International Communications in Heat and Mass Transfer 137 (2022) 106266
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International Communications in Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ichmt
A numerical study of double-diffusive convection in the anisotropic porous
layer under rotational modulation with internal heat generation
Samah A. Ali , Munyaradzi Rudziva , Precious Sibanda , Osman A.I. Noreldin *, Sicelo P. Goqo ,
Hloniphile Sithole Mthethwa
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa
A R T I C L E I N F O
A B S T R A C T
Keywords:
Rotation
Heat and mass transfer
Stability analysis
Block hybrid method
Porous Media
Double-diffusive convection in a non-uniformly rotating anisotropic fluid layer with internal heating is inves­
tigated. The normal mode technique is used to obtain the critical stationary and oscillatory Rayleigh numbers.
The analysis for the nonlinear case is based on minimal truncated double Fourier series which gives rise to the
nonlinear Lorenz type equations. A local quasilinearization block hybrid method (LQBHM) is employed to solve
the coupled nonlinear Lorenz type equations. The solution obtained using this method is compared with solutions
obtained using the ode45 solver. The numerical results indicate that the LQBHM is accurate, efficient, and
flexible. A weakly nonlinear analysis is used to investigate the rate of heat and mass transfer in the fluid system.
The effects of time varying rotation, internal heat generation, anisotropy parameters, concentration Rayleigh,
Vadasz, and Lewis numbers on the heat and mass transfer are shown graphically. Among other results, the
quantitative relationships for rotational modulation amplitude and internal heat generation are [Nu/Sh]δ1 =0.2 <
< [Nu/Sh]δ1 =1.1 and [Nu/Sh]Ri =5 << [Nu/Sh]Ri =30 respectively. Therefore, modulation amplitude and internal
heating have been found to enhance the rate of heat mass transfer hence advancing the onset of thermal con­
vection in the system.
1. Introduction
Recently, the study of rotating double-diffusive convection in inter­
nally heated porous media has received significant attention. The study
of this problem was prompted by its industrial and environmental ap­
plications including geothermal power usage and storage, food pro­
cessing, the movement of contaminants in lakes and underground water,
and atmospheric pollution, to name a few. When a solute is added to the
fluid layer, the temperature difference is not only the factor affecting
buoyancy force but also by the concentration difference. Horton and
Rogers [1] and Lapwood [2], were among the first to study thermal
convection in a saturated porous medium. The study of the influence of
rotation on such problems is also motivated by its scientific, engineer­
ing, and geophysical applications. Geophysical applications include
porous geological formation of earth rotation and magma movement in
the earth mantle [3]. Vadasz [4,5], Nield and Bejan [6,7] and Bejan [8]
have presented detailed industrial applications of the problem in
comprehensive reviews of porous media flows. Other notable
investigations on convection caused by thermal and concentration gra­
dients in porous media are given by [9–14]. There are also scenarios of
significant practical importance where a rotating porous media with a
solute dissolved in it provides its own source of heat. This case gives a
unique way in which a convective flow can be set up through the local
heat source within the rotationally affected porous media. The scenario
can occur through radioactive decay or through a relatively weak
exothermic reaction in the porous system. Problems in hydrogeology,
geothermal energy extraction, porous heat exchangers and cooling of
electronic equipment involve convective heat transmission mechanisms
through rotating porous media with internal heating. Yadav et al.
[15–17], Vadasz [18,19], Patil et al. [20] and Srivastava [21,22] re­
ported several effects for this problem under various conditions,
including rotation, heat-generating porous layer, chemical reaction ef­
fect, electric field effect, the double-diffusive effect, and anisotropic
porous media.
Many previous investigations on porous layer diffusive convection
mainly focused on homogeneous isotropic porous materials [12,23–26].
The study of the impacts of non-homogeneity and anisotropy of porous
* Corresponding author.
E-mail addresses: samahanwarali@gmail.com (S.A. Ali), munyarudziva@gmail.com (M. Rudziva), SibandaP@ukzn.ac.za (P. Sibanda), osman@aims.edu.gh
(O.A.I. Noreldin), goqos@ukzn.ac.za (S.P. Goqo), sitholeh@ukzn.ac.za (H.S. Mthethwa).
https://doi.org/10.1016/j.icheatmasstransfer.2022.106266
Received 6 April 2022; Received in revised form 14 June 2022; Accepted 12 July 2022
Available online 29 July 2022
0735-1933/© 2022 Elsevier Ltd. All rights reserved.
S.A. Ali et al.
International Communications in Heat and Mass Transfer 137 (2022) 106266
Nomenclature
Va
Latin Symbols
a
Wave number
d
Depth of the anisotropic porous layer
Da
Darcy number
→
g
Gravitational acceleration
̂
k
Unit vector in the z-direction
K
Permeability tensor
Le
Lewis number
N
Buoyancy ratio
Nu
Nusselt number
P
Pressure
Pr
Prandtl number
→
u
Velocity
R
Revised Rayleigh number
RaT
Thermal Rayleigh number
Salinity Rayleigh number
RaS
Ri
Internal Rayleigh number
S
Solute concentration
Sh
Sherwood number
T
temperature
t
Time
Ta
Taylor number
Ta*
Revised Taylor number
Greek symbols
Thermal expansion coefficient
β1
β2
Concentration expansion coefficient
Amplitude of modulation
δ1
ε
Perturbation parameter
κT
Thermal diffusivity
κS
Solute diffusivity
γ
Porosity
ς
Mechanical anisotropy parameter
η
Thermal anisotropy parameter
→
Ω
Angular velocity
μ
Fluid dynamic viscosity
ν
kinematic viscosity
ω
Frequency of modulation
Subscripts
γ
b
c
0
Vadasz number
Fluid volumetric heat capacity
basic state
critical value
reference value
Superscripts
perturbed quantities
*
non-dimensional quantities
′
layer has increased in recent years. Given the structure of the porous
material, there can be substantial anisotropy in permeability and ther­
mal diffusivity variables. Compaction, frost action, sedimentation, and
solid matrix reorientation are just a few of the factors that contribute to
the formation of natural anisotropic porous media [27]. Anisotropy may
be found in man-made porous materials including chemical engineering
pellets and fibrous materials used for insulation. Anisotropy is also
important in the mathematical modeling of fractured rocks in
geothermal systems. Castinel and Combarnous [28] were among the
first to investigate the convective flow in anisotropic porous media.
Many researchers have since studied the influence of anisotropy on the
onset of instability in porous media [29–33]. The onset of thermal
convection in a rotating Jeffrey fluid with rotation in an anisotropic
porous medium was investigated by Yadav [34]. Yadav concluded that
rotation and anisotropic parameters delay the onset of Jeffery convec­
tive motion significantly.
The study of internal heating in porous media is necessary due to its
relevance in a variety of applications such as fire and combustion in­
vestigations and radioactive material storage. Bhadauria [35] further
studied an anisotropic layer with internal heating. Bhadauria concluded
that internal heat generation has a positive effect on the heat and mass
transfer rates hence resulting in the advancement of the onset of con­
vection in the fluid system. For impermeable and isothermal boundaries,
the influence of the internal heat generation and anisotropy in a fluid
layer modeled using the modified Darcy equation were investigated by
Mahajan and Nandal [36]. In their results, they concluded that the
presence of internal heating and medium anisotropy raises the possi­
bility of subcritical instability. Several authors [37–40] have conducted
studies on the effect of internal heating in an anisotropic layer. Other
studies on the effect of internal heat source/sink on heat transfer can be
found in [39,41,42].
The investigation of double-diffusive convection in uniformly
rotating porous medium with internal heating is motivated by both
theoretical and practical engineerings applications. These include
molding and solidification of metals centrifugally, and petroleum in­
dustry processes. In 1981, Chakrabarti and Gupta [43] investigated
nonlinear thermohaline convection in a uniformly rotating porous layer.
Lombardo and Mulone [44] used the Lyapunov direct method to analyze
the effect of symmetric rotation on the heat exchange of a fluid with
temperature and solute gradients and a saturated porous medium.
Rudraiah et al. [45] investigated the uniform rotation effect on the
convection onset in a sporadically packed porous medium. Malashetty
and Heera [46] investigated rotation influence on the onset of in­
stabilities in an anisotropic porous layer. They concluded, among other
results, that the oscillatory convection is most preferable for a fluid
system with considerable to high Taylor numbers. Other investigations
on anisotropy effects in porous media including rotational effects has
been undertaken by Vaidyanathan et al. [47], Patil et al. [48] and Alex
and Patil [49]. Further, Govender and Vadasz [50,51] investigated the
effects of centrifugal forces on natural convection in an anisotropic
porous layer. In their studies, they used the Darcy model to describe the
flow and a modified energy equation. Among other results, they
observed that increasing the magnitude of the thermal anisotropy ratio
delays the onset of convection.
Over previous years, there has been a rise of interest in externally
regulated fluid systems. Venezian [52] was among the first to carry out
an investigation on the effect of infinitesimal thermal modulation on the
onset of fluid instabilities. He observed that temperature modulation is
capable of advancing or delaying the onset of convection in the system.
Venezian’s investigation was based on Donnelly’s [53] earlier experi­
mental investigation on the influence of modulation on the stability of
the flow located between rotating cylinders. In Donnelly’s experiments,
a fluid was confined in-between two cylinders, with the outer cylinder
held fixed while the inner cylinder rotated with a sinusoidal angular
speed. He showed that modulating the angular speed of the inner cyl­
inder delayed the onset of instability. Researchers who have used this
modulation in their investigations of convection instability in a hori­
zontal fluid layer include [54–60]. Recently, Siddheshwar et. al [61,62],
investigated the influence of different modes of gravitational modula­
tion in Newtonian liquids and nanoliquids. Their results revealed that all
modes of gravitational modulation affect the rate of heat and mass
transfer. Rudziva et al. [63] and Meghana and Pranesh [64] made
2
S.A. Ali et al.
International Communications in Heat and Mass Transfer 137 (2022) 106266
Fig. 1. Geometry of the problem.
comparisons between different modes of rotational modulations in
diffusive convection. They concluded that all modes of rotational
modulations affect the stability of the fluid system.
From the literature, the study of the time-dependent rotation rate has
not received enough attention. Therefore, the focus of the current study
focuses on the effect of time-varying rotation on onset of instability in a
porous layer with internal heating using a newly developed local qua­
silinearisation block hybrid method LQBHM. The block hybrid methods
are generally developed by combining interpolation and collocation
[65–67]. The current approach differs from what has been reported in
the literature in that it does not emphasize interpolation. We show how a
hybrid method can be derived using only collocation.
capacity ratio, γ is the porosity, P is the pressure, →
g = (0, 0, − g) ̂
k is the
gravitational acceleration, ρ is the density and ρ0 is the reference
density.
The fluid layer is kept at a temperature gradients and solute gradients
ΔT/d, ΔS/d as shown in the geometry configuration of the problem, with
the thermal boundary conditions are
2. Mathematical formulation
where δ1 denotes the modulation amplitude, ε is a small perturbation
parameter, ω is the modulation frequency.
The fluid is assumed to be quiescent at basic state and the corre­
sponding quantities are
T = T0 + ΔT, atz = 0, andT = T0 , atz = d,
S = S0 + ΔS, atz = 0, andS = S0 , atz = d.
The time-varying rotational modulation term is given as
→
Ω = Ω0 [1 + ε2 δ1 cos(ωt)]̂
k,
A rotating anisotropic porous layer which is saturated, confined
between infinitely extended two horizontal parallel planes with distance
d apart is considered. The horizontal planes extend in the x and y di­
rections. A Cartesian coordinate system is selected so that the origin is
on the bottom plane and the z-axis is perpendicular to the top, with the
gravitational force →
g acting vertically downwards. Adverse thermal and
solute gradients are applied across the porous layer, and the lower and
upper planes are kept at temperature T0 + ΔT, concentration S0 + ΔS,
and T0, S0, respectively, where ΔT and ΔS are temperature and con­
centration gradients, respectively. The system is rotating vertically with
→
a non-uniform angular velocity Ω . Across the permeable layer, tem­
perature and concentration variations are applied see, Fig. 1.
The Oberbeck-Boussinesq approximation takes into consideration
the effect of density changes. Under these assumptions, the generalized
Darcy model has been used for the momentum equation, see [31,37,39]
dPb
g,
= − ρb →
dz
κS
(9)
d2 Tb
+ Q(Tb − T0 ) = 0,
dz2
(10)
d2 Sb
= 0,
dz2
(11)
κ Tz
and
ρb = ρ0 [1 − β1 (Tb − T0 ) + β2 (Sb − S0 )].
χ
∂T
+ (→
q ⋅∇)T = ∇⋅(κT ⋅∇T) + Q(T − T0 ),
∂t
(3)
γ
∂S →
+ ( q ⋅∇)S = κS ∇2 S,
∂t
(4)
(
z)
Sb (z) = S0 + ΔS 1 −
.
d
ρ = ρ0 [1 − β1 (T − T0 ) + β2 (S − S0 )],
(8)
where the subscript b denotes the basic state. Using (8) in Eqns. (1) - (5),
gives
The basic state solutions are
√̅̅̅̅̅̅̅̅̅̅̅̅
/
sind Q κTz (1 −
√̅̅̅̅̅̅̅̅̅̅̅̅
Tb (z) = T0 + ΔT
/
sind Q κTz
→
ρ0 ∂→
q
Ω
μ→
q = − ∇P + ρ→
g −
+2 ×→
q,
γ
γ ∂t
K
(7)
→
q = (0, 0, 0), ρ = ρb (z), P = Pb (z), S = Sb (z), T = Tb (z),
(1)
∇⋅→
q = 0,
(6)
(2)
z
)
d
,
(12)
(13)
(14)
To investigate the behaviour of infinitesimal disturbances, we perturb
the basic state as
(5)
where →
q is the velocity, μ is the dynamic viscosity, Q is internal heat
source, K = Kx (̂îi + ̂ĵj) + Kz (̂
k̂
k) denotes the permeability tensor, κT
′
′
′
′
′
′
′
→
q =→
q = (u , v , w ), T = Tb + T , S = Sb + S , P = Pb + p ,
′
ρ = ρb + ρ ,
denotes the thermal diffusivity tensor, κTx (̂îi + ̂ĵj) + κTz (̂
k̂
k), T is the
temperature, β1 and β2 are thermal and concentration expansion co­
efficients respectively, κS is the concentration diffusivity, χ denotes heat
(15)
where the primes indicate infinitesimally small perturbations.
Substituting Eq. (15) into Eqs. (1) – (5) and using basic states (8), the
3
S.A. Ali et al.
International Communications in Heat and Mass Transfer 137 (2022) 106266
perturbed equations are obtained as
3. Linear stability analysis
∇⋅→
q = 0,
(16)
′
→
′
ρ0 ∂→
q
Ω
μ ′
′
′
′
q = − ∇P − →
k,
+2 ×→
q + (ρ0 β1 T − ρ0 β2 S )ĝ
γ
γ ∂t
K
(17)
For the investigation to be complete, we present a linear stability
analysis of the fluid flow. Eqs. (20) - (23) are linearized by neglecting the
nonlinear terms. Assuming that the solution satisfies the boundary
conditions (26), we use the normal mode technique with
(18)
Ψ = 𝒜1 eσt sin(παx)sin(πz),
(27)
ξ = 𝒜2 eσt sin(παx)cos(πz),
(28)
T = 𝒜3 eσt cos(παx)sin(πz),
(29)
S = 𝒜4 eσt cos(παx)sin(πz),
(30)
′
′
χ
′
∂T
′
′
′ ∂Tb
′
+ (→
q ⋅∇)T + w
= ∇⋅(κT ⋅∇T ) + QT ,
∂t
∂z
′
′
∂S
′
′ ∂Sb
′
γ
+ (→
q ⋅∇)S + w
= κS ∇2 S .
∂t
∂z
(19)
Two-dimensional disturbances are considered with the stream function
( ∂ψ ∂ψ )
′
′
∂z , − ∂x .
The pressure term in Eq. (17) is removed by taking the curl of the
equation. Eqs. (17) – (19) are non-dimensionalized using
ψ defined as (u , w ) =
where α is the wave number, σ = σ R + iσim represents complex growth
rate, and 𝒜1 , 𝒜2 , 𝒜3 , 𝒜4 are constants. Substituting Eqs. (27) – (30) in
the linearized form of Equations (20)–(23) and eliminate the constants
after applying the orthogonality of trial functions then solve for RaT we
obtain
d2 ∗
′
t , Tb = ΔT.Tb∗ , T = ΔT.T ∗ ,
κ Tz
′
κT ∗
′
′
Sb = ΔS.Sb∗ , S = ΔS.S∗ , →
q = z→
q , ψ = κ Tz ψ ∗ ,
d
κ T ξ∗
ξ= z .
d
′
′
′
(x , y , z ) = d(x∗ , y∗ , z∗ ), t =
RaT =
(4π2 − Ri )(σ + λ2 − Ri ) 2
Va2 Taςπ 2 π2 α2 Va LeRaS
},
+
{m σ + Vaλ1 +
4 2
ςσ + Va
4π α Va
Leσ + m2
(31)
Dropping the asterisk and setting γ and χ equal to unity for convenience,
we get the non-dimensionalized system of equations
[
) ( 2
)]
(
√̅̅̅̅̅̅
1 ∂ ∂2
∂2
∂
1 ∂2
∂ξ
+ 2 +
+
ψ + Ta(1 + ε2 δ1 cos(ωt))
2
2
2
Va ∂t ∂x
ς ∂z
∂z
∂z
∂x
= − RaT
(
[
[
∂T
∂S
+ RaS ,
∂x
∂x
(
η
∂2
∂2
+ 2 + Ri
2
∂x ∂z
(
∂
1 ∂2
∂2
+ 2
−
2
∂t Le ∂x ∂z
where Ta =
)]
( 2Ω0 Kz )2
ν
(20)
)]
T=
where
(21)
∂ψ
∂(ψ , T)
h(z) +
,
∂x
∂(x, z)
Δ1 =
∂ψ ∂(ψ , S)
S=−
+
,
∂x ∂(x, z)
+
(23)
Δ2 =
is the Taylor number, Pr = κνT is the Prandtl number,
z
is the concentration Rayleigh number, Ri = Qd
κT is the internal Rayleigh
number, ν = ρμ is the kinematic viscosity, Le =
0
dTb (z)
dz .
κTz
κS
is the Lewis number,
and h(z) =
The basic state temperature and concentration in dimensionless
forms are
√̅̅̅̅̅
sin( Ri (1 − z))
√̅̅̅̅̅
Tb (z) =
,
(24)
sin Ri
3.1. Stationary convection (σ im = 0, (Δ2 ∕
= 0))
Substituting σ im = 0 into Eq. (32), we get the stationary thermal
Rayleigh number RaT = RastT as
(25)
Sb (z) = 1 − z.
RastT =
It is interesting to observe that the Vadasz number Va in Eq. (20) in­
corporates the classic Prandtl number Pr and the Darcy number Da. The
plane surfaces are assumed to be stress-free and ideal heat and salt
conductors. The boundary conditions are
∂ξ ∂2 ψ
= 0 at z = 0, 1.
=
∂z ∂z2
(λ2 − Ri )(4π2 − Ri ){m2 (λ1 + ςπ 2 Ta) + π2 α2 LeRas }
.
4π4 α2 m2
(35)
When the rotational effect is absent, RastT reduces to
2
RastT =
2 2
π
(Ri − ηπ 2 α2 − π2 )(Ri − 4π 2 ) π α + ς
LeRas
{
+ 2 2
}.
2
2
2
4π
πα
π α + π2
(36)
which gives the result given by Bhadauria [35].
In the case of no rotational effect and internal heat source, RastT re
(26)
4
­
(34)
Considering that RaT is a physical quantity, it is then required to be real.
Therefore, it follows from Eq. (32) that either σ im = 0, (Δ2 ∕
= 0) or
Δ2 = 0,
(σ im ∕
= 0). Stationary convection corresponds to the case
σ im = 0, (Δ2 ∕
= 0), whereas Δ2 = 0, (σim ∕
= 0) leads to the oscillatory
mode of convection.
z
z dΔT
z dΔS
parameter, RaT = β1 gK
is the thermal Rayleigh number, RaS = β2 gK
νκT
νκ T
2
4π2 − Ri
π2 ςVa2 Ta(Va − ς(λ2 − Ri ))
{Vaλ1 + m2 (λ2 − Ri ) +
Va2 + ς2 σ2im
4π4 α2 Va
+
mechanical anisotropy parameter, η = κκTTx is the thermal anisotropy
z
(33)
π2 α2 Le Va(m2 − Le(λ2 − Ri ))RaS
}.
m4 + Le2 σ 2im
z
z
4π2 − Ri
π2 ςVa2 Ta(Va(λ2 − Ri ) + ςσ2im )
{Vaλ1 (λ2 − Ri ) − m2 σ2im +
4 2
4π α Va
Va2 + ς2 σ2im
π2 α2 Le Va(m2 (λ2 − Ri ) + Leσ 2im )RaS
},
m4 + Le2 σ 2im
(22)
Pr
, is the Vadasz number, ς = KKxz is the
Da = Kd2z is the Darcy number, Va = Da
ψ =T=S=
(32)
RaT = Δ1 + iσim Δ2 ,
)
√̅̅̅̅̅̅
1 ∂ 1
∂ψ
+ ξ = Ta(1 + ε2 δ1 cos(ωt)) ,
∂z
Va ∂t ς
∂
−
∂t
where m2 = π2(α2 + 1), λ1 = π2(α2 + ς− 1), λ2 = π2(ηα2 + 1).
To have a clear analysis of the onset of instability in the fluid system,
the real part of σ is set to zero and σ = iσ im in Eq. (31). Getting rid of the
complex terms in the denominator, Eq. (31) yields
S.A. Ali et al.
International Communications in Heat and Mass Transfer 137 (2022) 106266
duces to
RastT =
4. Weakly nonlinear stability analysis
2
2
π
1
(ηα + 1)LeRaS
(ηα2 + 1)(α2 + ) +
.
α2
ς
(α2 + 1)
The main focus of the study is on investigating the weakly nonlinear
stability with rotation modulation. The results for the unmodulated case
have been discussed by Altawallbeh et al. [39], Bhadauria et al. [37],
Gaikwad et al. [69]. The nonlinear analysis provides information on the
heat and mass transfer rates which the linear stability analysis cannot
provide.
(37)
which gives the result given by Malashetty and Swamy [27]
For RaS = 0, which is the case of single component saturated
permeable layer with no internal heat generation, rotational effect and a
thermal isotropic porous medium which is equal to unit, RastT reduces to
RastT =
π2 (α2 + 1)(α2 + ς− 1 )
,
α2
4.1. Lorenz type model
(38)
The nonlinear stability analysis is carried out using the system of Eqs.
(20) – (23) which satisfy boundary conditions (66) by means of a
truncated minimal Fourier series. The stream function, vorticity, tem­
perature and concentration distributions are represented as
which gives the result given by Storesletten [68]. Taking ς = 1, Eq. (38)
reduces to the usual result of Horton and Rogers [1] and Lapwood [2]
RastT =
π2 (α2 + 1)2
,
α2
(39)
whith the critical value
RastT
= 4π for αc = 1.
3.2. Oscillatory convection (σim ∕
= 0, Δ2 = 0)
Setting Δ2 = 0 gives a dispersion relation
(40)
2
Λ1 (σ2im ) + Λ2 σ2im + Λ3 = 0,
Λ1 = ς2 Le2 (4π2 − Ri )(Vaλ1 + m2 (λ2 − Ri )),
Λ2 = (4π2 − Ri )[(Vaλ1 + m2 (λ2 − Ri ))(Va2 Le2 + m4 ς2 )
+π2 ςVa2 Le2 Ta(Va − ς(λ2 − Ri ))
+π 2 α2 ς2 Le Va(m2 − Le(λ2 − Ri ))RaS ],
Λ3 = (4π2 − Ri )[Va2 m4 (Vaλ1 + m2 (λ2 − Ri ))
+m4 π 2 ςVa2 Ta(Va − ς(λ2 − Ri ))
+π2 α2 LeVa3 (m2 − Le(λ2 − Ri ))RaS ].
(41)
(42)
ξ = B1 (t)sin(αx)cos(πz) + B2 (t)sin(2αx),
(48)
T = C1 (t)cos(αx)sin(πz) + C2 (t)sin(2πz),
(49)
S = D1 (t)cos(αx)sin(πz) + D2 (t)sin(2πz),
(50)
X7 = − RπD2 (t), R =
(43)
4π2 − Ri
Raos
{Vaλ1 (λ2 − Ri ) − m2 σ 2im
T =
4π4 α2 Va
π2 ςVa2 Ta(Va(λ2 − Ri ) + ςσ2im )
+
Va2 + ς2 σ2im
(44)
π2 α2 Le Va(m2 (λ2 − Ri ) + Leσ 2im )RaS
+
}.
m4 + Le2 σ2im
Eq. (40) is quadratic in terms of σ 2im , which can give two distinct real
positive roots for some fixed parameter values. Thus, Eq. (40) is first
solved to get all positive values of σ2im that may exist. There will be no
possibility of oscillatory convection if positive solutions does not exist.
For the existence of two positive solutions, the minimum of Eq. (40)
gives the oscillatory Rayleigh number with σ 2im given by Eq. (44).
When the rotational effect is absent, we recover the result of Bha­
dauria [35].
4π2 − Ri
= 4 2 {Vaλ1 (λ2 − Ri ) − m2 σ 2im
4π α Va
π2 α2 Va(m2 (λ2 − Ri ) + Leσ 2im )LeRaS
+
}.
m4 + Le2 σ 2im
π α (m + Le(Ri − λ2 ))Rs
2
Le( m (RVai − λ2 )
− λ1 )
4
−
m
.
Le2
, τ = tδ2 .
dX1
Vaa21
=−
X1 − Va(1 + ε2 δ1 f )X2 + VaX4 − N VaX6 ,
dτ
δ2
(51)
dX2
Va
= − Ta★ Va(1 + ε2 δ1 f )X1 −
X2 ,
dτ
ςδ2
(52)
dX3
Va
= − 2 X3 ,
dτ
ςδ
(53)
dX4
1
= − 2RHX1 + 2 (Ri − a22 )X4 − X1 X5 ,
dτ
δ
(54)
dX5
1
X1 X4
= 2 (Ri − 4π4 )X5 +
,
dτ
2
δ
(55)
dX6
X6
= RX1 −
− X1 X7 ,
dτ
Le
(56)
dX7
4π 2 X7 X1 X6
=− 2
+
,
dτ
2
δ Le
(57)
δ2 = α2 + π2 , a21 = α2 +
and
2
δ6
where
(45)
where
2 2
α2c RaT
This leads to the following coupled nonlinear Lorenz-type system of
equations
From Eq. (32), when Δ2 = 0, we observe that oscillatory convection may
occur when RaT = Raos
T where
σ 2im =
(47)
where A1, Bi, Ci, Di, (i = 1, 2) are time dependent functions and α is the
wave number. The generalized Lorenz model is obtained by applying the
truncated Fourier series to non-dimensionalised Eqs. (20) - (23) and
define new variables as
√̅̅̅̅̅̅
√̅̅̅̅̅̅
απ
π2 α Ta
π2 αc Ta
X1 = 2 A1 (t), X2 =
B1 (t), X3 =
B2 (t),
6
δ
δ6
δ
X4 = − RπC1 (t), X5 = − RπC2 (t), X6 = − RπD1 (t),
where
Raos
T
ψ = A1 (t)sin(αx)sin(πz),
2
∫
(46)
1
h(z)sin2 (π z)dz.
H=
0
5
π2 ∗ δ1 2
, δ = 2 , a2 = ηα2 + π2 ,
ς
ε
S.A. Ali et al.
International Communications in Heat and Mass Transfer 137 (2022) 106266
points can be anticipated from the Jacobian matrix. The eigenvalues of
the Jacobian matrix are determined based on the parameter values.
Therefore, we use the general knowledge of the trace Tr and the deter­
minant Δ of the Jacobian matrix J to determine the nature of the fixed
points, where
]
[ 2
a1 Va 2Va 4π2 − Ri a22 − Ri
4π2
1
,
(61)
+ 2 +
+
+ 2 +
Tr = −
2
2
2
δ
ςδ
δ
δ
δ Le Le
2
RaS
The buoyancy ratio is denoted by N = Ra
and Ta★ = π δTa
is the revised
6
T
Taylor number.
Because this autonomous nonlinear system of equations cannot be
solved analytically, a numerical method is used. Qualitative predictions,
on the other hand, can be made, as detailed below. Eqs. (51) – (57) are
uniformly bounded in time. The equations must also be dissipative, just
like the original Eqs. (16) – (19). As a result, the volume in the phase
space is expected to contract with time. To prove volume contraction,
we show how the velocity field has a negative divergence. Thus the
uniform rate of contraction is given by
and
)
]
[ ( 2
a
2
1
1
1
4π 2
,
= − Va 21 + 2 + 2 (a22 − Ri ) + 2 (4π2 − Ri ) + +
Le Leδ2
∂Xn
δ
ςδ
δ
δ
Δ=
7
∑
∂X˙n
n=1
(58)
Based on the values of physical parameters, the trace and determinant
can be used to make general conclusions about the system’s stability.
From Eq. (61) it is observed that the trace Tr is always negative for all
n
where X˙n = dX
dτ .
The right-hand side of Eq. (58) is always negative if Ri ⩾(a22 + 4π2 )/2
provided all physically important parameters in the square bracket are
non-negative, suggesting that the system is bounded and dissipative.
Therefore, the trajectories are either attracted to a fixed point, a limit
attractor, or a strange attractor. After time τ > 0, the end-points of the
respective trajectories populate the volume
{ ( ( 2
)
a
2
1
1
1
V(τ) = V(0)exp − Va 21 + 2 + 2 (a22 − Ri ) + 2 (4π2 − Ri ) +
Le
δ
δ
δ
ςδ
)}
4π 2
+
.
Leδ2
(59)
4π 2 +a2
positive variables and provided 2 2 ⩾Ri and the determinant Δ de­
pends on parameter variables. It can be summarized as
• if T < 0, Δ > 0, characterize a stable fixed point.
• if T > 0, Δ < 0, characterize a saddle point.
4.3. Heat and mass transfer
The onset of diffusive convection is perceived through its effect on
heat and mass movements. Heat and mass movements are measured in
terms of Nusselt number Nu and Sherwood number Sh respectively. A
Nusselt number which has a numerical value close to unit represents
heat transfer mostly by conduction and this is probably in the steady
state. Larger Nu and Sh numbers characterize significant convection
with flow instabilities.
The rates of heat and mass transport per unit area are represented by
H and J respectively, where
Eq. (59) shows that the volume decreases in time exponentially. The
Vadasz number Va and the internal Rayeigh number Ri enhance the
dissipation rate.
4.2. Stability of the fixed points
The fixed point is the simplest kind of an equilibrium position. It is
essential to analyze the state of fixed points after a small disturbance.
The linearization of a nonlinear system can often be used to infer its
stability. Setting
dXi
= 0,
dτ
)(
))
(
(
4π2 (4π2 − Ri ) 2δ4 H R − a22 − Ri a21 + δ2 LeN R − ςδ4 f ∗ 2 Ta∗ Va3
.
12
2
2
ς δ Le
(62)
H = − κT 〈
J = − κS 〈
i = 1, …, 7
∂Ttotal
〉 ,
∂z z=0
∂Stotal
〉 ,
∂z z=0
(63)
(64)
where the angular brackets refer to the horizontal average at z = 0. The
expressions for Ttotal and Stotal are
in Eqs. (51) - (57) we have seven possible solutions of the system. One
obvious solution will be X1, …, X7 = 0 which relates to the fixed point.
Linearizing about this fixed point, we get the Jacobian matrix
⎡
⎤
Vaa21
− Vaf ∗ 0
Va
0
− NVa 0 ⎥
⎢ − δ2
⎢
⎥
⎢
⎥
⎢
⎥
Va
∗
∗
⎢ − VaTa f −
0
0
0
0
0 ⎥
⎢
⎥
ς
⎢
⎥
⎢
⎥
Va
⎢
⎥
⎢
0
0 − 2
0
0
0
0 ⎥
⎢
⎥
ςδ
⎢
⎥
⎢
⎥
⎢
⎥
1
2
⎢ − 2RH
⎥
0
0
(R
−
a
)
0
0
0
i
2
J =⎢
⎥,
δ2
⎢
⎥
⎢
⎥
1
⎢
⎥
2
0
0
0
0
(R− i− 4π ) 0
0 ⎥
⎢
2
⎢
⎥
δ
⎢
⎥
⎢
⎥
1
⎢
⎥
0
R
0
0
0
0
−
⎢
⎥
Le
⎢
⎥
⎢
⎥
2 ⎥
⎢
4π ⎥
⎢
0
0
0
0
0
0 − 2 ⎦
⎣
δ Le
Ttotal = T0 − (ΔT)z + (ΔT)T(t, x, z),
(65)
Stotal = T0 − (ΔS)z + (ΔS)S(t, x, z).
(66)
Substituting Eqs. (47) – (50) into Eqs. (65) and (66) then solve (63) –
(64), we arrive at
H = ΔTκT (1 − 2πC2 ),
(67)
J = ΔSκS (1 − 2π D2 ).
(68)
Nu and Sh are defined as
Nu =
H
= 1 − 2π C2 ,
κT ΔT
(69)
Sh =
J
= 1 − 2πD2 .
κS ΔS
(70)
Using the scaled variables (X5, X7) = − π R(C2, D2), we obtain
(60)
2
Nu = 1 + X5 ,
R
where f ∗ = 1 + δ∗1 cos(ωτ).
Using the trace-determinant method, the characteristics of the fixed
6
(71)
S.A. Ali et al.
International Communications in Heat and Mass Transfer 137 (2022) 106266
Fig. 2. Error norm for X1, X2, X4 and X6 for collocation points M = 2, 3, 4, 6.
2
Sh = 1 + X7 .
R
convergence. The accuracy is increased by adding extra off-step points
while keeping the grid size constant.
The continuous method is based on approximating the exact solution
x(τ) of the non-linear differential equation
(72)
To perform a substantial investigation into the effect of time-dependent
rotation on heat and mass transfer, the time-averaged Nusselt number
(mean Nusselt number) Nu and Sherwood number (mean Sherwood
number), Sh are defined as
Nu =
ω
2π
∫
2π
ω
Nu dt, and Sh =
0
ω
2π
∫
0
dx
= f (t, x(t)),
dt
by
2π
ω
Sh dt.
x(τ0 ) = x0 ,
(73)
x(τ) ≈ X(τ) =
2π
where [0, ω ] denotes the chosen interval to calculate the mean Nusselt
and Sherwood numbers.
M +1
∑
cn,k (τ − τn )k ,
(74)
k=0
where cn,k are unknown coefficients in the [τn, τn+1] block. The co­
efficients are obtained from a system of M + 2 equations with M + 2
unknowns generated from
5. Method of solution
To investigate the effect of various parameters on the Nusselt and
Sherwood numbers, we solved the system of Eqs. (51) – (57) using the
newly developed Local quasilinearisation block hybrid method LQBHM.
This is a novel method for solving coupled nonlinear initial value
problems. The development of block hybrid linear multistep method
considers off-step points. The additional off-step points enhance the
accuracy of the methods and ensure consistency, zero-stability, and
Ẋ n+pi = f (τn+pi , xn+pi ), i = 0, 1, 2, ..., M
X(τn ) = cn,0 = xn , n = 0, 1, ..., M − 1.
(75)
where pi, i = 1, 2, . . . , M − 1 are intra-step points, the dot denotes the
derivative with respect to time τ.
Solving the equations that arise in Eq. (75), gives cn,0 = Xn and
cn,1 = fn for all values of M.
7
S.A. Ali et al.
International Communications in Heat and Mass Transfer 137 (2022) 106266
In general, considering the linear first order equation of the form
ẋ = f (τ, x) = ϕ(τ) + υ(τ)x,
Vaa21
, 𝒩 1 = − Va(1 + ε2 δ1 f )X2 + VaX4 − N VaX6 ,
δ2
Va
ℒ2 = − 2 , 𝒩 2 = − Ta∗ Va(1 + ε2 δ1 f )X1 ,
ςδ
ℒ1 = −
τ0 < τ < τ T ,
where ϕ(τ) and υ(τ) are known functions of τ. The block hybrid method
with the intra-step points p1, p2, . . . , pM− 1, is expressed as
Xn+pi = Xn + hβi (ϕn + υn Xn ) + h
M
∑
αi,j (ϕn+pj + υn+pj Xn+pj ).
ℒ3 = −
(76)
ℒ4 =
The steps to be be considered for a general non-linear first order equa­
tion are summarized below.
A system of non-linear first order differential equation is assumed to
take the form
= f1 (τ,X1 ,X2 ,…,XM ) = ℒ1 (τ,X2 ,X3 ,…,XM )X1 +𝒩 1 (τ,X1 ,X2 ,…,XM ),
= f2 (τ,X1 ,X2 ,…,XM ) = ℒ2 (τ,X1 ,X3 …,XM )X2 +𝒩 2 (τ,X1 ,X2 ,…,XM ),
ℒ7 = −
Vaa21
, υ1 = − Va(1 + ε2 δ1 f )X2 + VaX4 − N VaX6 ,
δ2
Va
ϕ2 = − 2 , υ2 = − Ta∗ Va(1 + ε2 δ1 f )X1 ,
ςδ
= fM (τ,X1 ,X2 ,…,XM ) = ℒM (t,X1 ,…,XM− 1 ,XM )XM +𝒩 M (τ,X1 ,X2 ,…,XM ),
(77)
ϕ1 = −
where ℒk (τ) is the non-linear function component which is a coefficient
to Xk in the k-th equation and 𝒩 k (τ) is the remaining component which
may or may not be a non-linear function for each k = 1, 2, …, M.
We then consider the quasilinearisation method QLM iteration. The
quasilinearisation technique is based on Taylor series expansion of the
non-linear term 𝒩 k (τ, Xk ) based on the assumption that the difference
between the current and previous iteration (Xk,r+1 − Xk,r) is small. Thus,
ϕ3 = −
1,r+1 , Xk+1,r , …, XM,r )Xk,r+1
+𝒩 k (τ, X1,r+1 , X2,r+1 , …, Xk−
1,r+1 , Xk,r , …, XM,r )
, υ3 = 0,
1
(Ri − a22 ), υ4 = − 2RHX1 − X1 X5 ,
δ2
1
X1 X4
ϕ5 = 2 (Ri − 4π4 ), υ5 =
,
2
δ
1
, υ6 = RX1 − X1 X7 ,
ϕ6 = −
Le
∂𝒩 k
(Xk,r+1 − Xk,r ).
∂yk
= ℒk (τ, X1,r+1 , X2,r+1 , …, Xk−
Va
ςδ2
ϕ4 =
A quasilinearisation scheme [70–72] which has good convergence rate
is developed by applying sequential linearisation in Xk to obtain
Ẋ k,r+1
4π2
X1 X6
.
, 𝒩7 =
2
δ2 Le
The parameters for the LQBHM becomes
= fk (τ,X1 ,X2 ,…,XM ) = ℒk (τ,X1 ,…,Xk− 1 ,XM )Xk +𝒩 k (τ,X1 ,X2 ,…,XM ),
𝒩 k (t, Xk,r+1 ) ≈ 𝒩 (t, Xk,r ) +
, 𝒩 3 = 0,
1
(Ri − a22 ), 𝒩 4 = − 2RHX1 − X1 X5 ,
δ2
1
X1 X4
ℒ5 = 2 (Ri − 4π4 ), 𝒩 5 =
,
2
δ
1
ℒ6 = −
, 𝒩 6 = RX1 − X1 X7 ,
Le
i=1,j=1
X˙1
X˙2
⋮
X˙k
⋮
X˙M
Va
ςδ 2
ϕ7 = −
4π2
X1 X6
, υ7 =
.
2
δ2 Le
5.1. Validation and error analysis of the method
(78)
X˙1 = ℒ1 (τ, X2 , X3 , X4 , X5 , X6 , X7 )X1 + 𝒩 1 (τ, X1 , X2 , X3 , X4 , X5 , X6 , X7 ),
(80)
X˙2 = ℒ2 (τ, X1 , X3 , X4 , X5 , X6 , X7 )X2 + 𝒩 2 (τ, X1 , X2 , X3 , X4 , X5 , X6 , X7 ),
(81)
X˙3 = ℒ3 (τ, X1 , X2 , X4 , X5 , X6 , X7 )X3 + 𝒩 3 (τ, X1 , X2 , X3 , X4 , X5 , X6 , X7 ),
(82)
X˙4 = ℒ4 (τ, X1 , X2 , X3 , X5 , X6 , X7 )X4 + 𝒩 4 (τ, X1 , X2 , X3 , X4 , X5 , X6 , X7 ),
(83)
X˙5 = ℒ5 (τ, X1 , X2 , X3 , X4 , X6 , X7 )X5 + 𝒩 5 (τ, X1 , X2 , X3 , X4 , X5 , X6 , X7 ),
(84)
The corresponding error profiles for selected solutions of Xr are
displayed in Fig. 2 when h = 0.1, Nt = 1000 for 14 iterations and
collocation points M = 2, 3, 4, 6. It can be observed from the graphs that
the method converges at increasingly higher accuracy levels for larger
values of M as the number of iterations increases. This demonstrates that
as the number of collocation points increases, the LQBHM becomes more
stable. It can also be seen that the method converges fully within 14
iterations for all solutions of Xr. In addition, fewer iterations are needed
for higher values of M. The above factors justify the use of M = 6 in this
work.
Fig. 3 displays the pictorial comparison between the selected time
series LQBHM solution and the ode45 generated solution of the coupled
nonlinear Lorenz type equations (51) - (57). It is found that these two
results are in acceptable agreement, demonstrating the validity of the
LQBHM. It is to note that we have considered the value of RaT = 38
which is below the critical value to ensure the stability of the system.
Due to this reason, the amplitude of the profiles seems to die down as
time progress.
X˙6 = ℒ6 (τ, X1 , X2 , X3 , X4 , X5 , X7 )X6 + 𝒩 6 (τ, X1 , X2 , X3 , X4 , X5 , X6 , X7 ),
(85)
6. Results and discussion
X˙7 = ℒ7 (τ, X1 , X2 , X3 , X4 , X5 , X6 )X7 + 𝒩 7 (τ, X1 , X2 , X3 , X4 , X5 , X6 , X7 ),
(86)
The study of double-diffusive convection in a rotating anisotropic
porous fluid layer with a fraction of thermal heat being generated
internally involves the external regulation of heat and mass transfer
characteristics. The goal of this work is to focus on three mechanisms,
∂𝒩 k
(X
− Xk,r ).
∂Xk k,r+1
+
The LQBHM is now applied with
ϕ = ℒk +
∂𝒩 k
,
∂Xk
υ = 𝒩 k − Xk,r
∂𝒩 k
.
∂Xk
(79)
Eqs. (51) - (57) are now expressed as
with
8
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International Communications in Heat and Mass Transfer 137 (2022) 106266
Fig. 3. Comparison between the LQBHM and ode45 results for X1, X2, X6 and X7 profiles.
namely internal heating, the presence of a solute concentration, and
rotational modulation, for promoting or inhibiting convective heat and
mass transport. To analyze heat and mass transfer we use the nonlinear
stability theory. The Nusselt number Nu and Sherwood number Sh are
employed to represent heat and mass transmission, respectively.
The other interesting part of this work is the use of the newly
developed Local Quasilinearisation Block Hybrid Method LQBHM to
solve the nonlinear Lorenz-type system of equations and this method
necessitated the analysis of heat and mass transport in the fluid system.
The effect of rotational modulations is assumed to be of infinitesimal
Fig. 4. Effect of Ta, Ri and Va on the stability curves for stationary and oscillatory convection against the wave number α with fixed parameters Le = 2, Ri = 5, η = 1,
ς = 1, RaS = 10 and Ta = 15.
9
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International Communications in Heat and Mass Transfer 137 (2022) 106266
Fig. 5. Changes in Nu and Sh with revised Rayleigh number R for modulation amplitudes δ1 = 0.2, 0.5, 0.8 and 1.1 with fixed parameters Va = 2, Le = 4, Ri = 20,
ω = 2, η = 0.4, ς = 0.6, RaS = 100 and Ta = 25.
order O(ε). For this reason we only consider small values of amplitude
modulations δ1 ⩽ 0.5. For the parameters, internal Rayleigh number Ri,
Vadasz number Va, thermal anisotropy η, Lewis number Le, solute
Rayleigh number RaS and Taylor number Ta, we have used the values in
Altawallbeh et al. [39], Bhadauria et al. [37], Srivastava et al. [21] and
Gaikwad et al. [31].
The linear stability analysis is performed to determine the stability
criteria in terms of the critical Rayleigh number Ra, below which the
system is stable and above which it is unstable. The effects of rotation,
Vadasz number and internal heating on the marginal stability plots are
displayed in Fig. 4. From Fig. 4a, it is observed that rotation has the
effect of increasing both the critical stationary and oscillatory Rayleigh
numbers, which delays the onset of convection in the system. When
Ta = 0, Fig. 4a illustrates the same plots obtained by Bhadauria [35].
Fig. 4b shows that internal heating has the effect of reducing the critical
stationary and oscillatory Rayleigh numbers, implying that it can act as a
destabilizing agent. The effect of Vadasz number on oscillatory con­
vection is illustrated by Fig. 4c. It is observed that increasing the Vadasz
number increases the oscillatory Rayleigh number showing that Va de­
lays the onset of oscillatory convection. From the linear stability results,
it is observed that, for oscillatory instability to set in, the condition
Raosc ≤ Rast is established. The results obtained in this section are similar
to the results obtained by Bhadauria et al. [37]. We are not going to
spend more time explaining the linear stability of the problem as it has
been discussed in the literature by authors including Altawallbeh et al.
[39], Bhadauria et al. [37], Srivastava et al. [21] and Gaikwad et al.
[31].
In this study, the truncated double Fourier representation is used to
provide the quantitative information on heat and mass transfer across
the anisotropic porous layer. The effects of various parameters on the
non-linear stability of the fluid system are investigated through their
effects on the heat and mass transfer profiles. Fig. 5 shows the plots of
the amplitude of modulation with revised Rayleigh number. The figure
shows that increasing the amplitude of modulation increases both heat
and mass transfer, which advances the onset of instability in the system.
The result confirms the result of Bhadauria and Kiran [73]. In this study,
the effect of modulation frequency was found to be negligible, therefore,
the graphical analysis is not presented.
Fig. 6 shows the effect of the Vadasz number on the Nusselt and
Sherwood numbers against the revised Rayleigh number. It is observed
Fig. 6. Changes in Nu and Sh with revised Rayleigh number R for Vadasz numbers Va = 1.2, 1.4, 1.6 and 1.8 with fixed parameters δ1 = 2, Le = 4, Ri = 20, ω = 2,
η = 0.4, ς = 0.6, RaS = 100 and Ta = 25.
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S.A. Ali et al.
International Communications in Heat and Mass Transfer 137 (2022) 106266
Fig. 7. Changes in Nu and Sh with revised Rayleigh number R for internal Rayleigh numbers R1 = 5, 10, 20 and 30 with fixed parameters δ1 = 2, Le = 4, Va = 2,
ω = 2, η = 0.4, ς = 0.6, RaS = 100 and Ta = 25.
Fig. 8. Changes in Nu and Sh with revised Rayleigh number R for Lewis numbers Le = 2, 4, 6 and 8 with fixed parameters δ1 = 2, Va = 2, ω = 2, η = 0.4, ς = 0.6,
RaS = 100, N = 5 and Ta = 25.
that with an increase in the Vadasz parameter, both the Nusselt number
and Sherwood number increase, showing that the parameter advances
the onset of thermal convection in the system. The expressions of nondimensional parameters show that there is a direct proportionality be­
tween the Vadasz number and the Prandtl number, as well as an indirect
proportionality with the Darcy number. Increases in the Darcy number
are proportional to increases in the permeability of the porous media,
which tends to slow fluid flow and so needs more heating to initiate
diffusive convection. According to the aforementioned relationships, the
Darcy number delays the onset of thermal convection and therefore
stabilizes the porous system, whereas the Prandtl number accelerates
the onset of thermal convection and so destabilizes it.
In some industrial instances, it is discovered that the system creates
its source of heat, which opens up a new approach to establishing a
convective flow through local thermal production in the fluid layer. The
effect of internal heating on the Nusselt and Sherwood numbers with
revised Rayleigh number is shown in Fig. 7. It is observed that heat and
mass transport increases upon increasing internal heating. As a result,
internal heat generation enhances the onset of thermal convection in the
system. This confirms the results of Bhadauria et al. [74] and Bhadauria
et al. [35].
Fig. 8 indicates the effect of the Lewis number on Nusselt and
Sherwood numbers. It is observed that as the Lewis number increases,
more heat and mass transfer occurs in the porous layer. The increase in
heat and mass movements consequently expedites the onset of convec­
tion, resulting in the destabilization of the fluid system.
The influence of rotation on heat and mass transfer is depicted in
Fig. 9. It is seen that the heat and mass movements are decreased by
increasing rotation. Therefore, introducing rotation in the porous layer
will eventually delay the onset of convection and destabilize the fluid
system. This is in agreement with the results of Yadav [34].
The effects of the mechanical and thermal anisotropy parameters on
the transfer of heat and mass are shown in Figs. 10 and 11. It is observed
that as the values of both parameters are increased, there will be more
heat and mass transport in the fluid system. Therefore the two variables
promote the onset of convection hence destabilizing the fluid system.
Fig. 12 shows the effect of solute Rayleigh number on the Nusselt and
Sherwood numbers. The figure shows that the addition of salt in the
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International Communications in Heat and Mass Transfer 137 (2022) 106266
Fig. 9. Changes in Nu and Sh with revised Rayleigh number R for Taylor numbers Ta = 5, 15, 25 and 50 with fixed parameters δ1 = 2, Le = 4, Va = 2, ω = 2, η = 0.4,
ς = 0.6, RaS = 100, N = 5 and Le = 4.
Fig. 10. Changes in Nu and Sh with revised Rayleigh number R for the mechanical anisotropy parameter values ς = 0.2, 0.4, 0.6 and 0.8 with fixed parameters
δ1 = 2, Le = 4, Va = 2, ω = 2, η = 0.4, Ta = 25, RaS = 100, N = 5 and Le = 4.
porous layer stabilizes the system by reducing heat and mass transfer.
This is similar to the result obtained by Malashetty and Kollur [75]. In
each of the cases for different parameters, we observe that as the revised
Rayleigh number increases, the average Nusselt and Sherwood numbers
decrease sharply. It is again observed that in each case the average
Nusselt number is above the average Sherwood number, this is due to
the presence of internal heat source.
Fig. 13 shows the streamlines, isotherms, and isoconcentrations for
different buoyancy ratio term values. There are two distinct contours
observed. The negative and positive function values represent the
clockwise and anticlockwise flows, respectively. For double-diffusive
convection, concentration gradients are capable of producing a down­
ward buoyant force for negative N values, whereas temperature gradi­
ents produce an upward buoyant force. The combination of these two
opposite forces gives the driving power that dictates the nature and the
magnitude of the flow field. For N = − 5 it is observed that the stream
function values in the central contours are higher than for the positive
values of N, this is because of the thickness of the boundaries and this
shows that conduction is the major form of heat and mass transfer. As N
increases to positive values, the effects of the downward buoyancy
forces due to the concentration gradients are outweighed by the upward
buoyancy forces ascending from the temperature gradients. As a result,
the flow within the enclosure becomes very weak and the values of the
central contours decrease significantly. Buoyancy forces due to con­
centration and temperature differences are in the aiding mode for pos­
itive values of N.
Isotherms in the enclosures are shown in the middle row of Fig. 13.
At the lowest buoyancy ratio considered (N = − − 5), isotherms are
concentrated more spaced and the central contours are dense showing
that there is less thermal convection experienced in the system. As the
buoyancy ratio increases taking positive values, more light and closely
packed contours deviating from the boundaries are observed, this tells
that more convection is experienced. The analogous isoconcentration
maps for the cases discussed above are shown at the bottom of Fig. 13. It
is again observed that the distribution of isoconcentration lines over the
domain is affected by the positive changes in the values of N. At N = − 5
it is seen that the central contours have high magnitudes and they are
well spaced and closer to the boundaries highlighting that heat and mass
are transferred more by convection. As the buoyancy takes positive
values, the lighter contours which are closely packed and approaching
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International Communications in Heat and Mass Transfer 137 (2022) 106266
Fig. 11. Changes in Nu and Sh with revised Rayleigh number R for thermal anisotropy parameter values η = 0.4, 0.8, 1.2 and 1.6 with fixed parameters δ1 = 2,
Le = 4, Va = 2, ω = 2, ς = 0.6, RaS = 100, Ta = 25, N = 5 and Le = 4.
Fig. 12. Changes in Nu and Sh with revised Rayleigh number R for concentration Rayleigh numbers RaS = 50, 100, 200 and 300 with fixed parameters δ1 = 2, Le = 4,
Va = 2, ω = 2, ς = 0.6, η = 0.4, Ta = 25, N = 5 and Le = 4.
the center are observed.
• Graphical results for the time series solutions of the derived coupled
nonlinear Lorenz type equations confirmed that the LQBHM results
were in good agreement with the ode45 results. Further, increasing
the number of collocation points reduced LQBHM error.
7. Conclusion
The effect of time-varying rotation on the rate of transfer of heat and
mass in an anisotropic horizontal porous layer with internal heat gen­
eration and salted from below is studied analytically. The coupled Lor­
enz type of equations has been derived and solved numerically by the
newly developed local quasilinearisation block hybrid method LQBHM.
The following is a summary of the results of this study:
In summary we can show that the following relations hold:
1.
2.
3.
4.
5.
6.
7.
8.
• Increasing the Taylor number Ta, solutal Rayleigh number RaS, has
the effect of reducing the transfer of heat and mass hence delaying
the onset of convection.
• Effects of Vadasz number Va, internal Rayleigh number Ri, Lewis
number Le, anisotropy parameters η, ς and the rotational modulation
δ1 are to enhance the rate of heat and mass transfer as they are
augmented.
[Nu/Sh]δ1 =0.2 < [Nu/Sh]δ1 =0.5 < [Nu/Sh]δ1 =0.8 < [Nu/Sh]δ1 =1.1 .
[Nu/Sh]Va=1.2 < [Nu/Sh]Va=1.4 < [Nu/Sh]Va=1.6 < [Nu/Sh]Va=1.8.
[Nu/Sh]Ta=50 < [Nu/Sh]Ta=25 < [Nu/Sh]Ta=15 < [Nu/Sh]Ta=5.
[Nu/Sh]RaS =300 < [Nu/Sh]RaS =200 < [Nu/Sh]RaS =100 < [Nu/Sh]RaS =50 .
[Nu/Sh]Le=2 < [Nu/Sh]Le=4 < [Nu/Sh]Le=6 < [Nu/Sh]Le=8
[Nu/Sh]Ri =5 < [Nu/Sh]Ri =10 < [Nu/Sh]Ri =20 < [Nu/Sh]Ri =30 .
[Nu/Sh]ς=0.2 < [Nu/Sh]ς=0.4 < [Nu/Sh]ς=0.6 < [Nu/Sh]ς=0.8.
[Nu/Sh]η=0.4 < [Nu/Sh]η=0.8 < [Nu/Sh]η=1.2 < [Nu/Sh]η=1.6.
CRediT authorship contribution statement
Samah A. Ali: Visualization, Investigation, Writing - original draft.
13
S.A. Ali et al.
International Communications in Heat and Mass Transfer 137 (2022) 106266
Fig. 13. The distribution of streamlines (top), isotherms (middle), and isoconcentration (bottom) for different buoyancy ratio N values.
Munyaradzi Rudziva: Writing – original draft, Visualization, Investi­
gation. Precious Sibanda: Supervision, Writing – review & editing.
Osman A.I. Noreldin: Conceptualization, Methodology, Supervision,
Writing – review & editing. Sicelo P. Goqo: Conceptualization, Meth­
odology, Supervision. Hloniphile Sithole Mthethwa: Supervision.
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Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgements
The authors are thankful for the support they received from the
University of KwaZulu-Natal.
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