Numerical analysis of the onset of longitudinal convective rolls in a porous medium saturated by an electrically conducting nanofluid in the presence of an external magnetic field

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Numerical Analysis of the Onset of

Longitudinal Convective Rolls in a Porous Medium Saturated by an Electrically

Conducting...

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DOI: 10.1016/j.rinp.2017.06.003

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Numerical analysis of the onset of longitudinal convective rolls in a porous medium saturated by an electrically conducting nanofluid in the presence of an external magnetic field

Abderrahim Wakif ^⇑ , Zoubair Boulahia, Rachid Sehaqui

Hassan II University, Faculty of Sciences Aïn Chock, Laboratory of Mechanics, B.P. 5366 Mâarif, Casablanca, Morocco

a r t i c l e i n f o

Article history:

Received 24 February 2017

Received in revised form 18 May 2017 Accepted 2 June 2017

Available online 8 June 2017

Keywords:

Linear stability Nanofluid Porous medium Magnetic field Spectral method

a b s t r a c t

The effect of a uniform external magnetic field on the onset of convection in an electrically conducting nanofluid layer is examined numerically based on non-homogeneous two-phase model (i.e., classical Buongiorno’s mathematical model) which incorporates the effects of Brownian motion and thermophore- sis of nanoparticles in the thermal transport mechanism of nanofluids. In this investigation, we consider that the nanofluid is Newtonian, heated from below and confined horizontally in a Darcy-Brinkman por- ous medium between two infinite rigid boundaries, with different nanoparticle configurations at the hor- izontal boundaries (i.e., top heavy and bottom heavy nanoparticle distributions). The linear stability theory has been wisely used to obtain a set of linear differential equations which are transformed to an eigenvalue problem, so that the thermal Rayleigh number R

a

is the corresponding eigenvalue. The thermal Rayleigh number R

a

and its corresponding wave number a are found numerically using the Chebyshev-Gauss-Lobatto collocation method for each set of fixed nanofluid parameters. The marginal instability threshold (R

ac

, a

c

) characterizing the onset of stationary convection is computed accurately for wide ranges of the modified magnetic Chandrasekhar number Q, the modified specific heat increment N

B

, the nanoparticle Rayleigh number R

N

, the modified Lewis number L

e

, the modified diffusivity ratio N

A

and the Darcy number D

a

. Based on these control parameters and the notions of streamlines, isotherms and iso-nanoconcentrations, the stability characteristics of the system and the development of complex dynamics at the critical state are discussed in detail for both nanoparticle distributions.

Ó

2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction

With the rising demands to use specific fluids enhanced ther- mally, several theoretical and experimental studies have been recently carried out in order to develop homogeneous mixtures (solid–liquid) that are more effective in terms of performance and heat storage capacity, among these interesting works, we find those examined by S.A. Angayarkanni and J. Philip [1], they have been proposed to disperse small amounts of ultrafine solid parti- cles (e.g. stable metals, metal oxides, carbides, nitrides) in a base fluid (e.g. water, oils, ethylene glycol) for obtaining a non- homogeneous mixture from a microscopic point of view .The resulting mixture (nanoparticles + base fluid) is commonly known as nanofluid or nanoliquid.

The first test with nanofluids observed by S.U.S. Choi [2] and S.K.

Das et al.[3] gave more encouraging features about the abnormal

enhancement of thermal conductivity than they were thought to possess, they have been considered the nanofluid as a new advanced heat transfer fluid in the engineering applications.

J. Buongiorno et al. [4] showed experimentally that the thermal conductivity enhancement afforded by the tested nanofluids increased with increasing particle loading, particle aspect ratio and decreasing base fluid thermal conductivity.Keeping in mind the remarkable thermal properties of nanofluids, it is currently preferred to use these kinds of mixtures in many industrial and technological sectors, such as engineering, heat exchangers, nuclear reactors, electronics cooling technology, waste heat recov- ery, thermal insulations, geothermal power extraction and many others. In their analysis, K.V. Wong and O. De Leon [5] have noted that the nanofluids could be used as working fluids to extract energy from the earth’s core and aid in the process of creating energy within a power plant system producing large amounts of work energy.

Due to several advantages and widespread applications of mag- netic nanoparticles in biotechnology and medical areas, much

http://dx.doi.org/10.1016/j.rinp.2017.06.003

2211-3797/Ó2017 The Authors. Published by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑

Corresponding author.

E-mail address:wakif.abderrahim@gmail.com(A. Wakif).

Contents lists available at ScienceDirect

Results in Physics

j o u r n a l h o m e p a g e : w w w . j o u r n a l s . e l s e v i e r . c o m/ r e s u l t s - i n - p h y s i c s

attention has been focused to the synthesis of different kinds of biomagnetic nanoparticles that are able to overcome the existing problems in the process of diagnosing and treating cancer.

Recently, S. Aman et al. [6] and A.R. Nochehdehi et al. [7] have reported in their works that we can use the biomagnetic nanopar- ticles such as Gold (Au) or Iron oxide (Fe

3

O

4

) in the biomedical domain as delivery vehicles for drugs or radiation in cancer patients. W. Yu and H. Xie [8] have confirmed that we can use Zinc oxide nanoparticles (ZnO) as inorganic antibacterial materials, these kinds of nanoparticles exhibited impressive antibacterial properties against an important foodborne pathogen.

Due to the increased interest in the field of magnetohydrody- namics (MHD), several investigations have recently been per- formed and analyzed numerically or analytically using powerful methods in the purpose to control the heat transfer, thermal stabil- ity and flow characteristics under the effect of an external mag- netic field, and also to examine the simultaneous interactions among the thermal gradient and magnetic field applied to an elec- trically conducting fluid or nanofluid. In such contributions Muhammad Saqib et al. have studied the physics of slip effect at the boundary of a vertical plate in starting the flow of Casson fluid with the combined effect of radiative heat and mass transfer in the presence of first-order chemical reaction using the Laplace trans- form technique. Farhad Ali et al. [9–12] have applied the time frac- tional derivatives approach to study the MHD free convection flow of generalized Walter’s-B fluid over a static vertical plate, and the effects of magnetohydrodynamics on the blood flow in a horizontal cylinder. They have also used the Laplace transform technique to investigate the unsteady MHD flow of Brinkman type nanofluid over a vertical plate embedded in a porous medium with variable

surface velocity, temperature and concentration. Moreover, to carry out an exact analysis of the MHD free convection flow of a Walter’s-B fluid over an oscillating isothermal vertical plate embedded in a porous medium. Nadeem Ahmad Sheikh et al.

[13–15] have used the concept of Caputo-Fabrizio time fractional derivatives to MHD flow of a second-grade fluid together with radiative heat transfer. They have also performed a comparison between the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with chemical reac- tion in the presence or absence of heat generation source. A. Gul et al. [16] have investigated the heat transfer in MHD mixed con- vection flow of a ferrofluid along a vertical channel. B.M. Shankar et al. [17] have studied the stability of the conduction regime in an electrically conducting fluid saturated a porous vertical slab in the presence of a uniform external transverse magnetic field. D.

Yadav et al. [18–20] have showed that the presence of a uniform external magnetic field allows to delay the onset of convection in an electrically conducting nanofluid. T. Javed et al. [21] have trea- ted the natural convection in square cavity filled with ferrofluid saturated a porous medium in the presence of a uniform magnetic field. In summary, these researchers have demonstrate that the presence of an external magnetic field has generally a stabilizing effect, it allows to decrease both the heat transfer rate and flow velocity. From an energetic point of view, the MHD phenomenon has received considerable attention during the recent years due to its importance in MHD generators for power generation, MHD propulsion for propelling seagoing vessels, MHD pumps for pump- ing electrically conducting fluids and MHD flow meters.

The onset of convection in an electrically conducting nanofluid layer saturated a Darcy-Brinkman porous medium under the effect Nomenclature

Symbols

a

^⁄

Horizontal wave number (m

¹

) c Nanofluid specific heat (J kg

¹

K

¹

) D

B

Brownian diffusion coefficient (m

²

s

¹

) D

T

Thermophoresis diffusion coefficient (m

²

s

¹

) D

a

Darcy number

g Gravity acceleration (m s

²

)

H

^⁄

Induced magnetic field strength (A m

¹

) H

0

External magnetic field strength (A m

¹

) k

m

Effective thermal conductivity (W K

¹

m

¹

) K Permeability of the medium (m

²

)

L Layer depth (m) L

e

Modified Lewis number N

A

Modified diffusivity ratio N

B

Modified specific heat increment P

^⁄

Pressure (Pa)

P

r

Prandtl number

P

_rM

Magnetic Prandtl number

Q Modified magnetic Chandrasekhar number R

_a

Thermal Rayleigh number

R

M

Density Rayleigh number R

N

Nanoparticle Rayleigh number T

^⁄

Temperature (K)

t

^⁄

Time (s) V

a

Vadasz number

~ V

Darcy’s velocity

u

^⁄

, v

^⁄

, w

^⁄

Velocity components (m s

¹

) x

^⁄

, y

^⁄

, z

^⁄

Cartesian coordinates (m) Greek symbols

a

m

Effective thermal diffusivity (m

²

s

¹

)

b Thermal expansion coefficient (K

¹

)

e Porosity of the medium

g Effective magnetic diffusivity (m

²

s

¹

) k

^⁄

Growth rate of the disturbances (s

¹

)

l Dynamic viscosity (Pa s)

~

l Effective dynamic viscosity (Pa s)

l

e

Effective magnetic permeability (H m

¹

)

q Nanofluid density (kg m

³

)

q c Nanofluid heat capacity (J m

³

K

¹

)

r Effective electrical conductivity (S m

¹

) /

^⁄

Nanoparticle volume fraction

Superscripts

⁄ Dimensional variables

0

Perturbed quantities Subscripts

ac Critical number b Basic state

c Cold wall or upper boundary

f Base fluid

h Hot wall or lower boundary m Effective value

p Nanoparticle

Operators

r ~

Vector differential operator r ~

²

Laplacian operator

D First derivative with respect to z

of an imposed vertical magnetic field with different nanoparticle distributions at the horizontal boundaries (i.e., top heavy and bot- tom heavy nanoparticle distributions) has not been previously investigated accurately, despite its importance in many practical problems as noted above. J. Ahuja et al. [22] have studied the hydromagnetic stability of Cu-water and Ag-water nanofluids for the top heavy distribution of nanoparticles in a Darcy-Brinkman porous medium by the Galerkin weighted residuals method (GWRM) with a single order term, with this technique it is not pos- sible to get an exact analytical solution. A. Wakif et al. [23] have investigated the same problem by the power series method (PSM) with higher order terms to obtain a fourth-order accurate solution.

The present study is conducted through a linear stability analysis in a different way, and the resulting generalized eigen- value problem is solved numerically using a more accurate numerical method based on Chebyshev-Gauss-Lobatto tech- nique. The impact of the magnetic field and the influence of other physical parameters on the thermal stability of an electri- cally conducting nanofluid saturated a Darcy-Brinkman porous medium have been discussed in detail for both nanoparticle dis- tributions at the horizontal boundaries. At the onset of longitu- dinal convective rolls, the distributions of the velocity, the temperature and the nanoparticle volume fraction across a ver- tical section of the layer have been also analyzed via represen- tative streamlines, isotherms and iso-nanoconcentrations for different values of the modified magnetic Chandrasekhar num- ber and the Darcy number.

Mathematical formulation

We consider an infinite horizontal layer of incompressible nanofluid layer of depth L, saturated a Darcy-Brinkman porous medium, heated from below and subjected to both the gravita- tional field g ! ¼ ð 0 ; 0 ; g Þ and a uniform upward magnetic field

! H

0

¼ ð 0 ; 0 ; H

0

Þ , so that T

h

and T

c

(<T

h

) are the temperatures at the lower and upper boundaries, respectively, whereas /

h

and /

c

are their corresponding volume fractions of nanoparticles.

In this investigation, we take /

c

> /

h

for the top heavy distribu- tion of nanoparticles, and /

h

> /

c

for the bottom heavy distribu- tion of nanoparticles. The nanofluid used in this study is assumed to be dilute (i.e., the volumetric fraction of nanoparti- cles is only a few percent), electrically conducting, has a Newto- nian rheological behavior and containing fine spherical nanoparticles, whose the size of each one is less than those of the leaky pores of the medium. Further, the horizontal walls are chosen to be impermeable no-slip boundaries (i.e., rigid- rigid case), perfectly heat-conducting but electrically non- conductive. The schematic of the studied problem is well shown in Fig. 1.

The present study has been carried out using the Cartesian coor- dinate system (x

^⁄

, y

^⁄

, z

^⁄

), where x

^⁄

- axis and y

^⁄

- axis are horizontal and z

^⁄

- axis is directing vertically upwards .The thermophysical properties of nanofluid (viscosity, thermal conductivity, specific heat, magnetic permeability and electrical conductivity) are assumed to remain constant in the vicinity of the reference tem- perature T

_c

of the cold wall, except for the density in the momen- tum equation which is based on the Oberbeck-Boussinesq approximation. Furthermore, the mathematical equations describ- ing the physical model are based upon the following additional assumptions:

(i) The fluid phase and nanoparticles are in thermal equilibrium state.

(ii) The viscous dissipation is negligible.

(iii) The radiative heat transfer is negligible.

(iv) A laminar flow occurs into nanofluidic system at the onset of convection.

The asterisks are used to distinguish the dimensional variables from the non-dimensional variables (without asterisks).

Governing equations

According to the works cited in the references [24–29], the prin- cipal equations of conservation characterizing the magnetohydro- dynamic nanofluid flow in a Darcy-Brinkman porous medium are given as follows:

Conservation of mass;

! r

! V

¼ 0 ð1Þ

Conservation of momentum;

q

₀

e

o ot

þ 1

e ^{ð V}

!

r ^!

^Þ

! V

¼ r ^!

^P

^þ l ^~ _r ^!

²

! V

l

K ! V

þ q ! g þ l

_e

4 p ^{ð r}

!

! H

Þ ! H

ð2Þ Density equation;

q ¼ q

_p

/

þ q

₀

ð1 /

Þ½1 bðT

T

c

Þ ð3Þ where q

₀

is the nanofluid density at reference temperature T

c

and H !

ð H

_x

; H

_y

; H

_z

Þ is the induced magnetic field.

Conservation of energy;

ð q cÞ

m

o

ot

þ ð q cÞ

f

ð ! V

r ^!

Þ

T

¼ k

m

^! r

²

^T

þ e ð q cÞ

p

D

B

^! r

^/

^! r

^T

þ D

T

T

c

r ^!

^T

^! r

^T

ð4Þ

Conservation of nanoparticles;

o ot

þ 1

e ^{ð V}

!

r ^!

Þ

/

¼ D

B

^! r

²

^/

þ D

T

T

c

r ^!

²

^T

ð5Þ Modified Maxwell’s equations;

o ot

þ 1

e ^{ð V}

!

r ^!

Þ

! H

¼ 1

e ^ðH

!

^! r

Þ ! V

þ g _r ^!

²

! H

ð6Þ

! r

! H

¼ 0 ð7Þ

Here, g = 1/(4 pl

e

r ) is the effective magnetic diffusivity of the nanofluid, where, l

e

and r are the effective magnetic permeabil- ity and electrical conductivity of the nanofluid, respectively.

Electrically Conducting Nanofluid Layer

Darcy-Brinkman Porous Medium

Fig. 1.Geometrical configuration of the problem.

Taking into account that the temperature T

^⁄

and the volumetric fraction of nanoparticles /

^⁄

are constant on the horizontal rigid boundaries, the magnetohydrodynamic boundary conditions for electrically non-conducting walls can be written as follows:

w

¼

^ow_oz

¼ 0; T

¼ T

h

; /

¼ /

h

; ! H

¼ ! H

0

; at z

¼ 0 w

¼

^o_o^w_z

¼ 0; T

¼ T

c

; /

¼ /

_c

; ! H

¼ ! H

0

; at z

¼ L ð8Þ Considering the following non-dimensional variables:

ðx;y;zÞ ¼ ðx=L;

y

=L;

y

=LÞ;

t¼ ½ a

m=ðdL²Þt; !

V

¼ ðL=

a

mÞV!;

P¼ ½K=ð l a

mÞP;

T ¼ ðT

T

c

Þ=ðT

h

T

c

Þ; / ¼ ð/

/

h

Þ=ð/

c

/

h

Þ; ! H

¼ ð1=H

0

ÞH !

; The non-dimensional form of Eqs. (1)–(7) can be written as:

r ! ! V

¼ 0 ð9Þ

1 V

a

1 d o ot þ 1

e ^{ð V}

! r ^! Þ

! V

¼ ^! r ðPþR

M

zÞþD

a

r ^!

²

^{! V} ! V

þðR

a

TR

N

/Þ ! e

z

þ e ^P

^r

^Q

P

rM

ð r ^! ! H Þ ! H

ð10Þ o

ot þ ð ! V ^! r Þ

T ¼ r ^!

²

^T þ N

B

L

e

r ^{! /} r ^{! T} þ N

A

N

B

L

e

r ^{! T} r ^{! T} ð11Þ

e

d o ot þ ð ! V

: ^! r Þ

/ ¼ 1

L

e

r ^!

²

^/ þ N

A

L

e

^! r

²

^T ð12Þ

e

d o ot þ ð ! V

: ^! r Þ

! H

¼ ðH ! : r ^! Þ ! V

þ e P

r

P

rM

^! r

²

^{! H} ð13Þ r !

:H !

¼ 0 ð14Þ

where d ¼ ð q c Þ

m

=ð q c Þ

f

is the ratio of effective heat capacity of the nanofluid to heat capacity of the base fluid and a

m

¼ k

m

=ð q c Þ

f

is the effective thermal diffusivity of the nanofluid.

According to D.A. Nield and A.V Kuznetsov [26], Eq. (10) has been linearized under the Oberbeck-Boussinesq approximation by the neglect of a term proportional to the product of T and /.

The expressions of the principal non-dimensional parameters appearing in Eqs. (10)–(13) are given by:

V

a

¼ ð e L

²

=KÞP

r

; R

M

¼ ½ q

_p

/

_h

þ q

₀

ð1 /

_h

ÞðgKL= la

m

Þ;

R

a

¼ ½ q

₀

gKLb=ð la

m

ÞðT

h

T

c

Þ;

D

a

¼ð l ~ = l ÞðK=L

²

Þ;R

N

¼½gKL=ð la

m

Þð q

_p

q

₀

Þð/

c

/

h

Þ;

P

r

¼ l =ð q

₀

a

m

Þ;

P

rM

¼ l =ð q

₀

g Þ; Q ¼ ½ l

_e

K=ð4 pelg ÞH

²0

; N

B

¼ ð q cÞ

p

ð/

c

/

h

Þ=ð q cÞ

f

; L

e

¼ a

m

=ð e D

B

Þ;

N

A

¼ D

T

ðT

h

T

c

Þ=½D

B

T

c

ð/

c

/

_h

Þ:

In non-dimensional form, the boundary conditions (8) become:

w ¼ ow

oz ¼0; T ¼ 1; / ¼ 0; ! H ¼ ! e

z

at z ¼ 0 w ¼ ow

oz ¼0; T ¼ 0; / ¼ 1; ! H

¼ ! e

z

at z ¼ 1

ð15Þ

Basic state

The basic state of the system is quiescent and is described by:

! V

b

¼ ! 0

; P

b

¼ P

b

ðzÞ; T

b

¼ T

b

ðzÞ; /

_b

¼ /

_b

ðzÞ; ! H

b

¼ H

b

! e

z

: ð16Þ

Substituting the expressions (16) into Eqs. (9)–(14), we can obtain the following basic equations:

! r

ðP

b

þ R

M

zÞ ¼ ðR

a

T

b

R

N

/

_b

Þ ! e

z

þ e ^P

^r

^Q

P

rM

ð r ^! ! H

b

Þ ! H

b

ð17Þ

d

²

T

b

dz

²

þ N

B

L

e

d/

b

dz dT

b

dz

þ N

A

N

B

L

e

dT

b

dz

2

¼ 0 ð18Þ

d

²

/

b

dz

²

þ N

A

d

²

T

b

dz

²

¼ 0 ð19Þ

d

²

H

b

dz

²

¼ 0 ð20Þ

dH

b

dz ¼ 0 ð21Þ

The analytic expressions of T

b

, /

_b

and H

b

can be found by con- sidering the following basic boundary conditions:

T

b

¼ 1; /

b

¼ 0; H

b

¼ 1 at z ¼ 0

T

b

¼ 0; /

b

¼ 1; H

b

¼ 1 at z ¼ 1 ð22Þ Under the boundary conditions (22), the integration of Eqs. (18)–

(21) gives:

d

²

T

b

dz

²

ðN

A

1ÞN

B

L

e

dT

b

dz ¼ 0 ð23Þ

/

_b

¼ N

A

T

b

þ ð1 N

A

Þz þ N

A

ð24Þ

H

b

¼ 1 ð25Þ

According to J. Buongiorno [25], D.A. Nield and A.V. Kuznetsov [26], and D.Yadav et al. [28] we have for most of the nanofluids j N

A

j 1 10, L

e

10

²

10

³

, j N

B

j 10

⁴

10

²

, and consequently the ratio r ¼ ð N

B

= L

e

Þð N

A

1 Þ is very small, it is of the order of 10

⁷

10

³

. Hence, if we neglect the term r in Eq. (23), we will obtain the following solutions:

T

b

¼ 1 z ð26Þ

/

b

¼ z ð27Þ

Perturbed state

In order to investigate the stability of the system, we superim- pose infinitesimal disturbances on the basic solutions (25)–(27), so that:

! V

¼ ! V

b

þ ! V

₀

; ! H

¼ ! H

b

þ ! H

₀

; T ¼ T

b

þT

⁰

; P ¼ P

b

þ P

⁰

; / ¼ /

b

þ /

⁰

: ð28Þ where V !

₀

, H !

₀

, T

⁰

, P

⁰

and U

⁰

are the perturbed quantities over their equilibrium counterparts, they are assumed to be small, where

! V

b

¼ ! 0 and H !

b

¼ ! e

z

.

Introducing the expressions (28) into Eqs. (9)–(14), using the expressions (25)–(27) of the basic solutions and linearizing the resulting equations, we obtain the following linear perturbation equations:

! r : ! V

₀

¼ 0 ð29Þ

1 V

a

d

@ ! V

₀

@t ¼ ^! r ^P

⁰

þ D

a

^! r

²

^{! V}

⁰

! V

₀

þ ðR

a

T

⁰

R

N

/

⁰

Þ ! e

z

þ e ^P

r

Q P

rM

ð r ^! ! H

₀

Þ ! e

z

ð30Þ

@T

⁰

@t w

⁰

¼ r ^!

²

^T

⁰

þ ð1 2N

A

ÞN

B

L

e

@T

⁰

@z N

B

L

e

@/

⁰

@z ð31Þ

e

d

@/

⁰

@t þ w

⁰

¼ 1

L

e

^! r

²

^/

⁰

þ N

A

L

e

^! r

²

^T

⁰

ð32Þ

e

d

@H !

₀

@t ¼ @ V

^!0

@z þ e P

r

P

rM

^! r

²

^{! H}

⁰

ð33Þ

r ! : ! H

₀

¼ 0 ð34Þ

To eliminate the pressure term P

⁰

from the momentum Eq. (30) we apply the operator curl twice to this equation. Using the iden- tity ^! r ð r ^! ! X

Þ ¼ r ^! ð ^! r : ! X

Þ r ^!

²

^{! X} together with Eqs. (29) and (34), the z-component of both the resulting momentum equation and Maxwell’s equation can be obtained as:

1 V

a

d

@

@t ð r ^!

²

^w

⁰

Þ ¼ ðD

a

r ^!

²

1Þ r ^!

²

^w

⁰

þ R

a

r ^!

²2

T

⁰

R

N

^! r

²2

/

⁰

þ e ^P

^r

^Q

P

rM

@

@z ð ^! r

²

^H

⁰z

Þ ð35Þ

e

d

@H

⁰z

ot ¼ ow

⁰

oz þ e P

r

P

rM

r ^!

²

^H

⁰z

ð36Þ

where r ^!

²2

X ¼ o

²

X =o x

²

þ o

²

X =o y

²

is the horizontal Laplacian of a scalar function X ð x ; y ; z Þ .

Taking into account of Eq. (36), we can get after simplification of Eq. (35):

1 V

a

d

o

ot ð r ^!

²

^w

⁰

Þ ¼ ðD

a

r ^!

²

1Þ r ^!

²

^w

⁰

þ R

a

r ^!

²2

T

⁰

R

N

^! r

²2

/

⁰

þ Q o

o z

e

d oH

⁰z

ot o w

⁰

oz

ð37Þ In non-dimensional form, the resulting boundary conditions corresponding to the perturbation equations obtained in the case where the horizontal walls are electrically non-conducting are given as follows:

w

⁰

¼ ow

⁰

oz ¼ T

⁰

¼ /

⁰

¼ oH

⁰z

oz ¼ 0 at z ¼ 0; 1 ð38Þ

Linear stability analysis

For simplicity, we restrict our analysis to two dimensional rolls, so that all physical quantities are independent of y. Using the def- inition of stream function w

⁰

ð i : e :; u

⁰

¼ o w

⁰

=o z ; w

⁰

¼ o w

⁰

=o x Þ , the governing Eqs. (31), (32), (36) and (37) become:

1 V

a

d

o

²

otox ð r ^!

²

^w

⁰

Þ ¼ ðD

a

r ^!

²

1Þ ^! r

²

^ow _ox

⁰

þ R

a

r ^!

²2

T

⁰

R

N

^! r

²2

/

⁰

þ Q o oz

e

d oH

⁰z

ot o

²

w

⁰

ozox

!

ð39Þ

oT

⁰

ot ow

⁰

ox ¼ r ^!

²

^T

⁰

þ ð1 2N

A

ÞN

B

L

e

oT

⁰

oz N

B

L

e

o/

⁰

oz ð40Þ

e

d o/

⁰

ot þ ow

⁰

ox ¼ 1

L

e

r ^!

²

^/

⁰

þ N

A

L

e

^! r

²

^T

⁰

ð41Þ

e

d oH

⁰z

ot ¼ o

²

w

⁰

ozox þ e ^P

^r

P

rM

r ^!

²

^H

⁰z

ð42Þ

Here, ^! r

²2

X ¼ o

²

X =o x

²

and r ^!

²

^X ¼ o

²

X =o x

²

þ o

²

X =o z

²

.

The resulting Eqs. (39)–(42) are simplified in the usual manner by decomposing the solution in terms of normal modes. According to A.N. Borujerdi et al. [30] and A.Wakif et al. [31], we can take the perturbation quantities in the form:

w

⁰

T

⁰

/

⁰

H

⁰_z

2 66 64

3 77 75 ¼

wðzÞcosðaxÞ TðzÞsinðaxÞ u ðzÞsinðaxÞ

s ðzÞsinðaxÞ 2

66 64

3

77 75 expðktÞ ð43Þ

Substituting the expressions (43) into Eqs. (39)–(42), we get:

k

V

a

d aðD

²

a

²

Þ W _{¼ a½D}

_a

_ðD

²

a

²

Þ 1ðD

²

a

²

Þw R

a

a

²

T þ R

N

a

²

u þ QD e k

d s þ aD W

ð44Þ kT þ a W _{¼ ðD}

²

a

²

ÞT þ ð1 2N

A

ÞN

B

L

e

DT N

B

L

e

D u _ð45Þ

e k

d u a W ¼ 1

L

e

ðD

²

a

²

Þ u þ N

A

L

e

ðD

²

a

²

ÞT ð46Þ

e k

d s ¼ aD W _þ e P

r

P

rM

ðD

²

a

²

Þ s ð47Þ

where DX ¼ o X =o z, a is the dimensionless wave number along the x direction and k is the dimensionless growth rate of the disturbances.

The boundary conditions of the problem in view of normal mode analysis are:

w ¼ D W ¼ T ¼ u ¼ D s ¼ 0 at z ¼ 0; 1 ð48Þ For the stress-free boundary conditions, J. Ahuja et al. [22] and U. Gupta et al. [32] have shown that the oscillatory mode of heat transfer in porous or non-porous medium is possible for the bottom heavy distribution of nanoparticles (i.e., U

c

< U

h

) and is not possible for the top heavy distribution of nanoparticles (i.e., U

c

> U

h

). In this investigation, we will assume that the princi- ple of exchange of stabilities is valid and the oscillatory convection occurs only for the bottom heavy case. To study the stationary mode, we must take the dimensionless growth rate k of each per- turbation equal to zero (i.e., k ¼ 0) in all subsequent analyses.

Hence, we can reduce our problem to the following equations:

½D

a

D

⁴

þ ð2a

²

D

a

þ Q þ 1ÞD

²

a

²

ð1 þ D

a

a

²

Þw þ R

N

a u _¼ R

a

aT ð49Þ a W D

²

þ ð1 2N

A

ÞN

B

L

e

D a

²

T þ N

B

L

e

D u ¼ 0 ð50Þ

a W _þ ^N

^A

L

e

ðD

²

a

²

ÞT þ 1

L

e

ðD

²

a

²

Þ u _¼ 0 ð51Þ The above system of simplified equations must be solved sub- ject to the following boundary conditions:

w ¼ D W _{¼ T ¼} u _¼ 0 at z ¼ 0; 1 ð52Þ

Method of solution Numerical method

The stationary Eqs. (49)–(51) together with the boundary

conditions (52) constitute a linear eigenvalue problem with the

thermal Rayleigh number R

a

as the eigenvalue. The resulting sys-

tem is solved numerically using the spectral method [33–37] based

on Chebyshev-Gauss-Lobatto technique, for this purpose we must consider another spatial variable f defined by:

f ¼ 2z 1; D

ⁿ

¼ 2

ⁿ

D

ⁿ

: ð53Þ

Here, f 2 ½ 1 1 and D X ¼ d X = df, where n is a positive integer number which represents the derivative order.

Using the transformations X ð f Þ ¼ X ð z ð f ÞÞ for the variables w ð z Þ , Tð z Þ and u ð z Þ in Eqs. (49)–(51), we can obtain the following mod- ified equations:

½16D

a

D

⁴

þ 4ð2a

²

D

a

þ Q þ 1Þ D

²

a

²

ð1 þ D

a

a

²

Þ w þ R

N

a u _¼ R

a

a T ð54Þ a w 4 D

²

þ 2ð1 2N

A

ÞN

B

L

e

D a

²

T þ 2N

B

L

e

D u ¼ 0 ð55Þ

a w þ N

A

L

e

ð4 D

²

a

²

Þ T þ 1

L

e

ð4 D

²

a

²

Þ u _¼ 0 ð56Þ We point out that, the resulting eigenvalue problem con- structed from Eqs. (54)–(56) is given by:

A

w

Z A

u

B

w

B

T

B

u

C

w

C

T

C

_u

0

B @

1 C A

w T u 0 B @

1 C A ¼ R

a

Z E

T

Z Z Z Z Z Z Z 0

B @

1 C A

w T u 0 B @

1

C A ð57Þ

Here:

A

w

¼ 16D

a

D

⁴

þ 4ð2a

²

D

a

þ Q þ 1Þ D

²

a

²

ð1 þ D

a

a

²

Þ; Z ¼ 0;

A

u

¼ R

N

a; B

w

¼ a;

B

T

¼ 4 D

²

2ð1 2N

A

ÞðN

B

=L

e

Þ D þa

²

; B

u

¼ 2ðN

B

=L

e

Þ D; C

w

¼ a;

C

T

¼ ðN

A

=L

e

Þð4 D

²

a

²

Þ; C

u

¼ ð1=L

e

Þð4 D

²

a

²

Þ; E

T

¼ a:

The Dirichlet-Neumann boundary conditions associated to the sys- tem (57) are given by:

w ¼ D w ¼ T ¼ u ¼ 0 at f ¼ 1; 1 ð58Þ The eigenvalue problem (57) obtained in this investigation is discretized spatially using the Chebyshev spectral collocation method based on N collocation points of Gauss-Labatto, which are defined by:

f

_i

¼ cos p ^{i 1}

N 1

; 1 6 i 6 N: ð59Þ

According to C.Canuto et al. [34], the first order Chebyshev dif- ferentiation matrix D ~ ¼ ð d

_ij

Þ

16i;j6N

is given by:

d

ij

¼

2ðN1Þ²þ1

6

for i ¼ j ¼ 1

fi

2ð1f²iÞ

for i ¼ j–1; N

cið1Þ^iþj

cjðfifjÞ

for i–j

2ðN1Þ²1

6

for i ¼ j ¼ N 8 >

> >

> >

> >

<

> >

> >

> >

> :

ð60Þ

Here, the coefficients c

i

are given as follows:

c

i

¼ 2 for i ¼ 1; N 1 for i–1; N

ð61Þ The other derivatives are obtained from the first derivative matrix (60) as follows:

D ~

^ðmÞ

¼ ð DÞ ~

^m

¼ ð d ~

^m_ij

Þ

_16i;j6N

ð62Þ

Here, the m order derivative D ~

^ðmÞ

is equal to the product of the matrix D m times. ~

Considering the boundary conditions (58) and the expressions (59)–(62), the eigenvalue problem (57) in its discretized form can be written as follows:

A

^ij_w

Z

^ij

A

^ij_u

B

^ij

w

B

^ij_T

B

^ij_u

C

^ij_w

C

^ij_T

C

^ij_u

0

B B B @

1 C C C A

w

_j

T

j

u

j

0 B @

1 C A ¼ R

a

Z

^ij

E

^ij_T

Z

^ij

Z

^ij

Z

^ij

Z

^ij

Z

^ij

Z

^ij

Z

^ij

0

B @

1 C A

w

_j

T

j

u

j

0 B @

1

C A; 26 i;j 6 N 1:

ð 63 Þ The matrix elements appearing in the system (63) are given by:

A

^ij_w

¼ 16D

a

D

⁴_ij

þ 4 ð 2a

²

D

a

þ Q þ 1 Þ D

²_ij

a

²

ð 1 þ D

a

a

²

Þ d

ij

; Z

^ij

¼ 0 ; A

^ij_u

¼ R

N

ad

ij

; B

^ij_w

¼ ad

ij

; B

^ij_T

¼ 4 d ~

²_ij

2 ð 1 2N

A

Þð N

B

= L

e

Þ d

ij

þ a

²

d

ij

; B

^ij_u

¼ 2 ð N

B

= L

e

Þ d

ij

; C

^ij_w

¼ ad

ij

; C

^ij_T

¼ ðN

A

=L

e

Þ 4 ~ d

²_ij

a

²

d

ij

; C

^ij_u

¼ ð1=L

e

Þ 4 d ~

²_ij

a

²

d

ij

; E

^ij_T

¼ ad

ij

ð64Þ where d

ij

is the Kronecker delta symbol.

The modified matrix elements D

²_ij

and D

⁴_ij

used above in the expression of the elements A

^ij_w

are given by L.N. Trefethen [33] as follows:

D

²_ij

¼ 1

ð1 f

²_i

Þ ð1 f

²_i

Þ ~ d

²_ij

4f

_i

d

ij

2d

ij

h i

ð65Þ

D

⁴_ij

¼ 1

ð1 f

²_i

Þ h ð1 f

²_i

Þ ~ d

⁴_ij

8f

i

~ d

³_ij

12 ~ d

²_ij

i

ð66Þ The system of algebraic equations (63) is a generalized linear eigenvalue problem for the thermal Rayleigh number R

a

, whose the characteristic matrices are real of order ð 3N 6 Þ ð 3N 6 Þ . The real eigenvalue R

a

is determined when the other control parameters are specified. For this purpose, we vary the wave num- ber a, and then determine for each value given to this number a set of ð 3N 6 Þ eigenvalues, so that among these eigenvalues, we retain only the positive real ones. Furthermore, the corresponding thermal Rayleigh number R

_a

sought for each wave number a is the smallest positive one. Hence, for a range of very close values to each other for the wave number a we find a set of values for the thermal Rayleigh number R

_a

which allows to find the marginal instability threshold ð R

ac

; a

c

Þ characterizing the onset of convec- tion for the stationary mode.

Validation and convergence of the method

In order to check on the exactness of the results obtained in this

investigation by the Chebyshev-Gauss-Lobatto spectral method

(CGLSM), we will authenticate it by performing a simple compar-

ison between this procedure and other powerful methods for two

limiting cases. The first test computations are carried out for a

non-porous medium filled with an electrically conducting fluid

which is subjected to a uniform vertical magnetic field. The results

obtained with the spectral method (CGLSM) are displayed in

Table 1 and compared with those of the analytical method used

by S. Chandrasekhar [24] .The second tests are reserved to a non-

porous medium filled with an electrically conducting nanofluid

(e.g., alumina-water nanofluid) which is subjected to a uniform

vertical magnetic field. The results obtained in the case of alu-

mina-water nanofluids (i.e., N

B

¼ 0 : 00075, R

N

¼ 0 : 122 and

N

A

¼ 5) are clearly shown in Table 2 for different values of the

magnetic Chandrasekhar number Q and the Lewis number L

e

,

these results are compared with those of the Galerkin weighted

residuals technique (GWRT) used by D.Yadav et al. [18] .The

spectral method (CGLSM) is examined another time with the same nanofluid (i.e., N

B

¼ 0 : 00075, R

N

¼ 0 : 1, L

e

¼ 5000 and N

A

¼ 5) for different values of the magnetic Chandrasekhar number Q . The results obtained are presented in Table 3 and compared with those of the Galerkin weighted residuals technique (GWRT) used by D.

Yadav et al. [18], and also with those of the power series method (PSM) used by A.Wakif et al. [23].

The results shown in Tables 1–3 prove that there is a very good agreement between our results and those of S.Chandrasekhar [24], D.Yadav et al. [18] and Wakif et al. [23]. Hence, the accuracy of the present method is verified. We note that to obtain a fourth-order accurate approximation for the computational results obtained in this investigation, it is enough to choose a number of collocation points N equal to 30 (i.e., N ¼ 30).

Results and discussion

The effect of a uniform external magnetic field on the criterion for the onset of thermal convection in an electrically conducting nanofluid saturated a Darcy-Brinkman porous medium is

investigated for different nanoparticle distributions (i.e., top heavy and bottom heavy cases). The nanofluid layer is confined between two isothermal rigid surfaces with a volume fraction of nanoparticles imposed on the horizontal boundaries. The stability equations established in this investigation reveal that the thermal stability of such nanofluid depends on the modified magnetic Chandrasekhar number Q , the modified specific heat increment N

B

, the nanoparticle Rayleigh number R

N

, the modified Lewis number L

e

, the modified diffusivity ratio N

A

and the Darcy number D

_a

.

To study numerically the effect of a control parameter Q , N

B

, R

N

, L

e

, N

A

or D

a

on the onset of convection, we must fix the others and represent the variations of the critical stability parameters R

ac

and a

c

as a function of the modified magnetic Chandrasekhar number Q for different values of this parameter, these variations have been plotted graphically as shown in Figs. 2–6.

To give the credibility to this investigation, the different numer- ical evaluations examined for an electrically conducting nanofluid are performed and validated respecting the ranges of all nanofluid parameters which are reported previously by J. Buongiorno [25], D.

A. Nield and A.V. Kuznetsov [26], and D.Yadav et al. [28]. For this purpose, we have taken the values of the modified specific heat increment j N

B

j in the order of 10

⁴

10

²

, the modified Lewis number L

e

is of the order of 10

²

10

³

, the modified diffusivity ratio j N

A

j is taken in the order of 1 10. The value of the nanoparticle Rayleigh number j R

N

j is not more than 10 and the Darcy number D

a

is selected in such a way that it is less than 1. Furthermore, we note that the sign of all nanofluid parameters is positive for the top heavy case ð i : e :; /

_c

> /

_h

Þ and negative for the bottom heavy case ð i : e :; /

_h

> /

_c

Þ , except for the modified Lewis number L

e

, the modified magnetic Chandrasekhar number Q and the Darcy num- ber D

a

.

Fig. 2(a) illustrates the influence of the modified specific heat increment N

B

on the thermal stability of an electrically conducting nanofluid characterized by j R

N

j ¼ 0 : 1, L

e

¼ 1000, j N

A

j ¼ 1 and D

a

¼ 0 : 8 in both nanoparticle distributions. Whatever the value taken for the modified magnetic Chandrasekhar number Q , it is observed from Fig. 2(a) and its corresponding Table 4 that the modified specific heat increment N

B

has no effect on the onset of convection. Hence, the contribution of Brownian motion and ther- mophoresis of nanoparticles in the non-dimensional thermal energy equation can be neglected. Rather, the effects of Brownian motion and thermophoresis directly enter in the non- dimensional equation expressing the conservation of nanoparti- cles, this result may be explained by the weak value of the terms ð N

B

= L

e

Þ and ð 1 2N

A

Þð N

B

= L

e

Þ which appear in the non- dimensional thermal energy equation.

For both nanoparticle distributions,the variation of the critical thermal Rayleigh number R

ac

with the modified magnetic Chan- drasekhar number Q and the nanoparticle Rayleigh number R

N

for an electrically conducting nanofluid characterized by j N

B

j ¼ 0 : 01, L

e

¼ 1000, j N

A

j ¼ 1 and D

a

¼ 0 : 8 is shown in Fig. 3(a).

Whatever the value taken for the modified magnetic Chan- drasekhar number Q , it is found from Fig. 3(a) that an increase in the absolute value of the nanoparticle Rayleigh number j R

N

j allows

Table 1

Comparison of the present results with the Chandrasekhar’s results in the case of an electrically conducting fluid, for different values of the magnetic Chandrasekhar number Q .

S. Chandrasekhar[24]

(Analytical method)

Present study (CGLSM)

Q Rac ac Rac ac

0 1707.8 3.13 1707.7617 3.1163

10 1945.9 3.25 1945.7456 3.2652

50 2802.1 3.68 2802.0058 3.6792

100 3757.4 4.00 3757.2301 4.0120

200 5488.6 4.45 5488.5332 4.4458

500 10110 5.16 10109.7658 5.1647

1000 17103 5.80 17102.8378 5.8139

Table 2

Comparison between the present results and Yadav’s results for alumina–water nanofluids for different values of the parameters Q and Le, in the case where NB¼0:00075, RN¼0:122 and NA¼5.

D. Yadav et al.[18]

(GWRT)

Present study (CGLSM)

Q Le Rac ac Rac ac

0 2000 1463.151 3.136 1463.1517 3.1163

4000 1219.152 3.136 1219.1517 3.1163

6000 975.151 3.136 975.1517 3.1163

8000 731.152 3.136 731.1517 3.1163

100 2000 3512.621 4.012 3512.6201 4.0120

4000 3268.621 4.012 3268.6201 4.0120

6000 3024.620 4.012 3024.6201 4.0120

8000 2780.622 4.012 2780.6201 4.0120

200 2000 5243.933 4.446 5243.9232 4.4458

4000 4999.933 4.446 4999.9232 4.4458

6000 4755.934 4.446 4755.9232 4.4458

8000 4511.937 4.446 4511.9232 4.4458

Table 3

Comparison of the present results with Yadav’s and Wakif’s results for alumina-water nanofluid, in the case where NB¼0:00075, RN¼0:1, Le¼5000 and NA¼5, for different values of the parameter Q.

D. Yadav et al.[18](GWRT) A. Wakif et al.[23](PSM) Present study (CGLSM)

Q Rac ac Rac ac Rac ac

0 1207.262 3.136 1207.2617 3.1163 1207.2617 3.1163

100 3256.731 4.012 3256.7301 4.0120 3256.7301 4.0120

200 4988.042 4.446 4988.0332 4.4458 4988.0332 4.4458

Fig. 2.The effects of parameters Q and NBon the critical stability parametersðRac;acÞof an electrically conducting nanofluid for both nanoparticle distributions, in the case wherejRNj ¼0:1, Le¼1000,jNAj ¼1 and Da¼0:8.

Fig. 3.The effects of parameters Q and RNon the critical stability parametersðRac;acÞof an electrically conducting nanofluid for both nanoparticle distributions, in the case wherejNBj ¼0:01, Le¼1000,jNAj ¼1 and Da¼0:8.

Fig. 4.The effects of parameters Q and Leon the critical stability parametersðRac;acÞof an electrically conducting nanofluid for both nanoparticle distributions, in the case wherejNBj ¼0:01,jRNj ¼0:1,jNAj ¼1 and Da¼0:8.

Fig. 5.The effects of parameters Q and NAon the critical stability parametersðRac;acÞof an electrically conducting nanofluid for both nanoparticle distributions, in the case wherejNBj ¼0:01,jRNj ¼0:1, Le¼1000 and Da¼0:8.

Fig. 6.The effects of parameters Q and Daon the critical stability parametersðRac;acÞof an electrically conducting nanofluid for both nanoparticle distributions, in the case wherejNBj ¼0:01,jRNj ¼0:1, Le¼1000 andjNAj ¼1.

to decrease the critical thermal Rayleigh number R

_ac

in the top heavy case, and increase it in the bottom heavy case. The observa- tions noted for the nanoparticle Rayleigh number are mainly due to the fact that, as we increase the absolute value of the nanoparticle Rayleigh number j R

N

j , the volumetric fraction of nanoparticles increases, and thus the Brownian motion and thermophoresis of nanoparticles also increase. Hence, these changes can cause a

destabilizing effect on the system for the top heavy case, and a stabilizing effect for the other case.

Fig. 4(a) exhibits the influence of the modified Lewis number L

e

on the stability of a an electrically conducting nanofluid character- ized by j N

B

j ¼ 0 : 01, j R

N

j ¼ 0 : 1, j N

A

j ¼ 1 and D

a

¼ 0 : 8 for various values of the modified magnetic Chandrasekhar number Q in both nanoparticle distributions. Whatever the value taken for the mod-

Table 4

The critical stability parametersðRac;acÞof an electrically conducting nanofluid characterized byjRNj ¼0:1, Le¼1000,jNAj ¼1 and Da¼0:8, for various values of the parameters Q and NB.

Q¼10 Q¼50 Q¼100

Case NB Rac ac Rac ac Rac ac

Top heavy

0 1546.4252 3.2989 2383.4989 3.7686 3310.6141 4.1315

0.0001 1546.4252 3.2989 2383.4989 3.7686 3310.6141 4.1315

0.001 1546.4252 3.2989 2383.4989 3.7686 3310.6141 4.1315

0.01 1546.4253 3.2989 2383.4991 3.7686 3310.6143 4.1315

Bottom heavy

0 1746.4252 3.2989 2583.4989 3.7686 3510.6141 4.1315

0.0001 1746.4252 3.2989 2583.4989 3.7686 3510.6141 4.1315

0.001 1746.4252 3.2989 2583.4989 3.7686 3510.6141 4.1315

0.01 1746.4253 3.2989 2583.4991 3.7686 3510.6143 4.1315

Table 5

The critical stability parametersðRac;acÞof an electrically conducting nanofluid characterized byjNBj ¼0:01,jRNj ¼0:1, Le¼1000 and Da¼0:8, for various values of the parameters Q and NA.

Q¼10 Q¼50 Q¼100

Case NA Rac ac Rac ac Rac ac

Top heavy

1 1546.4253 3.2989 2383.4991 3.7686 3310.6143 4.1315

5 1546.0253 3.2989 2383.0991 3.7686 3310.2143 4.1315

8 1545.7253 3.2989 2382.7991 3.7686 3309.9143 4.1315

10 1545.5253 3.2989 2382.5991 3.7686 3309.7143 4.1315

Bottom heavy

1 1746.4253 3.2989 2583.4991 3.7686 3510.6143 4.1315

5 1746.0253 3.2989 2583.0991 3.7686 3510.2143 4.1315

8 1745.7253 3.2989 2582.7991 3.7686 3509.9143 4.1315

10 1745.5253 3.2989 2582.5991 3.7686 3509.7143 4.1315

Table 6

The critical stability parametersðRac;acÞof an electrically conducting nanofluid and fluid for various values of the parameters Q , RN, Le, NAand Da, in the case where NB¼0.

Nanofluids case Fluids case

Case Q RN Le NA Da R_ac;n a_c;n R_ac;f a_c;f

Top heavy

0 0.1 1000 1 0.8 1310.5531 3.1212 1410.6531 3.1212

10 0.1 1000 1 0.8 1546.4252 3.2989 1646.5252 3.2989

50 0.1 1000 1 0.8 2383.4989 3.7686 2483.5989 3.7686

100 0.1 1000 1 0.8 3310.6141 4.1315 3410.7141 4.1315

100 0.3 1000 1 0.8 3110.4141 4.1315 3410.7141 4.1315

100 0.5 1000 1 0.8 2910.2141 4.1315 3410.7141 4.1315

100 0.1 3000 1 0.8 3110.6141 4.1315 3410.7141 4.1315

100 0.1 5000 1 0.8 2910.6141 4.1315 3410.7141 4.1315

100 0.1 1000 5 0.8 3310.2141 4.1315 3410.7141 4.1315

100 0.1 1000 10 0.8 3309.7141 4.1315 3410.7141 4.1315

100 0.1 1000 1 0.2 1972.9391 5.1165 2073.0391 5.1165

100 0.1 1000 1 0.5 2690.4921 4.4285 2790.5921 4.4285

Bottom heavy

0 0.1 1000 1 0.8 1510.5531 3.1212 1410.6531 3.1212

10 0.1 1000 1 0.8 1746.4252 3.2989 1646.5252 3.2989

50 0.1 1000 1 0.8 2583.4989 3.7686 2483.5989 3.7686

100 0.1 1000 1 0.8 3510.6141 4.1315 3410.7141 4.1315

100 0.3 1000 1 0.8 3710.4141 4.1315 3410.7141 4.1315

100 0.5 1000 1 0.8 3910.2141 4.1315 3410.7141 4.1315

100 0.1 3000 1 0.8 3710.6141 4.1315 3410.7141 4.1315

100 0.1 5000 1 0.8 3910.6141 4.1315 3410.7141 4.1315

100 0.1 1000 5 0.8 3510.2141 4.1315 3410.7141 4.1315

100 0.1 1000 10 0.8 3509.7141 4.1315 3410.7141 4.1315

100 0.1 1000 1 0.2 2172.9391 5.1165 2073.0391 5.1165

100 0.1 1000 1 0.5 2890.4921 4.4285 2790.5921 4.4285

ified magnetic Chandrasekhar number Q , it is noticed from Fig. 4(a) that an increase in the modified Lewis number L

e

quicken the onset of convection in the top heavy case, and delayed it in the bottom heavy case.

Fig. 5(a) represents the influence of the modified diffusivity ratio N

A

on the stability of a an electrically conducting nanofluid charac- terized by j N

_B

j ¼ 0 : 01, j R

_N

j ¼ 0 : 1, L

_e

¼ 1000 and D

_a

¼ 0 : 8 for various values of the modified magnetic Chandrasekhar number Q in both nanoparticle distributions. Whatever the value taken for the modi- fied magnetic Chandrasekhar number Q , it is found graphically from Fig. 5(a) that the modified diffusivity ratio N

A

has no effect on the onset of convection in both nanoparticle distributions. A quantitative analysis of the influence of the modified diffusivity ratio N

A

presented below in Table 5 shows that an increase in the absolute value of the modified diffusivity ratio j N

A

j allows to desta- bilize somewhat the system in both nanoparticle distributions.

For both nanoparticle distributions,the variation of the critical thermal Rayleigh number R

ac

with the modified magnetic Chan- drasekhar number Q and the Darcy number D

a

of an electrically con- ducting nanofluid characterized by j N

_B

j ¼ 0 : 01, j R

_N

j ¼ 0 : 1, L

_e

¼ 1000

and j N

A

j ¼ 1 is presented in Fig. 6(a). Whatever the nanoparticle dis- tributions at the horizontal boundaries and the value taken for the modified magnetic Chandrasekhar number Q , it is seen from Fig. 6 (a) that the critical thermal Rayleigh number R

ac

increases as the Darcy number D

a

also increases, indicating that an increase in the Darcy number D

a

allows to delay the onset of convection.

Whatever the nanoparticle distributions at the horizontal boundaries and the values taken for the nanofluid parameters N

B

, R

N

, L

e

, N

A

and D

a

, it is found generally from Figs. 2(a)–6(a) that an increase in the modified magnetic Chandrasekhar number Q tends to increase the critical thermal Rayleigh number R

ac

, this result is due to the fact that, as we increase the modified magnetic Chandrasekhar number Q , the effect of the Lorentz forces also increases, and thus the effect of the buoyancy forces due to the presence of a vertical temperature gradient is reduced. Hence, an increase in the modified magnetic Chandrasekhar number Q allows to delay the onset of convection.

Taking into account the weak effect of the modified specific heat increment N

B

on the thermal stability of an electrically conducting nanofluid, we can seek numerically a general

Fig. 7.The predicted evolution of streamlines (a), isotherms (b) and iso-nanoconcentrations (c) of an electrically conducting nanofluid for various modified magnetic Chandrasekhar number Q and nanoparticle distributions, in the case wherejNBj ¼0:01,jRNj ¼0:1, Le¼1000,jNAj ¼1 and Da¼0:8.

expression of the critical thermal Rayleigh number R

ac

that explains the different results found previously in this investigation for both nanoparticle distributions. For this purpose, we carry out a set of numerical computations to find the critical stability param- eters ð R

ac

; a

c

Þ for an electrically conducting nanofluid for various values of the parameters Q , R

N

, L

e

, N

A

and D

a

. The results obtained for an electrically conducting nanofluid are presented in Table 6 for a zero value of the modified specific heat increment (i.e., N

B

¼ 0), and compared with the case of an electrically conducting fluid.

From the results displayed in Table 6, it is fairly clear that the critical thermal Rayleigh number R

_ac;n

of a nanofluid can be expressed as follows:

R

_ac;n

¼ R

_ac;f

R

N

ðL

e

þ N

A

Þ ð67Þ where R

_ac;f

is the critical thermal Rayleigh number corresponding to the base fluid.

In the expression (67), the critical thermal Rayleigh number R

_ac;f

of the base fluide depends only on the parameters Q and D

a

. The numerical results presented in Table 6 show that the critical wave number of the nanofluid a

_c;n

is equal to that of the base fluid a

_c;f

. The numerical and graphical results presented previously in this paper are in good agreement with the expression (67) established numerically. Furthermore, these results confirm that the nanoflu- ids are thermally more stable in the bottom heavy case than the

Fig. 8.The predicted evolution of streamlines (a), isotherms (b) and iso-nanoconcentrations (c) of an electrically conducting nanofluid for various Darcy numbers Daand nanoparticle distributions, in the case where Q¼100,jNBj ¼0:01,jRNj ¼0:1, Le¼1000 andjNAj ¼1.

other case. Hence, the addition of nanoparticles in a base fluid with a bottom heavy distribution can return this latter more stable com- pared with the other distribution, such that:

D R

_ac;n

¼ 2jR

N

jL

e

ð68Þ where D R

ac;n

is the critical difference between the bottom heavy case and the top heavy case for a nanofluid.

Figs. 2(b)–6(b) clarify the influence of the parameters Q , N

B

, R

N

, L

e

, N

A

and D

a

on the critical wave number a

c

. From these figures, it is found for both nanoparticle distributions that the critical wave number a

c

can increase only in the case where the modified mag- netic Chandrasekhar number Q is increased or the Darcy number D

a

is decreased .On the contrary, it is observed that the modified specific heat increment N

B

, the nanoparticle Rayleigh number R

N

, the modified Lewis number L

e

and the modified diffusivity ratio

N

A

have no significant effect on the critical wave number a

c

.What- ever the value taken for the parameters Q , N

B

, R

N

, L

e

, N

A

and D

a

, it is seen from Figs. 2(b)–6(b) and Tables 4–6 that there is an equality between the critical wave numbers of the top heavy case and the bottom heavy case.

For both nanoparticle distributions, the zero boundary condi- tions for w, D w, T and u at the lower and upper bounding surfaces provide a sufficient number of boundary conditions to find exactly the thermal Rayleigh number R

a

and its corresponding wave num- ber a, and then conclude the critical stability parameters character- izing the transition point ð R

ac

; a

c

Þ of the system. However, it is difficult to obtain some typical streamlines, isotherms and iso- nanoconcentrations at the onset of convection, because this prob- lem requires one more suitable condition to find the values of the amplitude functions w ð f Þ , Tð f Þ and u _ð f Þ in each collocation point f

_i

Fig. 9.Variations of the Darcy’s velocity V, the temperature T and the volumetric fraction of nanoparticlesUas a function of the spatial variableffor different values of the modified magnetic Chandrasekhar number Q and nanoparticle distributions, in the case wherejNBj ¼0:01,jRNj ¼0:1, Le¼1000,jNAj ¼1, Da¼0:8 and x¼

p

=ð2acÞ.

which allow to plot in Figs. 7 and 8 the streamlines, the isotherms and the iso-nanoconcentrations of an electrically conducting nano- fluid for various values of the modified magnetic Chandrasekhar number Q and the Darcy number D

a

. According to S. Savithiri et al. [38], we can suggest the following additional condition:

wðx;fÞ ¼ 10

³

at x ¼ L

c

2 and f ¼ 0 ð69Þ

where L

c

¼ 2 p ₌ a

c

is the size of the convection cells and w ð x ; f Þ is the dimensionless stream function.

Taking into account the condition (69), we can determine numerically the representative streamlines, isotherms and iso- nanoconcentrations at the onset of convection using the dimen- sionless expressions of the stream function w ð x ; f Þ , the temperature T ð x ; f Þ and the volumetric fraction of nanoparticles / ð x ; f Þ , which are given by:

wðx;fÞ ¼ wðfÞcosða

c

xÞ ð70Þ

Tðx;fÞ ¼ 1 2 1

2 f þ TðfÞsinða

c

xÞ ð71Þ

U _{ðx;fÞ ¼} ¹ 2 þ 1

2 f þ u _ðfÞsinða

c

xÞ ð72Þ From the expression (70), we can calculate the Darcy’s velocity V ð x ; f Þ as follows:

Vðx;fÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a

²_c

w

²

þ cos

²

ða

c

xÞ 4 o w

of

2

a

²_c

w

²

" #

v u

u t ð73Þ

For both nanoparticle configurations, we can use the expres- sions (71)–(73) to obtain the Darcy’s velocity profiles V ð x ; f Þ , the temperature distributions T ð x ; f Þ and the nanoparticle volume frac- tion distributions / ð x ; f Þ for different values of the modified mag- netic Chandrasekhar number Q and the Darcy number D

a

, these distributions are plotted in Figs. 9 and 10 along the spatial variable f in the section x ¼ p _=ð 2a

c

Þ .

Fig. 10.Variations of the Darcy’s velocity V, the temperature T and the volumetric fraction of nanoparticles/as a function of the spatial variableffor different values of the Darcy number Daand nanoparticle distributions, in the case where Q¼100,jNBj ¼0:01,jRNj ¼0:1, Le¼1000,jNAj ¼1 and x¼

p

=ð2acÞ.