85815
G4 15714 0.68001 0.70155 0.76132 0.86482
G5 32794 0.68044 0.70255 0.76338 0.86801
G6 41404 0.68040 0.70249 0.76330 0.86793
Base fluid nanofluid
N Kefayati [16] Present study Error Kefayati [16] Present study Error % |𝛹𝑚𝑎𝑥| |𝛹𝑚𝑎𝑥| |𝛹𝑚𝑎𝑥| |𝛹𝑚𝑎𝑥|
0.2 0.0134 0.0131 -2.29 0.0152 0.0149 -2.013
0.4 0.0249 0.0233 -6.95 0.0275 0.0265 -3.77
0.6 0.029 0.0306 5.2 0.038 0.0370 -2.7
0.8 0.043 0.0419 -2.625 0.047 0.046 -2.1
1 0.052 0.050 -4 0.056 0.055 -1.81
Nu.ave H=30 ,Ri=0.01,φ=0.09
GH. R. Kefayati [16] Present study Error %
0.2 5.7776 5.8937 1.9
0.4 6.4814 6.7953 4.6
0.6 8.9584 8.6588 3.4
0.8 9.935 9.9550 0.2
1 10.6409 10.3236 -3.0
, Icase on hot surface at different power index n , Average Nusselt numberGrid testing for . 2Table
0.06.Ha=30.= φ , 510= Ra Ri=1,
Table 3A. Comparison of the max. stream function between Kefayati [20] and present study for
different power law index at Ha=30, Ri=0.01, φ=0.09.
Table 3B. Comparison of the average Nusselt number between Kefayati [16] and present study
for different power law index.
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Ha=0
Stremlines Isotherms Stremlines Ha=30
Ha=60
Fig. 3. Comparison of the streamlines and isotherms between GH.R. Kefayati [20] and present study for
various power-law indexes, Ri = 0.01, Ha=30 ,φ = 0, 0.09, a) n=0.2, b) n=0.4, c) n=0.6, d) n=0.8, e) n=1
Streamlines Isotherms
Present Study Kefayati[20] Kefayati[20] Present Study
a
b
c
d
e
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Fig.(4) Streamlines, Isotherms for different Hartman number and power-law index at
Ri=0.001, Case I.
Stremlines Isotherms
Isotherms
Fig.(5) Streamlines, Isotherms for different Hartman number and power-law index at
Ri=0.001 Case II.
N=0.2
N=0.6
N=1
N=1.4
N=0.2
N=0.6
N=1
N=1.4
Ha=0
Stremlines Isotherms Ha=30 Ha=60
Stremlines Stremlines Isotherms Isotherms
Stremlines Isotherms
|𝚿|=0.02 |𝚿|=0.007 |𝚿|=0.006
|𝚿|=0.072 |𝚿|=0.029 |𝚿|=0.021
|𝚿|=0.1 |𝚿|=0.052 |𝚿|=0.038
|𝚿|=0.109 |𝚿|=0.072 |𝚿|=0.054
|𝚿|=0.016 |𝚿|=0.009 |𝚿|=0.007
|𝚿|=0.074 |𝚿|=0.033
Ha=0
|𝚿|=0.023
|𝚿|=0.106
Ha=30
|𝚿|=0.054
Ha=60
|𝚿|=0.039
Isotherms
|𝚿|=0.118
Stremlines
|𝚿|=0.052 |𝚿|=0.07
Stremlines Isotherms
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Fig.(6) Streamlines, Isotherms for different Hartman number and power-law index at
Ri=0.001 Case III.
Fig.(7) Streamlines, Isotherms for different Hartman number and power-law index at
Ri=0.001 Case IV.
Stremlines Isotherms
Ha=0 Ha=30
Stremlines Isotherms Isotherms Stremlines
Ha=60
N=0.2
N=0.2
N=0.6
N=1
N=1.4
N=0.6
N=1
N=1.4
|𝚿|=0019 |𝚿|=0.007 |𝚿|=0.006
|𝚿|=0.068 |𝚿|=0.029 |𝚿|=0021
|𝚿|=0.099 |𝚿|=0.072
|𝚿|=0.091 |𝚿|=0.052 |𝚿|=0.038
|𝚿|=0.054
|𝚿|=0.018 |𝚿|=0.007 |𝚿|=0.006
|𝚿|=0.062 |𝚿|=0.029 |𝚿|=0.021
|𝚿|=0.084 |𝚿|=0.052 |𝚿|=0.038
|𝚿|=0.09 |𝚿|=0.071 |𝚿|=0.054
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Fig.(8) Streamlines, Isotherms for different Hartman number and power-law index at
Ri=1 Case I.
Ha=0 Ha=30 Ha=60
Stremlines Isotherms Isotherms Isotherms Stremlines Stremlines
N=1.4
N=1
N=0.6
N=0.2
|𝚿|=0.044 |𝚿|=0.014 |𝚿|=0.009
|𝚿|=0.079 |𝚿|=0.034 |𝚿|=0.023
|𝚿|=0.094 |𝚿|=0.053 |𝚿|=0.038
|𝚿|=0.104 |𝚿|=0.071 |𝚿|=0.054
Fig.(9) Streamlines, Isotherms for different Hartman number and power-law index at
Ri=1 Case II.
Ha=0
Stremlines Isotherms Stremlines Stremlines Isotherms Isotherms
N=0.2
N=0.6
N=1
N=1.4
Ha=30 Ha=60
|𝚿|=0.035 |𝚿|=0.016 |𝚿|=0.011
|𝚿|=0.079 |𝚿|=0.037 |𝚿|=0.026
|𝚿|=0.099 |𝚿|=0.054 |𝚿|=0.04
|𝚿|=0.109 |𝚿|=0.069 |𝚿|=0.053
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Ha=0 Ha=30
Ha=60
Fig.(10) Streamlines, Isotherms for different Hartman number and power-law index
at Ri=1 Case III.
Fig.(11) Streamlines, Isotherms for different Hartman number and power-law index at Ri=1 Case IV.
Stremlines Isotherms
Isotherms
IsothermsIsotherms
Isotherms IsothermsStremlines Stremlines Stremlines
N=1.4
N=1
N=0.6
N=0.2
N=1.4
N=1
N=0.6
N=0.2
|𝚿|=0.04 |𝚿|=0.014 |𝚿|=0.009
|𝚿|=0.072 |𝚿|=0.034 |𝚿|=0.023
|𝚿|=0.086 |𝚿|=0.053 |𝚿|=0.038
|𝚿|=0.095 |𝚿|=0.071 |𝚿|=0.054
|𝚿|=0.031 |𝚿|=0.015 |𝚿|=0.01
|𝚿|=0.025 |𝚿|=0.064 |𝚿|=0.036
|𝚿|=0.078 |𝚿|=0.053 |𝚿|=0.038
|𝚿|=0.087 |𝚿|=0.067 |𝚿|=0.051
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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 114
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Fig.(12) Average Nusselt number with law-index number for different Richardson
number and Hartmann number at φ=0.06.
Fig.(13) Average Nusselt number with volume fraction for different law-index
number.
Fig.(14) Average Nusselt number with Hartmann number for different law-index
number.
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