Cu-water micropolar nanofluid natural convection within a porous enclosure with heat generation

Accepted Manuscript Cu-water micropolar nanofluid natural convection within a porous enclosure with heat generation

Heidar Hashemi, Zafar Namazian, S.A.M. Mehryan PII: DOI: Reference:

S0167-7322(17)30262-3 doi: 10.1016/j.molliq.2017.04.001 MOLLIQ 7151

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

19 January 2017 23 March 2017 2 April 2017

Please cite this article as: Heidar Hashemi, Zafar Namazian, S.A.M. Mehryan , Cu-water micropolar nanofluid natural convection within a porous enclosure with heat generation. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Molliq(2017), doi: 10.1016/j.molliq.2017.04.001

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ACCEPTED MANUSCRIPT Cu-water micropolar nanofluid natural convection within a porous enclosure with heat generation Heidar Hashemi*, Zafar Namazian, S. A. M. Mehryan Department of Mechanical Engineering, Yasooj Branch, Islamic Azad University, Yasooj, Iran *

[email protected]

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Postal address: 75911-81459, Tell/ Fax: +98 743 331 3001

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Abstract

This investigation is concerned with natural convection within a porous enclosure occupied by

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Cu-water micropolar nanofluid at the presence of the heat generated in both solid and fluid phases of the porous medium. Darcy model is applied to simulate macroscopic flow dynamics. A

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vector equation, angular velocity describing microelements rotation, is added to common equations to satisfy the conservation of angular momentum. The governing equations are

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reduced to a non-dimensional form and then solved numerically using the Galerkin finite element method using a non-uniform structured grid with quadratic elements. The influence of

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dimensionless parameters like external Darcy-Rayleigh number RaE=1-103, internal DarcyRayleigh number RaI=1-104, Darcy number Da=10-4-10-1, porosity ε=0.1-0.9, nanoparticles

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volume fraction φ=0.0-0.1, vortex viscosity number Δ=0-2, the ratio of heat generation within the solid phase to fluid phase qr=0-2 on the velocity, temperature, and angular momentum fields

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are investigated. Additionally, the rate of heat transfer through porous medium are studied as these key parameters are varied. It is found that an increment in Darcy number leads to a slight

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decline in the strength of fluid flow and micro-rotation of particles. When vortex viscosity number Δ increases, the strength of vortices rotating in porous medium and micro-rotation of particles grow and reduce, respectively. The effect of RaI on thermal and dynamic characteristics of flow is more essential at high values of Δ.

Keywords: natural convection, micropolar nanofluid, porous medium, heat generation

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ACCEPTED MANUSCRIPT Nomenclature Latin Symbols heat capacity in constant pressure (kJ kg-1K-1)

g

gravitational acceleration vector (m s-2)

H

square cavity size (m)

k

thermal conductivity (W m-1 K-1)

K

permeability of the porous medium (m2)

j

micro-inertia density (m2)

N

dimensional micro-rotation, s-1

Nu

Nusselt number

p

pressure (Pa)

q

heat production per unit volume (W m-3)

RaE

external Darcy-Rayleigh number

RaI

internal Darcy-Rayleigh number

T

temperature (K)

u, v

velocity components along x, y directions, respectively (m s-1)

V

Darcian velocity vector

x, y

Cartesian coordinates (m)

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thermal diffusivity (m2 s-1)

β

thermal expansion coefficient (K-1)

Δ

vortex viscosity number

ε

porosity of the porous medium

κ

vortex viscosity (kg m-1 s-1)

μ

dynamic viscosity (kg m-1 s-1)

ρ

density (kg m-3)



nanoparticles volume fraction



stream function

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α

Subscripts avg

average

bf

base-fluid

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Greek symbols

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Cp

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ACCEPTED MANUSCRIPT c

cold

h

hot

m

effective

nf

nanofluid

np

nanoparticle

s

solid porous matrix

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dimensional parameters

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*

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superscripts

1. Introduction

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Micropolar non-Newtonian fluid has been introduced to portray the influences of microrotational and local structures of fluid particles. The Navier-Stokes theory cannot describe

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behaviors of this kind of fluid. Hence, the model proposed by Eringen [1, 2] is applied as a basis to study flow dynamics of these fluids. The stress tensor of micropolar fluids is asymmetrical and

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deformation of fluid particles is neglected. In the theory formulated by Eringen [1, 2], the vector

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equation of angular velocity of the microelements rotation is applied to represent the conservation law of the local angular momentum. Practical examples of micropolar fluids are observed in polymeric fluids, ferrofluids, blood flow of animals, bubbly liquids, liquid crystals,

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and so on; all of them comprise internal polarities. Comprehensive reviews on the theory and applications of micropolar fluids can be seen in papers published by Eringen [3], Lukaszewicz

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[4] and Ariman et al [5]. In an experimental study, Hoyt and Fabula [6] represented that fluids with polymeric additives as a micropolar fluid significantly decrease the skin friction. Power [7]

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showed that Cerebrospinal Fluid (CSF), which is found in the brain, behaves as a micropolar fluid. Ramzan et al. [8] studied the flow related to a micropolar magnetic fluid over a stretching sheet in the presence of joule heating, radiative phenomena and partial slip on the surface. Many of published articles have focused on natural or force convection of micropolar fluid in closed containers [9-16]. Aydin and Pop [9] studied the thermal and dynamic characteristics of natural convection related to a micropolar fluid in a square enclosure. Analysis of natural convection of a micropolar magnetic fluid in a two dimensional square enclosure with a uniform and non-uniform heated thin plate is conducted by Periyadurai et al. [10]. Gibanov et al. [11]

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ACCEPTED MANUSCRIPT studied unsteady natural convection of micropolar fluid flowing in a complex enclosure with the wavy hot wall. The effects of a local heat source on free convection related to a micropolar fluid flowing within a trapezoidal cavity performed by Miroshnichenko et al. [16]. Moreover, a number of published papers have studied the effects of dispersing solid nanoparticles on flow and heat transfer of micropolar fluids [17-20]. Nanofluid is a combination of a base fluid and suspended nanometer-sized particles. Although dispersing nanoparticles in the base fluid is done

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in order to increase the heat transfer, heat transfer increment using nanofluids is a controversial

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issue. Bourantas and Loukopoulos [17] proposed a micropolar model to study natural convection

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of nanofluids. In another study [18], they also performed an investigation concerning unsteady and laminar natural convection of Al2O3/water micropolar magnetic nanofluid in a partially

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heated inclined enclosure. Mixed convection in a lid-driven square enclosure filled by nanofluid with two vertical moving walls has been examined numerically by Saleem et al. [19].

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Several applications can be mentioned to micropolar fluid flowing in a porous medium, such as porous rocks, foams and foamed solids, aerogels, alloys, polymer mixtures and micro-emulsions.

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Sharma and Gupta [21] investigated the effect of porous medium permeability on thermal convection in micropolar fluid. Sharma and Kumar [22] analyzed the thermal instability of

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micropolar magnetic fluids heated from below under the influence of uniform vertical magnetic field in porous medium. It is understood that coupling between magnetic field and permeability

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can cause oscillatory motions in the system. Numerical study concerning a porous medium occupied by a micropolar fluid flowing over a permeable stretching/ shrinking sheet is performed

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by Rosali et al. [23]. Tripathy et al. [24] analyzed the influences of the volumetric non-uniform heat generation and chemical reaction on mixed convection flow, heat and mass transfer of a

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micropolar magnetic nanofluid along a stretching sheet in a porous medium. Siddiqa et al. [25] performed the analysis of natural convection of an electrically conducting micropolar fluid in the simultaneous presence of radiation and periodic magnetic field. As observed, most studies associated with micropolar fluid flow in porous medium focus on free and force convection flow along vertical and horizontal sheets embedded in the porous medium (see [26-27]). Hence, there is not any investigation that examines the flow and heat transfer concerning a micropolar nanofluid/pure fluid in a porous enclosure. So, the main goal of the present work is to investigate the thermal and dynamic characteristics of a micropolar nanofluid within a square enclosure in the presence of heat generation in the fluid and solid phases of the porous medium. Therefore, it 4

ACCEPTED MANUSCRIPT could be said that this work is completely original and the authors believe that this study would attract attentions of researchers in this field.

2. Statement of the problem and derivation of the governing equations In this study, a square porous enclosure with the length of H and occupied by Cu-water

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micropolar nanofluid is taken into account as shown in Fig. 1. In the enclosure, the length of the

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side perpendicular to the surface of the paper is very large compared to the other two sides. Hence, there is no flow in z* direction. As seen, the top and bottom bounds of the porous

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enclosure are insulated thermally, nevertheless, the left and right bounds have been kept constant at high and low temperatures, respectively. The high and low values of temperatures are Th* and

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Tc*, correspondingly. Furthermore, it is considered that all bounds of the porous enclosure are impervious to the penetration of the fluid and also the solid nanoparticles. The aluminum foam is

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taken into account for the solid matrix of the porous medium. The solid and fluid phases of the porous medium are locally at the same temperature so that no heat is exchanged between these

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phases. Since the temperature difference between two vertical bounds is restricted, it can be declared that all properties of the nanoparticles and base fluid, expect for density in the

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momentum equation, remain constant during the convection flow. Variations of the density in the momentum equation can be approximated by Boussinesq model. Table 1 provides the

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thermophysical properties of water, Cu nanoparticles and aluminum foam. There are dynamic and thermal equilibriums between the solid nanoparticles and base fluid. The Darcy model is

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employed to study and simulate the micropolar nanofluid flowing inside the porous enclosure. To prevent settling and agglomeration of nanoparticles, the surface charge technology or

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surfactant can be utilized.

Applying above statements to derive the governing equations leads to the equations given below which are the conservation of continuity, linear momentums, angular momentum and energy, respectively [18, 21-22, 30].

  V*  0

0  p * 

nf   K

V*    nf T * T c*  g   N *

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(1)

(2)

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  nf j  V* *  N *   nf   j * N *  *  V*  2 N * 2 2



V* *T * 



k m ,nf

* T *  2

 C 

 C p nf

(4)

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p nf

q m ,nf

(3)

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In the above equations, the bold letters are the representations of vectors. Additionally, j is called micro-inertia density and equals H2. km,nf is the effective heat conduction of the porous medium

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and is written as follows [30]:

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k m ,nf   k nf  1    k s

(5)

where knf and ks are the thermal conductivity of the micropolar nanofluid and solid matrix of the

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porous media, respectively. Additionally, ε represents the porosity of the medium. The inner heat

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generated per unit volume of the medium is defined as [30]:

(6)

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q m ,nf  1    q s   q nf

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q s and q nf are the rate of heat generation in the solid and fluid phases of the porous media. The

impact of nanoparticles on the density of micropolar nanofluid can be modeled using the

nf  1    bf  np

(7)

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followed relation:

μnf is the effective dynamic viscosity of the micropolar nanofluid which can be achieved using the model proposed by Brinkman [31]:

nf 

bf

1   

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2.5

(8)

ACCEPTED MANUSCRIPT Thermal expansion coefficient is the other property of the micropolar nanofluid employed in the following relation:

  nf  1      bf     np

(9)

Further, in order to apply the influence of nanoparticles on the effective capacitance heat of the

 C nf  1    C bf p

   C p np

(10)

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p

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micropolar nanofluid, the following relation is utilized:

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Finally, we utilize the Maxwell model [32] to achieve the effective thermal conductivity of the micropolar nanofluid as follows:

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  k np  2k bf   2  k bf  k np   k nf  k bf     k np  2k bf     k bf  k np  

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(11)

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Since the flow is incompressible, it cannot define a state equation for the nanofluid and, hence, the pressure term in the equations is known as an undefined variable. So, it is better to eliminate

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the pressure term of the momentum equations. The pressure terms are eliminated by cross differentiation. In addition, a new variable named stream function is utilized so that it satisfies

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the continuity equation and at the same time, the components of the velocity can be achieved using its partial differentials. The stream function is represented as the following:

 *  * * , v   nf y * x *

(12)

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u *nf 

Using the stream functions and also, cross differentiating between the x and y momentum equations, the governing equations of Eq. (1)–(2) can be written in Cartesian coordinates as

nf     2 * K

 *2  x



 2N * 2N *   2 *  T *  g        x *2  y *2  nf y *2  x *  

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(13)

ACCEPTED MANUSCRIPT It is worth to mention that here N is the component of micro-rotation vector along z* axis. To do a dimensionless parametric analysis, the dimensional derived equations can be transferred to the dimensionless coordinate of x and y using the dimensionless parameters below:





x  x * / H , y  y * / H ,    * / m ,bf ,  = T * T c*  T h* T c*  and N  H 2 m ,bf N *

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        2N 2N    1  T   2   Da     nf nf  x y 2       x        bf   bf 

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2

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   nf    2  RaE  2     x y bf  2

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Finally, the dimensionless forms of the governing equations are:

(14)

 bf   N   2 Pr      nf 

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     N  N     2N  2N   Pr  bf   nf     y x x y 2   x 2 y 2   nf   bf    v u    Pr  bf  nf  nf  y    nf   x

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     m ,nf   2  2    C p bf      y x x x  m ,bf  x 2 y 2    C p nf 

 RaI  1    q r       RaE 

(15)

(16)

(17)

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Here, respectively, RaE, RaI, Pr and qr are:

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g bf T h* T c*  KH

m ,bf bf

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RaE 

, RaI 

g bf q nf KH 3  q , Pr  bf , q r = s ,    m ,bf q nf bf m ,bf bf k m ,bf

(18)

The boundary conditions subjected to the bounds are:

  0, N  0, T  1   0, N  0, T  0   0, N  0,   0, N  0,

on x  0 and 0  y  1 on x  1 and 0  y  1

T  0 on y T  0 on y

y  0 and 0  x  1 y  1 and 0  x  1

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(19)

ACCEPTED MANUSCRIPT To measure the rate of heat transfer through the fluid and solid phases of the porous cavity, the average Nusselt number is defined as follows:

k m ,nf T nf , k m ,bf x

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Nu avg 

1 Nu local dy L 0

(20)

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Nu local  

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3. The method of solution, examination of grid independency and verification Since the partial differential equations Eqs. (15)-(17) are non-linear and coupled, the use of a

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numerical approach is essential. Hence, the finite element method is utilized to solve these partial equations subjected the boundary conditions (19). In this numerical approach, firstly the strong

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forms of the equations, which have been represented in Eqs. (15)-(17), are rewritten in equivalent forms namely weak forms. The details of this method have been explained completely

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in [33-34]. Fig. 2 displays the type of grid used (non-uniform structured grid) to discrete the computation domain to subdomains. The outcomes of mathematical simulating should not be

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dependent on the grid size utilized to make subdomains. Hence, it is necessary to achieve an optimal grid which is sufficiently fast and accurate. From Table 2, as found, a grid with the

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number of elements 100×100 can satisfy these demands. For grid size 100×100, the error percentage of mesh size is much less than 1 %, exactly 0.218 and 0.022 % for Nuavg and |ψ|max,

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respectively. It is worth saying that the parameters used to achieve an optimal grid are RaE=103, RaI=104, Da=10-2, ε=0.5, Δ=1, φ=0.02 and qr=0.5.

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The authenticity of the outcomes of the present work is assessed by re-calculating consequences reported by [9, 28 and 35]. At the first verification, the problem of natural convection in a square

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enclosure filled with pure micropolar fluid (done by Aydin and Pop [9]) is re-simulated. The results of this verification are given in Table 3. In the other verification [35], the natural convection within an enclosed porous medium by the sides of a triangular is studied. The data represented in Table 4 illustrates this verification. The excellent matching between the results of this study and those reported in reference papers [9, 28 and 35] ensures the accuracy of our results.

4. Results and discussion

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ACCEPTED MANUSCRIPT The survey of natural convection of Cu-water micropolar nanofluid within a porous cavity with aluminum foam solid matrix at the presence of the internal heat generation is conducted numerically. In the present study, the key parameters are external Darcy-Rayleigh number RaE=1-103, internal Darcy-Rayleigh number RaI=1-104, Darcy number Da=10-4-10-1, vortex viscosity number Δ=0-2, porosity ε=0.1-0.9, the ratio of heat generated in solid to fluid phases of porous media qr=0-2 and nanoparticles volume fraction φ=0.0-0.1.

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Dependency of the velocity, temperature and angular momentum fields to external Darcy-

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Rayleigh number RaE are depicted in Fig. 3. Here, the other parameters are assumed to be

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constant so that RaI=103, ε=0.5, qr=1, Δ=2, Da=10-2, φ=0.05. As observed, when RaE is low (RaE=10), all the fields are approximately symmetric with respect to the vertical line passing

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through the center of porous enclosure. These symmetries as well as the counter-clockwise vortices rotating in the adjacent half of cold wall gives predominance of RaI to RaE in controlling

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the flow direction and heat transfer. As external Darcy-Rayleigh number RaE increases, theses symmetries are disturbed. The size and strength of the vortex rotating in the right half of the cavity become larger than one formed in the left half of the enclosure when RaE is 102.

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Eventually, when RaE=103, this clockwise vortex covers the whole of the porous cavity. In

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pursuance of this flow field, the isotherms are elongated along the horizontal bounds and the particles rotate counterclockwise in the whole of flow domain. This means that the control of

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flow and heat transfer is conducted by RaE. To justify these observations, it can be expressed that the strength of the flow where is flowing due to the difference of the temperature of cold and hot

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walls increases with an increment in RaE. The outcomes show that increasing RaE leads to a reduction in temperature of porous medium. In addition to all of this, it can be found that the

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micro-rotation strength of both particles of solid and fluid is enhanced with increasing RaE. It is notable that the lines of micro-rotation contours indicate the particle rotation along with the micropolar nanofluid flow inside the pores of porous medium. To investigate the dependency of velocity, temperature and angular momentum fields to internal Darcy-Rayleigh number RaI, streamlines, isotherms and lines of micro-rotation contours are represented at different values of RaI in Fig. 4. During this survey, the other parameters are considered to be constant at RaE=103, ε=0.5, qr=1, Δ=2, Da=10-2, φ=0.05. The mobility and internal rotation of Cu-water micropolar nanofluid augments when the value of internal DarcyRayleigh number RaI grows. Meanwhile, it is interesting to say that an increase in RaI leads to 10

ACCEPTED MANUSCRIPT the velocity reduction in the vicinity of the hot wall. The density of the streamlines in this region shows this allegation. Also, it is worth mentioning that an increase in RaI leads to decrease elongation of the streamlines and lines of micro-rotation contours and move the center of the vortices and micro-rotation isolines towards the cold bound. Additionally, a second eddy is created by achieving RaI to 104. In addition, it is found that thermal boundary layer gets thicker by increasing RaI and hence, the temperature of porous medium increases. To interpret these

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results, it can be argued that the flow created by thermal difference between the cold and hot

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walls (driven by RaE) rises up alongside the hot wall and descends alongside the cold wall. On

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the other hand, the flow induced by RaI tends to move down along the vertical walls. So, it can be concluded that these Rayleigh numbers reinforce and undermine each other in the vicinity of

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the cold and hot walls, respectively.

Dependency of velocity, temperature and angular momentum fields to nanoparticles volume

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fraction φ is exhibited in Fig. 5 at fixed parameters RaE=102, RaI=103, ε=0.5, qr=Δ=2, Da=10-2. As nanoparticles volume fraction increases, the maximum values of streamlines in both

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clockwise and counter-clockwise vortices slightly diminish, which means the strength reduction of micropolar nanofluid flow. This occurs because of an increase in internal resistance of the

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fluid against the motion, as a result of increasing dynamic viscosity by augmentation of nanoparticles concentration. Also, the increment in temperature of porous medium can occur for

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this reason. Eventually, micro-rotation isolines show that the maximum value of the particles orientation declines with dispersing nanoparticles in base micropolar fluid.

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The influences of vortex viscosity number Δ on the streamlines, isotherms, and micro-rotation isolines are depicted in Fig. 6 when RaE=102, RaI = 103, ε=0.5, qr=1, Da=10-1, φ=0.05. In case of

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Δ=0, Newtonian nanofluid flow is observed. Regardless of the value of Δ, a second eddy is formed in the vicinity of the hot bound. An increment in Δ leads to becoming the larger size of this second vortex rotating in the vicinity of left bound of the cavity. In fact, when Δ increases, the internal resistance of the micropolar nanofluid is amplified against the fluid flow. Hence, the strength of the clockwise primary vortex decreases and the effects of heat generation in porous media become more visible. Further, it can be concluded that temperature of the porous medium and the strength of particles orientation increase as a result of increasing Δ. Dependency of streamlines, isotherms and micro-rotation contours to porosity ε for two different values of qr=0.5 and 2 have been displayed in Fig. 7. When qr is less than 1 (qr=0.5), an 11

ACCEPTED MANUSCRIPT amplification in convective flow of the micropolar nanofluid is resulted by augmentation of the value of porosity ε. Additionally, it can be observed that in this case, increasing ε causes an increment in temperature of medium and strength of particles orientation rotating in the pores. In contrast, in the case of qr>1 (qr=2), an increment in porosity declines the convective flow in the porous enclosure. From the definition of qr, for qr<1, the rate of heat generated in fluid phase is more that of solid phase. Hence, increasing occupied volume by the micropolar nanofluid with

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an increase in ε leads to augmentation of heat generated in porous enclosure which intensifies the

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mobility of fluid flow. A revers explanation justifies the variations of governing fields with ε.

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Regardless of the ε values, when qr augments, the strength of flow, the temperature of medium and the strength of particles orientations increase. It is interesting to note that how much ε gets

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lower, the effects of qr on the governing fields gets more.

Dependency of velocity, temperature and angular momentum fields to Darcy number Da is

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exhibited in Fig. 8, at fixed parameters of RaE=102, RaI = 103, ε=0.5, qr=1, Δ=2, φ=0.05. As presented in Fig. 8, an increase in Darcy number Da results a slight decrease in the intensity of

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the primary vortex rotating in the porous enclosure. To justify this observation, it should be cited to the term including Darcy in the momentum equation. This term states that an increase in

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Darcy number ultimates an increase of effectiveness of particles orientation on the flow field of micropolar nanofluid. It is therefore expected that intensity of vortices rotating decreases with

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increasing Da. For the reasons mentioned, the temperature of porous medium and the strength of particles orientation slightly enhance and decline with increasing Da number.

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Figs. 9 (a) and (b) show the variations of Nulocal on hot wall for various values of RaI and qr, respectively, at the fixed parameters of RaE=103, Da=10-2, Δ=2, ε=0.5, φ=0.05. The drastic

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increment at local Nusselt number at the beginning of the hot boundary, next to the insulated bottom bound, is observed. The very thin thermal boundary layer in this region has the lowest thermal resistance against heat transfer, so the maximum heat transfer rate locally occurs at the lowest part of the hot boundary. By moving along y axis, the thickness of thermal boundary layer increases, hence, the rate of heat transfer decreases along the y axis. A negative value of Nulocal refers to heat transfer from micropolar nanofluid to the hot bound. Also, it can be concluded that decreasing trend Nulocal along the wall with increasing RaI is more essential for high values of RaI.

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ACCEPTED MANUSCRIPT The changes of Nuavg according to porosity ε for the different values of qr is displayed in Fig. 10 at fixed parameters of RaE=103, RaI = 104, Da=10-2, Δ=2, φ=0.05. According to the findings, this is obvious that the variations of Nuavg are entirely dependent on the ratio of the heat generated in solid phase to fluid phase of the porous medium. When qr is less than 1 (qr<1), a decreasing trend can be observed as porosity increases. On the contrary, for qr>1, the increment of porosity augments the heat transfer rate through porous medium. As mentioned before, for qr<1,

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increasing porosity ε eventuates additional heat generation in porous medium that decreases

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Nuavg with an increase in the thickness of thermal boundary layer. When qr>1, A reverse

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explanation can be used to justify increasing trend of Nuavg with ε.

Fig. 11 depicts behavior of Nuavg with Darcy number Da for different values of Δ at RaE=103,

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RaI=104, ε=0.5, qr=0.5, φ=0.05. The increment in the thermal boundary layer thickness with increasing medium temperature as a result of increasing vortex viscous number Δ can lead to a

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decline in the rate of heat transfer. This fact is observed in Fig. 11. This figure also displays that decreasing trend of Nuavg with increasing Da is more visible for high values of Δ. The behavior of Nuavg versus internal Darcy-Rayleigh number RaI for different values of Δ at

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RaE=103, ε=0.5, Da=10-2, qr=1.5, φ=0.05 is depicted in Fig. 12 (a) and (b). Once again, as Δ

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increases, the rate of heat transfer decreases. Additionally, the outcomes display that the increment in RaI declines Nuavg. The increase in thickness of thermal boundary layer, as a result

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of an increment in RaI, results in a decline in Nuavg. From Fig. 12 (a), it can be clearly seen that decreasing trend of Nuavg with increasing Δ declines as Δ increases. As shown in Fig. 12 (b), the

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reduction of Nuavg with increasing of RaI increases when Δ increases. This result is due to this fact that an increment in Δ declines the effects of RaE in controlling flow. On the other hand, the

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reduction of the effects of RaE means the enhancement of impacts of RaI in controlling the flow. Figs. 13 (a) and (b) show changes of Nuavg and the ratio of Nuavg of nanofluid to Nuavg of pure micropolar fluid versus nanoparticles volume fraction for different values of RaE at RaI=102, ε=0.5, Δ=1, Da=10-2, qr=1.5. As seen, for RaE=101, dispersing the nanoparticles in Cu-water micropolar nanofluid results in an enhancement in Nuavg while at values of RaE=102 and 103, nanoparticles reduce the rate of heat transfer through porous medium. When RaE is low (RaE=101), the conduction heat transfer mechanism is dominant and thus, the changing of the viscosity does not have a significant effect on the heat transfer rate. On the other hand, the increase of the heat transfer due to the increase of the thermal conductivity as a result of 13

ACCEPTED MANUSCRIPT dispersing nanoparticles is dominant in contrast with decreasing that due to the reduction of the mobility of the fluid arising from dispersing the nanoparticles. However, at the high values of RaE (RaE=102 and 103), A reverse explanation can be expressed to justify decreasing trend of Nuavg with using the nanoparticles of Cu. Moreover, it is clearly seen that the increase in RaE strongly enhances Nuavg. The effects of dispersing Cu nanoparticles in micropolar nanofluid are more essential when RaE=102.

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Figs. 14 (a) and (b) show changes of Nuavg and the ratio of Nuavg of nanofluid to Nuavg of pure

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micropolar fluid versus Darcy number Da for different values of nanoparticles volume fraction φ

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when RaE=103, RaI=104, ε= qr=0.5, Δ=1. Once again, when RaE=103, dispersing Cu nanoparticle in the pure micropolar fluid leads to a reduction of Nuavg. Fig. 14 (b) illustrates that the effect of

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Da on the decrease of Nuavg becomes more visible with an increment in φ.

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5. Conclusion

This investigation is concerned with natural convection within a porous square enclosure

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occupied by Cu-water micropolar nanofluid with heat generation in both solid and fluid phases of porous medium. Darcy model is applied to simulate macroscopic flow dynamic and vector

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equation of angular velocity describing microelements rotation is added to common equations to satisfy the conservation of angular momentum. The governing equations are reduced to a non-

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dimensional form and then solved numerically using the Galerkin finite element method using a non-uniform structured grid with quadratic elements. The most important observations can be



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summarized as follows:

As external Darcy-Rayleigh number RaE increases, both the strength of vortices formed

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and micro-rotation of particles increases and the maximum value of temperature decreases. 

An increment in internal Darcy-Rayleigh number RaI increases the temperature of porous medium, the strength of flow and micro-rotation of both particles. Decreasing trend of Nulocal along the hot wall (or Nuavg) with increasing RaI is more essential for high values of RaI.



At RaE=102, RaI=103, ε=0.5, qr=Δ=2, Da=10-2, dispersing Cu nanoparticles in micropolar water-base fluid declines strength of vortices formed and micro-rotation particles. It is found that at low value of RaE (RaE=10), using Cu nanoparticles in micropolar water14

ACCEPTED MANUSCRIPT based leads to an augmentation in Nuavg. However, for high values of RaE (RaE=102 and 103), the presence of nanoparticles reduce the rate of heat transfer. 

When qr<1, an increase of porosity ε augments the strength of Cu-water micropolar fluid flowing and particles rotating in pores. A revers results can be found for qr>1.



As vortex viscosity number Δ increases, the strength of rotating vortices and micro-

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rotation of particles increase and decreases, respectively. The effect of RaI on thermal and

An increment in Darcy number leads to a slight decline in the strength of fluid flow and

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

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dynamic characteristics of flow is more essential at high values of Δ.

micro-rotation of particles. The effect of Darcy number Da is more essential for high

Finally, it is concluded that the effect of Da on the decrease of Nuavg becomes more

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visible with an increment in φ.

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

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values of Δ.

15

ACCEPTED MANUSCRIPT References [1] A.C. Eringen, Simple micropolar fluids, Int. J. Eng. Sci. 2, pp. 205-217. [2] A.C. Eringen, Theory of micropolar fluids, Journal of Mathematics and Mechanics, 16 (1966) 1–18. [3] T. Ariman, M.A. Turk, N.D. Sylvester, Microcontinuum fluid mechanics – a review, Int. J. Engng. Sci. 11 (1973) 905–930. [4] G.Łukaszewicz, Micropolar Fluids: Theory and Application, Birkhäuser, Basel, 1999. [5] A.C. Eringen, Microcontinuum Field Theories. II: Fluent Media, Springer, New York, 2001.

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[6] J.W. Hoyt, A.G. Fabula, The effect of additives on fluid friction, Tech. rep., US Naval Ordnance Test Station

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Report, 1964.

[7] H. Power, Micropolar fluid model for the brain fluid dynamics, Int. Conf. Bio-Fluid Mechanics, U.K., 1998.

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[8] M. Ramzan a, M. Farooq,T.Hayat, Jae Dong Chung, Radiative and Joule heating effects in the MHD flow of a micropolar fluid with partial slip and convective boundary condition, Journal of Molecular Liquids 221 (2016) 394–

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400.

[9] O. Aydın, I. Pop, Natural convection in a differentially heated enclosure filled with a micropolar fluid, International Journal of Thermal Sciences 46 (2007) 963–969.

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[10] K. Periyadurai, M. Muthtamilselvan, Deog-Hee Doh, Influence of inclined Lorentz force on micropolar fluids in a square cavity with uniform and nonuniform heated thin plate, Journal of Magnetism and Magnetic Materials

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420 (2016) 343–355.

[11] N.S. Gibanov, M.A. Sheremet, I. Pop, Natural convection of micropolar fluid in a wavy differentially heated

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cavity, Journal of Molecular Liquids 221, (2016) 518–525. [12] S.G. Wang, T.H. Hsu, natural convection of micropolar fluids in an inclined rectangular enclosure, Mathematical and Computer Modelling 17 (10) (1993) 73-80.

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[13] N.S. Gibanov, M.A. Sheremet, I. Pop, Free convection in a trapezoidal cavity filled with a micropolar fluid, International Journal of Heat and Mass Transfer 99 (2016) 831–838

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[14] M. Saleem, S. Asghar, M.A. Hossain, Natural convection flow of micropolar fluid in a rectangular cavity heated from below with cold sidewalls, Mathematical and Computer Modelling 54 (2011) 508–518. [15] M. A. Sheremet, I. Pop, A. Ishak, Time-dependent natural convection of micropolar fluid in a wavy triangular

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cavity, International Journal of Heat and Mass Transfer 105 (2017) 610–622. [16] I. V. Miroshnichenko, M. A. Sheremet, I. Pop, Natural convection in a trapezoidal cavity filled with a micropolar fluid under the effect of a local heat source, International Journal of Mechanical Sciences 120 (2017) 182–189. [17] G.C. Bourantas , V.C. Loukopoulos, Modeling the natural convective flow of micropolar nanofluids, International Journal of Heat and Mass Transfer 68 (2014) 35–41 [18] G.C. Bourantas a, V.C. Loukopoulos, MHD natural-convection flow in an inclined square enclosure filled with a micropolar-nanofluid, International Journal of Heat and Mass Transfer 79 (2014) 930–944

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ACCEPTED MANUSCRIPT [19] S.E. Ahmed, M.A. Mansour, A.K. Hussein , S. Sivasankaran, Mixed convection from a discrete heat source in enclosures with two adjacent moving walls and filled with micropolar nanofluids, Engineering Science and Technology, an International Journal 19 (2016) 364–376 [20] S.K. Jena, L.K. Malla, S.K. Mahapatra, A.J. Chamkha, Transient buoyancy-opposed double diffusive convection of micropolar fluids in a square enclosure, International Journal of Heat and Mass Transfer 81 (2015) 681–694 [21] R.C. Sharma, U. Gupta, Thermal convection in micropolar fluids in porous medium, International Journal of

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Engineering Science 33 (1995) 1887–1892.

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[22] R.C. Sharma, P. Kumar, on micropolar fluids heated from below in hydromagnetics in porous medium, Czechoslovak Journal of Physics 47 (6) (1997).

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[23] H. Rosali, A. Ishak, I. Pop, Micropolar fluid flow towards a stretching/shrinking sheet in a porous medium with suction, International Communications in Heat and Mass Transfer 39 (2012) 826–829

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[24] R.S. Tripathy, G.C. Dash, S.R. Mishra, Mohammad Mainul Hoque, Numerical analysis of hydromagnetic micropolar fluid along a stretching sheet embedded in porous medium with non-uniform heat source and chemical reaction, Engineering Science and Technology, an International Journal 19 (2016) 1573–1581.

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[25] S. Siddiqa, A. Faryad, N. Begum, M.A. Hossain, R.S.R. Gorla, Periodic magnetohydrodynamic natural convection flow of a micropolar fluid with radiation, International Journal of Thermal Sciences 111 (2017) 215-222.

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[26] D. Pal, S. Biswas, Perturbation analysis of magnetohydrodynamics oscillatory flow on convective-radiative heat and mass transfer of micropolar fluid in a porous medium with chemical reaction, Engineering Science and

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Technology, an International Journal 19 (2016) 444–462.

[27] J.I. Oahimire, B.I. Olajuwon, Effect of Hall current and thermal radiation on heat and mass transfer of a chemically reacting MHD flow of a micropolar fluid through a porous medium, Journal of King Saud University –

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Engineering Sciences (2014) 26, 112–121

[28] M.A. Sheremet, T. Grosan, I. Pop, Free convection in a square cavity filled with a porous medium saturated by

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nanofluid using Tiwari and Das’ nanofluid model. Transport in Porous Media 106 (2015) 595–610. [29] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int J Heat Fluid Flow 29 (2008) 1326–1336.

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[30] D.A. Nield, A. Bejan, Convection in Porous Media (4 th edition). Springer, New York, 2013. [31] H. Brinkman, The viscosity of concentrated suspensions and solutions, J.Chem. Phys. 20 (1952) 571. [32] J.C. Maxwell, A Treatise on Electricity and Magnetism, Clarendon, Oxford, 1873. [33] S. Rao, "The finite element method in engineering. Butter-worth," ed: Heinemann publications, Amsterdam, Boston, Heidelberg, London, New York, Oxford Paris, San Diego, San Francisco, Singapore, Sydney, Tokyo, 2005. [34] P. Wriggers, Nonlinear finite element methods: Springer Science & Business Media, 2008. [35] Q. Sun, I. Pop, Free convection in a triangle cavity filled with a porous medium saturated with nanofluids with flush mounted heater on the wall. International journal thermal science. 50 (2011) 2141–2153.

17

ACCEPTED MANUSCRIPT Tables Table 1 thermal-physical properties of the base fluid, nanoparticles (see [28, 29]) and solid structure of the porous medium

Cu

Aluminum foam

4179 0.613 1.47 21 997.1 8.9

385 401 1163.1 1.67 8933 -

897 205 846.4 2.22 2700 -

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18

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water

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Physical properties cp (J/kg. K) k (W/m. K) α×10-7 (m2/s) β×10-5 (K-1) ρ (kg/m3) μ×10-4 (kg/m. s)

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Table 2 grid independency test for the pure fluid when Ra=103 |ψ|max

Grid size

Nuavg

Error (%)

Error (%)

50×50

8.139

100×100

8.227

1.081

13.180

0.121

150×150

8.245

0.218

13.177

0.022

200×200

8.252

0.008

13.176

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PT

ED

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13.196

19

0.007

ACCEPTED MANUSCRIPT Table 3 The average Nusselt number for this study and Aydin and Pop’s work [9] (given in bracket) Ra=103

Ra=104

Ra=105

Ra=106

0

1.118 [1.118]

2.245 [2.234]

4.522 [4.486]

8.828 [8.945]

0.5

1.059 [1.057]

1.977 [1.947]

4.077 [4.033]

8.030 [7.984]

1

1.035 [1.034]

1.797 [1.771]

3.765 [3.729]

7.468 [7.433]

2

1.016 [1.016]

1.566 [1.545]

3.344 [3.314]

6.700 [6.673]

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Δ

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ACCEPTED MANUSCRIPT Table 4 Values calculated for average Nusselt number in a porous triangular shaped enclosure occupied with Cuwater nanofluid φ

Sheremet et al [28]

Sun and pop [35]

Present work

500

0

9.65

9.66

9.64

1000

0.1

9.41

9.42

9.42

500

0

14.05

13.9

13.96

1000

0.2

12.84

12.85

12.85

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Ra

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Fig. 1 A simple view of the geometry of the problem

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-6

1

RaE=10 2 RaE=10 RaE=103

CR

Nuavg

Nuavg/Nuavg |

1

0.02

0.04



0.06

0.08

0.1

(b)

Fig. 13 variations of the average Nusselt number according to volume fraction φ for different values of Ra E when

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ED

M

RaI=102, ε=0.5, Δ=1, Da=10-2, qr=1.5

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ACCEPTED MANUSCRIPT

1

     

     

0.95

T

Nuavg

5.2

0.9

Nuavg/Nuavg |

5.6

4.4 -5 10

10

-4

10

-3

10

-2

10

-1

10

-5

10

US

Da

CR

0.85

IP

4.8

(a)

-4

10

-3

10

-2

10

-1

Da (b)

Fig. 14 variations of the average Nusselt number according to Darcy number Da for different values of φ when

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CE

PT

ED

M

AN

RaE=103, Rai=104, ε= qr=0.5, Δ=1

35

ACCEPTED MANUSCRIPT Highlights

CE

PT

ED

M

AN

US

CR

IP

T

Natural convection of a micropolar nanofluid inside a porous medium is addressed. The effect of heat generation in solid and fluid phases of the porous medium is studied. An increment in Da leads to a slight decline in the strength of micro-rotation of particles. Depending on the value of qr, an increment in ε can increase or decrease Nuavg.

AC

   

36