Effect of Layer Thickness on Natural Convection in a Square … · 2019-06-27 · Non-Newtonian...

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]

[email protected] *Corresponding author: [email protected]

[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)

mailto:mohyousif%[email protected]

mailto:[email protected]





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



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



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



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)


𝑢𝑛𝑁

𝜕𝑢𝑛𝑁

𝜕𝑥+ 𝑣𝑛𝑁

𝜕𝑢𝑛𝑁

𝜕𝑦= −

1

𝜌𝑛𝑁

𝜕𝑝

𝜕𝑥+

1

𝜌𝑛𝑁

(𝜕𝜏̿̿ ̿

𝑥𝑥

𝜕𝑥+

𝜕𝜏̿̿ ̿𝑥𝑦

𝜕𝑦) + 𝜌𝑛𝑁𝛽𝑛𝑁𝑔(𝑇𝑝𝑛 − 𝑇𝑐) (6)


𝑢𝑛𝑁

𝜕𝑣𝑛𝑁


𝜕𝑣𝑛𝑁

𝜕𝑦= −

1

𝜌𝑛𝑁

𝜕𝑝

𝜕𝑦+

1

𝜌𝑛𝑁

(𝜕𝜏̿̿ ̿

𝑥𝑦

𝜕𝑥+

𝜕𝜏̿̿ ̿𝑦𝑦

𝜕𝑦) + 𝜌𝑛𝑁𝛽𝑛𝑁𝑔(𝑇𝑝𝑛 − 𝑇𝑐) (7)


𝑢𝑛𝑁

𝜕𝑇𝑛𝑁


𝜕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:


𝜕𝑈𝑝𝑛

𝜕𝑋+

𝜕𝑉𝑝𝑛

𝜕𝑌= 0 (17)




𝑈𝑝𝑛

𝜕𝑈𝑝𝑛

𝜕𝑋+ 𝑉𝑝𝑛

𝜕𝑈𝑝𝑛

𝜕𝑌= −

𝜌𝑓𝑙

𝜌𝑝𝑛

𝜕𝑃

𝜕𝑋+ 𝑃𝑟 (

𝜕2𝑈𝑝𝑛

𝜕𝑋2+

𝜕2𝑈𝑝𝑛

𝜕𝑌2) +

(𝜌𝛽)𝑝𝑛

𝜌𝑝𝑛

𝑅𝑎𝑃𝑟𝜃𝑝𝑛 −𝜇𝑝𝑛

𝜇𝑓𝑙

𝑃𝑟𝑈𝑝𝑛

𝐷𝑎 (18)


𝑈𝑝𝑛

𝜕𝑉𝑝𝑛


𝜕𝑉𝑝𝑛

𝜕𝑌= −

𝜌𝑓𝑙

𝜌𝑝𝑛

𝜕𝑃

𝜕𝑌+ 𝑃𝑟 (

𝜕2𝑉𝑝𝑛

𝜕𝑋2+

𝜕2𝑉𝑝𝑛

𝜕𝑌2) +

(𝜌𝛽)𝑝𝑛

𝜌𝑝𝑛

𝑅𝑎𝑃𝑟𝜃𝑝𝑛 −𝜇𝑝𝑛

𝜇𝑓𝑙

𝑃𝑟𝑉𝑝𝑛

𝐷𝑎 (19)


𝑈𝑝𝑛

𝜕𝜃𝑝𝑛


𝜕𝜃𝑝𝑛

𝜕𝑌=

𝛼𝑝𝑛

𝛼𝑓𝑙

(𝜕2𝜃𝑛

𝜕𝑋2+

𝜕2𝜃𝑛

𝜕𝑌2) (20)

And the set of governing dimensionless equations for the non-Newtonian fluid layer become:


𝜕𝑈𝑛𝑁

𝜕𝑋+

𝜕𝑉𝑛𝑁

𝜕𝑌= 0 (21)


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)


𝑈𝑛𝑁

𝜕𝑉𝑛𝑁


𝜕𝑉𝑛𝑁

𝜕𝑌= −

𝜕𝑃

𝜕𝑌+ 𝑃𝑟 [

𝜕

𝜕𝑋(𝜇𝑎

𝜎

𝜕𝑈𝑛𝑊

𝜕𝑌+

𝜇𝑎

𝜎

𝜕𝑉𝑛𝑁

𝜎) + 2

𝜕

𝜕𝑌(𝜇𝑎

𝜎

𝜕𝑉𝑛𝑁

𝜕𝑌)] + 𝑅𝑎𝑃𝑟𝜃𝑛𝑁 (23)


𝑈𝑛𝑁

𝜕𝜃𝑛𝑁


𝜕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)



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)



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



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.



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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extension of Darcys law: coupled parallel flow within a

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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.



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



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



0=max bf 𝜓3.5339, -= min bf 𝜓 0=max nf 𝜓3.7752, -= 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


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 nf 𝜓1.9942, -= min nf 𝜓

S=0.75



S=0.25

N=1


N=2

0.0026=max bf 𝜓5.9504, -= min bf 𝜓

0.0029=max nf 𝜓6.1772, -= min nf 𝜓

N=3


N=4



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




0.0034=max nf 𝜓10.88, -= min nf 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


0.00514=max nf 𝜓9.1737, -= min nf 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


0.0049=max nf 𝜓9.3203, -= min nf 𝜓

0.005=max bf 𝜓8.0483, -= min bf 𝜓 0=max nf 𝜓8.0447, -= min nf 𝜓


0.0032 =max nf 𝜓10.594, -= min nf 𝜓

0.0167=max bf 𝜓8.0965, -= min bf 𝜓 𝜓𝜓



N=1

S=0.25 S=0.5 S=0.75

N=2

N=3

N=4



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



0=max bf 𝜓 5.9346,-= min bf 𝜓 0=max nf 𝜓6.1345, -= min nf 𝜓








0=max bf 𝜓 3.2142,-= min bf 𝜓 0=max nf 𝜓 3.4387,-= min nf 𝜓


0=max bf 𝜓6.0479, -= min bf 𝜓 0=max nf 𝜓6.2488, -=min nf 𝜓




N=1

S=0.25 S=0.5 S=0.75

N=2

N=3

N=4



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).



=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 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





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).



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.



Fig. 14. Average Nusselt Number along hot wall with Power-law index with different Darcy number and four caces.

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