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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:03 26
I J E N S IJENS © June 2019 IJENS -IJMME-1717-031925
Effect of Layer Thickness on Natural Convection in a
Square Enclosure Superposed by Nano-Porous and
Non-Newtonian Fluid Layers Divided by a Wavy
Permeable Wall
Mohammed Y. Jabbar1 Saba Y. Ahmed1,* Hameed K. Hamzah1 Farooq Hassan Ali1
Salwan Obaid Waheed Khafaji1
1Mechanical Engineering Department, College of Engineering, University of Babylon, Babylon, Iraq.
mohyousif [email protected]
[email protected] *Corresponding author: [email protected]
Mobile:009647801275760
Abstract--- The aim of this work is to investigate numerically
the influence of layer thickness and wavy interface wall on the
heat transfer and fluid flow inside a differentially heated
square enclosure occupied with two portions, Ag/water
nanofluid-porous medium and non-Newtonian substance,
respectively. The numerical computations have been carried
out for Rayleigh number Ra= 103 to 105, number of undulation
N= 1 to 4 (four cases) Darcy number Da= 10-1 to 10-5, volume
fraction ϕ= 0 to 0.2 non-Newtonian index n= 0.6 to 1.4 and the
Prandtl number of water Pr= 6.2, for different interface
location S=0.25, 0.5 and 0.75. The governing equations were
solved numerically by using finite element techniques based
Glarkine approach solver to take out an expression of
streamlines and isotherms. The output results were compared
with previous work and it was observed that a good formal
equivalent between the results. Results demonstrated that as
power law index "n" increases the average Nusselt number
decreases due to high viscosity and shear force of pseudoplastic
fluid. Consequently, layer thickness has a major impact while
the number of undulation has a a minor impact on the heat
transfer rate across the layers.
Index Term--- Natural convection; Sinusoidal vertical
interface; Nanofluid porous medium; Non-Newtonian fluid;
Prandtl number; Average Nusselt number.
Nomenclature
Cp Specific heat at constant pressure (KJ/kg.K) N Number of undulation
D Rate of strain-tensor Nu Nusselt number
G Gravitational acceleration (m/s2) n Power-law index
K Thermal conductivity (W/m.K) U Dimensionless velocity component in x-direction
L Dimensionless Length and Height of the
cavity 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
Da Darcy numbers y Cartesian coordinate in vertical direction (m)
S Layer thickness
Greek symbols
Θ Dimensionless temperature (T-Tc/ΔT) ∅ Nanoparticle volume fraction (%)
Ψ Dimensional stream function (m2/s) ε Permeability of porous medium
Ψ Dimensionless stream function β Volumetric coefficient of thermal expansion (K-1)
Μ Dynamic viscosity (kg.s/m) ρ Density (kg/m3)
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:03 27
I J E N S IJENS © June 2019 IJENS -IJMME-1717-031925
Ν Kinematic viscosity (μ /ρ)(Pa. s) β Thermal expansion (1/K)
Α Thermal diffusivity coefficient (m2/s) σ The consistency coefficient
Subscripts
C Cold pn Porous-nanofluid
Fl Base fluid nN non-Newtonian fluid
H Hot sn Solid-nanoparticle
1. INTRODUCTION
The wavy interface, undulation number and the
layer thickness were important roles that govern the energy
transformation and fluid motion in a very wide practical
application. The main applications related with wavy
interface included the thermal insulation treatments, a
depuration of sludgy water by separating the sinter which
passing horizontally over the wavy plate and the dispersion
of cooling of emigrated heat. The undulation number
investment in a very important field to increase the
efficiency of heat transfer, such as radiator design in cooling
and heating in electronic equipment. Thereafter the
thickness of the layer has attracted the attention of many
engineering applications such as solidification and melting
process and cooling tower filling design. After spaciously
meditation, the cavity wall in the literature either straight or
curved lines. The straight wall cavities defined as square,
rectangular or trapezoidal and the curved wall cavities
founded similitude to wavy sinusoidal or semicircular.
Besides that, the cylindrical and irregular solid shapes
cavities were arranged at the end of the literature.
The rectangular and square shaped cavities were
arranged from [1 to 10]. Al-Zamily [1 and 2] the square and
partitioned cavities were performed numerically. The first
[1], the partition lines are vertical forming three adjacent
layers filled with a nanofluid and one porous layer in the
middle. The results indicate that the average Nusselt number
increases with increases of the thickness of the middle
porous layer. The second [2], the layers are two and a
horizontal one of them is porous and the other is nanofluid
with the presence of a magnetic field. It was founded at high
Rayleigh number, and Darcy number increases the effect of
Hartmann number on the Nusselt number increase. Similar
to the three vertical layers that were mentioned in [1], Gao
et al. [3] Used a modified Lattice Boltzmann method to
study a conjugate heat transfer in these three mediums
again. Chamkha and Ismael [4] consolidated two vertical
porous layers in a rectangular cavity occupied by a
nanofluid, to study numerically the natural conduction. They
concluded that at a certain Rayleigh number and Nusselt
number is the maximum the critical porous layer thickness
was denoted. Iman [5] utilized a heatline visualization
method to explain the heat transport path for buoyancy-
driven flow inside a rectangular porous cavity filled with
nanofluids. The results show that the Cu-water nanofluid
having the higher rates of heat transfer in all cases. Such as
cavity of reference [5], Homman [6] add the vector of
energy flux determination as the viscosity variations with
temperature was considered. It was found that at a certain
temperature of evaluation of fluid properties evaluation, was
a function of the shape parameter of porous media and other
considered parameters. The same rectangular porous
cavities of references [5 and 6] but with different boundary
conditions were investigated by Yasin et al. [7]. The
sinusoidal profile of temperatures on the bottom cavity wall
was considered and the others were adiabatic. It was noticed
that the amplitude increasing of the sinusoidal temperature
distribution the rate of heat transfer increases.
Muthtamilselvan and Sureshkumar [8], proposed a square
porous cavity with lid-driven filled with nanofluid and
different heat sources. The results reveal a reduction in the
flow intensity as the solid volume fraction increases.
Sheikholeslami et al. [9] Presented a square porous cavity
with centered hot circular obstacle filled with nanofluid and
subjected to Lorentz forces to study the forced convection
by using the Lattic Boltzmann technique. The results show
that the Nusselt number increases of Darcy and Rayleigh
numbers. A triangular abstraction was fixed at the lower left
corner of a nanofluid porous cavity by Muneer et al. [10] To
investigate numerically the heat transfer and entropy
generation. They predicted the entropy generation increases
with nanoparticles addition.
As a complement of the straight wall cavities the
trapezoidal cavities were presenting from [11 to 16].
Alsabery et al. [11 and 12] studied numerically the heatline
visualization of the natural convection of natural convection
in trapezoidal cavities filled partially with nanofluid porous
layer and partially with non-Newtonian fluid layer [12]. The
results showed that the variation of cavity inclination angle
effect on the heat transfer rate. Yasin [13] performed a
numerical study to analyze the behavior of heat
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:03 28
I J E N S IJENS © June 2019 IJENS -IJMME-1717-031925
transportation and the fluid flow due to natural convection
in a trapezoidal cavity with the presence of a horizontal
solid partition and filled with fluid-saturated porous
medium. It was observed that the conduction mode became
dominant inside the cavity for higher partition thickness.
Ratish and Bipin [14] coupled a numerical finite element
method with a Krylov subspace solver to study the natural
convection in a trapezoidal porous cavity. Nusselt number
was noted to increase with increasing of Rayleigh and
Grashof numbers. Tanmay et al. [15] performed a numerical
investigation of natural convection in trapezoidal, porous
cavities with uniform and non-uniform heated bottom wall
was illustrated using heatlines. They found that minimum
local natural convection was near the corners of the bottom
wall. Mohammad et al. [16] investigated numerically the
natural convection inside a trapezoidal cavity occupied by a
nanofluid. The results show at low value of Rayleigh
number 104 the average Nusselt number decreases.
The curved walls cavities founded similitude to a
sinusoidal wave from [17 to 26]. Sheikholeslami and
Sadoughi [17] demonstrated the effect of the magnetic field
on a cavity filled with a nanofluid in porous media. Results
proved that the values of Nusselt number decreases when
the Lorentz forces increase. Sheikholeslami et al. [18 and
19] used the same cavity but with differ boundary
conditions, then the numerical study were occurring under
the effect under the magnetic field and electrical field,
respectively. The results approved that the heat transfer rate
reduces with rise of Hartmann number and the convection
mode was boosted by electric field, respectively.
Sheikholeslami and Shamlooei [20] simulated the effect of
magnetic field on the flow of nanofluid in the cavity with
permeable medium. The influence of nanoparticles shape
was taken into account. The results detected the velocity of
the fluid and Nusselt number was decreasing due to increase
in Hartmann number. Sheikholeslami and Shehzad [21]
studied the nanofluid migration using Darcy model inside a
permeable medium by control volume based finite element
method. The results illustrated that the distribution of the
isotherms becomes more complex with increasing of
buoyancy force. Same cavity, but with different boundary
conditions subjected to a magnetic field was investigated by
Sheikholeslami and Houman [22]. The results show that the
heat transfer rate reduces with increasing of buoyancy force.
Minh et al. [23] investigated the effect of a horizontal wavy
interface on the natural convection inside a nanofluid porous
medium by using ISPH formulation. It was depicted that the
average Nusselt number reduces due to increase in
amplitude, height and the number of undulation of the
sinusoidal interface. The natural convection in a permanent
cavity was investigated numerically with two adiabatic
horizontal wavy walls and the other sides subjected a
variable temperature by Sherment and Pop [24]. It was
found that the average Nusselt number were increasing with
Lewis number decreases. Khalil et al. [25] investigated
numerically the natural convection inside a porous cavity
with a left, vertical, isothermal and wavy side wall. This
investigation shows that the wavy surface amplitude and
undulation number affect isotherms. Salam [26] by using
heatline visualization technique, investigated numerically
the natural convection and the generation of entropy inside
porous, two sides wavy walls tilted cavity with the presence
of a magnetic field. Moreover, the results depicted that the
generation of entropy due to the horizontal magnetic field
were higher than the vertical.
The curved walls similitudes to a semi-circular
cavity were presented from [27 to 31]. Ali et al. [27] has
been analyzed numerically the conjugate unsteady, natural
convection and entropy generation inside a semi-circular
permanent cavity. It was concluded that the happening of
entropy generation along the internal interface of solid-
porous at high Rayleigh number. Sheikholeslami and Ganji
[28] examine the impact magnetic field external source on
Fe3O4 nanofluid in a porous medium. It was indicated that
the increase in Hartmann number leads to decrease in
transformation of nanofluid. Sheikholeslami and Shehzad
[29] proposed a simulation of convective flow in the porous
nanofluid gap between hot elliptical inner wall and cold
circular outer wall. The outputs results demonstrated that the
maximum streamline values enhance with an increase of
Rayleigh number. The semi-circular cavity filled with a
nanofluid and faced with heating source, was carried out by
Ali and Amin [30] to investigate the natural convection and
entropy generation.it was clear from the results the rate of
heat transfer enhanced with rise of Rayleigh number. Same
cavity that subjected to a heat flux in reference [30] was
subjected to a magnetic field by Ali [31] to check its main
effect on natural convection. The results detected the rate of
heat transfer diminution with a progress of Hartmann
number. An annulus porous cavity with conjugate heat
transfer was studied by Irfan et al. [32]. Same as vertical
annulus porous cavity, but with different boundary
conditions have been analyzed numerically by Sankar et al.
[33]. It was denoted that the lower half of the inner wall was
the perfect place for heater to produce highest heat transfer
rate. T-shaped inclined cavity with various types of
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:03 29
I J E N S IJENS © June 2019 IJENS -IJMME-1717-031925
nanofluids subjected to a uniform heat flux source and the
magnetic field was examined numerically by Ahmed et al.
[34]. It was concluded the mean Nusselt number decreases
with increases of the heat source length and Hartmann
number.
Finally, Ahmed et al. [35] studied natural
convective heat transfer within trapezoidal cavity having a
horizontal porous media partition and filled with TiO2-water
nanofluid. Their results showed that the maximum value of
the percentage enhancement of heat transfer was 4.22%
occurred at porous position equal to 0.5. The wavy word
came in the literature either refers to the external cavity
walls or the interface between two or more adjacent
mediums. Therefore, it was obvious from the above
literature the rarity of information's concerning with the
effect of vertical sinusoidal interface and its location on the
fluid migration and heat transportation inside a cavity with
partially permeable medium and partially occupied by a
non-Newtonian fluid as well as, the complexity due to
geometry and numerical simulation, therefore, this work
was execution.
2. Physical model explanation
The square cavity is represented by four
cases shown schematically in Fig. 1. The
cavity with the height of (L) is heated from the
left vertical wall and is cooled from the right
wall whereas the other two walls are
considered adiabatic. This cavity is filled with
two layers of nanofluid porous medium and
non-Newtonian fluid. These layers are
vertically separated by the sinusoidal vertical
interface where the interface shape is changed
depending on undulation numbers, namely
one undulations (case1), two undulations
(case2), three undulations (case3) and four
undulations (case4). Furthermore, some
assumptions are offered:
1. The fluid flow is incompressible, steady
and laminar.
2. The density in the buoyancy term
depends only on temperature, which is
approximated by the standard
Boussinesq model, whereas other
physical properties of non-Newtonian
nanofluid are taken constant.
3. The thermal conductivity of the
nanofluid is assumed to equal the
effective thermal conductivity of the
porous medium.
4. Thermal equilibrium is taken between
the water and silver nanoparticles.
5. The rigid boundaries of cavity are
impermeable (no momentum exchange)
and the velocity components are zero
because of no-slip condition, while the
boundary of sinusoidal interface is
assumed to be permeable.
6. The equations of extended Darcy–
Brinkman model and the energy
transport are adopted to depict the flow
of non-Newtonian fluid and the heat
transfer process in the Ag-water
nanofluid and porous layer.
3. Governing equations and boundary conditions
The two dimensions governing equations of the porous medium-nanofluid layer are:
𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦
𝜕𝑢𝑝𝑛
𝜕𝑥+
𝜕𝑣𝑝𝑛
𝜕𝑦= 0 (1)
𝑋 − 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑢𝑝𝑛
𝜕𝑢𝑝𝑛
𝜕𝑥+ 𝑣𝑛
𝜕𝑢𝑝𝑛
𝜕𝑦= −
1
𝜌𝑝𝑛
𝜕𝑝
𝜕𝑥+
𝜇𝑝𝑛
𝜌𝑝𝑛
(𝜕2𝑢𝑛
𝜕𝑥2+
𝜕2𝑢𝑛
𝜕𝑦2) (2)
𝑌 − 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:03 30
I J E N S IJENS © June 2019 IJENS -IJMME-1717-031925
𝑢𝑝𝑛
𝜕𝑣𝑝𝑛
𝜕𝑥+ 𝑣𝑝𝑛
𝜕𝑣𝑝𝑛
𝜕𝑦= −
1
𝜌𝑝𝑛
𝜕𝑝
𝜕𝑦+
𝜇𝑝𝑛
𝜌𝑝𝑛
(𝜕2𝑣𝑝𝑛
𝜕𝑥2+
𝜕2𝑣𝑝𝑛
𝜕𝑦2) +
(𝜌𝛽)𝑝𝑛
𝜌𝑝𝑛
𝑔(𝑇𝑝𝑛 − 𝑇𝑐) −𝜇𝑝𝑛𝜈𝑝𝑛
𝜌𝑝𝑛ε (3)
𝐸𝑛𝑒𝑟𝑔𝑦
𝑢𝑝𝑛
𝜕𝑇𝑝𝑛
𝜕𝑥+ 𝑣𝑝𝑛
𝜕𝑇𝑝𝑛
𝜕𝑦= 𝛼𝑝𝑛 (
𝜕2𝑇𝑛
𝜕𝑥2+
𝜕2𝑇𝑛
𝜕𝑦2) (4)
The two-dimensional governing equations for the non-Newtonian fluid are:
𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝜕𝑢𝑛𝑁
𝜕𝑥+
𝜕𝑣𝑛𝑁
𝜕𝑦= 0 (5)
𝑋 − 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑢𝑛𝑁
𝜕𝑢𝑛𝑁
𝜕𝑥+ 𝑣𝑛𝑁
𝜕𝑢𝑛𝑁
𝜕𝑦= −
1
𝜌𝑛𝑁
𝜕𝑝
𝜕𝑥+
1
𝜌𝑛𝑁
(𝜕𝜏̿̿ ̿
𝑥𝑥
𝜕𝑥+
𝜕𝜏̿̿ ̿𝑥𝑦
𝜕𝑦) + 𝜌𝑛𝑁𝛽𝑛𝑁𝑔(𝑇𝑝𝑛 − 𝑇𝑐) (6)
𝑌 − 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑢𝑛𝑁
𝜕𝑣𝑛𝑁
𝜕𝑥+ 𝑣𝑛𝑁
𝜕𝑣𝑛𝑁
𝜕𝑦= −
1
𝜌𝑛𝑁
𝜕𝑝
𝜕𝑦+
1
𝜌𝑛𝑁
(𝜕𝜏̿̿ ̿
𝑥𝑦
𝜕𝑥+
𝜕𝜏̿̿ ̿𝑦𝑦
𝜕𝑦) + 𝜌𝑛𝑁𝛽𝑛𝑁𝑔(𝑇𝑝𝑛 − 𝑇𝑐) (7)
𝐸𝑛𝑒𝑟𝑔𝑦
𝑢𝑛𝑁
𝜕𝑇𝑛𝑁
𝜕𝑥+ 𝑣𝑛𝑁
𝜕𝑇𝑛𝑁
𝜕𝑦= 𝛼𝑛𝑁 (
𝜕2𝑇𝑛𝑁
𝜕𝑥2+
𝜕2𝑇𝑛𝑁
𝜕𝑦2) (8)
Non-Newtonian fluid is used which follow the power law model, therefore the shear stress tensor is Ostwald-DeWaele
[36]
�̿�𝑖𝑗 = 2𝜇𝑎𝐷𝑖𝑗 = 𝜇𝑎 (𝜕𝑢𝑖
𝜕𝑥𝑗+
𝜕𝑢𝑗
𝜕𝑥𝑖) (9)
Where Dij is the rate of strain tensor for the two-dimensional Cartesian coordinate and μa is the dimensional Cartesian
coordinate as:
𝜇𝑎 = 𝜎 {2 [(𝜕𝑢𝑛𝑁
𝜕𝑥)
2
+ (𝜕𝑢𝑛𝑁
𝜕𝑦)
2
] + (𝜕𝑣𝑛𝑁
𝜕𝑥+
𝜕𝑢𝑛𝑁
𝜕𝑦)
2
} 𝑛−1
𝑛 (10)
Where σ and n are power-law model constants. (σ) is the consistency coefficient and (n) is the power-law index. Where
(n<1), it represents a pseudoplastic such as condensed milk, toothpaste, ketchup, etc. and (n>1), it represents a dilatant
fluid such as cornstarch and water mixtures. When (n=1) a Newtonian fluid is obtained. The two layers (porous-nano
layer and non-Newtonian fluid layer) is considered separately by sinusoidal permeable interface with corresponding join
conditions at that interface. These conditions include continuity, normal stress, shear stress, heat rate, and temperature.
These conditions can be expressed as equations [11-15] at the permeable interface.
𝑇𝑝𝑛|𝑥=𝐼− = 𝑇𝑛𝑁|𝑥=𝐼+ (11)
𝑘𝑝𝑛
𝜕𝑇𝑝𝑛
𝜕𝑥|
𝑥=𝐼−= 𝑘𝑛𝑁
𝜕𝑇𝑛𝑁
𝜕𝑥|
𝑥=𝐼+ (12)
𝑢𝑝𝑛|𝑥=𝐼− = 𝑢𝑛𝑁|𝑥=𝐼+ , 𝑣𝑝𝑛|
𝑥=𝐼− = 𝑣𝑛𝑁|𝑥=𝐼+ (13)
𝑝𝑝𝑛|𝑥=𝐼− = 𝑝𝑛𝑁|𝑥=𝐼+ (14)
𝜇𝑝𝑛(𝜕𝑣𝑝𝑛
𝜕𝑥+
𝜕𝑢𝑝𝑛
𝜕𝑦)|
𝑥=𝐼−
= 𝜇𝑛𝑁(𝜕𝑣𝑛𝑁
𝜕𝑥+
𝜕𝑢𝑛𝑁
𝜕𝑦)|
𝑥=𝐼+
(15)
Equation (15) denotes addition of the shear stress corresponding condition [37]. To convert the systems of equations (1-
10) to the dimensionless form, one can suppose the following assumptions:
𝑋 =𝑥
𝐿, 𝑌 =
𝑦
𝐿, 𝑈 =
𝐿𝑢
𝛼𝑓𝑙, 𝑉 =
𝐿𝑣
𝛼𝑓𝑙, 𝑃 =
𝐿2𝑝
𝜌𝛼2 , 𝜃𝑝𝑛 =𝑇−𝑇𝑐
𝑇𝑝𝑛−𝑇𝑐, 𝜃𝑛𝑁 =
𝑇−𝑇𝑐
𝑇𝑛𝑁−𝑇𝑐 (16)
The set of governing dimensionless equations for the porous- nanofluid layer become:
𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦
𝜕𝑈𝑝𝑛
𝜕𝑋+
𝜕𝑉𝑝𝑛
𝜕𝑌= 0 (17)
𝑋 − 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:03 31
I J E N S IJENS © June 2019 IJENS -IJMME-1717-031925
𝑈𝑝𝑛
𝜕𝑈𝑝𝑛
𝜕𝑋+ 𝑉𝑝𝑛
𝜕𝑈𝑝𝑛
𝜕𝑌= −
𝜌𝑓𝑙
𝜌𝑝𝑛
𝜕𝑃
𝜕𝑋+ 𝑃𝑟 (
𝜕2𝑈𝑝𝑛
𝜕𝑋2+
𝜕2𝑈𝑝𝑛
𝜕𝑌2) +
(𝜌𝛽)𝑝𝑛
𝜌𝑝𝑛
𝑅𝑎𝑃𝑟𝜃𝑝𝑛 −𝜇𝑝𝑛
𝜇𝑓𝑙
𝑃𝑟𝑈𝑝𝑛
𝐷𝑎 (18)
𝑌 − 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑈𝑝𝑛
𝜕𝑉𝑝𝑛
𝜕𝑋+ 𝑉𝑝𝑛
𝜕𝑉𝑝𝑛
𝜕𝑌= −
𝜌𝑓𝑙
𝜌𝑝𝑛
𝜕𝑃
𝜕𝑌+ 𝑃𝑟 (
𝜕2𝑉𝑝𝑛
𝜕𝑋2+
𝜕2𝑉𝑝𝑛
𝜕𝑌2) +
(𝜌𝛽)𝑝𝑛
𝜌𝑝𝑛
𝑅𝑎𝑃𝑟𝜃𝑝𝑛 −𝜇𝑝𝑛
𝜇𝑓𝑙
𝑃𝑟𝑉𝑝𝑛
𝐷𝑎 (19)
𝐸𝑛𝑒𝑟𝑔𝑦
𝑈𝑝𝑛
𝜕𝜃𝑝𝑛
𝜕𝑋+ 𝑉𝑝𝑛
𝜕𝜃𝑝𝑛
𝜕𝑌=
𝛼𝑝𝑛
𝛼𝑓𝑙
(𝜕2𝜃𝑛
𝜕𝑋2+
𝜕2𝜃𝑛
𝜕𝑌2) (20)
And the set of governing dimensionless equations for the non-Newtonian fluid layer become:
𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦
𝜕𝑈𝑛𝑁
𝜕𝑋+
𝜕𝑉𝑛𝑁
𝜕𝑌= 0 (21)
𝑋 − 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑈𝑛𝑁
𝜕𝑈𝑛𝑁
𝜕𝑋+ 𝑉𝑛𝑁
𝜕𝑈𝑛𝑁
𝜕𝑌= −
𝜕𝑃
𝜕𝑋+ Pr [2
𝜕
𝜕𝑋(𝜇𝑎
𝜎
𝜕𝑈𝑛𝑁
𝜕𝑋) +
𝜕
𝜕𝑌(𝜇𝑎
𝜎
𝜕𝑈𝑛𝑁
𝜕𝑌+
𝜇𝑎
𝜎
𝜕𝑈𝑛𝑁
𝜕𝑋)] + 𝑅𝑎𝑃𝑟𝜃𝑛𝑁 (22)
𝑌 − 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑈𝑛𝑁
𝜕𝑉𝑛𝑁
𝜕𝑋+ 𝑉𝑛𝑁
𝜕𝑉𝑛𝑁
𝜕𝑌= −
𝜕𝑃
𝜕𝑌+ 𝑃𝑟 [
𝜕
𝜕𝑋(𝜇𝑎
𝜎
𝜕𝑈𝑛𝑊
𝜕𝑌+
𝜇𝑎
𝜎
𝜕𝑉𝑛𝑁
𝜎) + 2
𝜕
𝜕𝑌(𝜇𝑎
𝜎
𝜕𝑉𝑛𝑁
𝜕𝑌)] + 𝑅𝑎𝑃𝑟𝜃𝑛𝑁 (23)
𝐸𝑛𝑒𝑟𝑔𝑦
𝑈𝑛𝑁
𝜕𝜃𝑛𝑁
𝜕𝑋+ 𝑉𝑛𝑁
𝜕𝜃𝑛𝑁
𝜕𝑌=
𝜕2𝜃𝑛𝑁
𝜕𝑋2+
𝜕2𝜃𝑛𝑁
𝜕𝑌2 (24)
The subscript (po) refers to porous-nanofluid and (nN) refers to the non-Newtonian fluid layer.
Where
𝑅𝑎𝑓𝑙 =𝑔 𝑘𝜌𝑓𝑙𝛽𝑓𝑙(𝑇ℎ−𝑇𝑐)𝐿
𝑣𝑓𝑙𝛼𝑓𝑙 (𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ 𝑛𝑢𝑚𝑏𝑒𝑟) , 𝑃𝑟𝑝𝑛 =
𝜈𝑝𝑛
𝛼𝑝𝑛 is the Prandtl number for both porous-nanofluid layer and
non-Newtonian fluid layer, 𝐷𝑎 =𝑘
𝐿2 is the Darcy number for the porous-nano fluid layer. The thermophysical properties
properties of the nanoparticle with base fluid (water Pr=6.2) are described in Table 1.
(αpn) is the thermal diffusivity of the nanofluid in porous layer, (ρpn) is the density of the nanofluid, (μpn) is the dynamic
viscosity of the nanofluid and (ρcp)pn is the heat capacitance of the nanofluid which are expressed [38] as:
𝜌𝑝𝑛 = (1 − ∅)𝜌𝑓𝑙 + ∅𝜌𝑠𝑛 , 𝛼𝑝𝑛 =𝑘𝑝𝑛
(𝜌𝑐𝑝)𝑝𝑛
, (𝜌𝑐𝑝)𝑝𝑛 = (1 − ∅)(𝜌𝑐𝑝)𝑓𝑙
+ ∅(𝜌𝑐𝑝)𝑠𝑛 (25)
The dynamic viscosity of the nanofluid is expressed in [39] as
𝜇𝑝𝑛 =𝜇𝑓𝑙
(1 − ∅)2.5 (26)
The thermal expansion factor of the nanofluid can expressed as
𝛽𝑝𝑛 = (1 − ∅)(𝛽)𝑓𝑙 + ∅𝛽𝑠𝑛 (27)
(𝜌𝛽)𝑝𝑛 = (1 − ∅)(𝜌𝛽)𝑓𝑙 + ∅(𝜌𝛽)𝑠𝑛 (28)
The thermal conductivity of nanofluid constructed by Maxwell-Garnet’s [40] is
𝐾𝑝𝑛 = 𝐾𝑓𝑙[𝑘𝑠𝑛+2𝑘𝑓𝑙−2∅(𝐾𝑓𝑙−𝐾𝑠𝑛)
𝑘𝑠𝑛+2𝑘𝑓𝑙+∅(𝐾𝑓𝑙−𝐾𝑠𝑛) ] (29)
The suitable dimensionless boundary conditions for dimensionless governing equation of the present study are:
Left vertical wall U=V=0, θ=1, X=0
Right vertical wall U=V=0, θ=0, X=1
Top horizontal wall U=V=0 𝜕𝜃
𝜕𝑌= 0 Y=1
Bottom horizontal wall U=V=0 𝜕𝜃
𝜕𝑌= 0 Y=0
The non-dimensional equation of the heat function for the porous-nanofluid layer can be expressed as [41]:
𝜕2𝐻
𝜕𝑋2= 𝑈𝜃 −
𝛼𝑝𝑛
𝛼𝑓𝑙
𝜕𝜃
𝜕𝑋, −
𝜕𝐻
𝜕𝑋= 𝑉𝜃 −
𝛼𝑝𝑛
𝛼𝑓𝑙
𝜕𝜃
𝜕𝑌 (30)
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Which produces the following equation.
𝜕2𝐻
𝜕𝑋2+
𝜕2𝐻
𝜕𝑌2=
𝜕
𝜕𝑌(𝑈𝜃) −
𝜕
𝜕𝑋(𝑉𝜃) (31)
The non-dimensional equation of the heat function for the non-Newtonian fluid layer expressed as[42]:
𝜕𝐻
𝜕𝑌= 𝑈𝜃 −
𝜕𝜃
𝜕𝑋, −
𝜕𝐻
𝜕𝑋= 𝑉𝜃 −
𝜕𝜃
𝜕𝑌 (32)
Which produce the following equation.
𝜕2𝐻
𝜕𝑋2+
𝜕2𝐻
𝜕𝑌2=
𝜕(𝑈𝜃)
𝜕𝑌−
𝜕(𝑉𝜃)
𝜕𝑋 (33)
The Neuman boundary condition for heat function for both thermal vertical walls from the equation (32) and the
normal deviation (�̂�∇𝐻) are defined as below.
(�̂�∇𝐻) = 0 𝑎𝑡 𝑡ℎ𝑒 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑤𝑎𝑙𝑙𝑠 (34)
𝑛.̂ ∇𝐻 =𝑘𝑝𝑛
𝑘𝑓𝑙
∫ (𝜕𝜃𝑝𝑛
𝜕𝑋)𝑥=0𝑑𝑌
1
0
𝑎𝑡 𝑡ℎ𝑒 𝑎𝑑𝑖𝑎𝑏𝑎𝑡𝑖𝑐 𝑡𝑜𝑝 𝑤𝑎𝑙𝑙 (35)
The adiabatic bottom wall may be expressed by the Dirichelet boundary condition attained from the equation (32),
which reduced to the (𝜕𝐻
𝜕𝑋= 0) for and adiabatic wall. A mention amount of heat function is supposed as at X=0, Y=0
and therefore heat function (H=0) is accurate for Y=0. H=αpn/αfl Nupn is found from equation (32) at X=0, Y=0. The total
amount of heat transfer rate within the enclosure can expressed by the average Nusselt number equations along the
heated wall.
𝑁𝑢𝑎𝑣𝑒̅̅ ̅̅ ̅̅ ̅̅ = ∫ [−(
𝑘𝑝𝑛
𝑘𝑓𝑙
)𝜕𝜃𝑝𝑛
𝜕𝑋] 𝑑𝑌
1
0
(36)
4. Numerical procedure and code validation
Dimensionless governing equations with dimensionless boundary conditions in the earlier section have been given
are described using finite element method. The following assumption were considered, laminar flow, two-dimensional,
steady state, incompressible and Newtonian fluid with Boussing approximation is used to describe the buoyancy effect.
Galerkine finite element least-square method is used to discretization the partial differential equation and to ensure the
stability. The nonlinear equations which produce from Galarkine finite element method solved by using the damped
Newtonian method. The following assumptions are used incompressible fluid, two dimensional, laminar flow, steady
state and the Poisson's equation are used for the stream function
𝜕2Ψ
𝜕𝑋2+
𝜕2Ψ
𝜕𝑌2=
𝜕𝑈
𝜕𝑌−
𝜕𝑉
𝜕𝑋 (37)
This study examined at various Darcy numbers, volume fraction, Prandtl number and power-law index. The current
program was verified for grid independence by computing the average Nusselt number along the left hot wall as reported
in table 2. It was found that a grid size (180*180) confirms a grid independence explanation. To ensure the accurate
results, the current program results are compared with a published article by [11].
In this work, rectangular elements with mapped mesh shown in Fig. 2, is descrtrized in the trapezoidal cavity.
Rectangular finite elements of variable order are taken for every term of the partial differential equation. The error for
the relative convergence solution for each of the variables in governing equation convenience the following criteria
|𝜁𝑖+1−𝜁𝑖
𝜁𝑖| ≤ 10−6 , where i signify the iteration number. The results are validated in Fig. 3, as the stream and isothermal
lines have a good convenient with the published article by [11].
The weak formula of the dimensionless governing equations is achieved, after approaching flow variables are replaced
in the dimensionless governing equations, Residual Approach Consequences (RAC) and the weighted average of
residual approach will be compelled to equal to zero as in Equation below. For more information on the finite element
approach can be gotten in [43, 44].
∫ 𝑊 ∗ (𝑅𝐴𝐶)𝑑𝐴 = 0𝐷𝑜𝑚𝑎𝑖𝑛
(38)
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5. Results and Discussion The numerical solution was carried out to investigate the effect
of wavy vertical interface on the natural convection in a
differentially heated cavity filled with two mediums, porous on
the left and non-Newtonian fluid on the right. For all
calculations, the pure water with Pr=6.2 was a base fluid,
Ra=103 to 105, Da=10-1 to10-5 and ϕ=0 to 0.2
1- Effect of Rayleigh number
Figs. 4 to 1 show the effect of the Rayleigh number on the
streamlines and isotherms. It was clear from the figures of the
streamlines the circulation was obviously enhanced with Ra
put up to 105. Consequently, the centrifugal force increases
pushing the convergent streamlines from the core of the large
one vortex to the boundaries in the right non-Newtonian
medium. Additionally, the weakness of circulation strength at
low Ra, defeat the fluid to permeate through the left porous
medium, so the streamlines were very difficult to penetrate.
While at high Ra, the streamlines penetrated significantly at
the left medium. Commonly, the streamlines values were rises
with Ra rises. The results of streamlines contours showed,
after mixing the nanoparticles, red lines (ϕ=0.05), the stream
lines magnitude (Ψmax nf and Ψmin nf) was overall rises if
compared with a pure water, green lines (ϕ=0).
The isotherms contours display different heat flow
schemes in the porous and non-Newtonian mediums as Ra
increase. At low Ra=104, the isotherms trend to be arranged
vertically not overall cavity, but especially at left porous
medium, therefore, the isotherms demonstrated to be
transitional case and unclaimed to say a pure conduction heat
transfer regime. While, at the right non-Newtonian medium,
the isotherms patterns obliquity is perfect for the dominated
convection heat transfer flow. As the Ra increases to 105, the
vortex circulation intensity was increasing. Therefore, the
obliquity of the isothermlines gotten clear more, thereby, the
convection mode was the dominant on the left and right
mediums of the cavity. Due to low permeability at DA=105 the
conduction regime keeps dominating in the left porous
medium.
Fig. 12 shows the relation between Darcy number and
average Nusselt number on the left hot wall. It was obvious the
average Nusselt number was increasing with increasing of Ra
from 103 to 105 overall values of Da. Uniquely case it was
observed at S=0.75, the average Nusselt number doesn’t
enhance with Ra increases at low Da= 10-5 to 10-4, but rises
from Da=10-3 to 10-1 due to dominant conduction heat transfer
at low permeability in the left porous medium.
2- Effect of Darcy number:
Figs. 4 to 11 shows the streamlines and isotherms for Ra=104
and 105, Da=10-3 and 10-5 and ϕ= 0 and 0.05. The streamlines
contours explain the decreasing Da from 10-3 to 10-5 prevent
the nanofluid penetration through the porous medium due to
the permeability decreases. In most cases this weakens the
vortices circulation. The thermal behavior changes the
mechanism of heat transfer from weak convection to
conduction with decreasing Da in the left porous medium. But
the isotherms distribution on the right non-Newtonian medium
becomes more than before and maintain to convection mode.
At low Da=10-5, the high hydrodynamic resistances
displayed by the left porous medium, this lead to low
nanofluid penetration in this medium. The non-Newtonian
fluid enclosed by left porous medium and cold right side wall
enjoys with dominated convection heat transfer mode.
Generally, at high Da=10-3 the effect of nanoparticles
addition ϕ=0.05 (red lines) on the streamlines contours, figs. 4
and 6, was obvious if it was compared with low Da=10-5, figs.
8 and 10, because the nanoparticles adding increase the fluid
revolving in the two mediums occupied the cavity. Inversely,
at low Da=10-5 the streamlines patterns in the left porous
medium were almost appear as a result of permeability
reduction and the nanoparticles addition ϕ=0.05 affect slightly
on the non-Newtonian fluid part. The augmentation of water
thermal conductivity due to nanoparticle additive enhances the
conductivity of it, the augment with heat transfer rate.
Fig. 12 explains the relation between Da and average
Nu for different cases. As Da built up, the permeability was
increased in the left porous medium and the domination heat
transfer was convection, hence more heat transfer rate.
Generally, a directly proportional relation between these two
parameters was denoted.
3- Effect of the interface location:
Figs. 4 to 11explain the width S effect of the porous
layer on the stream function and isotherms for Ra=104 and
105, Da=10-3 and 10-5 and ϕ=0 to 0.05.the fluid revolving
intensity decreases dramatically with the left porous
medium width increases from 0.25 to 0.75. Whenever, the
interface aboard from the right wall the porous layer
expanded, therefore, the stream function values were
decreases and the streamlines compressed toward the cold
side wall. The addition of nanoparticles enhanced the
stream function in both porous and non-Newtonian
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mediums. In respect to isotherms, the addition had no
noticeable affects.
Generally, when the width of the left porous layer
increases, the isothermlines inclination reduces (change
from horizontal distribution to vertical). Consequently, the
conduction regime was overcome and the convection
reduces gradually with width of porous medium increases
from 0.25 to 0.75.
Fig. 12 shows the relation between average Nu and
Da. After focusing, the descent of the average Nu values
at any Ra, were detected from S=0.25 to 0.75. The cause
of this descent was reducing of fluid circulation that is
reflected on the nature of heat transfer.
4- Effect of the interface undulation number (F):
The undulation number of the wavy interface is
interesting geometrical criterion were demonstrated. Figs.
8 and 10 show the cavity stream function and isotherms
for Ra=104, 105, Da=10-5 and S=0.75. The contour maps
of streamline show the core of a single elongated vortex at
F=1 located nearest to the upper adiabatic wall. As F
increases from 2 to 4, the streamlines obeys and follow
the sinusoidal shape of the interface and splits into 2, 3
and 4 individual, small and weak vortices that was
contribute to reduce the stream function values. It should
be noticed, at F=3 and 4 the length of the interface
becomes longer (more contact area) allow to some of
streamlines to penetrate slightly through the left porous
layer. But at S=0.25 and 0.5, there was no dramatic
influence occurred on streamlines only it was follow the
interface shape too, and the lonely dominates vortex
became smaller at S=0.5.
For other cases figs. 4 and 6when Ra=104 and 105 and
Da=10-3 there was no important change that was draw
attention were observed. In respect to the isotherms, due
to increase with number of undulation from 1 to 4, the
isotherms patterns bending were reduces and a large
interface area that was increasing with F, attempt to
obtrudes the conduction regime through the non-
Newtonian fluid medium.
If we consider the Da=104, S=0.25 to 0.75 and
Ra=103 to 105 of fig. 12, two things were noticed, the first
one was the average Nu were decreases with increasing of
F from F=1 (case 1) to F=4 (case 4) when S=0.25 and 0.5,
because of the idleness of non-Newtonian fluid
circulation. The second, at S=0.75, the maximum average
Nu obtained when F=3 (yellow lines), the others were
compacted to each other and almost don’t change with
varying of F.
Fig. 13 show the relation between average Nu and
volume fraction ϕ at a constant Ra=105, Da=10-3, n=0.7,
F=1 to 4 and S=0.25 to 0.75. The average Nu increases at
S=0.25with volume fraction increases because of the fluid
thermal conductivity increase. Inversely, when the S
becomes 0.5 and 0.75 the behavior of the relation was
changed from increases to decreases due to the circulation
idleness, and then the convection regime reduces
gradually as the sinusoidal interface nears the right
vertical cold wall. As well as, it was clear from fig. 13,
when S=0.25, the value of average Nu were reduces as F
increases except F=3 (case 3) the average Nu values were
higher than the other cases, due to the matching between
the concavity of the interface and the streamlines
contours. Hence, more uniform fluid motion and uniform
heat transfer rate than the other cases (1, 2 and 4).
As the porous layer thickness increase the uniform
fluid circulation and heat transfer regime occurs too in the
(cases 2 and 3) and (cases 2, 3 and 4) when S=0.5 and
S=0.75 fig. 13, respectively. Therefore, when S=0.75 the
values of the average Nu for (cases 2, 3 and 4) were
convergent forming almost one line parallel up to the
average Nu of (case 1), due to circulation velocity was
slow down and governs the situation as the thickness of
the porous medium increases. Under these conditions the
increases of S don’t effect on the values of average Nu.
Two important things can be concluded from fig. 14
show the relation between the average Nu and power law
index n. The first, the average Nu was decreases with
rising the n because as n increases the circulation intensity
slows down, due to high viscosity and shear force of
pseudo-plastic flow. Therefore, decreases heat transfer
rate and average Nu. The second, clearly, the convection
heat transfer enhanced at high Da (0.0001 to 0.1) which
lead to increase the average Nu. But at low Da (0.00001 to
0.0001) the convection remains weak, accordingly, the
average Nu decrease slightly.
Finally, it was appeared from fig. 14, the increasing
of F from 1 to 4 have not effect on average Nu values,
then thee four figures of the four cases were similar.
Conclusion
1- The streamlines values increased with Ra increase.
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2- The strength of streamlines enhanced with addition of
nanoparticles.
3- At lowest Da=10-3 the porous medium becomes likes
a solid material, hence, plug the permeability ahead
the nanofluid to penetrate through the porous
medium, consequently, the conduction heat transfer
was dominating.
4- The nanoparticle additive enhance the stream
function at high Da=10-3 and overall augment the heat
transfer flow through the cavity.
5- Generally, whenever the width of the porous layer
increase the conduction mode overcome and the
convection mode recedes.
6- With increasing the number of undulations F, the
conduction regime to force through the non-
Newtonian fluid circulation nears the interface.
7- As power law index n increases, the average Nu
decreases due to high viscosity and shear force of
pseudo plastic fluid.
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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:03 37
I J E N S IJENS © June 2019 IJENS -IJMME-1717-031925
Table I
Thermo-physical of Ag/water nanofluid
Physical properties Water Ag
Cp (J/kg.K) 4179 235
ρ (kg/m3) 997.1 10500
k (w/m.K) 0.6 429
β*10-5 (W/m.K) 21 5.4
Fig. 1. simplified diagram of physical of the present study
S S
S S
Case 1. Case 2.
Case 3.
y
cT
L
L
nfL nwL
hT
Adiabatic wall
Adiabatic wall
Nan
ofl
uid
Non
-New
tonia
n
x
y
cT
L
L
nfL nwL
hT
Adiabatic wall
Adiabatic wall
Nan
ofl
uid
Non
-New
tonia
n
x
y
cT
L
L
nfL nwL
hT
Adiabatic wall
Adiabatic wall
Nan
ofl
uid
No
n-N
ewto
nia
n
x
y
cT
L
L
nfL nwL
hT
Adiabatic wall
Adiabatic wall
Nan
ofl
uid
No
n-N
ewto
nia
n
x
Case 4.
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Table II
Grid testing for Average Nusselt number at different mesh number for Pr = 6.2, Ra = 105; Da = 10-3; n = 0.7; φ=0.05, Case 4.
Test number Mesh number Average Nusselt number
Domain elements Boundary elements
𝟓𝟎 × 𝟓𝟎 2000 230 3.9278
𝟕𝟎 × 𝟕𝟎 4900 350 3.9603
𝟗𝟎 × 𝟗𝟎 8100 450 3.9662
𝟏𝟎𝟎 × 𝟏𝟎𝟎 10000 500 3.9680
𝟏𝟐𝟎 × 𝟏𝟐𝟎 14400 600 3.9703
𝟏𝟒𝟎 × 𝟏𝟒𝟎 19600 700 3.9718
𝟏𝟔𝟎 × 𝟏𝟔𝟎 25600 800 3.9728
𝟏𝟖𝟎 × 𝟏𝟖𝟎 32400 900 3.9730
Fig. 2. Mesh Distribution of the Model
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Streamlines
Streamlines
Isotherms
Isotherms
918.- nf )min(Ψ81, 8.-= bf)min(Ψ
Fig. 3. Comparision of the present study and A.I. Alsabery et al [5] for Streamlines and Isotherms contours: , for Ra = 105; Da = 10-3; n = 0.7; S = 0.5, φ =
16.7, Pr=13.4 and 4.4; ϕ= 0 (green line) and ϕ= 0.05 (red lines).
Pr=6.2
8.84- nf )min(Ψ,8.75 -= bf)min(Ψ
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0=max bf 𝜓3.5339, -= min bf 𝜓 0=max nf 𝜓3.7752, -= min nf 𝜓
0=max bf 𝜓1.4127, -= min bf 𝜓 0=max nf 𝜓1.3099, -= min nf 𝜓
0=max bf 𝜓3.7031, -= min bf 𝜓 0=max nf 𝜓3.8457, -= min nf 𝜓
0=max bf 𝜓1.3921, -= min bf 𝜓 0.0011=max nf 𝜓1.3006, -= min nf 𝜓
0=max bf 𝜓3.4333, -= min bf 𝜓 0=max nf 𝜓 3.5534,-= min nf 𝜓
0=max bf 𝜓1.3428, -= min bf 𝜓 0=max nf 𝜓1.2286, -= min nf 𝜓
Fig. 4. Streamlines contours: , for Ra = 104; Da = 10-3; n = 0.7, Pr=6.2 for different positions of porous surface and four cases; ϕ= 0 (green line) and ϕ= 0.05 (red lines).
0=max bf 𝜓4.2896, -= min bf 𝜓
0=max nf 𝜓 4.4567,-= min nf 𝜓
S=0.5
0=max bf 𝜓2.0045, -= min bf 𝜓
0=max nf 𝜓1.9942, -= min nf 𝜓
S=0.75
0=max bf 𝜓5.8605, -= min bf 𝜓
0=max nf 𝜓6.0655, -= min nf 𝜓
S=0.25
N=1
0=max bf 𝜓6.1730, -= min bf 𝜓 0=max nf 𝜓6.4423, -= min nf 𝜓
N=2
0.0026=max bf 𝜓5.9504, -= min bf 𝜓
0.0029=max nf 𝜓6.1772, -= min nf 𝜓
N=3
0=max bf 𝜓5.8823, -= min bf 𝜓 0=max nf 𝜓6.0948, -= min nf 𝜓
N=4
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Fig. 5. Isotherms contours: , for Ra = 104; Da = 10-3; n = 0.7, Pr=6.2 for different positions of porous surface and four cases; ϕ= 0 (green line) and ϕ= 0.05
(red lines).
S=0.25 S=0.5 S=0.75
N=1
N=2
N=3
N=4
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0=max bf 𝜓10.746, -= min bf 𝜓
0.0034=max nf 𝜓10.88, -= min nf 𝜓
0=max bf 𝜓9.5019, -= min bf 𝜓 0.004=max nf 𝜓9.5002, -= min nf 𝜓
0=max bf 𝜓8.8876, -= min bf 𝜓
0.00576=max nf 𝜓8.8950, -= min nf 𝜓
0.0031=max bf 𝜓10.292, -= min bf 𝜓
0.00323=max nf 𝜓10.515, -= min nf 𝜓
0=max bf 𝜓9.2049, -= min bf 𝜓
0.00514=max nf 𝜓9.1737, -= min nf 𝜓
0=max bf 𝜓7.9933, -= min bf 𝜓
0.00488=max nf 𝜓7.8569, -= min nf 𝜓
0.0047=max bf 𝜓10.592, -= min bf 𝜓
0.00453=max nf 𝜓10.426, -= min nf 𝜓
0=max bf 𝜓9.2563, -= min bf 𝜓
0.0049=max nf 𝜓9.3203, -= min nf 𝜓
0.005=max bf 𝜓8.0483, -= min bf 𝜓 0=max nf 𝜓8.0447, -= min nf 𝜓
0=max bf 𝜓10.473, -= min bf 𝜓
0.0032 =max nf 𝜓10.594, -= min nf 𝜓
0.0167=max bf 𝜓8.0965, -= min bf 𝜓 𝜓𝜓
0=max bf 𝜓8.1169, -= min bf 𝜓 0.0034=max nf 𝜓8.0620, -= min nf 𝜓
Fig. 6. Streamlines contours: , for Ra = 105; Da = 10-3; n = 0.7, Pr=6.2 for different positions of porous surface and four cases; ϕ= 0 (green line) and ϕ= 0.05 (red lines).
N=1
S=0.25 S=0.5 S=0.75
N=2
N=3
N=4
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Figure (7). Isotherms contours: , for Ra = 105; Da = 10-3; n = 0.7, Pr=6.2 for different positions of porous surface and four cases; ϕ= 0 (green line) and ϕ= 0.05 (red lines).
N=1
S=0.25 S=0.5 S=0.75
N=2
N=3
N=4
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0=max bf 𝜓 5.9346,-= min bf 𝜓 0=max nf 𝜓6.1345, -= min nf 𝜓
0=max bf 𝜓4.003, -= min bf 𝜓 0=max nf 𝜓 4.2368,-= min nf 𝜓
0=max bf 𝜓1.1788, -= min bf 𝜓 0=max nf 𝜓1.3141, -= min nf 𝜓
0=max bf 𝜓6.4433, -= min bf 𝜓 0=max nf 𝜓6.6704, -= min nf 𝜓
0=max bf 𝜓3.5298, -= min bf 𝜓
0=max nf 𝜓3.7819, -= min nf 𝜓
0=max bf 𝜓0.54763, -= min bf 𝜓 0=max nf 𝜓0.61715, -= min nf 𝜓
0=max bf 𝜓6.1841, -= min bf 𝜓 0=max nf 𝜓 6.3917,-= min nf 𝜓
0=max bf 𝜓 3.2142,-= min bf 𝜓 0=max nf 𝜓 3.4387,-= min nf 𝜓
0=max bf 𝜓0.26922, -= min bf 𝜓 0=max nf 𝜓0.30565, -= min nf 𝜓
0=max bf 𝜓6.0479, -= min bf 𝜓 0=max nf 𝜓6.2488, -=min nf 𝜓
0=max bf 𝜓3.0932, -= min bf 𝜓 0=max nf 𝜓3.2986, -=min nf 𝜓
0=max bf 𝜓0.15744, -= min bf 𝜓 0=max nf 𝜓0.17851, -=min nf 𝜓
Fig. 8. Streamlines contours: , for Ra = 104; Da = 10-5; n = 0.7, Pr=6.2 for different positions of porous surface and four cases; ϕ= 0 (green line) and ϕ= 0.05 (red lines).
N=1
S=0.25 S=0.5 S=0.75
N=2
N=3
N=4
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Fig. 9. Isotherms contours: , for Ra = 104; Da = 10-5; n = 0.7, Pr=6.2 for different positions of porous surface and four cases; ϕ= 0 (green line) and ϕ= 0.05 (red lines).
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=0.002 min bf 𝜓10.354, -= min bf 𝜓
0.0027=max nf 𝜓10.677, -= min nf 𝜓
𝜓max bf =0.002
0.0032=max bf 𝜓8.7847, -= min bf 𝜓 0.0038=max nf 𝜓9.2066, -= min nf 𝜓
0=max bf 𝜓6.5548, -= min bf 𝜓 =0max nf 𝜓6.9979, -= min nf 𝜓
0.0042=max bf 𝜓11.189, -= min bf 𝜓 =0.0045max nf 𝜓 11.495,-= min nf 𝜓
0=max bf 𝜓10.429, -= min bf 𝜓 =0max nf 𝜓10.849, -= min nf 𝜓
0=max bf 𝜓4.6670, -= min bf 𝜓 =0max nf 𝜓5.0718, -= min nf 𝜓
0=max bf 𝜓11.058, -= min bf 𝜓 =0.001max nf 𝜓11.360, -= min nf 𝜓
0=max bf 𝜓 9.5003,-= min bf 𝜓 =0.001max nf 𝜓 9.9651,-= min nf 𝜓
0=max bf 𝜓3.6888, -= min bf 𝜓 =0.001max nf 𝜓4.0359, -= min nf 𝜓
0.0028=max bf 𝜓10.860, -= min bf 𝜓 =0.001max nf 𝜓11.157, -= min nf 𝜓
0.0028=max bf 𝜓9.2651, -= min bf 𝜓
=0.0052max nf 𝜓9.6597, -= min nf 𝜓
0=max bf 𝜓2.9584, -= min bf 𝜓 =0max nf 𝜓3.2589, -= min nf 𝜓
Fig. 10. Streamlines contours: , for Ra = 105; Da = 10-5; n = 0.7, Pr=6.2 for different positions of porous surface and four cases; ϕ= 0 (green line) and ϕ= 0.05 (red lines).
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Fig. 11. Isothers contours: , for Ra = 105; Da = 10-5; n = 0.7, Pr=6.2 for different positions of porous surface and four cases; ϕ= 0 (green line) and ϕ= 0.05 (red lines).
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Fig. 12. Average Nusselt Number along hot wall with Darcy number with different Rayleigh number and four cases.
Fig. 13. Average Nusselt umber long hot wall with Volume Fraction with Four Cases.
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Fig. 14. Average Nusselt Number along hot wall with Power-law index with different Darcy number and four caces.