J. Nig. Soc. Phys. Sci. 5 (2023) 1137

Journal of the
Nigerian Society

of Physical
Sciences

Electrohydrodynamics Convection in Dielectric Rotating
Oldroydian Nanofluid in Porous Medium

Pushap Lata Sharmaa, Mohini Kapaltaa, Ashok Kumara, Deepak Bainsa, Sumit Guptab, Pankaj
Thakurc,

aDepartment of Mathematics & Statistics, Himachal Pradesh University, Summer Hill, Shimla, India
bRajiv Gandhi Govt college Chaura Maidan, Shimla, India

cFaculty of Science and Technology, ICFAI University, Baddi, Solan, India

Abstract

An electrically conducting nanofluid saturated with a uniform porous medium has been tested to determine how rotation affects thermal convection.
Utilizing the Oldroydian model, which incorporates the specific effects of the electric field, Brownian motion, thermophoresis and rheological
factors for the distribution of nanoparticles that are top-heavy and bottom-heavy, one may use linear stability theory to ensure stability. Analysis
and graphical representation of the effects of the AC electric field Rayleigh number, Taylor number, Lewis number, modified diffusivity ratio,
concentration Rayleigh number and medium porosity are provided for both bottom-heavy and top-heavy distribution.

DOI:10.46481/jnsps.2023.1231

Keywords: Convection, dielectric, electric field, nanofluid, Oldroydian, porous medium, rotation

Article History :
Received: 24 November 2022
Received in revised form: 18 January 2023
Accepted for publication: 28 January 2023
Published: 02 March 2023

c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: B. J. Falaye

1. Introduction

Nanofluids are fluids that include particles with a diame-
ter of less than 100 nm and can stay suspended in them. Choi
[1] was the first to coin the term. Nanofluid is a suspension
of normal nano-size particles such as metals (Al, Cu), oxides
ceramics (Al2O3, CuO), metal carbides (S iC), nitrides and car-
bon nanotubes in an aqueous or non-aqueous dispersion me-
dia. Because of their unique chemical and physical features,
nanofluids are now considered the next-generation heat trans-
fer fluid. Nanofluids can be used for a variety of things, in-
cluding nanocomposites, electrical cooling, bio-medicine and

Email address: pankaj_thakur15@yahoo.co.in (Pankaj Thakur )

nanostructure production transportation because of their capa-
bilities.

Many researchers have investigated the properties of
nanofluids as well as potential application scenarios. Nanoflu-
ids are perfect for heat transfer applications since they have
better thermal conductivity. Because of their superior thermal
conductivity, nanofluids are perfect for transferring heat. The
effects of particle size, pH and zeta potential on the thermal
conductivity of nanofluids have been the subject of several in-
vestigations. In nanofluids with a low concentration of metal
oxides, the addition of 4 to 5 percent metal oxides by volume
results in an increase in thermal conductivity of between 10 and
20 percent. Several models have been used to estimate the ther-
mal conductivity of nanofluids.

1



Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1137 2

The first, which relates to the thermal conductivity of par-
ticles and fluids was created by Maxwell to ascertain the ther-
mal conductivity of colloidal suspension which is analyzed by
Nield and Kuznetsov [2]. Sheu [3] has studied the thermal in-
stability in a porous medium layer saturated with a viscoelastic
nanofluid and demonstrated that oscillatory instability is possi-
ble in both bottom-heavy and top-heavy nanoparticle distribu-
tions. The results indicate a conflict between the thermophore-
sis, Brownian diffusion and viscoelasticity processes, leading to
oscillatory rather than stationary modes of convection. Sharma
et al. [4] have investigated the onset of thermal convection in an
Oldroydian nanofluid layer saturating a porous medium using a
Darcy-Brinkman model revolving vertically with a uniform an-
gular velocity. Yu and Choi [5] changed the Maxwell model by
considering the influence of nanolayers on the electromagnetic
field. Another appealing strategy for heat transfer development
in industrial systems is the use of porous media in nanofluids.

Porous media are normally saturated, hard-open cells that
are often filled with fluid to allow fluid to pass through the
voids. Porous media improve heat conductivity by increas-
ing the contact area between liquid, solid and nanofluid, hence
boosting the efficiency of conventional thermal systems. Due
to its use in many practical applications such as chemical re-
actors, heat exchangers and fluid filters, the increase in ther-
mal conductivity of nanofluid with the use of porous media has
attracted many researchers for the use of materials with high
porosity in the current era for many technological problems.

Natural convection in an AC/DC electric field was studied
by Jones [6] and Chen et al. [7] for electrically accelerated heat
movement in fluids and its applications. Convective heat trans-
fer through polarized dielectric liquids was studied by Stiles et
al. [8]. They found that the convection pattern identified by
the electric field is similar to that of the well-known Bénard
cell convection. In their study of the effects of electro-thermal
convection on dielectric rotating fluid, Shivakumara et al. [9]
discovered that an AC electric field accelerates convection ini-
tiation and boosts heat transfer. The dielectric nanofluid may
be used for instrument transformers, regulating and converter
transformers and other electrical devices. Sharma et al. [10]
have researched of heat convection in a dielectric rheological
nanofluid layer employed an AC electric field.

A Maxwellian model was used to explain the rheology of
the nanofluid and it was shown that for bottom-heavy nanopar-
ticle distributions, the electric field and the stress relaxation
parameter destabilize both stationary and oscillatory modes.
Sharma et al. [11] investigation into thermosolutal convection
of an elastic-viscous nanofluid in a porous medium with ro-
tation and magnetic field led them to the conclusion that sta-
tionary convection is stabilized by the magnetic field and the
Taylor number, while stationary convection is destabilized by
the solutal Rayleigh number, nanoparticle Rayleigh number,
thermo-nanofluid Lewis number and modified diffusivity ratio
The Rivlin-Ericksen fluid issue in a Darcy-Brinkman porous
medium was studied by Sharma et al. [12].

Thermosolutal convection in a porous medium and its rela-
tionship to the rotation was examined by Sharma et al. [13].
Kumar et al. [14] researched thermosolutal convection in Jef-

frey nanofluid with porous medium and Sharma et al. [15]
studied electrohydrodynamics convection in dielectric Oldroy-
dian nanofluid layer in porous media. For free-free, rigid-free
and rigid-rigid boundaries, Poonam et al. [16] looked into the
electrohydrodynamic convection in thermal instability of Jef-
frey nanofluid in porous media. Shivakumara et al. [17] in-
vestigated on the problem Darcy-Brinkman model of electrical
convection in a porous dielectric fluid. Chand et al. [18] exam-
ined the convection of an electric field saturated by nanofluid
in a porous medium. Ramanuja et al. [19] researched MHD
SWCNT-blood nanofluid flow through porous medium in the
presence of viscous dissipation and radiation effects. Akinpelu
et al. [20] studied hydromagnetic double exothermic chemical
reactive flow with convective cooling through a porous medium
using bimolecular kinetics. This concise survey of the literature
shows that research has been done on the current issue, the ini-
tiation of thermal convection.

2. Mathematical formulation of problem

An infinitely extending electrically conducting horizon-
tal layer of an incompressible rotating non-Newtonian Ol-
droydian nanofluid of thickness d is taken under gravity
g (0, 0, −g) . The rotating vertical angular velocity is Ω(0, 0, Ω)
. The temperatures T and volumetric fractions of nanoparticles
φ are taken to be T0 and φ0 at z = 0 and T1 and φ1 at z = d
(T0 > T1 and φ1 > φ0). This dielectric nanofluid layer is domi-
nated by a uniform vertical electric field.

Figure 1: Physical Configuration

3. Governing equations

The conservation equations for mass and momentum using
Boussinesq approximation are

∇.q D = 0, (1)

ρ f

ε

(
1 + λ

∂

∂t

) [
∂

∂t
+

1
ε

q D.∇
]

q D

=


[−∇p +

(
φρp + (1 −φ) ρ f {1 −β (T − T1)}

)
g

+ fe +
2ρ
ε

(Ω× q D)]
(
1 + λ ∂

∂t

)
−

µ
k1

(
1 + λ0

∂
∂t

)
q D

 ,
(2)

where q D, p,ε,λ,λ0,φ,ρ f ,ρp,β,µ and k1 are the Darcy ve-
locity, pressure, porosity, relaxation time, retardation time,

2



Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1137 3

nanoparticles volume fraction, the density of the base fluid, the
density of nanoparticles, coefficient of volume expansion, coef-
ficient of viscosity and medium permeability respectively. fe is
the electrical force given by

fe = ρe E −
1
2

E2∇K +
1
2

(
ρ
∂K
∂t

E2
)
,

where ρe is the density of charge, K is the dielectric constant,
E is the electric field. The term ρe E is due to the free charge
known as Coulomb force. Here, the term ρe E is neglected as
compared to the term −12 E

2∇K for most dielectric fluids. The
modified pressure term is

P = p −
1
2

(
ρ
∂K
∂t

E2
)
, (3)

where p is the hydrodynamical pressure.
Maxwell Equations are

∇. (K E) = 0,∇× E = 0. (4)

From equation (4), E can be shown as

E = −∇ϕ, (5)

where ϕ is a measure of electric potential’s root mean square. It
is also assumed that

K = K0
[
1 −γ (T − T1)

]
. (6)

γ > 0, where, 0 < γ∆T << 1. Thus, the modified equations
of motion for rotating Oldroydian nanofluid saturating a porous
medium in the presence of an electric field become

ρ f

ε

(
1 + λ

∂

∂t

) [
∂

∂t
+

1
ε

q D.∇
]

q D

=


(
1 + λ

∂

∂t

)
[−∇p +

(
φρp + (1 −φ) ρ f {1 −β (T − T1)}

)
g

−
1
2

(E.E)∇K +
2ρ
ε

(Ω× q D)] −
µ

k1

(
1 + λ0

∂

∂t

)
q D

 .
(7)

The nanoparticles’ equation of continuity is[
∂

∂t
+

1
ε

(q D.∇)
]
φ = DB∇

2φ +

(
DT
T1

)
∇

2T. (8)

A porous medium’s saturating nanofluid’s heat-energy equation
is

(ρc)m
∂T
∂t

+ (ρc) f q D.∇T =

km∇
2T + ε(ρc)p

[
DB∇φ.∇T +

(
DT
T1

)
∇T.∇T

]
.

(9)

The boundary conditions appropriate to the problem are

w = 0, ∂ϕ
∂z = 0, T = T0, φ = φ0 at z = 0

w = 0, ∂ϕ
∂z = 0, T = T1, φ = φ1 at z = d

}
. (10)

Using the non-dimensional variables


(x∗, y∗, z∗) = (x,y,z)d , t

∗ =
t αm
σ d2 , q

∗
D =

q D d
αm

,

p∗ = pk1
µαm

,φ∗ =
φ−φ0
φ1−φ0

,ϕ∗ =
ϕ

γE0 ∆T d
,

T∗ = T−T1T0 −T1 , E
∗ = E

γE0 ∆T d
, K∗ = KK0 ,

where σ = (ρc)m(ρc) f and αm =
km

(ρc) f
.

The non-dimensional forms of equations (1) and (5) - (9)
are (asterisk is removed for convenience):

∇.q D = 0, (11)

1
Va

(
1 + λ1

∂

∂t

) [
1
σ

∂

∂t
+

1
ε

q D.∇
]

q D =
(
1 + λ1

∂
∂t

)
[−∇p − Rnφêz − Rmêz + RaT êz + ReT êz

−Re
∂ϕ
∂z +

√
Ta(vêx − uêy)] −

(
1 + λ2

∂
∂t

)
q D

 ,
(12)

1
σ

∂φ

∂t
+

1
ε

q D.∇φ =
1
Le
∇

2φ +
NA
Le
∇

2T, (13)

∂T
∂t

+ q D.∇T = ∇2T +
NB
Le
∇φ.∇T +

NA NB
Le
∇T.∇T, (14)

E = −∇ϕ, (15)

K =
[
1 −γT (T0 − T1)

]
, (16)

where the non-dimensional parameters are λ1 =
λαm
σd2 is the

Deborah number, λ2 =
λ0αm
σd2 is the Strain-Retardation time pa-

rameter, Pr =
µ

ρ f αm
is the Prandtl number, Dr =

k1
d2 is the

Darcy number, Va =
εPr
Dr

is the Vadasz number, Le =
αm
DB

is
the Lewis number, Ra =

ρ f gβdk1 ( T 0−T1 )
µαm

is the thermal Darcy-

Rayleigh number, Rm =
[φ0ρp +(1−φ0 )ρ f ]gdk1

µαm
is the basic density

Rayleigh number, Rn =
(ρp−ρ f )(φ1−φ0 )gdk1

µαm
is the concentration

Rayleigh number, NA =
DT ( T 0−T1 )

DB T 1 (φ1−φ0 )
is the modified diffusivity

ratio, NB =
ε(ρc)p (φ1−φ0 )

(ρc) f
is the modified particle-density incre-

ment,
√

Ta =
2Ωρk1
εµ

is Taylor number and Re =
Kγ2 E20 (T0−T1 )

2 k1 d2

µαm
is the AC electric Rayleigh number.

In terms of non-dimensional form, boundary conditions
(10) transform to

w = 0, ∂ϕ
∂z = 0, T = 1, φ = 0 at z = 0

w = 0, ∂ϕ
∂z = 0, T = 0, φ = 1 at z = 1

}
. (17)

4. Basic state solution

The basic state is stated as{
q D = (u, v, w) = (0, 0, 0) , p = pb (z) , T = Tb (z) ,
φ = φb (z) , K = Kb (z) , E = Eb (z) , ϕ = ϕb (z) .

(18)

When there is no motion, equations (13) and (14) require the
temperature and the volumetric fraction of nanoparticles to sat-
isfy the equations

d2φb
dz2

+ NA
d2Tb
dz2

= 0, (19)

d2Tb
dz2

+
NB
Le

dφb
dz

dTb
dz

+
NA NB

Le

dTb
dz

dTb
dz

= 0. (20)

Using the boundary conditions (17), equation (19) can be inte-
grated to give

φb (z) = −NATb + (1 − NA) z + NA. (21)
3



Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1137 4

Substituting φb from equation (21) into equation (20), we get

d2Tb
dz2

+
(1 − N A)NB

Le

dTb
dz

= 0. (22)

Equation (22) along with the boundary condition (14) gives the
solution as

Tb = e
−(1− N A )NB z/Le

(
1 − e−(1−NA )NB (1−z)/Le

1 − e−(1−NA )NB/Le

)
. (23)

The terms of second and higher order in the expansion of an
exponential function in equation (23) are neglected as they are
small and so one gets the best approximate initial stationary
state solutions as{

Tb = 1 − z, φb = z, Kb = 1 + γ∆T z, Eb =
E0

γ∆T (1+γ∆T z) , ϕb = −
E0

(γ∆T )2
log (1 + γ∆T ) , (24)

where, E0 =
−γ∆Tϕ

log(1+γ∆T ) at z = 0.

5. The formulae for perturbations

By adding minute disturbances to the state variables, we can
gently perturb the initial condition indicated by equation (24) so
that 

q D = (0, 0, 0) + q
′

D (u
′, v′, w′) , T = Tb + T ′,

φ = φb + φ
′, p = pb + p′, K = Kb + K′,

E = Eb + E′,ϕ = ϕb + ϕ′.
(25)

Using these perturbations given by equation (25) and neglecting
the terms of higher powers and products of perturbations (i.e.,
applying linear stability theory) in equations (11) - (16), the
resulting linearized non-dimensional perturbed equations are:

((
1 + λ2

∂

∂t

)
+

1
σVa

(
1 + λ1

∂

∂t

)
∂

∂t

)2
∇

2
+ Ta

(
1 + λ1

∂

∂t

)2
∂2

∂z2


w′ =


[(

1 + λ2
∂
∂t

)
+ 1

σVa

(
1 + λ1

∂
∂t

)
∂
∂t

] (
1 + λ1

∂
∂t

)[
−Rn∇2Hφ

′ + (Ra + Re)∇2H T
′ − Re∇2H

∂ϕ′

∂z

]  ,
(26)

1
σ

∂φ′

∂t
+

w′

ε
=

1
Le
∇

2φ′ +
NA
Le
∇

2T′, (27)

(28)

∂T′

∂z
−∇

2ϕ′ = 0. (29)

The boundary conditions (17) for the infinitesimal perturba-
tions become

w′ = 0, ∂
2 w′

∂z2
= 0, ∂ϕ

′

∂z = 0, T
′ = 1, φ′ = 0 at z = 0

w′ = 0, ∂
2 w′

∂z2
= 0, ∂ϕ

′

∂z = 0, T
′ = 0, φ′ = 1 at z = 1

 . (30)

6. The normal mode analysis

For the system of equations (26) - (29), the analysis can be
made in terms of two-dimensional periodic waves of assigned
wave numbers. Thus, we assign the quantities describing the
dependence on x, y, t of the form

exp
(
ikx x + iky x + st

)
,

where kx and ky are the wave numbers in x-direction and y-
direction, respectively and a2 = k2x + k

2
x is the resultant wave

number, s is the growth rate, which is a complex constant. The
above consideration allows us to suppose

(w′, T′,φ′,ϕ′) = (W, Θ, Φ, Ψ ) exp
(
ikx x + ikyy + st

)
. (31)

Using expression (31), the equations (26) - (29), reduces to
(
(1 + λ2 s) +

1
σVa

(1 + λ1 s) s
)2 (

D2 − a2
)

+ Ta(1 + λ1 s)
2 D2


W =


(
(1 + λ2 s) +

1
σVa

(1 + λ1 s) s
)

(1 + λ1 s)(
a2RnΦ − a2 (Ra + Re) Θ + a2Re DΨ

)  ,
(32)

s
σ

Φ +
W
ε

=
1
Le

(
D2 − a2

)
Φ +

NA
Le

(
D2 − a2

)
Θ, (33)

sΘ − W =
(
D2 − a2

)
Θ +

NB
Le

(DΘ − DΦ) −
2NA NB

Le
DΘ, (34)

DΘ −
(
D2 − a2

)
Ψ = 0, (35)

where a =
√

k2x + k2y.
The equations (30) for a free-free boundary are:

W = D2W = Θ = Φ = DΨ = 0 at z = 0 and z = 1. (36)

Therefore{
W = A1sinπz, Θ = A2sinπz,
Φ = A3sinπz, Ψ = A4cosπz,

(37)

where A1, A2, A3 and A4 are the constants.
Substituting (37) in equations (32) - (35) and using the

boundary conditions (36), we get
A B C D
1 −J − s 0 0
1
ε

NA J
Le

J
Le

+ s
σ

0
0 −π 0 −J




A1
A2
A3
A4

 =


0
0
0
0

 , (38)
where

A = M2 J + π2Ta(1 + λ1 s)2, B = −a2 (1 + λ1 s) M(Ra + Re),
C = a2(1 + λ1 s) MRn, D = −a2π(1 + λ1 s) MRe,

J =
(
π2 + a2

)
, and M =

(
(1 + λ2 s) +

1
σVa

(1 + λ1 s) s
)
.

where, the thermal Darcy-Rayleigh number,

Ra =


−

a2

π2 +a2
Re −

σLe
σ(π2 +a2)+sLe

[
π2 +a2 +s

ε
+

(π2 +a2)NA
Le

]
Rn +

π2 +a2 +s
a2

[ π
2 +a2

1+λ1 s

(
(1 + λ2 s) +

1
σVa

(1 + λ1 s) s
)

+
σVaπ2

σVa (1+λ2 s)+(1+λ1 s)s
(1 + λ1 s) Ta]

 . (39)
4



Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1137 5

7. Stationary Convection

For the validity of the principle of exchange of stabilities
(i.e., steady case), we have s = 0 ( s = r + iω = 0 ⇒ r = ω = 0 )
at the marginal stability.

Putting s = 0 in equation (39), we get the thermal Darcy-
Rayleigh number at which marginally stable steady mode ex-
ists, as

Ra =


(π2 +a2)2

a2
−

a2

π2 +a2
Re −

(
Le
ε

+ NA
)

Rn

+
π2(π2 +a2)

a2
Ta

 , (40)
which expresses the stationary thermal Darcy-Rayleigh number
Ra as a function of the dimensionless wave number a, electric
Rayleigh number Re, Taylor number Ta, nanofluid Lewis num-
ber Le, modified diffusivity ratio NA, concentration Rayleigh
number Rn and medium porosity ε. It is clear from the equation
(40) that Ra is independent of stress relaxation time λ1, strain
retardation time λ1 for stationary modes since these vanish with
the vanishing of s (growth rate).

The minimum value of Ra is obtained by putting
∂Ra
∂a2 = 0

and which on simplification implies that{
(ac)

8
+ 2π2(ac)

6
−

(
π2Re + π2Ta

)
(ac)

4

−2π2 (1 + Ta) (ac)
2
−π2 (1 + Ta)

}
= 0. (41)

Therefore, the critical wave number ac shows a substantial
increase when the electric Rayleigh number Re increases and is
independent of nanoparticles.

To obtain the critical wave number we substitute the electric
field i.e., Re = 0 and we get

ac
2

= π2
√

1 + Ta, (42)

which is the critical wave number for stationary Rayleigh num-
ber in the absence of AC electric Rayleigh number Re.

8. Results and discussion

To study Re, Ta, Le, NA, Rn,ε on the stationary convection,
we examine the behavior ∂Ra

∂Re
, ∂Ra
∂Ta

, ∂Ra
∂Le

, ∂Ra
∂NA

, ∂Ra
∂Rn

and ∂Ra
∂ε

analyt-
ically.

From equation (40) we obtain

∂Ra
∂Re

= −
a2(

π2 + a2
), (43)

which is always negative for all wave numbers. Thus, AC elec-
tric field has destabilizing effect for both bottom-heavy and top-
heavy distribution.

Equation (40) gives

∂Ra
∂Ta

=
π2

(
π2 + a2

)
a2

, (44)

which is always positive for all wave numbers. Thus, the Taylor
number has stabilizing effect for both bottom-heavy and top-
heavy distribution in the system.

Equation (40) further yields

∂Ra
∂Le

= −
Rn
ε
,
∂Ra
∂NA

= −Rn. (45)

It is clear from equation (45), the nanofluid Lewis number
Le and the modified diffusivity ratio NA enhance the stationary
convection if Rn < 0 and postpone the stationary convection if
Rn > 0.

Equation (40) also depicts that

∂Ra
∂Rn

= −

( Le
ε

+ NA
)
, (46)

which is always negative for
(

Le
ε

+ NA
)
> 0. The nanoparticle

Rayleigh number postpones the stationary convection for both
bottom-heavy and top-heavy configurations.

∂Ra
∂ε

=
LeRn
ε2

. (47)

If Rn < 0, thus the medium porosity delays the stationary
convection for bottom-heavy configuration and if Rn > 0 the
medium porosity advances the stationary convection.

9. Numerical discussion

The variation of thermal Darcy-Rayleigh number with re-
spect to wave number has been plotted using equation (40)
for stationary case, whereas the experimental values and the
fixed permissible values of the dimensionless parameters are
Re = 100, Ta = 100, Le = 200, NA = −5 and NA = 5,
Rn = −0.1 and Rn = 0.1 and ε = 0.6 . The stationary ther-
mal Rayleigh number does not depend upon stress relaxation
time and strain retardation time, since it vanishes with the van-
ishing of s (growth rate). Thus, the Oldroydian nanofluid acts
like a Newtonian nanofluid.

Figures 2 and 3 show the variation of Ra for stationary con-
vection with respect to the non-dimensional wave number for
three different values of Re = 100, 300, 500 for bottom-heavy
and top-heavy distribution and fixed permissible values. The
graph indicates that the value of Ra drops as Re rises, indicating
that Re has a destabilizing influence on both bottom-heavy and
top-heavy configurations.

Figure 2: Variations of Ra for distinct values of the Re for bottom-heavy
distribution

Figures 4 and 5 show that Ra increases with an increase in
Ta which implies that Ta has a stabilizing effect on stationary

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Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1137 6

Figure 3: Variations of Ra for distinct values of Re for top-heavy distri-
bution

Figure 4: Variations of Ra for distinct values of Ta for bottom-heavy
arrangement

Figure 5: Variations of Ra for distinct values of Ta for top-heavy ar-
rangement

convection for both bottom-heavy and top-heavy pattern of the
system.

From figures 6 and 7, It is found from the graphs that with
an increase in the values of Le, Ra increases for bottom-heavy
distribution, whereas Ra decreases for top-heavy configuration
with increase in the values of the Lewis number. Hence, Ra
stabilizes the bottom-heavy arrangement and destabilizes the
top-heavy arrangement.

Figures 8 and 9 show that Ra increases slightly with increase
in NA for bottom-heavy arrangement and Ra decrease slightly

Figure 6: Variations of stationary Ra for different values of Le for
bottom-heavy distribution

Figure 7: Variations of Ra for different values of Le for top-heavy dis-
tribution

with increase in the modified diffusivity ratio for top-heavy ar-
rangement. Hence NA stabilize the system for bottom-heavy
arrangement and destabilize the system for top-heavy arrange-
ment.

Figure 8: Variations of Ra for different values of NA for bottom-heavy
arrangement

Figures 10 and 11 show the variation of Ra for stationary
convection with respect to the non-dimensional wave number
for different values of Rn. It is depicted from the graphs that
for the cases of bottom-heavy and top-heavy configuration, Ra
decreases with the increase in Rn which causes the destabilizing

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Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1137 7

Figure 9: Variations of Ra for different values of NA for top-heavy ar-
rangement

effect on the system.

Figure 10: Variations of Ra for different values of Rn for bottom-heavy
arrangement

Figure 11: Variations of Ra for different values of Rn for top-heavy ar-
rangement

The effect of the medium porosity ε on Ra is displayed in
figures 12 and 13. It is found that with an increase in the ε, Ra
decreases and increases, respectively for bottom-heavy and top-
heavy configurations. Thus, a porous medium destabilizes the
system for the bottom-heavy pattern and stabilizes the system
for the top-heavy pattern.

Figure 12: Variations of Ra for three different values of ε bottom-heavy
distribution

Figure 13: Variations of Ra for three different values of ε top-heavy
distribution

10. Conclusions

The effect of rotation on thermal convection in an electri-
cally conducting nanofluid saturated by porous medium has
been studied using linear stability theory by employing an Ol-
droydian model which incorporates the effects of the electric
field, Brownian motion, thermophoresis and rheological param-
eters for bottom-heavy and top-heavy distribution of nanoparti-
cles. The conclusions of the present study are given below:

1. AC electric field has destabilizing for both bottom-heavy
and top-heavy distribution of nanoparticles.

2. The Taylor number Ta has stabilizing for both bottom-
heavy and top-heavy distribution of nanoparticles.

3. The effect of Lewis number (non-dimensional parameter
accounting for Brownian motion parameter DB) tends to
stabilize the stationary convection for bottom-heavy dis-
tribution and destabilizes for top-heavy configuration.

4. Modified diffusivity ratio has stabilized the system for
bottom-heavy and destabilized the system for top-heavy
configuration.

5. The concentration Rayleigh number postpones the sta-
tionary convection for both bottom-heavy and top-heavy
distribution.

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Sharma et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1137 8

6. Medium porosity has destabilizing effect for bottom-
heavy distribution and stabilizing effect for top-heavy
distribution on stationary convection.

Acknowledgments

The third author gratefully acknowledges the financial as-
sistance of CSIR-HRDG for JRF.

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