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Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19
Contents lists available at ScienceDirect
Journal of the Taiwan Institute of Chemical Engineers
journal homepage: www.elsevier.com/locate/jtice
Thermo-solutal buoyancy-opposed free convection of a binary
Ostwald–De Waele fluid inside a cavity having partially-active
vertical walls
Sofen K. Jena a,∗, Swarup K. Mahapatra b, Amitava Sarkar a, Ali J. Chamkha c
a Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, Indiab School of Mechanical Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar 751013, Indiac Department of Mechanical Engineering, Prince Mohammad Bin Fahd University (PMU), P.O. Box 1664, Al-Khobar 31952, Saudi Arabia
a r t i c l e i n f o
Article history:
Received 15 July 2014
Revised 6 January 2015
Accepted 10 January 2015
Available online 18 February 2015
Keywords:
Power-law fluid
Double-diffusive convection
Partially-active zones
Critical buoyancy ratio
a b s t r a c t
The present study investigates double-diffusive free convection of a non-Newtonian fluid inside a cavity
having discretely active walls of source and sink for heat and solute mass. The Ostwald–De Waele model has
been adopted for non-Newtonian fluid behavior. Five discrete heating and salting arrangements in the vertical
walls are considered in the present analysis. A numerical solution is obtained using the control volume-based
integration technique. The Modified Marker and Cell (MMAC) method is used for obtaining the numerical
solution of the governing equations. A gradient-dependent consistent hybrid upwind scheme of second order
(GDCHUSSO) is used for the discretization of the convective terms. The effect of the power-law index and the
buoyancy ratio for different heating and salting arrangements has been analyzed to determine the heat and
mass transfer characteristics. Interesting results of stability at a critical buoyancy ratio is noticed for dilatant
fluids. The isotherm, isoconcentration and flow line contour maps are provided to bring the clarity in the
understanding of flow, temperature and mass distributions for different arrangements. Graphical and tabular
records of heat and mass transfer characteristics are provided for understanding the various effects of the
pertinent parameters.
© 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
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. Introduction
In the past few decades; experimental and numerical research on
nternal flows inside a cavity got an overwhelming attention from
esearchers round the globe for its significant applications in natural
nd artificial systems. Most of the works on natural convection are
ased on imposed thermal gradients. But most of the geophysical and
ndustrial flows are induced by the simultaneous action of thermal
nd solutal gradients. Such kinds of flows, onset by combined mecha-
ism of thermal and solutal gradients are called double diffusive free
onvective flow. The literature on double diffusive free convection is
mmense and well documented to highlight various significant fea-
ures of the phenomenon (Bejan [1], Gebhart and Pera [2], Ostrach
3], Trevisan and Bejan [4], Viskanta et al. [5], Chamkha and Al-Naser
6,7]). The oscillating flow described by Bénard et al. [8] and Nishimura
t al. [9] at unit buoyancy ratio and the Lewis number range predicted
y Sezai and Mohamad [10], are the two important aspects of double
∗ Corresponding author.
E-mail addresses: [email protected] (S.K. Jena), [email protected]
A.J. Chamkha).
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ttp://dx.doi.org/10.1016/j.jtice.2015.01.007
876-1070/© 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All righ
iffusive convection. Experimental studies on thermosolutal convec-
ion within rectangular cavities with horizontal temperature and
oncentration gradients have been explained by several researchers
Kamotani et al. [11], Lin et al. [12], Ranganathan and Viskanta [13]).
The classical literature discussed above is restricted to flow of
ewtonian fluids. But applications like polymer processing, biological
ows, petroleum product extraction, chemicals and fertilizer produc-
ion, medicine and food processing, printing ink, paint and emulsion
anufacturing and packing require the knowledge of non-Newtonian
ows. Acrivos [14] is considered to be the pioneer in this field, who
as analytically derived the flow and heat transfer characteristics for
on-Newtonian fluids. Pittman et al. [15] have experimentally stud-
ed the natural convection between vertical plates for both Newtonian
nd non-Newtonian fluids, and have shown how the free convection
ubstantially gets affected by the rheological behavior of fluids. Non-
ewtonian fluids are mainly classified into two categories depend-
ng upon the developed shear stress in the fluid. Many polymeric
ystems, emulsions, suspensions and foams exhibit shear-thinning
ehavior. In these the fluid shear stress decreases with increase in
hear rate. But in many thick pastes, paints and organic oils, the shear
tress increases with an increase in the shear rate, this type of fluid
s known as a shear-thickening or a dilatant fluid. Many theories are
ts reserved.
10 S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19
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Nomenclature
C solute concentration (kg/m3)
Cp specific heat capacity (J/kg K)
D solute mass diffusivity (m2/s)
g acceleration due to gravity (m/s2)
H length of active region (m)
i, j nodal index
k thermal conductivity (W/m K)
K consistency coefficient (N sn/m2)
L length of the cavity (m)
Le Lewis number
Nu Nusselt number
Nu average Nusselt number
n power-law index
n̂ unit normal vector
P dimensionless pressure
Pr system-wide Prandtl number
Ra system-wide thermal Rayleigh number
Ras system-wide solutal Rayleigh number
Sh Sherwood number
Sh average Sherwood number
T absolute temperature (K)
t time (s)
u, v dimensional velocity component (m/s)
U, V dimensionless velocity component
x, y dimensional coordinate (m)
X, Y dimensionless coordinate
Greek letters
α thermal diffusivity (m2/s)
βT coefficient of thermal expansion (K−1)
βC coefficient of compositional expansion (m3/kg)
� dimensionless concentration
μ dynamic viscosity (kg/m s)
ν kinematic viscosity (m2/s)
� dimensionless temperature
ρ density (kg/m3)
τ dimensionless time
τij dimensionless stress tensor
℘ buoyancy ratio
F dimensionless length of active region
L starting point of active zone
Subscripts
H higher
L lower
0 reference
provided by researchers for these rheologically complex fluids. But the
Ostwald–De Waele fluid behavior or power-law fluid behavior is a
very promising theory for complex rheological flows. This theory pro-
vides a flexibility to characterize the behavior of the fluid in terms of
its power index. A detailed description on the power-law fluid is found
in the works of Chen and Wollersheim [16], Emery et al. [17] and
Shulman et al. [18]. Dale and Emery [19] have studied free convec-
tion heat transfer from a vertical plate to non-Newtonian pseudo-
plastic fluids. Ng and Hartnett [20] have studied natural convection
of power-law fluids. They have recalculated the experimental data
of (Gentry [21], Gentry and Wollersheim [22]) by their transforma-
tion method and compared the same with the analytical results of
Acrivos [14]. They have noticed the good agreement of both the ex-
perimental and the analytical data for non-Newtonian fluids in terms
of the power-law index. Kim et al. [23] have studied the transient
buoyant convection of a power-law fluid in an enclosure. They have
oticed the qualitative agreement of the scaling analysis results with
he numerical results, and have proposed a correlation for the steady-
tate Nusselt number. Recently, Turan et al. [24] have studied laminar
ree convection of a power-law fluid inside a square enclosure with
ifferentially-heated side walls. They have noticed that the mean
usselt number increases for a shear-thinning fluid and decreases
or a shear-thickening fluid compared to that of the corresponding
ewtonian fluid. They have also noticed that the mean Nusselt num-
er marginally affected by the increase in the Prandtl number for
ower-law fluids.
All the literature discussed above is about a single component
riven flow of non-Newtonian power-law fluid. But in many situa-
ions, convection onsets in the binary non-Newtonian fluid by the
ombined mechanism of thermal and solutal gradients. Thus, the
tudy of double-diffusive flow of a non-Newtonian fluid is essen-
ial. A literature review on the subject shows the existence of few
orks on double-diffusive convection of power-law fluids. Rastogi
nd Poulikakos [25] are the first to study double-diffusive convection
n a power-law fluid. They have studied the heat and mass transfer
cross a vertical plate embedded in a porous matrix saturated with
power-law fluid, through a scaling analysis and a numerical solu-
ion. They have noticed the variation of heat and mass transfer with
he power-law index and the buoyancy ratio with both isothermal
nd constant heat flux boundary conditions. Getachew et al. [26] are
he pioneer to study the double-diffusive convection inside a rect-
ngular porous cavity saturated by non-Newtonian fluids. They have
btained the heat and mass transfer rates for a wide range of param-
ters, through a scaling analysis and a numerical solution. Jumah and
ujumdar [27] have studied free convection heat and mass transfer of
non-Newtonian power-law fluid in an embedded saturated porous
edium around a vertical flat plate. They have concluded that the
nset of convection is possible when a certain inequality is satisfied,
hich strongly depends upon the yield stress. Benhadji and Vasseur
28] have used the Darcy model with the Boussinesq approximation
o study double-diffusive convection in a shallow porous cavity sat-
rated with a non-Newtonian power-law fluid. They have used an
nalytical method using parallel flow approximation for the solution.
hey have obtained the Nusselt and Sherwood numbers explicitly in
erms of the governing parameters. Cheng [29] has studied free con-
ective heat and mass transfer from a vertical plate in porous media
aturated with a non-Newtonian power-law fluid. In his study, active
alls are subjected to variable heat and mass fluxes. The governing
quations are transformed into a dimensionless form by the similarity
ransformation method and are solved by the cubic spline collocation
ethod. He has noticed the existence of a threshold pressure gradient
n a power-law fluid, which decreases the fluid velocity as well as the
ocal Nusselt and Sherwood numbers. Makayssi et al. [30] have stud-
ed double-diffusive free convection of a power-law fluid in a shallow
orizontal rectangular cavity with uniformly heated and salted side
alls. The study is limited to the buoyancy cooperating case. They
id the numerical experiment to predict the effect of the aspect ratio,
ewis number, buoyancy ratio and the power-law index on the con-
ection heat and mass transfer. They have also derived an analytical
xpression based on the parallel flow approximation and noticed a
ood agreement of the same with the numerical results. Makayssi
t al. [31] have extended their work (Makayssi et al. [30]) to the
pposing buoyancy condition with equal strength of thermal and so-
utal buoyancy forces. They have concluded that compared to New-
onian fluid, the shear-thinning fluid enhances the fluid circulation.
heng [32,33] has studied heat and mass transfer for a non-Newtonian
ower-law fluid through a porous medium from a vertical wavy sur-
ace and cone, respectively. Kairi and Murthy [34] have studied the ef-
ect of viscous dissipation and Soret effect on free convective heat and
ass transfer from a cone in a non-Darcy porous material saturated
ith a non-Newtonian power-law fluid. They have numerically solved
he governing equation by local non-similarity method. They have
S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19 11
TL
CL
TL
CL
TL
CL
TH
CH
TH
CH
TH
CH
TL
CL
TL
CL
TH
CH
TH
CH
Middle Middle (MM) Top Top (TT) Bottom Bottom (BB)
Bottom Top (BT) Top Bottom (TB)
X, U
Y, V
X, U
Y, V
X, U
Y, V
X, U
Y, V
X, U
Y, V
H
L
L
L
L
L
H
H
H
H
g g g
g g
Fig. 1. Schematic arrangements.
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ρ
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β
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T
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�
t
F
p
[
τ
τ
τ
oticed a significant change in heat and mass transfer coefficients
ith the power-law index. Recently, Khelifa et al. [35] have stud-
ed free convection in a horizontal porous cavity filled with a non-
ewtonian binary power-law fluid. They have derived an analytical
xpression on the basis of parallel flow approximation for a shallow
nclosure. They have predicted the criteria for subcritical and super-
ritical onset of motions.
The previous studies are limited to boundary layer type analy-
is or flow inside a cavity having complete active walls. But many
ndustrial and real life applications like liquid metal flow through
endritic structures in alloy casting, paint and emulsion preparation
nd mixing, involve convection subjected to discrete active walls.
atural convection in a discretely heating and cooling chamber is a
ell established study and the literature available on it, provides the
nformation about the relative positions for heating and cooling to op-
imize heat transfer. Valencia and Frederick [36] have done a classical
enchmark work on natural convection in a discretely heated and
ooled chamber. Recently, Sankar et al. [37] have extended the study
o porous media filled discretely heated and cooled chamber and have
rovided the flow and heat transfer characteristics for various prop-
rties of the porous media. Applications like organic fluid reservoirs,
aramedical material and food processing, liquid medicine, paint, ink
nd emulsion preparation process involve double diffusive convec-
ion of non-Newtonian fluids with discrete solute and heat sources.
rom an extensive literature review, it is noticed that no such study
eems to be available at present. Therefore, the present study is con-
erned with double diffusive buoyancy-opposed free convection of a
on-Newtonian power-law fluid inside a discretely-active enclosure.
ive different arrangements of discrete active heat and solute sources
nd sinks along the vertical walls are considered in the present study.
he study is focused on the effect of rheological properties of the
ower-law fluid on fluid flow, heat and mass transfer at different
uoyancy ratios and different arrangements. An extensive paramet-
ic study has been done numerically to depict the effects of pertaining
arameters on the flow and heat and mass transfer mechanisms.
. Mathematical formulation and governing equations
A two-dimensional square enclosure having a length L is filled
ith a non-Newtonian fluid. A portion H of both the vertical walls of
he enclosure are thermally and solutally active, while the remaining
ortions of the vertical walls and the horizontal walls are considered
o be adiabatic and impermeable. H = L/2 is kept fixed and five differ-
nt arrangements with partially heating and salting zones have been
onsidered as shown in Fig. 1 for the analysis purpose. The flow of
he non-Newtonian fluid is assumed to be incompressible and lam-
nar in the working range of parameters. All of the thermo-physical
roperties of the fluid are assumed to be constant except the density
hose variation is assumed as per the Boussinesq approximation. The
quation of state for the fluid is expressed as follows:
= ρ0(1 − βT(T − TL)− βC(C − CL)) (1)
here
T = − 1
ρ0
[∂ρ
∂T
]p,C
and βC = − 1
ρ0
[∂ρ
∂C
]p,T
ith βT > 0 and βC < 0. All the thermo-physical properties are
stimated at a reference temperature TL and concentration CL.
he Ostwald–De Waele (power-law) model is taken for the non-
ewtonian fluid. The following relationships are used to normalize
he variables (X, Y) = (x,y)L , F = H
L , (U, V) = (u,v)ν̄/L
, τ = tL2/ν̄
, � = T−TLTH−TL
,
= C−CLCH−CL
with TH > TL and CH > CL, μ̄a = μa
K(
ν̄L2
)(n−1) .
The dimensionless form of the mass continuity and the momen-
um equations are expressed as
∂U
∂X+ ∂V
∂Y= 0 (2)
∂U
∂τ+ ∂
∂X(U2 + P − τXX)+ ∂
∂Y(UV − τXY) = 0 (3)
∂V
∂τ+ ∂
∂X(UV − τYX)+ ∂
∂Y(V2 + P − τYY)− Ra
Pr(� + ℘�) = 0 (4)
or an Ostwald–De Waele power-law fluid, the stress tensors are ex-
ressed as (Acrivos [14], Chen and Wollersheim [16], Dale and Emery
19], Emery et al. [17], Pittman et al. [15])
XX = 2μ̄a∂U
∂X(5)
YY = 2μ̄a∂V
∂Y(6)
XY = τYX = μ̄a
(∂U
∂Y+ ∂V
∂X
)(7)
12 S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19
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The constitutive equation for the apparent viscosity for an
Ostwald–De Waele fluid is expressed in dimensionless form as
μ̄a ={
2
(∂U
∂X
)2
+ 2
(∂V
∂Y
)2
+(
∂U
∂Y+ ∂V
∂X
)2} n−1
2
(8)
The constitutive equation for the apparent kinematic viscosity for
the power-law fluid is derived based on the physical rationalizations
and a trial and error method, as discussed in Pittman et al. [15] and
Kim et al. [23].
ν̄ ≡(
K
ρ0
) 1(2−n)
L2(1−n)(2−n) (9)
Neglecting viscous dissipation and internal energy generation, the
energy conservation equation is expressed as
∂�
∂τ+ ∂
∂X
(U� − 1
Pr
∂�
∂X
)+ ∂
∂Y
(V� − 1
Pr
∂�
∂Y
)= 0 (10)
The species transport equation neglecting all cross gradient effects
is expressed as
∂�
∂τ+ ∂
∂X
(U� − 1
Pr Le
∂�
∂X
)+ ∂
∂Y
(V� − 1
Pr Le
∂�
∂Y
)= 0 (11)
The dimensionless numbers used in the above equations are ex-
pressed as
Le = α
D, Pr = 1
α
(K
ρ0
) 1(2−n)
L2(1−n)(2−n) , Ra = gβT L3(TH − TL)
α(
Kρ0
) 1(2−n)
L2(1−n)(2−n)
,
Ras = gβCL3(CH − CL)
D(
Kρ0
) 1(2−n)
L2(1−n)(2−n)
, ℘ = Ras
RaLe
No slip and no penetration velocity boundary conditions are as-
sumed at all solid walls. They can be expressed in dimensionless form
as
U = V = 0 at X = 0, 1 and Y = 0, 1
The horizontal walls are assumed to be perfectly insulated and im-
permeable. The governing boundary conditions for the temperature
and the concentration in dimensionless form are expressed as
∂�
∂Y= ∂�
∂Y= 0, Y = 0, 1
The boundary conditions for the temperature and the concentra-
tion at the vertical walls are expressed as
� = 1� = 1
∣∣∣∣X=1
Y ∈ [L,L + F]
� = 0� = 0
∣∣∣∣X=0
Y ∈ [L,L + F]
∂�
∂X= 0
∂�
∂X= 0
∣∣∣∣∣∣∣∣X=0,1
Y ∈ (1 − [L,L + F])
where L is the starting point of the respective active zone, and is
expressed as
L = 0 Bottom
L = 0.25 Middle
L = 0.50 Top
The initial conditions are taken as follows:
τ = 0 : U = V = 0, � = � = 0, 0 < (X, Y) < 1
The heat and mass fluxes across the discretely heated and salted
avity, are expressed in terms of the dimensionless Nusselt Number
Nu) and the Sherwood Number (Sh). They are defined as
Nu = −∂�
∂X
∣∣∣∣X=0,1,Y∈[L,L+F]
(12)
Sh = −∂�
∂X
∣∣∣∣X=0,1,Y∈[L,L+F]
(13)
The average heat and mass transfer rates across the cavity are
btained by integrating the heat and mass fluxes over the active walls,
nd are expressed as
u = 1
F
∫ L+F
L
−∂�
∂XdY
∣∣∣∣X=0,1
(14)
h = 1
F
∫ L+F
L
−∂�
∂XdY
∣∣∣∣X=0,1
(15)
. Solution technique and validation
It is observed from the previous section that the governing equa-
ions are highly nonlinear in nature and are coupled through their
ource term. As no closed-form solution exists, their solutions through
umerical schemes are reported in the present work. The modified
arker and Cell (MMAC) method developed by Hirt and Cook [38]
s used to solve the mass continuity and momentum equations. The
nvestigation by Kim and Benson [39] suggests the significance of
he MAC method for accuracy and computational effort. A partially-
taggered grid arrangement described by Vanka et al. [40] is taken for
he pressure and velocity correction. In the partially-staggered grid,
he pressure is calculated at the center of the control volume, and
ll other scalar and vector variables are calculated at the cell corners.
he finite-difference forms of the governing equations are derived
y integrating the equations over an elementary control volume. The
quation for pressure is obtained by taking the divergence of velocity
cross the elementary control volume such that there is no net mass
ow in or out of the control volume. The solution of the pressure Pois-
on equation involves simultaneous iteration on pressure and velocity
omponents to obtain a divergence-free velocity field as described by
horin [41]. In the pressure correction equation, the pressure correc-
ion in the neighboring cells is taken as zero, as described by Biswas
Muralidhar and Sundararajan [42]). This is equivalent to the solution
f the full Poisson equation for the pressure as shown by Brandt et al.
43]. The operator-splitting algorithm (Issa [44]; Muralidhar et al.
45]) is used to solve the energy and concentration equations. The
perator-splitting algorithm properly identifies the mixed mathe-
atical character of the governing differential equation and splits the
quation into a series of homogeneous components. The advection
iffusion is hyperbolic in nature when the diffusion part is neglected
hereas it becomes parabolic in nature when the advection part is ne-
lected. The operator-splitting algorithm neglects the diffusion term
nd solves the advection part to get a predicted field, again it neglects
he advection part and solves the diffusion part taking the predicted
alue to obtain the correct field. The false diffusion is absent in the
ase of operator-splitting algorithm as there is no upwinding in the
ethod. The governing differential equations (2)–(8) have been dis-
retized over an elementary control volume. The physical domain is
iscretized into a non-uniform Tchebycheff grid (Haldenwang [46]),
ith denser grid clustering near the boundaries, and a coarser mesh
ystem in the core region. The resulting sets of the discretized equa-
ions for each variable are solved by the Alternating Direction Implicit
ADI) scheme with a line-by-line procedure, combining the LU decom-
osition for matrix inversion. SSOR pre-conditioning (Hackbush [47])
s used for accelerating the convergence rates. In the present study, a
radient-dependent consistent hybrid upwind scheme of second or-
er (GDCHUSSO) is used for the discretization of the advection term
S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19 13
Table 1
Grid sensitivity study (Ra = 105, Pr = 10, Le = 3, ℘ = −0.4,
n = 1, MM).
Mesh size 40 × 40 52 × 52 60 × 60 72 × 72 80 × 80
Nu 6.221 6.213 6.206 6.202 6.199
Sh 9.321 9.310 9.301 9.294 9.287
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Fig. 2. Validation with Sezai and Mohamad [10].
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n the flow equations. In a five-point stencil, the nonlinear convective
erm, discretized using the GDCHUSSO scheme is as follows:
∂U2
∂X
∣∣∣∣i,j
= 1
4δX[(Ui,j + Ui,j+1)(Ui,j + Ui,j+1)
+�|(Ui,j + Ui,j+1)|(Ui,j − Ui,j+1)
− (Ui,j−1 + Ui,j)(Ui,j−1 + Ui,j)
−�|(Ui,j−1 + Ui,j)|(Ui,j−1 − Ui,j)] (16)
here � is a weighted average central and an upwind differencing
actor. When � = 0, and � = 1, Eq. (16) is space-centered and fully
pwind, respectively. The upwind scheme helps to suppress the os-
illations but suffers from false diffusion, while the central difference
cheme helps to improve the accuracy but suffers from instability.
he hybrid scheme discussed above is the combination of the cen-
ral difference and the upwind schemes (Gentry et al. [48], Raithby
nd Torrance [49], Roache [50]). The latter is employed in the region
here the solution gradients are high and the former is used in the
ow gradient region. The value of � in the domain is selected based on
he gradient. The second-order central difference scheme is adopted
or the diffusion terms. The overall accuracy of the solution is second
rder in space. The Simpson’s 1/3rd rule of integration is used to eval-
ate Eqs. (14) and (15). A global tolerance of 10−7 is taken between
wo successive iterations for convergence. After a detailed discussion
f the adopted numerical schemes and their solution procedure, the
enerated code is subscribed with a grid-independence test along
ith validations against relevant earlier works.
A grid sensitivity analysis is carried out, and the results obtained
re presented in Table 1. As per the grid-independence test, a mesh
ize of 60 × 60 has been adopted, in order to optimize the compu-
ational effort with the desired accuracy. Prior to discussion of the
btained results, the numerical code has been tested and validated
ith existing benchmark literature results. The code is validated with
alencia and Frederick [36], for natural convection of a Newtonian
uid in a partially-active cavity and the results are shown in Table 2.
he code for double diffusion is validated with Sezai and Mohamad
10] and the results are provided in Fig. 2. The code has been vali-
ated with Turan et al. [24] for the convection of a power-law fluid,
nd the results are provided in Table 3. The results out of the current
Table 2
Comparison of average Nusselt number for cavity ha
Frederick [36].
Ra = 105
TB MM
Valencia and Frederick [36] Nu 3.295 6.3
Present study Nu 3.263 6.3
Table 3
Comparison of average Nusselt number for power law fluid
Nu
n = 0.6 n = 0.8 n
Ra = 105 Turan et al. [24] 14.49 7.63 4
Present study 14.55 7.68 4
Ra = 106 Turan et al. [24] 34.05 16.21 9
Present study 34.12 16.26 9
umerical code are found to be in good agreement with the earlier
ited works.
. Results and discussion
In the present study, numerical simulations have been performed
o investigate the effect of various parameters on double-diffusive
onvection of a power-law fluid in a partially-active enclosure. In
rder to identify the sole effect of the parameters, some parame-
ers are kept fixed, while the remaining parameters are varied with
eaningful combination. The system-wide thermal Rayleigh number
Ra = 105) is kept fixed. Turan et al. [24] have concluded that convec-
ion of a power-law fluid is marginally affected by the Prandtl num-
er. In the current study, the system-wide Prandtl number (Pr = 10) is
ept fixed. Sezai and Mohamad [10] have shown the range 0 < Le < 4
or two-dimensional flows. In the present study, Le = 3 is also kept
xed. The power-law index is varied from 0.6 ≤ n ≤ 1.6 to character-
ze the effect of the power-law index on the convection of the non-
ewtonian fluid. Results are depicted for shear-thinning (pseudo-
lastic, n < 1), shear-thickening (dilatants, n > 1) and Newtonian
n = 1) fluids. The buoyancy ratio ℘ signifies the relative strength
f solutal to thermal buoyancy forces. It can be either positive or
egative. The positive value indicates the buoyancy-assisting situa-
ion, while the negative value indicates the buoyancy-opposing con-
ition. The current study deals with buoyancy counter-acting flow, so
hree discrete values ℘ ∈ {−0.4,−1,−4} are taken for the numerical
ving partially active walls with Valencia and
Ra = 106
BT TB MM BT
83 5.713 4.505 12.028 11.569
52 5.705 4.496 11.987 11.558
with Turan et al. [24] for Pr = 100.
= 1 n = 1.2 n = 1.4 n = 1.6 n = 1.8
.77 3.40 2.48 2.02 1.68
.83 3.44 2.51 2.06 1.71
.23 6.03 4.31 3.28 2.60
.27 6.08 4.36 3.33 2.65
14 S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19
a n b n c n
d n e n f n
g n h n i n
Fig. 3. Isotherms, isoconcentrations and flow lines distribution for Middle–Middle
(MM) arrangement and ℘ = −0.4.
a n b n c n
d n e n f n
g n h n i n
Fig. 4. Isotherms, isoconcentrations and flow lines distribution for Middle–Middle
(MM) arrangement and ℘ = −4.
b
d
t
fl
a
s
n
a
a
f
(
i
l
s
c
f
i
fl
c
a
l
t
−p
t
t
a
v
s
r
l
simulations. The flow transition at the critical buoyancy ratio ℘ = −1
is also described. Five discrete heating and salting locations are taken
for simulation, as shown in Fig. 1. They are Middle–Middle (MM), Top–
Top (TT), Bottom–Bottom (BB), Bottom–Top (BT) and Top–Bottom
(TB). For each arrangement, an extensive parametric study has been
done to delineate the flow, heat and solute transport mechanisms of
a non-Newtonian power-law fluid.
Fig. 3 shows the steady-state isotherms, isoconcentrations and
flow lines for Middle–Middle (MM) active regions for ℘ = −0.4. For
℘ = −0.4, the buoyancy force due to temperature gradient dominates.
For all power-law indices, the flow rises along the right heated wall,
and falls along the left cold wall (Fig. 3(c), (f), (i)). The entire cavity is
filled with a counter-clockwise fluid circulation. For a shear-thinning
fluid i.e. n = 0.6, the circulation is very strong and secondary vor-
tices are formed at the core (Fig. 3(c)). The isotherms are stratified
at the core (Fig. 3(a)), while the isoconcentrations are distorted in
the core (Fig. 3(b)). Both the isotherm and isoconcentration lines are
closely packed to the active boundary indicating the narrow ther-
mal and solutal boundary layer thicknesses for shear-thinning fluids.
When the power-law index is equal to unity n = 1, the fluid is New-
tonian and the flow becomes straight with a reduced hydrodynamic
boundary layer thickness. The formed secondary vortices are close
to each other with diminished size (Fig. 3(f)). The thermal and so-
lutal boundary layer thicknesses marginally enhance at the vicinity
of the active walls and the isotherms and isoconcentrations are a
little bit more smoothly placed in the core (Fig. 3(d) and (e)). For a
shear-thickening fluid n = 1.6, the flow loses its strength, and the
hydrodynamic boundary layer becomes relatively thick. The center
of rotation of the fluid coincides with the center of the cavity and
the secondary circulations do not appear (Fig. 3(i)). The thermal and
solutal boundary layer thicknesses are sufficiently thick, which are
indicated by the wide spreading distribution of the isothermal and
the isoconcentration lines at the vicinity of the active walls. The strat-
ification of the isotherms in the core is comparatively less and the
distortion in concentration appears to be less in the cavity (Fig. 3(g)
and (h)).
For ℘ = −4, the solutal buoyancy strength exceeds the thermal
uoyancy strength and the flow is solutally dominated. Fig. 4 eluci-
ates the steady-state results of a solutally-driven flow for middle
o middle configuration. It is seen that, for all power indices, the
ow forms a clockwise rotation throughout the cavity. The fluid falls
gainst the solutally-enriched right vertical wall and rises along the
olutally-deficient left vertical wall. For a pseudo-plastic fluid, i.e.,
= 0.6, the flow directly connects the source wall to the sink wall
t the mid of the cavity, and strong hydrodynamic boundary layers
re formed beside the vicinity of active walls. Secondary cells are
ormed at the right top and left bottom portions of the enclosure
Fig. 4(c)). The isotherm and isoconcentration distributions are strat-
fied and compact at the core (Fig. 4(a) and (b)). The secondary circu-
ations form at the corners due to the absence of steep thermal and
olutal gradients in the region. For a Newtonian fluid, the secondary
irculation at the corner diminishes, but the secondary vortices are
ormed in the vicinity of the active walls (Fig. 4(f)). The isotherm and
soconcentration distributions are a little bit spread for Newtonian
uids (Fig. 4(d) and (e)). For a dilatants fluid, the corner secondary
irculation disappears, but strong secondary circulations are formed
t the core (Fig. 4(i)). The isotherms are oriented uniformly along the
eading diagonal in a stratified manner (Fig. 4(g)), while the stratifica-
ion of the isoconcentrations is limited to the core region (Fig. 4(h)).
The steady-state solution for a thermally-dominated flow (℘ =0.4) for the Top–Top (TT) configuration is shown in Fig. 5. For a
seudo-plastic fluid (n = 0.6), the flow activities are intensive at the
op portion of the cavity (Fig. 5(c)). The rising flow along the right
op active wall gets obstructed by the top horizontal wall. The flow is
lmost parallel along the top horizontal wall, until it reaches the cold
ertical wall, where it falls downward and spreads out. A whirlpool
tructure is formed in the vicinity of the cold active wall. The cor-
esponding isotherm and isoconcentration distributions indicate the
imitation of intensive flow activities at the top portion (Fig. 5(a)
and (b)). The features for a Newtonian fluid flow remain almost the
same, but the flow activities gradually start spreading throughout the
cavity (Fig. 5(d)–(f)). All three boundary-layer thicknesses gradually
S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19 15
a n b n c n
d n e n f n
g n h n i n
Fig. 5. Isotherms, isoconcentrations and flow lines distribution for Top–Top (TT) ar-
rangement and ℘ = −0.4.
a n b n c n
d n e n f n
g n h n i n
Fig. 6. Isotherms, isoconcentrations and flow lines distribution for Top–Top (TT) ar-
rangement and ℘ = −4.
i
c
t
fl
a
a n b n c n
d n e n f n
g n h n i n
Fig. 7. Isotherms, isoconcentrations and flow lines distribution for Bottom–Top (BT)
arrangement and ℘ = −0.4.
i
p
s
a
d
c
a
a
o
(
d
fl
l
r
p
T
B
s
p
t
c
T
F
fl
a
s
s
i
i
T
(
r
c
ncrease. For a dilatant fluid, the formed whirlpool shifts towards the
enter of the cavity (Fig. 5(i)).
Steady-state results for a compositionally-dominated flow for
he Top–Top arrangement is shown in Fig. 6. For a pseudo-plastic
uid, the flow falls downward along the solutally-enriched right
ctive wall (Fig. 6(c)). But at the bottom right corner of the cav-
ty, a counter-clockwise rotating thermally-driven flow exists. This
revents the solutally-driven flow to reach the bottom; as a re-
ult, mechanical coupling occurs between the thermally-driven flow
nd the compositionally-driven flow (Fig. 6(c)). The compositionally-
riven flow occupies the left bottom portion of the cavity. The
ompositionally-driven flow mainly connects the source and the sink
t the top portion of the cavity. The isotherms and isoconcentrations
re regulated by the compositionally-driven flow at the top and are
riented in a stratified manner at the top of the cavity (Fig. 6(a) and
b)). For a Newtonian fluid, the thermally-driven flow at the bottom
isappears (Fig. 6(f)). For a dilatant fluid the compositionally-driven
ow occupies the entire cavity (Fig. 6(i)). For the BB arrangement, flow
ine, the isotherm and isoconcentration distributions maintain mir-
or symmetry of the TT arrangement, about a horizontal hyper-plane
assing through the center.
Figs. 7 and 8 show results for the Bottom–Top (BT) arrangement.
he thermally-driven flow structure of a pseudo-plastic fluid with a
T arrangement is shown in Fig. 7(c). The flow diagonally connects the
ource and sink of the vertical walls. For a solutal-dominated flow of a
seudo-plastic fluid the flow rotates individually with two loops near
he respective active zones. The isoconcentrations are stratified at the
enter (Fig. 8(b)) and the isotherms are diagonally stratified (Fig. 8(a)).
he feature remains unaltered for Newtonian and dilatant fluids.
For the Top–Bottom (TB) arrangement, the results are elucidated in
igs. 9 and 10. Fig. 9(c) shows the flow lines for a thermally-dominated
ow of a pseudo-elastic fluid in the TB arrangement. The main flow
ppears to be bifurcated into two cellular flows. The flow retains
ymmetry about the leading diagonal. The resulting isotherms are
tratified at the core (Fig. 9(a)), and the isoconcentrations make an
nteresting structure (Fig. 9(b)). With an increase in the power-law
ndex, the eyes of rotation approach each other and finally get merged.
his results in a diagonally-oriented flow structure for a dilatant fluid
Fig. 9(i)). The isotherms and isoconcentrations are also accordingly
e-oriented.
The most interesting feature of the flow is observed at the
ritical buoyancy ratio (℘ = −1). For ℘ = −1, the thermal and
16 S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19
a n b n c n
d n e n f n
g n h n i n
Fig. 8. Isotherms, isoconcentrations and flow lines distribution for Bottom–Top (BT)
arrangement and ℘ = −4.
a n b n c n
d n e n f n
g n h n i n
Fig. 9. Isotherms, isoconcentrations and flow lines distribution for Top–Bottom (TB)
arrangement and ℘ = −0.4.
a n b n c n
d n e n f n
g n h n i n
Fig. 10. Isotherms, isoconcentrations and flow lines distribution for Top–Bottom (TB)
arrangement and ℘ = −4.
t
s
W
fl
s
e
n
a
h
m
l
f
c
m
o
(
i
c
(
t
p
a
l
w
f
v
i
f
w
c
c
l
i
M
w
compositional buoyancy forces act opposite to each other with the
same order of magnitude, which exhibit an oscillatory behavior for
a Newtonian fluid as described by Bénard et al. [8] and Nishimura
et al. [9]. But for a non-Newtonian power-law fluid, the flow features
are noteworthy. Fig. 11, shows the dynamical evolution of the average
heat and mass transfer across the left active wall. It is seen from Fig. 11,
hat the flow shows an oscillatory behavior for all arrangements for
hear-thinning (pseudoplastic, n < 1) and Newtonian (n = 1) fluids.
hereas this oscillation feature disappears for a shear-thickening
uid, (dilatant fluid, n > 1) and leads to steady-state converged re-
ults. These oscillations are resulted as the phenomenon is driven by
qual strength of thermal and solutal buoyancy forces in an opposing
ature. The amplitude and frequency of oscillations for Newtonian
nd pseudo-plastic fluids are different for different arrangements of
eating and salting, which are shown in Fig. 11. The frequency do-
ain analysis has been done to understand the nature of the oscil-
ation. The results for the TT arrangement are shown in Fig. 12. Five
requencies (10.17, 20.35, 30.52, 40.69, and 50.83) are noticed at the
ritical buoyancy ratio for n = 0.6. Among which, 10.17 is the funda-
ental frequency, and the rest frequencies are the integral multiple
f the fundamental frequency. For n = 0.8, a set of four frequencies
3.942, 7.884, 11.83, 15.78) and for n = 1, a single frequency (2.718)
s noticed for the same arrangement. For shear-thinning fluids like
arboxy-methyl-cellulose (CMC series) and carboxy-polymethylene
Carbopol-series), the effective buoyancy force is higher compared
o the shear-thickening fluids like ethylene glycol, Oobleck and silly
utty at the critical buoyancy ratio. Mathematically, the effective or
pparent Rayleigh number increases with the decrease in the power-
aw index (n) of the power-law fluid at the same value of the system-
ide Rayleigh and Prandtl numbers. This intensifies the buoyancy ef-
ect, which plays a key role in controlling the fluid flow behavior. The
iscous force is unable to counter-balance the buoyant force, which
s intensive in the case of a pseudo-plastic fluid. But the buoyancy
orce progressively gets counter-balanced by the viscous resistance
ith the increase in the power-law index, which leads to steady-state
onverged results for dilatant fluids. The magnitude of the velocity
omponent increases significantly with the decrease in the power-
aw index.
The steady-state features of a dilatant fluid (n = 1.6) at the crit-
cal buoyancy ratio (℘ = −1) is shown in Fig. 13. For the Middle–
iddle (MM) arrangement, the thermally-driven flow occupies the
hole cavity except two tiny compositionally-driven circulations at
S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19 17
0 5 10 15 20 25 300
1
2
3
4
5
6
Sh
n...............................................0.6 - 0.8 - 1.0 - 1.2 - 1.4 -1.6
TB
0 5 10 15 20 25 30 350
1
2
3
4
5
6
7
8
9
10
Nu
n................................................0.6 - 0.8- 1.0 - 1.2 - 1.4 -1.6
MM
0 5 10 15 20 25 30 350
2
4
6
8
10
12
14
Sh
n................................................0.6 - 0.8- 1.0 - 1.2 - 1.4 -1.6
MM
0 5 10 15 20 25 30 350
1
2
3
4
5
6
7
8
Nu
n................................................0.6 - 0.8- 1.0 - 1.2 - 1.4 -1.6
TT
0 5 10 15 20 25 30 350
2
4
6
8
10
12
Sh
n................................................0.6 - 0.8- 1.0 - 1.2 - 1.4 -1.6
TT
0 10 20 30 400
1
2
3
4
5
6
7
Nu
n...............................................0.6 - 0.8 - 1.0 - 1.2 - 1.4 -1.6
BB
0 10 20 30 400
1
2
3
4
5
6
7
8
9
Sh
n................................................0.6 - 0.8 - 1.0 - 1.2 - 1.4 -1.6
BB
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
9
Nu
n...................................................0.6 - 0.8- 1.0 - 1.2 - 1.4 -1.6
BT
0 5 10 15 20 25 300
2
4
6
8
10
12
14
Sh
n.............................................0.6 - 0.8 - 1.0 - 1.2 - 1.4 - 1.6BT
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
Nu
n...............................................0.6 - 0.8 - 1.0 - 1.2 - 1.4 -1.6
TB
(a)
(c)
(e)
(g)
(i) (j)
(h)
(f)
(d)
(b)
Fig. 11. Left wall Nu and Sh variation with τ for ℘ = −1.
t
T
m
(
t
c
T
s
B
a
d
m
fl
101
102
103
104
105
X: 40.69Y: 3337
Frequency
Po
we
r sp
ect
rum
X: 30.52Y: 1.262e+04
X: 20.35Y: 6.665e+04
X: 10.17Y: 4.159e+05
101
103
104
105 X: 3.942
Y: 2.415e+05
Frequency
Pow
er
spe
ctru
m
X: 7.884Y: 4.394e+04
X: 11.83Y: 1.171e+04
X: 15.78Y: 5372
100
101
102
103
104
105
X: 2.718Y: 7.193e+04
Frequency
Po
we
r sp
ect
rum
Fig. 12. Power spectrum of Sh (TT, ℘ = −1).
T
c
d
a
t
a
b
a
s
a
I
r
d
he right-top and the left-bottom corners of the cavity (Fig. 13(c)).
he isotherm and the isoconcentration distributions reveal the for-
ation of a thick thermal and solutal boundary layer (Fig. 13(a) and
b)). For the Top–Top (TT) arrangement, the same feature is also no-
iced. The entire flow is divided into a centrally forming large thermal
irculation and few tiny solutal circulations at the corners (Fig. 13(f)).
he feature for the Bottom–Bottom (BB) arrangement is just a mirror
ymmetric to the Top–Top (TT) arrangement (Fig. 13(g)–(i)). For the
ottom–Top (BT) arrangement, a big thermally-driven flow appears
t the center of the cavity being oriented in the direction of the leading
iagonal of the cavity (Fig. 13(l)). For the Top–Bottom (TB) arrange-
ent, a tri-cellular flow structure is seen. A compositionally-driven
ow directly connects the source to the sink in a diagonal manner.
wo thermally-driven cells appear at the right-bottom and left-top
orners of the cavity (Fig. 13(o)). All thermally- and compositionally-
riven circulations are mechanically coupled with each other to form
tri-cellular structure. The corresponding isotherms and isoconcen-
rations are oriented in the trailing diagonal direction (Fig. 13(m)
nd (n)).
In the preceding section, the effects of active wall orientations,
uoyancy ratio and the power-law index on the flow, isotherm
nd isoconcentration distributions have already been discussed. The
teady-state heat and solute transport records are reported in Tables 4
nd 5, respectively, to elucidate the anticipating effects of parameters.
rrespective of the heating or salting arrangements and the buoyancy
atios, the average Nusselt and Sherwood numbers monotonically
ecrease with the increase in the power-law index of the fluid. The
18 S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19
Table 4
Average Nusselt number (Nu) record.
Middle–Middle (MM) Nu Top–Top (TT) Nu Bottom–Bottom (BB) Nu Bottom–Top (BT) Nu Top–Bottom (TB) Nu
℘ = −0.4 ℘ = −1 ℘ = −4 ℘ = −0.4 ℘ = −1 ℘ = −4 ℘ = −0.4 ℘ = −1 ℘ = −4 ℘ = −0.4 ℘ = −1 ℘ = −4 ℘ = −0.4 ℘ = −1 ℘ = −4
n = 0.6 11.644 – 6.389 10.566 – 5.985 9.992 – 6.367 11.149 – 1.968 4.949 – 10.629
n = 0.8 8.171 – 5.145 7.150 – 4.709 6.811 – 5.027 8.081 – 1.855 4.112 – 8.597
n = 1.0 6.206 – 4.106 5.276 – 3.640 5.058 – 3.861 5.943 – 1.757 3.519 – 6.273
n = 1.2 5.063 4.154 3.466 4.2214 3.316 2.999 4.067 3.221 3.155 4.695 3.968 1.689 3.098 1.647 4.836
n = 1.4 4.324 3.636 3.078 3.563 2.907 2.619 3.447 2.835 2.737 3.912 3.326 1.642 2.790 1.545 3.963
n = 1.6 3.801 3.272 2.824 3.117 2.629 2.446 3.028 2.570 2.468 3.383 2.897 1.615 2.557 1.513 3.397
Table 5
Average Sherwood number (Sh) record.
Middle–Middle (MM) Sh Top–Top (TT) Sh Bottom–Bottom (BB) Sh Bottom–Top (BT) Sh Top–Bottom (TB)Sh
℘ = −0.4 ℘ = −1 ℘ = −4 ℘ = −0.4 ℘ = −1 ℘ = −4 ℘ = −0.4 ℘ = −1 ℘ = −4 ℘ = −0.4 ℘ = −1 ℘ = −4 ℘ = −0.4 ℘ = −1 ℘ = −4
n = 0.6 18.282 – 19.624 17.393 – 18.085 15.816 – 19.183 17.812 – 4.781 9.990 – 22.154
n = 0.8 12.691 – 14.391 11.367 – 13.326 10.596 – 14.370 12.505 – 4.189 7.648 – 17.229
n = 1.0 9.301 – 10.641 8.219 – 9.595 7.749 – 10.272 9.109 – 3.702 6.159 – 12.231
n = 1.2 7.514 5.839 8.434 6.503 4.651 7.428 6.184 4.465 7.893 7.100 5.763 3.365 5.197 2.847 9.290
n = 1.4 6.394 5.119 7.099 5.449 4.056 6.129 5.213 3.921 6.472 5.870 4.796 3.129 4.545 2.145 7.537
n = 1.6 5.389 4.621 6.203 4.558 3.663 5.088 4.542 3.555 5.535 5.042 4.156 2.797 4.065 1.945 6.376
Isotherms Isoconcentrations Flow Lines
0.1 0.2
0.3
0.4
0.5
0.60.7
0.8
0.9
0.1
0.2 0.3
0.4
0.5
0.6
0.7
0.8
0.9
( )a ( )b ( )c
MM
( )d ( )e ( )f
TT
( )g ( )h ( )i
BB
( )j ( )k ( )l
BT
( )m ( )n ( )o
TB
Fig. 13. Isotherms, isoconcentrations and flow lines distribution for ℘ = −1, n = 1.6.
s
d
i
F
m
m
r
(
a
l
o
F
i
t
i
n
T
t
s
l
b
t
s
b
i
T
o
s
c
a
t
t
r
c
t
5
v
d
h
i
c
driving potential of the fluid decreases with an increase in the power-
index of the fluid. The hydrodynamic, thermal and solutal boundary
layer thicknesses increase with the increase in the power-law index
of the fluid, which results in sufficient decreases in the fluid velocity,
and average heat and solute mass transfer rates. For a pseudo-plastic
fluid, the heat and solute mass transfer are higher compared to the
ame for a Newtonian fluid, whereas the same is lower in the case of
ilatant fluids. The heat and solute mass transfer rates are also signif-
cantly affected by the heating and salting arrangements of the cavity.
or a thermally-dominated flow (℘ = −0.4), both the heat and solute
ass transfer are maximum for the Middle–Middle (MM) arrange-
ent. The Bottom–Top (BT) arrangement is the next maximum ar-
angement for the thermal buoyancy-augmented flow. The Top–Top
TT) and Bottom–Bottom (BB) arrangements are the next optimum
rrangements, respectively. The Top–Bottom (TB) arrangement is the
east effective arrangement for a thermally-dominated flow. In case
f the solutally-dominated flow (℘ = −4), the trend gets reversed.
or a solutally-dominated flow, the Top–Bottom (TB) arrangement
s the optimum arrangement for heat and solute mass transporta-
ion. The maximum amounts of heat and solute mass are transported
n this arrangement. The Middle–Middle (MM) arrangement is the
ext effective arrangement, and the Bottom–Bottom (BB) and the
op–Top (TT) arrangements are respectively the third and fourth op-
imum arrangements for the heat and solute mass transport. For a
olutally-dominated flow, the Bottom–Top (BT) arrangement is the
east effective for the heat and solute mass transport. At the critical
uoyancy ratio (℘ = −1), the flow does not reach to steady state for
he pseudo-plastic and Newtonian fluids. However, the flow becomes
teady for the dilatant fluids. For a shear-thickening fluid at the critical
uoyancy ratio, the Middle–Middle (MM) arrangement leads to max-
mum heat and solute mass transfer rates. The Bottom–Bottom (BB),
op–Top (TT) and the Top–Bottom (TB) arrangements are the next
ptimum arrangements, respectively. The least amount of heat and
olute mass are transported for the Bottom–Top (BT) arrangement at
ritical buoyancy ratio for shear-thickening fluids. The average heat
nd solute mass transfer rates bear a nonlinear complicated rela-
ionship with the buoyancy ratio. The average heat and solute mass
ransfer rates gradually decrease with the decrease in the buoyancy
atio, and then this attains a minimum value somewhere nearby the
ritical buoyancy ratio. The same increases with a further decrease in
he buoyancy ratio.
. Conclusions
A comprehensive numerical study on double-diffusive free con-
ection of a power-law non-Newtonian fluid inside a cavity having
iscretely active walls of source and sink for heat and solute mass
as been investigated. Five discrete heating and salting arrangements
n the vertical walls have been considered in the present analysis. A
omprehensive set of numerical results has been reported to elucidate
S.K. Jena et al. / Journal of the Taiwan Institute of Chemical Engineers 51 (2015) 9–19 19
t
o
fl
a
R
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
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he effects of various governing parameters such as the orientation
f active wall region, buoyancy ratio and the power-law index on the
ow behavior of a non-Newtonian fluid. The following conclusions
re outlined.
• The hydrodynamic, thermal and solutal boundary layers get
thicker with the increase in the power-law index of the fluid.• The heat and solute mass transfer rates are maximum for a pseudo-
plastic fluid compared to the same for Newtonian and dilatant
fluids.• A dilatant fluid gives a steady and convergent result at the critical
buoyancy ratio, while the flow for pseudo-plastic and Newtonian
fluids becomes oscillatory.• The fundamental frequency of oscillation decreases with the in-
crease in the power-law index of the fluid.• Both the heat and solute mass transfer rates decrease with the de-
crease in the buoyancy ratio and attain minimum values nearby
the critical buoyancy ratio, then they further increase with a fur-
ther decrease in the buoyancy ratio.• For a thermally-dominated flow, the Middle–Middle (MM) ar-
rangement is the optimum arrangement for maximum heat and
solute mass transfer rates, while the Top–Bottom (TB) arrange-
ment is the least effective arrangement for all considered kinds of
fluids.• For a solutally-dominated flow, the Top–Bottom (TB) arrangement
is the optimum arrangement and the Bottom–Top (BT) is the least
effective arrangement for heat and solute mass transfer for all
considered kinds of fluids.• At the critical buoyancy ratio, the Middle–Middle (MM) arrange-
ment is the optimum arrangement and the Bottom–Top (BT) ar-
rangement is the least effective arrangement for heat and solute
mass transfer for a dilatant fluid.
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