Mixed convection from a discrete heat source in enclosures with two adjacent moving walls and filled with micropolar nanofluids

Full Length Article

Mixed convection from a discrete heat source in enclosures with two

adjacent moving walls and filled with micropolar nanofluids

Sameh E. Ahmed

a,

*, M.A. Mansour

b

, Ahmed Kadhim Hussein

c

, S. Sivasankaran

d

a_{Department of Mathematics, South Valley University, Faculty of Science, Qena, Egypt} b_{Department of Mathematics, Assuit University, Faculty of Science, Assuit, Egypt}

c_{College of Engineering, Mechanical Engineering Department, Babylon University, Babylon City, Hilla, Iraq} d_{Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur 50603, Malaysia}

A R T I C L E I N F O

Article history: Received 23 May 2015 Received in revised form 26 July 2015

Accepted 24 August 2015 Available online 11 September 2015 Keywords:

Micropolar nanofluid Mixed convection Uniform heat source Square enclosure Lid-driven

A B S T R A C T

This paper examines numerically the thermal and flow field characteristics of the laminar steady mixed convection flow in a square lid-driven enclosure filled with water-based micropolar nanofluids by using the finite volume method. While a uniform heat source is located on a part of the bottom of the enclo-sure, both the right and left sidewalls are considered adiabatic together with the remaining parts of the bottom wall. The upper wall is maintained at a relatively low temperature. Both the upper and left side-walls move at a uniform lid-driven velocity and four different cases of the moving lid ordinations are considered. The fluid inside the enclosure is a water based micropolar nanofluid containing different types of solid spherical nanoparticles: Cu, Ag, Al2O3, and TiO2. Based on the numerical results, the effects of the dominant parameters such as Richardson number, nanofluid type, length and location of the heat source, solid volume fractions, moving lid orientations and dimensionless viscosity are examined. Com-parisons with previously numerical works are performed and good agreements between the results are observed. It is found that the average Nusselt number along the heat source decreases as the heat source length increases while it increases when the solid volume fraction increases. Also, the results of the present study indicate that both the local and the average Nusselt numbers along the heat source have the highest value for the fourth case (C4). Moreover, it is observed that both the Richardson number and moving lid ordinations have a significant effect on the flow and thermal fields in the enclosure.

Copyright © 2015, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

1. Introduction

Mixed (free and forced) convection flow and heat transfer in en-closures has received much attention in the recent years due to its many important practical applications such as thermal design of buildings, electronics cooling, solar collectors, commercial refrig-eration and float glass production. During the mixed free and forced convection, it is necessary to add the contributions of the free and forced convection in assisting flows and to subtract them in op-posing flows. Also, the mixed convection phenomena becomes very important when the forced velocity induced by a mechanical device like a fan has an effect equal to the free stream velocity induced by the buoyancy force which appears due to the density variation[1,2]. From the other hand, nanofluids are dilute liquid suspensions of nanoparticles with at least one critical dimension smaller than 100 nm which is firstly utilized by Choi[3]. Nowadays, more

attention was considered to this new type of composite materials because of its enhanced properties and behavior associated with the heat transfer. They have received more attention as a new gen-eration of heat transfer fluids in building heating, heat exchangers, plants and automotive cooling applications, because of their ex-cellent thermal performance. Many studies indicate that the dispersion of a small amount of solid nanoparticles in convention-al fluids increase remarkably their thermconvention-al conductivity. Compared to the existing techniques for enhancing heat transfer, the nanofluids show a superior potential for increasing heat transfer rates in a variety of cases. Various advantages of nanofluid applications include: improved heat transfer, heat transfer system size reduction, micro channel cooling and miniaturization of systems[4]. Various nu-merical and experimental studies related with the mixed convection in lid-driven square or rectangular cavities with and without nanofluids have been studied extensively in the literature. Iwatsu et al.[5]studied the mixed convection in a driven cavity with a stable vertical temperature gradient. Khanafer and Chamkha[6] ex-tended the work of Iwatsu et al.[5]and studied the mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium in the presence of the heat generation effects. Sivasankaran

* Corresponding author. Tel.:+201009450564, fax:+20965211279. E-mail address:[email protected](S.E. Ahmed). Peer review under responsibility of Karabuk University. http://dx.doi.org/10.1016/j.jestch.2015.08.005

2215-0986/Copyright © 2015, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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et al.[7]investigated numerically by using the finite volume method the mixed convection in a square lid-driven cavity with a corner heating and an internal heat generation or absorption. The results were obtained for different lengths of the heaters, Richardson numbers and internal heat generation or absorption parameters. It was observed that the heat transfer rate was enhanced at forced convection dominated regime when the vertical length of the heater was higher than that of the horizontal length. Tiwari and Das[8]

investigated numerically the heat transfer augmentation in a two sided lid-driven differentially heated square cavity utilizing copper– water nanofluid. They considered different cases characterized by the direction of wall movement. They concluded that both the Rich-ardson number and the direction of moving walls influenced the fluid flow and thermal behaviors. Muthtamilselvan et al.[9]studied numerically the mixed convection in a lid-driven enclosure filled with copper–water nanofluids for various aspect ratios. It was found that both the aspect ratio and the solid volume fraction affected the fluid flow and the heat transfer in the lid-driven enclosure. Waheed

[10]studied the mixed convective heat transfer in rectangular en-closures driven by a continuously moving horizontal plate while Ouertatani et al.[11]studied the mixed convection in a double lid-driven cubic cavity. Mahmoudi et al.[12]studied numerically using the finite volume approach the mixed convection flow and tem-perature fields in a vented square cavity subjected to an external copper–water nanofluid. In order to investigate the effect of inlet and outlet locations, four different placement configurations of the inlet and outlet ports were considered. In each of them, both the inlet and outlet ports were alternatively located either on the top or the bottom of the sides and external flow entered into the cavity through an inlet opening in the left vertical wall and exits from another opening in the opposite wall. The remaining boundaries were considered adiabatic. Their study had been carried out for Reyn-olds number in the range of 50≤Re≤1000, Richardson number in the range of 0≤Ri≤10 and solid volume fraction in the range of 0≤φ≤0.05. Results were presented in the form of streamlines, iso-therms and average Nusselt number. Talebi et al.[13]investigated numerically the mixed convection flows in a lid-driven square cavity utilizing the copper–water nanofluid. They showed that at a given Reynolds number and Rayleigh number, solid concentration had a positive effect on the heat transfer enhancement. Abu-Nada and Chamkha[14]solved numerically the steady laminar mixed con-vective flow and heat transfer of a nanofluid made up of water and Al2O3in a lid-driven inclined square enclosure using a

second-order accurate finite-volume method. It was found that the heat transfer mechanisms and the flow characteristics inside the cavity were strongly dependent on the Richardson number. Shahi et al.[15]

executed a numerical investigation of the mixed convection flows through a copper–water nanofluid in a square cavity with inlet and outlet ports. Their study had been carried out for Reynolds number in the range of 50≤Re≤1000, Richardson number in the range of 0≤Ri≤10 and for solid volume fraction in the range of 0≤φ≤0.05. Results were presented in the form of streamlines, isotherms, average Nusselt number and average bulk temperature. The results indi-cated that the increase in the solid concentration caused to increase the average Nusselt number at the heat source surface and a de-crease in the average bulk temperature. Hussein et al. [16]

investigated numerically the mixed convection in a rectangular in-clined lid-driven cavity filled with water-based nanofluids. While, a uniform heat source was located on a part of the left inclined side-wall of the cavity, the right inclined sideside-wall was considered adiabatic together with the remaining parts of the left inclined sidewall. The top and bottom walls were maintained at a low temperature and the top wall moved from left to right with a uniform lid-driven ve-locity. It was found that the local Nusselt number was decreased as the inclination angle and the solid volume fraction were in-creased. Moreover, it was observed that the shape of the circulation

vortex was sensitive to the inclination angle and the addition of nanofluids. Mansour and Ahmed[17]investigated numerically by using the finite volume method the mixed convection in double lid-driven enclosures with the heat source embedded in the left sidewall and filled with Al2O3–water nanofluid. Four cases were

consid-ered depending on the direction of the movement of the walls. Their study had been carried out for Richardson number (Ri=0.1–100), nanoparticles volume fraction (φ=0–4%), the heat source length (B=0.2, 0.4, 0.6, 0.8) and Reynolds number (Re=10–316.2278). It was found that the fluid flow and heat transfer characteristics de-pended strongly on the direction of the horizontal wall movement. Also, a significant heat transfer enhancement was obtained by using the Al2O3–water nanofluid. Esfe et al.[18]studied numerically the

mixed convection within nanofluid-filled cavities with two adja-cent moving walls by using the finite volume approach. It was found that by adding nanoparticles to the fluid, the strength of the vor-tices was increased. Also, they concluded that the average Nusselt number was increased by increasing the volume fraction of nanoparticles. Another useful research had been conducted to sim-ulate the mixed convection heat transfer using nanofluid under different conditions[19–23].

A number of experimental studies have been conducted to in-vestigate the flow field and heat transfer characteristics of lid-driven cavity flow in the past several decades. Kuhlmann et al.[24]

presented a numerical and experimental investigation of a steady flow in rectangular two sided lid-driven enclosures. Their results indicated that the basic two dimensional flow was not always unique. For low Reynolds numbers it consist two separate co-rotating vor-tices adjacent to the moving walls. The problem of an incompressible fluid flow in a rectangular container driven by two facing side walls which move steadily in anti-parallel for Reynolds numbers up to 1200 was studied experimentally by Blohm and Kuhlmann[25]. The moving sidewalls are realized by two rotating cylinders of large radii tightly closing the cavity. They found that beyond a first threshold robust, steady, three-dimensional cells bifurcate supercritically out of the basic flow state. If both side walls move with same velocity (symmetrical driving) the oscillatory instability was found to be tricritical.

Recently, much attention are devoted to the micropolar fluids due to their many practical applications such as liquid crystal, low-concentration suspension flow, blood rheology, the presence of dust or smoke, exotic lubricants and the effect of dirt in a journal bearing. In fact, micropolar fluids are those with microstructure belonging to a complex class of the Newtonian fluids with a non-symmetrical stress tensor. Physically, they represent fluids consisting of random particles suspended in a viscous medium[26]. The theory of the micropolar fluid was first suggested by Eringen[27]to take into account the effects of the local structure and micro-motions of the fluid particles which cannot be described by the classical models. Recently, a micropolar model for nanofluidic suspensions is proposed by Bourantas and Loukopoulos[28]and Bourantas et al.

[29]in order to investigate theoretically the natural convection of nanofluids. They observed that by keeping the microrotation number K constant, the average Nusselt number increases with increasing Rayleigh number. Anyway, limited investigations are considered the mixed convention in enclosure filled with micropolar fluid such as Hsu and Wang[30]. A literature survey indicates that no studies have been done on the mixed convection in a lid-driven square enclo-sure partially heated from its bottom wall and filled with a micropolar nanofluid. Most of the published papers are related with the mixed convection heat transfer in enclosures filled with the New-tonian nanofluids. The extension to the non-NewNew-tonian or micropolar nanofluids are not considered previously anywhere. To reach this goal, the objective of the present work is to examine the effects of Richardson number, nanofluid type, length and location of the heat source, solid volume fractions, moving lids orientations and

dimensionless viscosity on the mixed convection in a two-dimensional lid-driven square enclosure filled with micropolar nanofluids.

2. Mathematical model

2.1. Governing equations and geometrical configuration

Fig. 1shows a schematic diagram of a two-dimensional square enclosure of height (H) and width (w). The fluid inside the enclosure is micropolar water based nanofluid containing four different types of solid spherical nanoparticles: Cu, Ag, Al2O3, and TiO2. The nanofluid

is assumed to be incompressible and the flow is assumed to be two-dimensional, laminar and steady. It is assumed that the base fluid (i.e. water) and nanoparticles are in a thermal equilibrium state. The parameters of (Pr) and (Gr) are considered fixed during the calcu-lation at values of 6.2 and 104_{respectively. The solid volume fractions}

(φ) have been varied as 0%, 5%, 8%, while the dimensionless length of the heat source (B) is varied as 0.2, 0.4, 0.5 and 0.8 re-spectively. The dimensionless location of the heat source (D) is varied as 0.3, 0.5 and 0.7, while the dimensionless viscosity (K) is varied as 0 and 0.5 respectively. The range of the Richardson number is taken as 0.01≤Ri≤10. The thermo-physical properties of the base fluid and nanoparticles are tabulated inTable 1. It is assumed that both the left and the upper lids are moving with a constant velocity (Uo) and four various cases of the moving orientations are considered.

The right and left vertical walls are assumed to be thermally insu-lated, while the top lid wall is kept at a low temperature. In order

to create the buoyancy effect, a uniform heat source with a constant volumetric rate (q′′) is located in a part of the bottom wall while the remaining parts of the same wall are considered adiabatic. The thermo-physical properties of both the base fluid and nanofluids are assumed to be constant except for the density variation, which is modeled using Boussinesq model. Under these assumptions, the dimensionless governing continuity, momentum, angular momen-tum and energy equations are, respectively, expressed as:

∂ ∂ + ∂ ∂ = U X V Y 0 (1) U U X V U Y P X K U X U Y f nf nf f ∂ ∂ + ∂ ∂ = − ∂ ∂ + ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂ + ∂ ∂ 1 2 2 2 Re

ρ

ρ

μ

μ

22 ⎛ ⎝⎜ ⎞⎠⎟+ ⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂ K N Y f nf Re

ρ

ρ

(2) U V X V V Y P Y K V X V Y f nf nf f ∂ ∂ + ∂ ∂ = − ∂ ∂ + ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂ + ∂ ∂ 1 2 2 2 Re

ρ

ρ

μ

μ

22 ⎛ ⎝⎜ ⎞⎠⎟ − ⋅⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂ + ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟

( )

⋅ ⎛ ⎝⎜ ⎞ K N X Ri f nf f nf nf f f Re

ρ

ρ

ρ

ρ

ρβ

ρ β

⎠⎠⎟

θ

(3) U N X V N Y K N X N Y f nf nf f ∂ ∂ + ∂ ∂ = ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂ + ∂ ∂ ⎛ ⎝⎜ ⎞ 1 2 2 2 2 2 Re

ρ

ρ

μ

μ

⎠⎠⎟ − ⋅⎛ ⎝⎜ ⎞ ⎠⎟ − ∂ ∂ − ∂ ∂ ⎛ ⎝⎜ ⎞⎠⎟ ⎛ ⎝⎜ K N V X U Y f nf Re

χ

ρ

ρ

2 (4) U X V Y X Y nf f ∂ ∂

θ

+ ∂∂

θ

= ⎛⎝⎜ ⎞⎠⎟

α

α

⎛⎝⎜∂∂ + ∂∂ ⎞⎠⎟

θ

θ

1 2 2 2 2 Pr .Re (5)

In the present study, the relations that depend on the nanoparticles volume fraction only are adopted. These relations are used in many previous studies[31–33]and can be written as follows:

γ

nf

μ

nf k

j

=⎛_⎝⎜ + ⎞_{⎠⎟ ⋅}

2 (6)

The effective density, the heat capacitance, the thermal expan-sion coefficient and the thermal diffusivity of the nanofluid are given by respectively:

ρ

nf= −

(

1

φ ρ

)

f+

φρ

p (7)

ρ

c_{p nf}

φ ρ

c_{p f}

φ ρ

c_{p p}

( )

= −

(

1

)

( )

+

( )

(8)

ρβ

φ ρβ

φ ρβ

( )

nf= −

(

1

)( )

f+

( )

p (9)

α

ρ

nf nf p nf k c =

( )

(10)

while the thermal conductivity of the nanofluid is given by[34–36]:

k k k k k k R k k k k R nf f p f p f p f p f =

(

+ +

(

−

)

(

+

)

+ −

(

−

)

(

+

)

(

2 2 1 2 2 1 3 3

φ

φ

where R==

[

]

0 1. for Cu nanoparticles (11) k k nf f =

(

0 9508 0 9692. + .

φ

)

[

for Ag nanoparticles

]

(12) k k Al O nf f = +

(

1 7 47.

φ

)

[

for 2 3nanoparticles

]

(13) k k T O nf f i = +

(

1 2 92 −11 99 2

)

[

]

2 .

φ

.

φ

for nanoparticles (14) ∂T ∂x ∂T ∂x

Micropolar nanofluid

∂T ∂y

Fig. 1. Physical model and the coordinates system.

Table 1

Thermo-physical properties of water and nanoparticles. Property Water Copper

(Cu) Silver (Ag) Alumina (Al2O3) Titanium (TiO2) ρ 997.1 8933 10.500 3970 4250 Cp 4179 385 235 765 686.2 k 0.613 401 429 40 8.9538 β 21 10_× −5 _{1 67 10. ×} −5 _1.89×10−5 _0.85×10−5 _0.9×10−5

and the effective dynamic viscosity of the nanofluid is given by [34–36]:

μ

μ

φ

nf f = −

(

1

)

2 5.

[

for Cu nanoparticles

]

(15)

μ

nf=

μ

f

(

1 005 0 49. + .

φ

−0 1149.

φ

2

)

[

for Ag nanoparticles

]

(16)

μ

nf=

μ

f

(

1 39 11+

φ

+533 9

φ

)

[

Al O

]

2 2 3 . . for nanoparticles (17)

μ

nf=

μ

f

(

1 5 45+

φ

+108 2

φ

)

[

T Oi

]

2 2 . . for nanoparticles (18) 2.2. Boundary conditions

The dimensionless form of the boundary conditions is ex-pressed as follows: U V at wall Y X U V at wall X Y Y k k D B f nf = = = ≤ ≤ = = = ≤ ≤ ∂ ∂ = − − 0 0 0 1 0 1 0 1 0 5 , , , .

θ

_{≤≤ ≤ +} ∂ ∂ = = ∂ ∂ = = = X D B Y otherwise at wall Y X at walls X a 0 5 0 0 0 0 1 0 . , , ,

θ

θ

θ

tt wall Y N at all walls = = 1 0 (19)

The special boundary conditions for each four considered cases are described as follows:

1 Case 1 (C1): The upper wall moves at a constant velocity (Uo)

from left to right, while the left adiabatic sidewall moves down-wards at the same velocity.

C U V X Y U V Y X 1 0 1 0 0 1 1 0 1 0 1 : , , , , , , = = − = ≤ ≤ = = = ≤ ≤ ⎧ ⎨ ⎩

2 Case 2 (C2): The upper wall moves at a constant velocity (Uo)

from right to left, while the left adiabatic sidewall moves upwards at the same velocity.

C U V X Y U V Y X 2 0 1 0 0 1 1 0 1 0 1 : , , , , , , = = = ≤ ≤ = − = = ≤ ≤ ⎧ ⎨ ⎩

3 Case 3 (C3): The upper wall moves at a constant velocity (Uo)

from left to right, while the left adiabatic sidewall moves upwards at the same velocity.

C U V X Y U V Y X 3 0 1 0 0 1 1 0 1 0 1 : , , , , , , = = = ≤ ≤ = = = ≤ ≤ ⎧ ⎨ ⎩

4 Case 4 (C4): The upper wall moves at a constant velocity (Uo)

from right to left, while the left adiabatic sidewall moves down-wards at the same velocity.

C U V X Y U V Y X 4 0 1 0 0 1 1 0 1 0 1 : , , , , , , = = − = ≤ ≤ = − = = ≤ ≤ ⎧ ⎨ ⎩ (20)

2.3. Local and average Nusselt numbers

The local Nusselt number along the heat source surface can be written as: Nu X s s =

θ

( )

1 (21)

while the average Nusselt number is given by:

Nu B Nu dX m s D B D B = − +

∫

1 0 5 0 5 . * . * (22)

whereθs

( )

X is the dimensionless local temperature along the heat

source

3. Numerical procedure and validation

Because of the non-linear interactions among equations (1–5), the solution for these equations with the boundary conditions (19 and 20) can be obtained numerically by using the collocated finite volume method. In order to explain this solution, let us take Eq.(2)

as an example. This equation can be re-written as:

∂ ∂ + ∂ ∂ = − ∂ ∂ + ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂ ∂ ∂ ⎛ ⎝⎜ ⎞⎠ U X UV Y P X K X U X f nf nf f 2 ₁ Re

ρ

ρ

μ

μ

⎟⎟ + ∂ ∂ ∂ ∂ ⎛ ⎝⎜ ⎞⎠⎟ ⎛ ⎝⎜ ⎞⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂ Y U Y K N Y f nf Re

ρ

ρ

(23)

Integrating this equation over the control volume shown inFig. 2

∂ ∂ + ∂ ∂ ⎡ ⎣⎢ ⎤ ⎦⎥ = − ∂ ∂ + ⋅⎛_⎝⎜ ⎞_⎠⎟

∫

∫

UX

∫

∫

UV Y dXdY P XdXdY f nf nf f 2 1 Ω Ω Re

ρ

ρ

μ

μ

++ ⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂ ∂ ∂ ⎛ ⎝⎜ ⎞⎠⎟ ⎛ ⎝⎜ ⎡ ⎣⎢ + ∂ ∂ ∂ ∂ ⎛ ⎝⎜ ⎞⎠⎟⎞⎠⎟⎤⎦⎥ +

∫

∫

K X U X Y U Y dXdY K Ω Ree

ρ

ρ

f nf N YdXdY ⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂

∫

∫

Ω (24) The upwind differencing scheme and the central difference scheme are taken for the convective terms and the diffusion terms, respectively. Evaluating these integrals and re-arranging, the fol-lowing algebraic equation is obtained:

a UP a U S U P i U i i U =

∑

+ (25)

where i=E, W, N, S. Similar treatments are used for equations(3)–(5). The resulting algebraic equations have been solved iteratively, through the alternate direction implicit procedure (ADI), by using the SIMPLE algorithm[37]. The velocity correction has been made

using the Rhie and Chow interpolation. For convergence, the under-relaxation technique has been employed. The iteration is carried out until the normalized residuals of the mass, momentum, angular mo-mentum and temperature equations become less than10−6_{. In order}

to choose the suitable grid for these calculations, an accurcy test by using five sets of grids [31 31 41 41 61 61 81 81 101 101× , × , × , × , × ] is made for case one (C1), Cu–water nanofluid and [Ri=1, K=0.5, D=0.5, B=0.5 and φ =5%]. This test is clearly shown inTable 2. A (81 81× ) uniform grid is found to meet the requirements of both the grid independency study and the computational time limits. The numerical method was implemented in a FORTRAN language. The obtained results are plotted in 2D graphs and contour maps by using ORIGIN 6 software and SURFER 8 software, respectively. The com-puter program of the present work is found to be suitable and gives results that are close to the results obtained by Iwatsu et al.[5]and Khanafer and Chamkha[6].Table 3shows a very good agreement between the accuracy of the average Nusselt number in the special cases of the present study for different values of the Reynolds number and the results obtained by Iwatsu et al.[5]and Khanafer and Chamkha[6]. These favorable comparisons lend confidence in the numerical results to be reported subsequently.

4. Results and discussion

The mixed convection heat transfer of water-based micropolar nanofluids in a square lid-driven enclosure with a uniform heat source located on a part of its bottom wall has been investigated numerically in this study. The results of the present study are shown inFigs. 3–14. The fluid inside the enclosure is a water-based micropolar nanofluid containing four various types of solid spher-ical nanoparticles: Cu, Ag, Al2O3, and TiO2.

4.1. Effect of the Richardson number (Ri)

The streamlines (first row) and isotherms (second row) in the enclosure for the micropolar water-based copper nanofluid case are shown inFig. 3for various cases of the moving lids and when the Richardson number varies from 0.01 to 10 and at [Gr=104_{, K=0.5,}

D=0.5, B=0.5 and φ =5%].

4.1.1. The forced convection effect (Ri=0.01)

Fig. 3ashows the streamlines and isotherms when the forced convection effect is dominated (i.e., Ri=0.01) for various moving lid orientations. In this case, the shear force due to the moving lids has a stronger influence compared with the buoyancy force. It can be clearly seen that when the upper wall moves to the right and when the left sidewall moves downwards (C1), two large separate clockwise and anti-clockwise vortices with perfectly

symmetrical patterns about the diagonal of the enclosure can be generated which occupy most of the size of the enclosure. This is because the convection currents begin from the hot region in the bottom wall and move toward the upper cold wall due to both the buoyancy and shear forces. The same behavior can be ob-served when the direction of the moving lids converges (i.e., when the upper wall moves to the left and the left sidewall moves upwards (C2)). The only difference is that vortices become larger in the second case (C2). Now, when the upper wall moves from left to right, the left adiabatic sidewall moves upwards (C3). The flow field consists from a single large anti-clockwise vortex at the center of the enclosure while a similar single large clockwise vortex is observed when the upper wall moves to the left, while the left adiabatic sidewall moves downwards (C4). The maximum stream function for these cases can be found in the fourth case (C4) which is about 0.13. With respect to isotherms, there is a clear accumulation of them adjacent to the heat source location at the bottom wall of the enclosure. This is due to the strong temperature gradient in the vertical direction which gives an in-dication that the convection heat transfer becomes more significant compared to the conduction heat transfer. Also, it can be seen fromFig. 3athat the clustering of isotherms adjacent to the cold upper and hot region of the bottom wall increases when the di-rection of the moving lids converges [i.e., (C2)]. In this case, the isotherms begin to shift toward the diagonal of the enclosure. This is due to the strong effect of both the shear and buoyancy forces. By comparing the results of these various cases, it can be seen that the fourth case (C4) provides a high circulation intensity of the flow field compared with the other considered cases. The reason of this behavior is because in this case both the shear and buoyancy forces act in the gravity force direction. In general, when the Richardson number is very low (i.e., Reynolds number is high), the isotherm contours take a uniform shape and a slight distur-bance can be seen in it where the conduction heat transfer is dominant.

4.1.2. The mixed convection effect (Ri=1.0)

Fig. 3billustrates the streamlines and isotherms when the forced convection effect is approximately equivalent to the natural con-vection effect (i.e., Ri=1) for various moving lid orientations. In this case, the shear force has the same influence with the buoyan-cy force. It can be seen from the results that the behavior of the flow field for both cases one and two (C1 and C2) are approximate-ly similar to that presented inFig. 3afor the same cases. The only difference between these cases and those obtained at Ri=0.01 is that the two circulating vortices begin to decrease in size. With respect to the third and fourth cases (C3 and C4), the flow field becomes more intense especially toward the moving lids location, while the core of the rotating vortices begin to diminish and becomes more adjacent to the moving lids location. Also, the circulation in-tensity increases slightly due to the decrease in the shear force and the increase in the buoyancy force when the Richardson number increases from 0.01 to 1.0. This increase in the Richardson number reduces the influence of the moving lids and causes to increase the circulation intensity. The maximum stream function for these cases can be found in the fourth case (C4) which is about 0.135. Regarding the isotherms, for all moving lid orientations, the dis-tributions of the isotherms increase gradually at Ri=1 comparing with the case of Ri=0.01. This is because the intensity of circula-tion increases due to increase in the effect of the vortices coming from the buoyancy force effect when the Richardson number is unity. The thermal energy inside the enclosure is transferred by the convection. This can be observed from the non-uniform iso-therm lines instead of the conduction at the case when the Richardson number is very low.

Table 2

Grid independency test for case one (C1), Cu–water nanofluid and at Ri=1, K=0.5, D=0.5, B=0.5 and φ =5%.

Grid 31 31× 41 41× 61 61× 81 81× 101 101×

Num 22.43438 22.868230 22.840540 22.795980 22.518420

Table 3

Comparisons of the mean Nusselt number at the upper wall for different values of the Reynolds number and at Gr=102_{, Pr=0.71 andφ =0%.}

Re Iwatsu et al.[5] Khanafer and Chamkha[6] Present study

100 1.94 2.01 1.93

400 3.84 3.91 3.91

C1 C2 C3 C4

a. (Ri = 0.01)

b. (Ri = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

4.1.3. The natural convection effect (Ri=10)

Fig. 3cshows the streamlines and isotherms when the forced convection effect is less than the natural convection effect (i.e., Ri=10) for various moving lid orientations. In this case, the shear force has a slight influence compared with the buoyancy force. It can be noticed again from the results that the flow field for both the first and second cases (C1 and C2) are similar to that pre-sented inFig. 3bfor the same cases. The only difference between these cases and those obtained at Ri=1 is that the core of the upper vortices begin to increase in its size. The same observation can be noticed in the third and fourth cases (C3 and C4). The maximum stream function for these cases can be found in the fourth case (C4) which is about 0.135. With respect to isotherms, it is notice-able that they are distributed irregularly inside the enclosure as curved lines and the convection heat transfer is dominant. This is because when the Richardson number is high (i.e., Reynolds number is low) the temperature gradient increases and as a result the rate of heat transfer increases also. Furthermore, when the Richardson number is high, the buoyancy force effect due to the natural con-vection increases and the thermal energy transferred by it also increases which leads to increase in the rate of heat transfer and making the isotherms similar to that encountered in the convec-tion heat transfer.

4.2. Effect of the dimensionless length of the heat source (B)

Fig. 4shows contours of streamlines (upper) and isotherms (lower) for different heat source lengths at [D=0.5, K=0.5, Ri=1] and for case 1 (C1) where solid lines refer to the pure water (φ =0%) and dash lines refer to the micropolar Cu–water nanfluid (φ =8%). The results show that as the heat source length increases from

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

c. (Ri = 10)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 3.(continued)

B = 0.2

B = 0.8

B = 0.2

B = 0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 4. Contours of streamlines (upper) and isotherms (lower) for different heat source lengths at D=0.5, K=0.5, Ri=1 and C1 (solid lines refer to pure water (_{φ =}0%) and dashed lines refer to Cu–water nanfluid (_{φ =}8%)).

[B=0.2] to [B=0.8], the size of the two rotating vortices within the enclosure is intensified and high temperature patterns are observed. The reason of this behavior is due to the increase in the heat generation rate which is produced by the longer heat source. This leads to increase in the buoyancy force effect and increases the flow circulation and the convection effects in the enclosure. This be-havior can be noticed for both pure and micropolar nanofluids. Moreover, the increase in the heat generation rate increases the surface temperature of the heat source and as a result the effect of the natural convection becomes stronger when the heat source length increases. Furthermore, it can be seen that the rotating vor-tices inside the enclosure remain symmetrical in shape in spite of the increase of the heat source length from [B=0.2] to [B=0.8]. This behavior is due to the constant location of the heat source in the middle of the enclosure bottom wall (i.e., D=0.5). Also, as the heat source length increases, the isotherms begin to distribute uni-formly and horizontally especially in the heat source location.

4.3. Effect of the dimensionless heat source location (D)

Fig. 5illustrates contours of streamlines (upper) and isotherms (lower) for different heat source locations on the bottom wall at [B=0.4, Ri=1, φ =5%] and C1, where solid lines refer to [K=0 (i.e., ordinary nanofluid)] and dash lines refer to [K=0.5 (i.e., micropolar nanofluid)]. In general, the flow field consists two unsymmetrical upper and lower rotating vortices in the enclosure. When the heat source is located adjacent to the left sidewall of the enclosure (i.e., D=0.3), the lower vortex size becomes more than the upper one. But as the heat source locations increase to D=0.7, a reverse be-havior can be seen. In fact, when the heat source moves toward the left sidewall (i.e., D=0.3), the combined effects of the heat source and the downwards movement of the adiabatic left sidewall lead

to the increased size of the lower vortices, while the upper vorti-ces are affected by the movement of the upper wall from left to right only. On the other hand, when the heat source moves toward the right sidewall (i.e., D=0.7), the combined effects of the heat source and the left to right movement of the upper wall lead to the in-creased size of the upper vortices, while the lower vortices are affected by the downwards movement of the adiabatic left side-wall only. This behavior can be noticed for both ordinary and micropolar nanofluids. Moreover, the results ofFig. 5show that the isotherms are accumulated strongly near the heat source location and their shape becomes more uniform as the heat source loca-tion increases. This can be explained by the distance that the fluid needs to travel in the rotating vortices in order to exchange the energy between the heat source and the upper cold wall. So, as the heat source becomes adjacent to the right wall (i.e., D=0.7), this distance becomes larger and the heat removal increases while the heat source maximum temperature decreases. Therefore, the iso-therms become more uniform as the heat source location increases, which indicates that the heat is transferred by the conduction in this case. Also, it can be seen from the results ofFig. 5that the iso-therms follow the location of the heat source and their distribution in the enclosure becomes more noticeable and uniform as the heat source moves far away from the moving adiabatic left sidewall.

4.4. Velocity profiles

In order to have a better understanding of the flow behavior in the present work, the horizontal velocity profiles along the mid-section of the enclosure (X=0.5) are presented inFig. 6for Cu– water micropolar nanofluid with different moving lid cases and (Ri) at [B=0.5, D=0.5, Gr=104_{, K=0.5, φ =}

5%]. Different velocity pro-files are observed which indicate the direction of the flow rotation. It is shown that for case one (C1) the flow begins from zero, then decreases and after that begins to increase. A reverse behavior can be seen in case 2 (C2). This behavior is logical, since the directions of both upper and left moving lids are opposite in these cases. With respect to case 3 (C3), again the flow begins from zero, then de-creases and after that begins to increase. An opposite behavior can be seen in case 4 (C4). The reason of this behavior is due to the op-posite directions of both upper and left moving lids for cases 3 and 4 respectively. Also, the results indicate that when the Richardson number is low (i.e., Ri=0.01), the velocity profile varies signifi-cantly across the mid-section of the enclosure which indicates a high circulation zone in the enclosure.

4.5. Local Nusselt number results

Fig. 7shows the profiles of the local Nusselt number along the heat source for Cu–water micropolar nanofluid with different moving lid cases and [B=0.5, D=0.5, Ri=1, K=0.5, φ =5%]. The results il-lustrate that the fourth case (C4) provides the highest local Nusselt number compared with other considered cases. The reason of this behavior is due to the high circulation intensity of the flow field in this case as discussed previously inFig. 3.Fig. 8displays the pro-files of the local Nusselt number along the heat source for Cu– water micropolar nanofluid for different values of the Richardson number (Ri) at [B=0.5, D=0.5, K=0.5,φ =5%] and for case one (C1). It can be observed that the local Nusselt number has a large value when the Richardson number is low (Ri=0.01) [i.e., in the forced convection-dominant region]. The reason of this behavior is due to the high effect of the Reynolds number when the Richardson number is low. Therefore, the velocity of the moving lids increases signifi-cantly and leads to increase the forced convection effect on the local Nusselt number. From the opposite side, when the Richardson number is high (Ri=10) [i.e., in the natural convection-dominant region], the velocity of the moving lids begins to decrease

D = 0.3 D = 0.7

D = 0.3 D = 0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 5. Contours of streamlines (upper) and isotherms (lower) for different heat source locations at B=0.4, Ri=1,_{φ =}5%and C1 [solid lines refer to K=0 and dashed lines refer to K=0.5].

significantly and leads to decrease the forced convection and the moving lid effects on the local Nusselt number results.Fig. 9 illus-trates the profiles of the local Nusselt number along the heat source for Cu–water micropolar nanofluid with different values of dimen-sionless length of the heat source (B) and solid volume fraction (φ) at [D=0.5, K=0.5, Ri=1] and for case one (C1). It can be ob-served from the results that the local Nusselt number increases when the micropolar nanofluid (i.e., φ =8%) is used compared with the pure fluid (i.e., φ =0%) for all values of the dimensionless length of the heat source. This result indicates that the nanofluid has a sig-nificant effect to increase the local Nusselt number along the heat source. In addition, the results indicate that the local Nusselt number along the heat source decreases as the heat source length in-creases from (B=0.2) to (B=0.8) for both pure and micropolar nanofluids. The reason of this behavior is due to the increase in the maximum temperature of the heat source by increasing its length which leads to a decrease in the local Nusselt number.Fig. 10depicts the profiles of the local Nusselt number along the heat source for Cu–water nanofluid with different values of the dimensionless heat source location (D) and dimensionless viscosity (K) at [B=0.4,φ =5%, Ri=1] and for case one (C1). It is observed that the local Nusselt number decreases with an increase in the dimensionless heat source

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 C1

Y

U(0.5,Y)

Ri=0.01 Ri=1 Ri=10 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 C2

Y

U(0.5,Y)

Ri=0.01 Ri=1 Ri=10 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 C3

Y

U(0.5,Y)

Ri=0.01 Ri=1 Ri=10 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 C4

Y

U(0.5,Y)

Ri=0.01 Ri=1 Ri=10

Fig. 6.Profiles of the horizontal velocity component at the enclosure mid-section for Cu–water micropolar nanofluid for different moving lid cases and (Ri) at B=0.5, D=0.5, Gr=104_{, K=0.5, φ =5%.} 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10 20 30 40 50 60 70

Nu

s

X

C1 C2 C3 C4

Fig. 7. Profiles of the local Nusselt number along the heat source for Cu–water micropolar nanofluid with different moving lid cases and B=0.5, D=0.5, Ri=1, K=0.5, φ =5%.

location from (D=0.3) to (D=0.7). This behavior can be seen for both ordinary (K=0) and micropolar (K=0.5) nanofluids. The reason of this behavior is explained previously inFig. 5. With respect to the effect of the dimensionless viscosity (K) on the local Nusselt number profiles, it is found that its effect is insignificant for all con-sidered values of the dimensionless heat source location. The results show that its effect is insignificant for all considered values of the dimensionless heat source location.Fig. 11shows the profiles of the local Nusselt number along the heat source for different nanoparticles at [D=0.5, Ri=1, K=0.5, B=0.4, φ =5%] and for case one (C1). It is observed that the largest value of the local Nusselt number along the heat source can be obtained by adding copper (Cu), while the smallest value of it can be obtained by adding titanium (TiO2)

nanoparticles. The reason of this behavior is due to the thermal con-ductivity correlation used for Cu–water micropolar nanofluid. Moreover, symmetrical profiles of the local Nusselt number along the heat source are obtained for all nanoparticles with the lowest local Nusselt number toward the adiabatic right sidewall.

4.6. Average Nusselt number results

Fig. 12displays the profiles of the average Nusselt number along the heat source for Cu–water micropolar nanofluid with different values of the Richardson number (Ri) and moving lid cases at [B=0.5, D=0.5, K=0.5 andφ =5%]. The results indicate that the fourth case (C4) provides the highest average Nusselt number compared with other considered cases. The reason of this behavior is due to the high circulation intensity of the flow field in this case as discussed previously inFig. 3.Fig. 13displays the profiles of the average Nusselt number along the heat source (Num) for Cu–water micropolar

nanofluid at different values of the dimensionless length of the heat source (B) and the solid volume fraction (φ) when [D=0.5, Ri=1, K=0.5] and for case one (C1). The results indicate that the average Nusselt number along the heat source decreases as the heat source length increases from (B=0.2) to (B=0.8) for various values of thesolid volume fraction. The reason of this behavior is due to the increase in the maximum temperature of the heat source by in-creasing its length which leads to a decrease in the average Nusselt

0.2 0.3 0.4 0.5 0.6 0.7 0.8 10 20 30 40 50 60 70 80 Nu s X Ri=0.01 R=1 R=10

Fig. 8. Profiles of the local Nusselt number along the heat source for Cu–water micropolar nanofluid for different values of (Ri) at B=0.5, D=0.5, K=0.5,φ =5%and for case one (C1).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Nu

s

X

B=0.2, φ=0% B=0.2, φ=8% B=0.8, φ=0% B=0.8, φ=8%

Fig. 9. Profiles of the local Nusselt number along the heat source for Cu–water micropolar nanofluid with different values of dimensionless length of the heat source (B) and solid volume fraction (_φ) at D=0.5, K=0.5, Ri=1 and (C1).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 6 8 10 12 14 16 18 20 22 24 26 28 30

Nu

s

X

D=0.3, K=0 D=0.3, K=0.5 D=0.7, K=0 D=0.7, K=0.5

Fig. 10. Profiles of the local Nusselt number along the heat source for Cu–water nanofluid with different values of dimensionless heat source location (D) and di-mensionless viscosity (K) at B=0.4, _{φ =}5%, Ri=1 and (C1).

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 4 6 8 10 12 14 16 18 20 22 24 26 C1

Nu

s

X

Cu Al₂O₃ Ag TiO₂

Fig. 11. Profiles of the local Nusselt number along the heat source for different nanoparticles at D=0.5, Ri=1, K=0.5, B=0.4,φ =5% and for case one (C1).

number. On the other hand, it can be observed that the average Nusselt number along the heat source increases as the solid volume fraction increases. This result is due to the increase in the thermal energy transport when the solid volume fraction increases which leads to increase in the average Nusselt number along the heat source. Moreover, it can be seen fromFig. 13that the increase in the average Nusselt number along the heat source with the solid volume fraction becomes high for a short length of the heat source (i.e., B=0.2).Fig. 14illustrates the profiles of the average Nusselt number along the heat source for different values of the dimen-sionless location of the heat source (D) and dimendimen-sionless viscosity (K) for Cu–water micropolar nanofluid at [B=0.4, Ri=1,φ =5%] and for case one (C1). It can be seen that the average Nusselt number along the heat source decreases dramatically as the dimension-less location of the heat source increases from (D=0.3) to (D=0.7) for various values of the dimensionless viscosity. On the other hand, it can be seen fromFig. 14that there is a slight decrease in the

average Nusselt number along the heat source when the dimen-sionless viscosity increases. This behavior can be seen when the dimensionless location of the heat source is small (i.e., D=0.3) while an opposite behavior between the average Nusselt number along the heat source and the dimensionless viscosity can be observed for (D=0.7).

5. Conclusions

The following conclusions can be drawn from the results of the present work.

1 BothRiand moving lids ordinations have a significant effect on the flow and thermal fields in the enclosure.

2 WhenBincreases, both the flow circulation and the convec-tion heat transfer for pure and micropolar nanofluids increase. 3 Both the flow and thermal fields are affected significantly by increasingDand the heat is transferred by the convection whenDdecreases.

4 Different velocity profiles are observed depending on the moving lid cases andRi.

5 The local Nusselt number along the heat source decreases whenRiincreases.

6 The local Nusselt number along the heat source has the highest value for copper (Cu) nanoparticles while it has the lowest one for titanium (TiO2) nanoparticles.

7 The local Nusselt number along the heat source increases when the micropolar nanofluid is used compared with the pure fluid, while it decreases when the heat source lengthB

increases.

8 The local Nusselt number decreases with increasing the di-mensionless heat source location for both ordinary and micropolar nanofluids. No significant effect is found on the local Nusselt number by increasing the dimensionless viscosity.

9 The average Nusselt number along the heat source de-creases asBincreases while it increases whenφ increases. 10 The average Nusselt number along the heat source

de-creases asDincreases.

11 The average Nusselt number along the heat source de-creases slightly when the dimensionless viscosity inde-creases

0 2 4 6 8 10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Nu

m

Ri

C1 C2 C3 C4

Fig. 12. Profiles of the average Nusselt number along the heat source for Cu– water micropolar nanofluid with different values of the Richardson number (Ri) and moving lid cases at B=0.5, D=0.5, K=0.5, φ =5%.

0.00 0.02 0.04 0.06 0.08

φ

10 20 30 40 50 60 70 80 90 100 110 120

Nu

m B=0.2 B=0.8

Fig. 13. Profiles of the average Nusselt number along the heat source for different values of the dimensionless length of the heat source (B) and solid volume frac-tion (_φ) for Cu–water nanofluid at D=0.5, Ri=1, K=0.5 and C1.

0.0 0.2 0.4 0.6 0.8 1.0 22 24 26 28 30 32 34 36 38 40 42

Nu

m

K

D=0.3 D=0.7

Fig. 14. Profiles of the average Nusselt number along the heat source for different values of the dimensionless location of the heat source (D) and dimensionless vis-cosity (K) for Cu–water nanofluid at B=0.4, Ri=1, φ =5%and C1.

and D=0.3, while a slight increase between them can be ob-served at D=0.7.

12 Both the local and the average Nusselt numbers along the heat source have the highest value for the fourth case (C4).

Nomenclature

B Dimensionless length of the heat source (b/H)

Cp Heat capacitance, j/kg. °C

D Dimensionless distance of the heat source from the bottom wall (d/H)

Gr Grashof number

g Gravity acceleration, m/s2

H Height of the enclosure, m j Micro-inertia density, kg/m3 K Dimensionless viscosity, K f =k μ k Vortex viscosity, kg/ms

kf Thermal conductivity of the fluid, W/m. °C

Num Average Nusselt number

Nus Local Nusselt number

N Dimensionless angular velocity, N= ×n H

Uo

n Component of the micro-rotation vector normal to the

x y_ plane P Dimensionless pressure, P p U nf o = ρ 2 p Fluid pressure, N/m2 Pr Prandtl number, Pr₌ν α f f ′′

q Heat generation per unit area, W/m2

Ra Rayleigh number, Ra g q H k f f f f = ′′ ⋅ ⋅ β ν α 4 Re Reynolds number,Re=U H f 0 ν Ri Richardson number, Ri= Ra Pr Re2 T Fluid temperature, °C

U Dimensionless velocity component in x-direction,U= u Uo

Uo Uniform lid-driven velocity of the moving left and upper

walls, m/s

u Velocity component in x-direction, m/s

V Dimensionless velocity component in y-direction,V= v

Uo

v Velocity component in y-direction, m/s w Width of the enclosure, m

X Dimensionless coordinate in horizontal direction, X x w

=

x Cartesian coordinate in the horizontal direction, m Y Dimensionless coordinate in vertical direction, Y y

H

=

y Cartesian coordinate in the vertical direction, m

Greek symbols

α Thermal diffusivity, m2_/s

β Coefficient of thermal expansion, K−1

θ Dimensionless temperature, θ = − × ′′ T Tc H q kf μ Dynamic viscosity, kg/ms γ Spin-gradient viscosity, kg/ms

φ Solid volume fraction of nanofluid ν Kinematic viscosity of the fluid, m2_/s

ρ Density, kg/m3

Subscripts

c Cold

f Fluid

nf Nano fluid particle p Particle

s Heat source surface

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