mhd mixed convection in square enclosure filled with non...

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:[email protected]]

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



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 dimensionless 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



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



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)



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)



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


1

(1 − 𝜑)2.5[2

𝜕

𝜕𝑌(

𝜇𝑏𝑓𝑀

𝜕𝑉

𝜕𝑌) +

𝜕

𝜕𝑋(

𝜇𝑏𝑓𝑀

(𝜕𝑈

𝜕𝑌+

𝜕𝑉

𝜕𝑋))] +

(𝜌𝛽)𝑛𝑎𝜌𝑛𝑎𝛽𝑏𝑓

𝑅𝑖𝜃

+𝐻𝑎2

𝑅𝑒

𝜎𝑛𝑎𝜎𝑏𝑓


(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



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



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



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.

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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



].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.



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



Fig.(4) Streamlines, Isotherms for different Hartman number and power-law index at

Ri=0.001, Case I.

Stremlines Isotherms

Isotherms


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


|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





Ri=0.001 Case III.


Ri=0.001 Case IV.


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




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


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



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.


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



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.

mhd mixed convection in square enclosure filled with non...

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