fem solution

8/13/2019 Fem Solution

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Finite element based heatline approach to study mixed convection in a porous

square cavity with various wall thermal boundary conditions

Tanmay Basak a, P.V. Krishna Pradeep a, S. Roy b, I. Pop c,

a Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, Indiab Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, Indiac Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

a r t i c l e i n f o

Article history:

Received 28 June 2010

Received in revised form 27 November 2010

Accepted 21 December 2010

Keywords:

Finite element method

Mixed convection

Square cavity

Porous medium

Uniform and non-uniform heating

Heatlines

Streamlines

a b s t r a c t

A penalty finite element method based simulation is performed to analyze the influence of various walls

thermal boundary conditions on mixed convection lid driven flows in a square cavity filled with porous

medium. The relevant parameters in the present study are Darcy number (Da= 105 103), Grashof

number (Gr= 103 105), Prandtl number (Pr= 0.77.2), and Reynolds number (Re= 1102). Heatline

approach of visualizing heat flow is implemented to gain a complete understanding of complex heat flow

patterns. Patterns of heatlines and streamlines are qualitatively similar near the core for convection dom-

inant flow forDa = 103. Symmetric distribution in heatlines, similar to streamlines is observed irrespec-

tive ofDa at higherGr in natural convection dominant regime corresponding to smaller values ofRe. A

single circulation cell in heatlines, similar to streamlines is observed at Da = 103 for forced convection

dominance and heatlines are found to emanate from a large portion on the bottom wall illustrating

enhanced heat flow for Re = 100. Multiple circulation cells in heatlines are observed at higher Da and

GrforPr= 0.7 and 7.2. The heat transfer rates along the walls are illustrated by the local Nusselt number

distribution based on gradients of heatfunctions. Wavy distribution in heat transfer rates is observed

with DaP 104 for non-uniformly heated walls primarily in natural convection dominant regime. In gen-

eral, exponential variation of average Nusselt numbers with Grashof number is found except the caseswherethe side walls arelinearly heated. Overall, heatlines arefound to be a powerful tool to analyze heat

transport within the cavity and also a suitable guideline on explaining the Nusselt number variations.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

The study of fluid flow and heat transfer induced by the com-

bined effects of the mechanically driven lid and buoyancy force

within closed enclosures filled with fluid saturated porous medium

is of great interest due to high surface-area density. Various appli-

cations on convection in porous medium involve use of metal

foams for enhanced cooling in electronic equipment, foam filled

heat exchangers, open-cell metal foams, use of fibrous materials

in thermal insulation of buildings, solar energy collectors, crystal

growing, post-accidental heat removal in nuclear reactors to name

just a few of them [15]. In mixed convection flows, the forced

convection and the free convection effects are of comparable

magnitudes. In case of lid-driven cavity flows, the thermal non-

homogeneity gives rise to buoyancy force which in turn impacts

upon the coupled fields of velocity and temperature in the cavity.

The governing non-dimensional parameters for mixed convection

in a cavity filled with fluid saturated porous medium are Darcy

number (Da), Grashof number (Gr), Reynolds number (Re) and Pra-

ndtl number (Pr). Note that, Grand Rerepresent the strength of the

natural and forced convection flow effects, respectively. A compre-

hensive review on the fundamentals of the convective flow in por-

ous media can also be found in the literature[611].

Numerical and experimental studies on mixed convection in

porous media have received significant attention of investigators

due to various engineering applications [12,13]. The numerical

heat transfer characteristics of non-Darcy mixed convection flow

over a horizontal flat plate with porous medium was studied by

Chen [12]. DarcyBrinkmanForchheimer equation to model the

motion of fluid through porous medium has been used in this

study. Laminar transport processes in a lid driven porous square

cavity saturated with water was investigated by Al-Amiri [13]. A

few earlier investigations also involve detailed analysis of mixed

convection flow over vertical surface in porous medium [1418].

Oztop [14] investigated numerical heat transfer and fluid flow in

a porous lid driven cavity with isothermal moving top wall. The

effects of the flow governing parameters on the characteristics of

the flow and thermal fields on mixed convective heat transfer in

0017-9310/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2010.12.043

Corresponding author.

E-mail addresses: [email protected] (T. Basak), [email protected]

(P.V. Krishna Pradeep),[email protected](S. Roy), [email protected](I. Pop).

International Journal of Heat and Mass Transfer 54 (2011) 17061727

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

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2/22

rectangular enclosures driven by a continuously moving horizontal

plate was studied by Waheed[15]. Steady mixed convection flow

in a vented enclosure with an isothermal vertical wall and filled

with a fluid-saturated porous medium is investigated by Mahmud

and Pop[16]. Duwairi et al.[17]analyzed the effects of oscillating

plate temperature on transient mixed convection heat transfer

from a porous vertical surface embedded in a saturated porous

medium with internal heat generation or absorption. Jue [18]

investigated mixed convection flow caused by a torsionally oscilla-

tory lid with thermal stable stratification in an enclosure filled

with porous medium using semi-implicit projection finite-element

method.

Till date, most of the numerical investigations on lid drivenenclosures filled with fluid saturated porous medium are limited

to analysis based on streamlines and isotherms and the detailed

analysis of heat flow was not well understood. The present work

is carried out on visualization of heat flow to analyze optimal

thermal mixing and temperature distributions within porous

square cavities filled with different fluids in presence of a moving

top wall. Current work attempts for the first time to analyze heat

transfer, correlations and energy distributions using heatline

approach for mixed convection in a cavity filled with porous

medium.

The heatline is found to be the best numerical tool to visualize

the heat transport in two dimensional convective transport pro-

cess. Heatlines refer to trajectories of total heat transport involving

conductive as well as convective heat flux. In convective heattransport, the energy flow within various regimes can be best visu-

alized by heatlines as the isotherms are unable to give guideline for

energy flows. Heatlines are found via solving the governing equa-

tions of heatfunctions and each heatline contour corresponds to

constant heatfunction. It may be noted that the derivative of heat-

functions are defined as a combination of conductive and convec-

tive heat flux and various directional derivatives of heatfunctions

are obtained from energy balance equations. Proper dimensionless

forms of heatfunctions are closely related to Nusselt numbers. The

concept of heatline was first introduced by Kimura and Bejan

[19,20]. Over the years, heatlines have been employed as effective

tool to describe various physical phenomenon[2124].

A few earlier studies on heatlines were carried out for thermal

convection analysis and in analyzing heat flow in electroconduc-

tive melts [25,26]. Zhao et al. [27,28] studied natural convection

in a porous enclosure with heat and solute sources and illustrated

the flow characteristics via streamlines, heatlines, isotherms and

masslines. Heatline patterns for the fluid with temperature depen-

dent viscosity in a porous square cavity was reported by Hooman

and Gurgenci[29]. Heat flow visualization in a complicated cavity

has been studied by Dalal and Das[30]using the heatline concept.

Effects of wall-located heat barrier on conjugate conduction/natu-

ral-convection heat transfer and fluid flow in enclosures have been

studied using heatlines by Hakyemez et al. [31]. The concept of

masslines has been introduced, analogous to heatlines to visualize

mass transfer within the cavity[3237]. However, a detailed anal-

ysis of heat flow using heatline concept for lid driven flows in

square enclosures filled with porous medium is yet to appear inthe literature.

The aim of the current study is to analyze the heat flow due to

mixed convection in a square cavity filled with a fluid saturated

porous medium for various thermal boundary conditions as a first

attempt. The main objective of the present study is to examine the

extent of thermal mixing and heat transfer within the porous

cavity in the presence of a moving top wall. A square cavity with

four different thermal boundary conditions has been considered

in the current study. A penalty finite element approach using the

Galerkin method is applied to solve the non-linear coupled equa-

tions for flow and temperature fields. The Galerkin method is

further employed to solve the Poisson equation for streamfunc-

tions and heatfunctions. Finite discontinuity exists at the junction

of hot and cold walls leading to mathematical singularity. Solutionof heatfunction for such type of situation demands implementation

of exact boundary conditions. Each case is studied for a range of

parameters: Darcy number (Da = 105 103), Grashof number

(Gr= 103 105), Prandtl number (Pr= 0.77.2), and Reynolds num-

ber (Re= 1 100). Numerical results are obtained for velocity and

thermal fields within the cavity and are displayed using stream-

lines, isotherms and heatlines.

2. Mathematical formulation and simulation

The physical domain consists of a square cavity with the phys-

ical dimensions as shown in Fig. 1. The top wall is assumed to

move with a uniform velocity Uo. Four cases in the present study

are considered as follows: case 1: bottom wall is uniformly heatedwhere the side walls are maintained cold, case 2: bottom wall is

Nomenclature

Da Darcy numberg acceleration due to gravity, m s2

k thermal conductivity, W m1 K1

K permeability, m2

L height of the square cavity, m

Nu local Nusselt numberp pressure, PaP dimensionless pressurePr Prandtl numberRa Rayleigh numberRe Reynolds numberGr Grashof numberT temperature, KTh temperature of hot bottom wall, KTc temperature of cold wall, Ku xcomponent of velocityU xcomponent of dimensionless velocityv ycomponent of velocityV ycomponent of dimensionless velocity

X dimensionless distance alongx coordinateY dimensionless distance alongy coordinate

Greek symbolsa thermal diffusivity, m2 s1

b volume expansion coefficient, K1

c penalty parameterh dimensionless temperaturem kinematic viscosity, m2 s1

q density, kg m3

U basis functionsw streamfunctionP heatfunction

Subscriptsb bottom wallk node numbers side wall

T. Basak et al. / International Journal of Heat and Mass Transfer 54 (2011) 17061727 1707


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is valid at all boundaries. Also, there is no cross flow across the

boundary. Hence w = 0 is used as residual equations at the nodesfor the boundaries.

2.1.2. Nusselt number

The heat transfer coefficient in terms of the local Nusselt num-

ber (Nu) is defined by

Nu @h@n

; 12

wheren denotes the normal direction on a plane. The local Nusselt

numbers at bottom wall (Nub), left wall (Nul) and at the right wall

(Nur) are defined as:

NubX9i1

hi@Ui@Y

; NulX9i1

hi@Ui@X

; and Nur X9i1

hi@Ui@X

;

13

whereUi denotes the finite element basis set [39]. The average Nus-

selt numbers at the bottom and side walls are:

Nub R1

0NubdX

Xj10 Z

1

0

NubdX and Nus R1

0NusdY

Yj10 Z

1

0

NusdY:

14

Here, Nus may be referred as Nul and Nurfor the left and right walls,

respectively.

2.1.3. Heatfunction

The heat flow within the enclosure can be visualized using the

heatfunctionPobtained from conductive heat fluxes @h@X; @h

@Y

as

well as convective heat fluxes (Uh, Vh). The steady energy balance

equation(4)can be re-arranged as

@

@X Uh

1

RePr

@h

@X

@

@Y Vh

1

RePr

@h

@Y

0: 15

The heatfunction satisfies the above equation such that@P

@Y Uh

1

RePr

@h

@X

@P

@X Vh

1

RePr

@h

@Y

16

which yield a single equation

@2P

@X2

@2P

@Y2

@

@YUh

@

@XVh: 17

Using theabove definition of the heatfunction, the positive sign ofP

denotes anti-clockwise heat flow and the clockwise heat flowis rep-

resented by the negative sign ofP. Eq.(17)is solved using the sim-

ilar procedure for residuals of heatfunction as discussed by Kaluri et

al. [41]. The Neumann boundary conditions may be specified asfollows:

n rP 0 uniformly heated=cooled wall; 18

n rP p cospXRePr

sinusoidally heated bottom wall; 19

n rP 1

RePr linearly heated right wall and 20

n rP 1

RePr linearly heated left wall: 21

The top insulated wall may be represented by Dirichlet boundary

condition as obtained from Eq. (16)which is simplified into @P@X 0

for an adiabatic wall. A reference value ofP is assumed as 0 at

X= 0,Y= 1 and hence P= 0 is valid for Y= 1, "X. It may be noted

that, the unique solution of Eq. (17)is strongly dependent on thenon-homogeneous Dirichlet conditions. The following non-homo-

geneous Dirichlet boundary conditions are employed to obtain the

solution for Eq.(17).

P0; 0 1

RePrNul

P1; 0 1

RePrNur

22

3. Results and discussion

3.1. Numerical tests

The computational domain consists of 28 28 bi-quadratic ele-

ments which correspond to 57 57 grid points. The bi-quadratic

elements with lesser number of nodes smoothly capture the non-

linear variations of the field variables which are in contrast with fi-

nite difference or finite volume solution. In the current investiga-

tion, Gaussian quadrature based finite element method provides

the smooth solutions at the interior domain including the corner

regions as evaluation of residuals depends on the interior Gauss

points and thus the effect of corner nodes are less profound in

the final solution [41]. In cases 1 and 4, jump discontinuities at bot-

tom corner points exist due to hotcold junctions leading to math-

ematical singularities. The present finite element method based

approach offers special advantage on evaluation of local Nusselt

number at the left, right and bottom walls as the element basis

functions have been used here to evaluate the heat flux.

To assess the accuracy of the present numerical approach, we

have tested our algorithm based on the grid size (57 57) for dri-

ven cavity flow[42]and mixed convection [43]. The simulations

were carried out for 49 49, 57 57 and 61 61 grid points. It

is found that temperature and flow characteristics with 57 57

and 61 61 grids are identical and further simulation studies are

performed based on 57 57 grids. Validation results are not

shown for the brevity of the manuscript.

In order to validate heatfunction contours, we have carried out

simulations for all the cases with a range of Rayleigh numbers(Ra = 0,10,100,103) and Darcy numbers (Da = 105 103) at

Re = 0 which corresponds to natural convection. It may be noted

that, earlier works on heatlines have been reported for situations

involving natural convection and analysis on heatfunctions for

mixed convection problems within cavities are not yet reported till

date. Validation of the heatlines has been performed and the re-

sults are in good agreement with the earlier work [19]. The valida-

tion of heatfunctions for natural convection situation is already

discussed by Kaluri et al. [41] and hence the validation results

are not shown in this manuscript. The heat transfer in the cavity

is conduction dominant for lowvalues of the governing parameters

(Re, PrandGr) at any Da. Under these conditions, heatlines essen-

tially represent heat flux lines, which are commonly used for con-

ductive heat transport[44]. Also, heat flux lines are perpendicularto isothermal surfaces and parallel to adiabatic surfaces [45]. In

cases 3 and 4, some heatlines are found to emanate from the hot

portion of the wall and end on the relatively cold portion of the

same wall for higher Grashof numbers. The solution is strongly

dependent on a non-homogeneous Dirichlet boundary condition

and the sign of heatfunction is governed by the sign of a non-

homogeneous Dirichlet condition. In the current situation, a nega-

tive sign of heatlines represents a clockwise flow of heat while a

positive sign refers to an anticlockwise flow. The detailed discus-

sion on heat transport based on heatlines for various cases is pre-

sented in later sections.

Detailed computations have been carried out for various fluids

(Pr= 0.7 and 7.2), Da= 105 103, Gr= 103 105 corresponding

to lid velocity ranging withinRe= 1,10 and 100. Simulations werealso carried out for Pr= 0.026. It is observed that the heat transfer



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within the cavity for Da = 105 103 is conduction dominant irre-

spective of Gr andRe for Pr= 0.026 (figures not shown). Similar

trend is observed with Pr= 0.7at Gr= 103. However, some interest-

ing results are also discussed for Gr= 103 for higher Pr(Pr= 7.2) flu-

ids where convection dominant heat transfer is observed. It is

observed from figures that the fluid circulations are strongly

dependent onDa.

3.2. Case 1: Uniform heating of bottom wall

Dominant lid driven effect for lower Da is observed for Re = 1

andGr= 103 for higherPrfluids (Pr= 7.2) (figure not shown), as a

single streamline circulation span the entire cavity. Onset of natu-

ral convection is observed based on a secondary flow circulation at

the bottom left corner of the cavity. However, conduction domi-

nant heat transfer illustrated by heatlines perpendicular to isother-

mal surfaces are observed for Da6 104. Enhanced convective

effects are observed as the strength of secondary circulation in-

creases forDa = 103. This was also evident from slightly distorted

heatlines within the cavity.

Fig. 2 illustrates the streamlines, isotherms and heatlines for

Re = 10,Pr= 0.7 andGr= 105 for varyingDa(= 105

103). The ef-

fect of lid driven flow is observed at low Da(=105) as the primary

streamline circulation cells occupies more than 60% of the cavity

and a secondary circulation is observed to the bottom left portion

of the cavity. The isotherms are smoothly distributed and are sym-

metric (see Fig. 2(a)). Enhanced secondary flow circulation cells are

observed at Da = 104 due to less pore resistance and enhanced

buoyancy forces. The distribution of isotherms is similar to that

at smaller Da except for h 6 0.1. The effect of natural convection

is more pronounced and two flow circulation cells with similar

sizes are observed at higher Da (seeFig. 2(c)). The isotherms are

found to be compressed towards the side walls as well as the bot-

tom wall due to dominant natural convection.

The heatlines illustrate conduction dominant heat transfer

within the cavity for lowDa (=105) as the heatlines are perpen-

dicular to isothermal surfaces (Fig. 2(a)). Similar trend of heatlines

is also observed for Da = 104 (Fig. 2(b)). However, isotherms with

h 6 0.1 are distorted and more pronounced to the side walls at

Da= 104 compared to lower

Da. It may be noted that the thickness

of the thermal boundary layer is small towards the top portion of

the side walls for Da = 104 compared to lower Da. It may be noted

that the magnitude of heatlines that end towards the side walls is

slightly greater forDa = 104 signifying larger heat transport com-

pared to lower Da. Enhanced natural convective heat transfer with-

in the cavity is clearly illustrated by the heatline circulation cells

for Da = 103 (Fig. 2(c)). Dense heatlines corresponding to

0.0016P 6 0.07 emanate from the center of the bottom wall

and end to the top portion of the side walls signifying higher heat

transfer rates in this regime. Thus isotherms corresponding to

h 6 0.3 are more pronounced towards the side walls. Larger gradi-

ents in heatfunctions corresponding to 0 6 jPj6 0.2 are observed

along the side walls for Da = 103 whereas that varies within

06 jPj6 0.13 for Da6 104 over the length of side walls. Thus

thermal boundary layer is more compressed towards the side walls

for higherDa. Also, dense heatlines are emanated from the central

portion of the bottom wall of the cavity. Thus a highly thermal

mixing zone withh varying within 0.40.6 is observed to the cen-

tral portion of the cavity for Da = 103 (seeFig. 2(c)).


Re = 100,Pr= 7.2 and Gr= 103 for varying Da(=105 103) with

uniform heating of bottom wall and cold side walls. The dominant

effect of the moving lid is clearly seen from single flow circulation

cells irrespective ofDa. HighPrfluids correspond to low thermal

0.0001

0.001

0.001

0.003

(a)

0 0.5 10

0.5

1

0.02

0.01

0.005

0.00010.005

0.01

0.01

5

0.02

(b)

0 0.5 10

0.5

1

0.3

0.3

0.2

0.10.02

0.050.15

0.25

(c)

0 0.5 10

0.5

1

0.1

0.1

0.2

0.3

0.5

0.8

0 0.5 10

0.5

1

0.1

0.1

0.2

0.4

0.6

0.9

0 0.5 10

0.5

1

0.

1 0

.2

0.3

0.4

0.4

0.5

0.6

0.8

0.3

0.2

0.1

0 0.5 10

0.5

1

0.07

0.03

0.01

0.005

0.001

0.005

0.02

0.04

0.1

0 0.5 10

0.5

1

0.12

0.07

0.03

0.01

0.001

0.005

0.02

0.04

0.07

0.12

0 0.5 10

0.5

1

0.2

0.17

0.12

0.07

0.04

0.01

0.2

0.17

0.12

0.07

0.04

0.02

0 0.5 10

0.5

1

Fig. 2. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 1 with Re = 10, Pr= 0.7,Gr= 105 and (a) Da = 105, (b)Da = 104 and (c) Da = 103.

1710 T. Basak et al. / International Journal of Heat and Mass Transfer 54 (2011) 17061727


6/22

conductivity and due to weaker thermal diffusion, the isotherms

are compressed towards the bottom wall and a large isothermal

zone corresponding to h 6 0.1 is observed to the top portion of

the cavity forDa = 105 (Fig. 3(a)). Similar trend in the streamlinecells is observed with Da = 104. In addition, the isotherms are

more compressed towards the left wall compared to that of

Da = 105. Larger intense convection at higher Da is seen from

the increased magnitude of the streamline cells. It is interesting

to observe that the isotherms are more compressed towards the

left and bottom walls for Da = 103. A large region near the right

half becomes isothermally cooled and the effect of heating is con-

fined only near the bottom and the left walls of the cavity forming

a strong thermal boundary layer attached to the bottom wall with

h 6 0.1 in nearly 75% of the cavity (Fig. 3(c)). This is further ex-

plained based on heatlines.

Heatlines corresponding to jPj = 0.00010.001 end towards the

bottom portion of side walls. However, heatfunctions with very

small magnitudesP 6 0.0001 are observed in the upper half por-tion of the cavity, signifying no heat transfer in this regime and

the less heat transfer occurs to the cooler region for larger Prfluid.

Thus a large zone at the top portion of the cavity is maintained iso-

thermal with h 6 0.1 (Fig. 3(a)). Similar qualitative trend in heat-

functions in the upper half of the cavity is observed for Da = 104

with a weak heatline circulation cell (jPjmax= 0.0001). It is interest-

ing to observe that heatlines that emanate from 06X6 0.6 of the

bottom wall end towards the left wall signifying higher heat trans-

fer rates towards the left wall. Thus the thickness of thermal

boundary layer is less towards the left wall (Fig. 3(b)). At

Da = 103, it may be noted that heatlines from 0 6X6 0.8 of the

bottom wall end towards the left wall signifying larger heat trans-

fer to the left wall. Thus the thermal boundary layer is much com-

pressed towards the left wall compared to lower Da. Even thoughenhanced heatline circulation cells with jPjmax = 0.0005 are ob-

served near the adiabatic wall forDa = 103, significant heat trans-

fer is not observeddue to lesser magnitudes of heatfunction. Thus a

large regime near the top portion remains ath 6 0.1 (Fig. 3(c)). It is

interesting to observe that heatfunctions towards the right wallvary within 06 jPj6 0.0001 for Da = 105 whereas jPj varies with-

in 0 0.0004 along the right wall forDa = 103. Thus the thickness

of thermal boundary layer is large along the right wall and a large

isothermal zone with h 6 0.1 is observed near the right wall at

higher Da.

Distribution of streamlines, isotherms and heatlines for Re = 1,

Pr= 7.2 and Gr= 105 are illustrated in Fig. 4. Small effect of the

lid driven flow is observed atDa = 105, as a small amount of fluid

is being dragged to the top left corner of the cavity. However due to

high hydraulic resistance, the magnitude of streamlines is very low

signifying dominant conduction heat transfer (see Fig. 4(a)). This is

also represented by the smooth and symmetric isotherms. As Da

increases to 104, the effect of the moving wall tends to disappear

and the streamline circulations become almost symmetric. Herethe isotherms with h 6 0.5 are gradually compressed towards the

side walls illustrating dominant convection effect (see Fig. 4(b)).

It is observed that the streamlines stretch diagonally for high Da.

The larger intense flow enhances thermal mixing which results

in uniform temperature distribution at a larger portion in the cen-

tral core. The larger intensity of flow also causes smaller thickness

of boundary layer at the top portion of side walls (Fig. 4(c)).

Heat flow distribution inside the cavity is illustrated by the

heatlines. It is observed that the magnitudes of heatfunctions are

small implying small heat flow due to weak fluid flow at lower

Da(Da = 105) (seeFig. 4(a)). The top portion of the side walls re-

ceives heat mainly from the center of the bottom wall and less

dense heatlines corresponding to jPj = 0.0010.03 are observed

along 0.56 Y6 1 of the side walls signifying less heat absorptionfor Da= 105. It may be noted that due to the lid driven effect,

0.003

0.001

0.0001

(a)

0 0.5 10

0.5

1

0.005

0.0010.0001

(b)

0 0.5 10

0.5

1

0.001

0.005

0.01

0.02

(c)

0 0.5 10

0.5

1

0.1

0.2

0.4

0.70.9

0 0.5 10

0.5

1

0.1

0.2

0.4

0.6

0.9

0 0.5 10

0.5

1

0.1

0.2

0.3

0.5

0.8

0 0.5 10

0.5

1

0.0005

0.0002

5e05

1e05

0.000

1

0.00

02

0.000

5

0 0.5 10

0.5

1

0.001

.0005

0.0001

1e055e05

0.0001

0.00

01

0.00

03

0 0.5 10

0.5

1

.002

0.001

0.0005

0.0001

0.0005

0.000

4

0.000

3

0.000

0.0001

0

.0002

0 0.5 10

0.5

1

Fig. 3. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 1 with Re = 100,Pr= 7.2, Gr= 103 and (a) Da = 105, (b) Da = 104 and (c) Da = 103.



7/22

the top portion of the right wall receives slightly larger heat

(jPj = 0.0010.003) compared to top portion of left wall based on

heatlines (jPj = 0.0010.0025) (seeFig. 4(a)). Thus isotherms with

h 6 0.4 are non-symmetric and the thermal boundary layer is morecompressed towards top portion of the right wall. On the other

hand, symmetric rolls in the heatline circulation cells are observed

with jPjmax 0.25 for Da = 104 (seeFig. 4(b)). It is interesting to

observe symmetric heatline circulation cells similar to streamlines

illustrating convection dominant heat transfer. Denser heatlines

that emanate from the bottom wall end up in the top portion of

the side walls corresponding to jPj 00.18 at 0.56 Y6 1 on the

side walls, illustrating significant heat flow in this regime. It is

interesting to observe that the top portion of the side walls re-

ceives larger heat compared to bottom portion of the wall. Thus

thermal boundary layer is more compressed towards the top por-

tion of the side walls (see Fig. 4(b)). It is also observed that some

heatlines directly start from the hot bottom wall and end on cold

side walls near a small region of bottom corners due to conductiveheat transport in that region.

Heatlines (jPjmax = 0.7) similar to streamline cells are observed

except near the walls for higherDa(Da = 103) illustrating convec-

tion dominant heat transfer within the cavity (see Fig. 4(c)). Thus a

large regime at the central region corresponds to h = 0.50.6. Sim-

ilar to Da = 104, large amount of heat from the bottom wall is

transferred to the top portion of side walls resulting in smaller

thickness of thermal boundary layer at the top portion. Sparse

heatlines with 0.456 jPj6 0.5 are observed to the bottom portion

of the side walls. Thus the thickness of thermal boundary layer is

high near the bottom portion of the side walls. Denser heatlines

due to enhanced convective effects are found to be more intense

near the bottom wall and the thickness of the thermal boundary

layer is small along the bottom wall especially for Da = 103

(seeFig. 4(c)).

It is observed that the distribution in streamlines, isotherms

and heatlines forRe = 10 are qualitatively similar to that ofRe = 1

for Pr= 7.2 at Gr= 105 and Da = 105 103 and a qualitatively

similar explanation as that ofFig. 4follows.Forced convection is seen to be dominant for Re = 100,Gr= 105

andPr= 7.2 withDa = 105 103 (seeFig. 5). The isotherms are

compressed towards bottom and a large portion to the upper half

region of the cavity corresponds to h 6 0.1 as seen inFig. 5(a). En-

hanced secondary circulation cells in the streamlines are observed

to the bottom left corner of the cavity forDa = 104. The isotherms

are distorted and are more compressed towards the left wall

(Fig. 5(b)). An increase in the strength of streamline cells is ob-

served at Da= 103 and a primary circulation spans more than

90% of the cavity (Fig. 5(c)). It may be noted that the span of the

secondary circulation is decreased compared to that ofDa = 104.

Isotherms are highly compressed towards the left wall and a large

portion near the right wall is maintained isothermal with h 6 0.2

(Fig. 5(c)).Conduction dominant heat transfer is observed within the cav-

ity for Da= 105. However smaller magnitude of heatlines with

jPj6 0.0001 are observedto the top portion of the cavity signifying

less heat transfer rates in this zone and the heating effects are con-

fined to the bottom portion of the cavity (Fig. 5(a)) as also seen in

Fig. 3(a). Weak heatline circulation cells with jPjmax = 0.001 are ob-

served to the right half of the cavity for Da = 104 (Fig. 5(b)). Also

heatlines which start from 06X6 0.7 of the bottom wall, end to-

wards the left wall signifying larger heat transfer to the left wall. It

may be noted that significant variation in heatfunction is not ob-

served along the top portion of the right wall as jPj varies within

06 jPj6 0.0006 whereas that varies within 06 jPj6 0.001 to

the top portion of the left wall (Fig. 5(b)). Thus a large region of

the top portion of the right wall is maintained isothermal withh 6 0.1 and larger thermal boundary layer thickness is observed

0.02

0.015

0.01

0.005

0.001

0.003

0

.005 0

.01

0.015

0.02

(a)

0 0.5 10

0.5

1

0.35

0.30.25

0.2

0.10.05

0

.03

0.05

0.15

0.20.250.3

0.35

(b)

0 0.5 10

0.5

1

1.41.3

1.1

0.70.40.15 0.10

.4

0.7

1

1.3

1.51

.4

(c)

0 0.5 10

0.5

1

0.1

0.1

0.2

0.3

0.4

0.50.6

0.8

0 0.5 10

0.5

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5

0.4

0.3

0.2

0.

1

0 0.5 10

0.5

1

0.10.2

0.3

0.4

0.50.5

0.4

0.3

0.2

0.1

0.6

0 0.5 10

0.5

1

0.08

0.06

0.04

0.03

0.02

0.015

0.01

0.005

0.0010

.001

0.00

5

0.01

0.015

0.02

0.03

0.04

5

0.08

0 0.5 10

0.5

1

0.220.2

0.15

0.120.080.04

0.02

0.04

0.08

0.15

0.20.22

0.25

0 0.5 10

0.5

1

0.7

0.6

0.45

0.350.2

0.04

0.0

4

0.10

.2

0.4

0.6

0.7

0 0.5 10

0.5

1

Fig. 4. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 1 with Re = 1,Pr= 7.2, Gr= 105 and (a)Da = 105, (b) Da = 104 and (c)Da = 103.



8/22

near the right wall. Convection dominant heat transfer in the cavity

is clearly observed at Da = 103 as seen from the heatlines similar

to the streamlines. Larger gradients in jPj with 06 jPj6 0.005 are

observed along the left wall whereas that varies within

06 jPj6 0.0023 along the right wall signifying thinner thermalboundary layer thickness along the left wall (Fig. 5(c)). Also, largely

intense heatlines are observed along large portion of the bottom

wall as primary heatline circulation cells span most of the cavity.

Thus isotherms are more compressed towards the bottom wall

and left wall resulting in a large temperature gradient. whereas a

large portion at the right half of the cavity is maintained ath 6 0.3.

3.3. Case 2: Non-uniform heating of bottom wall

Streamlines, isotherms and heatlines in case of non-uniform

heating of bottom wall with sinusoidal variation of temperature

have also been studied. In case of uniform heating of bottom wall,

a finite discontinuity in the Dirichlet boundary conditions for the

temperature distribution occurs at the edges of bottom wall. Themathematical singularity at the edges of bottom wall is removed

by non-uniform heating and provides a smooth temperature distri-

bution in the entire cavity. Distribution of streamlines, isotherms

and heatlines are qualitatively similar to those of case 1 under

identical parameters and the maximum value of streamfunction

is found to be almost same for all Grashof numbers. Thus, illustra-

tive figures are not shown for the brevity of the manuscript and

similar to case 1, qualitative explanation may be drawn for identi-

cal parameters (Gr, Pr, Re) for varyingDa (Da = 105 103).

3.4. Case 3: Linearly heated side walls with uniformly heated bottom

wall

Weak streamline circulations are observed within the cavity forlow values of governing parameters (Re = 1,Pr= 0.7 andGr= 103),

irrespective ofDa. Isotherms with h P 0.5 are parallel to the bot-

tom wall and those with h 6 0.3 are compressed to the top corners

of the cavity. Also symmetric distribution in heatlines and iso-

therms is observed at low Re. Conduction dominant heat transfer

within the cavity is observed on smooth monotonic and paralleldistribution in heatlines irrespective ofDa(figure not shown). Figs.

610display the streamlines, isotherms and heatlines for various

Pr(Pr= 0.7 and 7.2), Gr and Re at various Darcy numbers in the

presence of linearly heated side walls and uniformly heated bot-

tom wall. A few cases are initially presented for Gr = 103 (Figs. 6

and 7).


Re = 10,Pr= 7.2 and Gr= 103 for various Da(=105 103). For lar-

gerDa, due to reduced flow resistance, convective flow and heat

transport is enhanced. The isotherms towards the left corner are

stretched while those to the right top corner are compressed and

the rest of the isotherms are smooth and span the entire cavity.

A single flow circulation cell with high flow intensity is observed

for high Da(=103

). The isotherms with h6 0.5 are distorted to-wards the top wall. It may be noted that isotherms with h 6 0.2

are more compressed towards the top corner of the right wall

(Fig. 6(c)).

Conduction dominant heat transfer is observed in the lower

portion of the cavity at low Da and the wall driven effect is found

at the top right corner. It may be noted that the magnitude of

heatlines that end on the top portion of the left wall is less

(jPj = 00.002) compared to those of right wall (jPj = 00.003).

Thus isotherms with h 6 0.3 are much compressed to the right cor-

ner forDa 6 104 (Fig. 6(a) and (b)). Heatline circulation cells with

jPjmax = 0.007 is observed to the top right portion of the cavity for

Da = 103 (Fig. 6(c)). Lower heat transfer rates are observed to the

top portion of the right wall as heatlines emanating from a small

portion of the bottom wall and lower portion of the right wallend near to Y= 1 on the right wall. Even though dense heatlines

5e05

0.0001

0.001

0.002

0.003

(a)

0 0.5 10

0.5

1

0.001

0.0001

0.001

0.005

0.007

(b)

0 0.5 10

0.5

1

0.03

0.025

0.02

0.01

0.003

0.

00

1

(c)

0 0.5 10

0.5

1

0.1

0.2

0.4

0.70.9

0 0.5 10

0.5

1

0.1

0.3

0.5

0.7

0.9

0 0.5 10

0.5

1

0.3

0.4

0.5

0.8

0.2

0.1

0.2

0 0.5 10

0.5

1

1e050.0001

0.0003

.00085

9e05

0.000

15

0.0

004

0 0.5 10

0.5

1

0.001

0.0008

0.0007

0.0005

.0026

.0015

0.0006

0.0001

0.0004

0.0007

0 0.5 10

0.5

1

0.003

0

.0035

0.005

0.007

0.004

0.002

0.0001

0.000

0 0.5 10

0.5

1

Fig. 5. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 1 with Re = 100,Pr= 7.2, Gr= 105 and (a) Da = 105, (b) Da = 104 and (c) Da = 103.



9/22

0.003

0.001

0.0001

(a)

0 0.5 10

0.5

1

0.005

0.001

0.0001

(b)

0 0.5 10

0.5

1

0.01

0.02

0.0050.0010.000

0

.0001

(c)

0 0.5 10

0.5

1

0.30.2

0.2

0.3

0.4

0.5

0.6

0.8

0 0.5 10

0.5

1

0.4

0.5

0.7

0.9

0.30.30

.2 0.2

0 0.5 10

0.5

1

0.3

0.4

0.5

0.7

0.9

0.20.2

0 0.5 10

0.5

1

0

.00

4

0.003

0.002

0.001

0.0003

0.0005

0.001

0

.002

0

.003

0

.004

0

.005

0 0.5 10

0.5

1

0.005

0.004

0.003

0.002

0.001

0.0003

0.001

0.002

0.0025

0.003

0

.004

50

0.

0

0 0.5 10

0.5

1

0.008

0.007

0.006

0.005

0.003

0.001

0.007

0.005

0.004

0.0030.002

0 0.5 10

0.5

1

Fig. 6. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 3 withRe = 10, Pr= 7.2, Gr= 103 and (a)Da = 105, (b)Da = 104 and (c)Da = 103.

0.0001

0.001

0.003

(a)

0 0.5 10

0.5

1

0.0001

0.001

0.005

(b)

0 0.5 10

0.5

1

0.001

0.005

0.01

0.02

(c)

0 0.5 10

0.5

1

0.1

0.2

0.3

0.5

0.7

0.9

0 0.5 10

0.5

1

0.1

0.2

0.3

0.5

0.7

0.9

0 0.5 10

0.5

1

0.2

0.3

0.5

0.7

0.9

0 0.5 10

0.5

1

.00025

0.00015

5e05

5e

05

0.00015

0

.00015

0 0.5 10

0.5

1

0

.0001

0.000

5

0.0007

0.0006

0.00

03 0.0001

0 0.5 10

0.5

1

0.003

0.002

0.001

.0018

0.001

0.0005

0.0003

0.0015

0 0.5 10

0.5

1

Fig. 7. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 3 with Re = 100,Pr= 7.2,Gr= 103 and (a) Da= 105, (b) Da = 104 and (c) Da= 103.



10/22

0.001

0.003

0.005

0.005

0.003

0.001

0

.0001

(a)

0 0.5 10

0.5

1

0.06

0.05

0.06

0.050.03

0.01

0.001.00

0.01

0.03

(b)

0 0.5 10

0.5

1

1.2

1

0.7

0.3

0.1

0

.05

0.3

0.7

1.2

1

(c)

0 0.5 10

0.5

1

0.2

0.3 0.3

0.2

0.4

0.5

0.7

0.9

0 0.5 10

0.5

1

0.2

0.3 0.3

0.2

0.4

0.5

0.7

0.9

0 0.5 10

0.5

1

0.2

0.3

0.4 0.4

0.3

0.20.5

0.6

0.7

0.9

0 0.5 10

0.5

1

0.

6

0.

5

0.4

0.3

0.2

0.1

0.01

0

.025

0

.1

0

.2

0

.3

0

.4

0

.5

0

.6

0 0.5 10

0.5

1

0.

6

0.5

0.4

0.3

0.2

0.1

0.01

0

.06

0

.15

0

.25

0

.35

0

.45

0

.55

0

.6

0 0.5 10

0.5

1

0.90.8.

63

0.63

0.4

0.10.20.4

0.9

0

.8

0

.7

0

.1

0.2

0.3 0.

0 0.5 10

0.5

1

Fig. 8. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 3 with Re = 1,Pr = 0.7,Gr= 105 and (a) Da = 105, (b)Da = 104 and (c) Da = 103.

0.0001

0.001

0.002

0.003

(a)

0 0.5 10

0.5

1

0.01

0.006

0.001

0.001

0.005

0.01

0.018

(b)

0 0.5 10

0.5

1

0.08

0.05

0.005

0

.03

0.06

0.1

0.03

0

.005

0.0

6

0.02

(c)

0 0.5 10

0.5

1

0.2

0.3

0.4

0.5

0.7

0.9

0.3

0.2

0 0.5 10

0.5

1

.2

0.4

0.5 0.4

0.3

0.6

0.8

0.9

0 0.5 10

0.5

1

0.4 0

.5 0.6

0.5 0

.4

0.7

0.9

0 0.5 10

0.5

1

0.004

0.003

0.002

0.001

0.0004

0

.002

0

.003

0

.004

0

.005

0 0.5 10

0.5

1

.005

0.007

0.008

0.01

0.006

0.005

0

.005

0

.00

0.00

3

0.0

01

.003

.0004

0

.00

2

0 0.5 10

0.5

1

0.03

0.05

0.050

.03

0.03

0.010

.02

.013

.005

0.0

0 0.5 10

0.5

1

Fig. 9. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 3 with Re = 10,Pr= 7.2, Gr= 105 and (a) Da = 105, (b) Da = 104 and (c) Da= 103.



11/22

are found near Y= 1, smaller gradients in heatlines

(0.0036 jPj6 0.0032) are observed at Y= 0.8 0.95. Thus lower

heat transfer rates are observed and the larger thermal boundarylayer thickness is observed compared to smaller Da (Fig. 6(c)).

However, slightly larger gradients in heatfunction are observed

along the left wall and isotherms with h 6 0.3 are more pro-

nounced to the top left corner atY= 0.8 1, compared to the right

wall.


Re = 100,Pr= 7.2 andGr= 103 for various Da(=105 103). Iso-

therms appear to be distorted to the top portion of the cavity

whereas they become parallel near the hot bottom wall

(Fig. 7(a)). Similar trend in streamlines is observed for Da = 104.

Single flow circulation cell with high flow intensity is observed

for high Da(= 103). It may be noted that isotherms with h 6 0.2

are largely compressed towards the bottom wall and a large iso-

thermal zone is observed at the top portion of the cavity withh 6 0.1 (Fig. 7(c)).

Smaller magnitudes in heatlines with jPj6 0.0001 are observed

in a large region within the cavity signifying less heat transfer at

Da = 105. Conduction dominant heat transfer based on parallel

heatlines is observed in the left portion of the cavity. It may be

noted that the magnitude of heatfunctions is very low signifying

less heat transfer rates, due to smaller heating effects and thus

the isotherms with hP 0.2 are parallel to the bottom wall. En-

hanced heatline circulation cells span in a large central region of

the cavity for Da = 104 103 (Fig. 7(b) and (c)). Thus a larger

zone with hP 0.2 is found to be compressed towards the bottom

wall. Dense heatlines with jPj = 00.002 emanate from the bottom

wall and end towards the left wall. Due to largely intense heatlines

towards the bottom portion of the cavity, isotherms with h P 0.2are compressed towards the bottom wall (Fig. 7(b) and (c)).

Fig. 8(a) illustrates the streamlines, isotherms and heatlines for

Re = 1,Pr= 0.7 andGr= 105 for varyingDa. The primary streamline

circulation is more stronger than the secondary cells at Da = 105.Significant effect of the buoyancy results in the onset of a second-

ary circulation in the cavity. The temperature contours for h 6 0.3

occurs symmetrically near the corners of the top wall. The other

temperature contours are smooth curves which span the entire

cavity and are symmetric with respect to vertical symmetric line.

At Da= 104 (Fig. 8(b)), the lid driven flow and the strength of both

circulations in streamlines are increased. The temperature distri-

bution is found to be similar to that of lower Da. AsDa tends to

103 (Fig. 8(c)), streamline circulation cells gradually become sym-

metric inside the cavity. The effect of natural convection becomes

dominant and the isotherms are observed to be distorted due to

strong fluid motion, but they are symmetric with respect to the

vertical center line. The strength of the streamlines is much higher

than those for low Darcy numbers.At low Da(Da = 105 and 104), dominant conductive heat

transfer is observed based on heatlines which are parallel to each

other and are perpendicular to the hot bottom surface. Higher

magnitudes of heatlines are observed as jPj varies within 0.01

0.6 along the side walls, due to larger heating effects. Thus, smooth

symmetric distribution in the isotherms is observed within the

cavity. Convection dominant heat transfer is observed at

Da = 103 as seen from the heatline circulation cells with

jPjmax= 0.9, occurring symmetrically within the cavity (Fig. 8(c)).

Largely intense heatlines that emanate from the bottom wall are

observed along the vertical center of the cavity. Thus temperature

along the central zone at Da= 103 is higher than that at Da6 104.

It is interesting to observe that heatlines with jPj = 0.010.6 are

observed along 0.86 Y6 1 of the side walls for Da = 103

whereasjPj varies within 0.010.3 in this region for lower Da. Thus

0

.0001

0.001

0.002

0.003

(a)

0 0.5 10

0.5

1

0.00

01

0.001

0.00

35

0.006

0.007

(b)

0 0.5 10

0.5

1

0.025

0.01

0

.005

0.0050.01

0.001

(c)

0 0.5 10

0.5

1

0.1

0.2

0.4

0.6

0.8

0 0.5 10

0.5

1

0.1

0.20.3

0.5

0.8

0 0.5 10

0.5

1

0.2

0.3

0.5

0.60.7

0.

8

0 0.5 10

0.5

1

0.000

2

0.0001

1e05

0.0001

0.0002

0.00016

5e05

0 0.5 10

0.5

1

0.00

05

0.00010

.00020

.0005

0

.0007

0.0008

0.0

00

1

4000

.0

0.001 0.0005

0 0.5 10

0.5

1

0.004

0.003

0.002

0.00

05

0.0005

0.002

0.005

0.008

0

.002

100

.0

0

.00

1

0 0.5 10

0.5

1

Fig. 10. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 3 with Re = 100,Pr= 7.2, Gr= 105 and (a)Da = 105, (b) Da = 104 and (c) Da = 103.



12/22

isotherms withh 6 0.4 are compressed towards the top corners of

the cavity at Da = 103. Dense heatlines from the bottom wall are

observed along the vertical center of the cavity illustrating larger

heat transfer. Thus isotherms with hP 0.5 are pulled towards

the top wall in this region.

The distribution in streamlines, isotherms and heatlines for

Re = 1,Pr= 7.2 andGr= 105 for various Da is qualitatively similar

to that for Re

= 10 with identical Pr

. It is observed that for

Re = 1, symmetric distribution in heatlines and isotherms is ob-

served due to weak lid driven flow compared to buoyancy forces

whereas isotherms tend to compress towards the left wall for

Re = 10 for Pr= 7.2. However, multiple heatline circulation cells

are found to be qualitatively similar for both Re = 1 and 10 with

Pr= 7.2 andGr= 105 at higherDa, and a representative case is ex-

plained next.


Re = 10, Pr= 7.2 andGr= 105 for various Da(=105 103). But at

low Da(Da = 105), natural convection is weak and lid velocity

dominates the circulation. Thus a single streamline circulation

spans the entire cavity even at highGr. The isotherms are smooth

and span the entire cavity (Fig. 9(a)). The buoyancy effects are

more dominant and a secondary circulation that spans half of the

cavity is observed for Da = 104 (Fig. 9(b)). But the strength of pri-

mary circulation is higher than the secondary one. The isotherms

are distorted but symmetric for hP 0.6. Multiple circulation cells

with two secondary cells are observed at higher Da (Da = 103)

near the top wall (Fig. 9(c)). The isotherms are distorted and small

effect of the moving lid is still observed. It is found that isotherms

with h 6 0.5 are compressed towards the top portion of the side

walls and an isothermal region with h = 0.5 0.6 is observed at

the central top portion of the cavity. Detailed analysis of the heat-

lines follows next.

At low Da, heatlines resemble conductive heat transfer as

shown in Fig. 9(a). The magnitude of clockwise heatline cells is

much greater than those to the left portion of the cavity for

Da = 104 (Fig. 9(b)). It is interesting to observe that jPj varies

within 00.005 along 0.6 6 Y6 1 of the left wall whereas that var-ies within 06 jPj 6 0.007 in the same regime on the right wall sig-

nifying larger heat transport to the top portion of the right wall.

Thus isotherms with h 6 0.4 are compressed towards the top cor-

ner of the right wall and the thickness of the thermal boundary

is observed to be smaller in this region compared to left wall

(Fig. 9(b)). Heatlines follow similar trend as that of streamlines

with multiple circulation cells even at high Da . Multiple heatline

circulation cells similar to flow circulation cells are observed illus-

trating dominant convection at Da = 103. Dense heatlines from

the bottom, along the heatlines circulation cells, are observed at

the central top portion of the cavity. Thus an isothermal zone with

h = 0.5 0.7 is observed at the core of the cavity ( Fig. 9(c)). It may

be noted that jPj varies within 0 0.01 near the top potion of the

right wall whereas lesser gradients in jPj = 00.005 are found inthe same regime on the left wall. Thus isotherms with h 6 0.4 are

largely compressed to the top portion of the right wall, compared

to left wall. Also, due to multiple circulation heatline circulation

cells, there are different levels of compression of isotherms along

the bottom wall and thus oscillatory and non-monotonic variation

in the isotherms is observed.


Re = 100, Pr= 7.2 andGr= 105 for various Da(=105 103). Iso-

therms appear to be distorted to the top portion of the cavity

whereas they become parallel near the hot bottom wall

(Fig. 10(a)). Similar patterns of isotherms were observed at

Gr= 103 (Fig. 7(a)). A secondary circulation cell is observed to the

bottom right portion of the cavity atDa = 104 (seeFig. 10(b)), sig-

nifying local dominance of buoyancy forces. The span of the sec-ondary circulation is found to be increased and the primary

streamline circulations are confined to the left and top portions

of the cavity at Da = 103. Highly distorted isotherms with an iso-

thermal zone to the top portion of the cavity, with isotherms being

compressed towards the bottom wall is observed at higher Da

(Fig. 10(c)).

It is found that the magnitude of heatlines is very less signifying

less heat transfer from the bottom wall at Da = 105. It is also ob-

served that due to weak fluid motion, some heatlines from the bot-

tom wall are observed to end directly to the bottom portion of the

right wall. Also, heatlines from the right wall end towards the rel-

atively cooler portion on the right wall (Fig. 10(a)). However, local

thermal mixing zone due to weak heatline circulation cells is ob-

served at the top right portion of the cavity, signifying an isother-

mal zone with h = 0.10.2. Also smaller magnitudes in heatlines,

signifying negligible heat transfer rates are observed to the bottom

left portion of the cavity. Thus smooth and parallel isotherms are

observedwithin the cavity. A secondary circulation cell in the heat-

lines is observed to the bottom right portion of the cavity at

Da = 104. But the magnitude of the heatlines is less as also seen

at lowDa. It is interesting to observe that larger thermal gradients

are observed near the left wall compared to the right wall. It may

be noted that heatlines that emanate from 0 6X6 0.7 of the bot-

tom wall end towards the left wall signifying larger heat transfer

towards the left wall. Heatline circulation cells with jPjmax =

0.0008 is observed near the adiabatic wall. Thus a local thermal

mixing zone with h 6 0.2 is observed in this region (Fig. 10(b)).

Enhanced secondary circulations in heatline cells occur at high-

er Da(Da = 103). It is found that due to the primary heatline circu-

lation cells, heatlines that endtowards the left wall are compressed

and dense heatlines are observed near the left wall with

06 jPj6 0.003 whereas that varies as 0 6 jPj6 0.0005 along the

right wall (Fig. 10(c)). Thus the thermal boundary layer is more

compressed towards the left wall. Also, dense compressed heat-

lines are observed along the heatline cells near X = 0.20.3 of the

bottom wall, due to enhanced secondary circulation. Thus iso-

therms withhP 0.5 are compressed to the bottom wall in this re-

gion and smaller thermal boundary layer is observed atX= 0.20.3.

3.5. Case 4: Linearly heated left wall with cooled right wall

As the left wall is linearly heated and bottom wall is uniformly

heated, hot fluid fromthe bottom rises to the top along the left wall

and flows down along the cooled right wall. Thus strong primary

circulation cells are found resulting in unidirectional flow in the

cavity. However, larger heat flux exists near the bottom right cor-

ner of the cavity where hotcold junction exists similar to case 1.

Conduction dominant heat transfer is clearly illustrated by the

smooth and parallel heatlines in the cavity for low governing

parameters (Re = 1,Pr= 0.026, 0.7 andGr= 103), irrespective ofDa

(figure not shown). Enhanced secondary circulations in streamlines

to the top left corner of the cavity are observed at Da = 103

. Dis-torted isotherms with h 6 0.4 are observed, compressed towards

the right wall and top corner of the left wall. Qualitatively, similar

trend is observed with Re = 10 andGr= 103 forPr= 0.7.


Re = 1,Pr= 7.2 andGr= 105 for variousDa(=105 103). The sec-

ondary cell is found to be gradually intense with increase in Da to

103. Dominant lid driven effect is observed irrespective ofDa at

high Gr for Pr= 7.2 even at low Re. A single circulation in the

streamline cells is observed and the isotherms with h 6 0.2 are

compressed towards the right wall and top portion of the left wall

(Fig. 11(a)). Isotherms are found to be distorted and isotherms with

h 6 0.4 are compressed towards the right wall and top portion of

the left wall atDa = 104 (Fig. 11(b)). Larger magnitude in flow cir-

culations, due to enhanced convective effects with primary circula-tion cells stretch diagonally to the cavity is observed for higher Da



13/22

(Fig. 11(c)). Isotherms with h 6 0.5 are much compressed towards

the right wall and a large portion at the core of the cavity is main-

tained isothermal withh = 0.5 0.7 (Fig. 11(c)).

Conduction dominant heat transfer is observed based on theheatlines perpendicular to isothermal surfaces at Da = 105. How-

ever, heatline circulation cells illustrating dominant convective

heat transfer are observed at Da = 104 (Fig. 11(b)). It may be noted

that variation in heatfunction is large (06 jPj6 0.1) at the top

portion of the right wall compared to left wall where jPj varies

within 0 0.05 near the adiabatic wall. Thus isotherms with

h 6 0.4 are much compressed to right wall. Heatline circulation

cells similar to streamline cells are observed at higher Da

(Fig. 11(c)). Largely intense heatlines from the bottom wall are

found to occur along the circulation cells and an isothermal zone

with h= 0.50.7 is observed at the core of the cavity. Enhanced

convective effects are observed atDa = 103. It may be noted that

jPj varies within 0 0.2 at Da= 104 whereas that varies within

jPj = 00.6 along 0.56 Y6 1 of right wall forDa = 103

. Thus ther-mal boundary layer is more compressed towards the right wall

with an increase inDa, and isotherms with h6 0.5 are compressed

towards the right wall at higherDa (Fig. 11(c)).

The distribution in streamlines, isotherms and streamlines for

Re = 10, at higher Gr are qualitatively similar to those with Re = 1

(Fig. 11), irrespective ofDaand a similar explanation may be given.

Also, the distribution in streamlines, isotherms and heatlines for

Re = 100, Gr= 105 and Pr= 7.2 for varying Da = 105 103 are

qualitatively similar to that of case 1 under identical parameters

as seen inFig. 5 and hence the detailed discussions are omitted

for the brevity of the manuscript. However, it is observed that lar-

ger gradients in heatfunctions are observed compared to those of

case 1, irrespective of Gr, illustrating larger heat transfer rates.

Thus smaller thermal boundary layer thickness towards the left

and bottom walls is observed in case 4 compared to case 1 under

identical parameters.

3.6. Heat transfer rates: local Nusselt numbers

Distributions of local Nusselt numbers demonstrate conduction

dominant mode for low governing parameters (Re = 1,Gr= 103,

Pr= 0.7) with Da6 104. It is observed that the distribution in local

heat transfer rates for Re = 10is similar to that ofRe = 1 due tosim-

ilar qualitative distribution in heatlines irrespective ofPrand Da.

Smaller distribution in local Nusselt numbers is observed along

the bottom and side walls for Pr= 0.7. We will discuss the test

cases for Re = 1 and 100 forPr = 7.2 andGr= 105 with Da ranging

within 105 103, where natural convection is dominant at

Re = 1 and forced convection is dominant at Re = 100.

Fig. 12(a)-(c) and Fig. 13(a)-(c) demonstrate the effect of Da

(105 103) on spatial distribution of the local Nusselt numbers

at the bottom, right and left walls (Nub, Nurand Nul), forGr= 105

,Pr= 7.2 at Re = 1 (solid line) and100 (dotted line).

3.6.1. Case 1: Uniform heating of bottom wall

The upper panel plots ofFig. 12(ac) illustrate the local Nusselt

number vs. distance along the bottom and side walls for case 1. It is

observed that the heat transfer rate (Nub) is very high at the edges

and that is gradually reduced towards the center of the bottom

wall at Re = 1 irrespective ofDa (upper panel ofFig. 12(a)). Sym-

metric distribution in Nub is observed at Re = 1, irrespective of

Da, with a minima at the X= 0.5. This is due to the symmetric dis-

tribution in heatlines caused by the weak lid driven effect. It is also

interesting to observe that distribution ofNub is almost flat over a

large region near the center for Da = 105 corresponding to smaller

gradients and uniform distribution in jPj with 06 jPj6 0.015along 0.46X6 0.6 of the bottom wall (see Fig. 4(a)). It is also

0

.03

0.02

0.01

0

.005

0

.002

(a)

0 0.5 10

0.5

1

0.3

0.20.1

50

.1

0

.01

0.03

.01

.001

(b)

0 0.5 10

0.5

1

0.70.50.250.1 0.1

0.5

1

1.5

1.8

(c)

0 0.5 10

0.5

1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2

0.1

0.2

0 0.5 10

0.5

1

.2

0.3

0.40.5

0.5

0.6

0.7

0.8

0.9

0.4

0.3

0.2

0.1

0 0.5 10

0.5

1

0.30.4

0.5

0.6

0.7

0.8

0.5

0.40.30

.2

0 0.5 10

0.5

1

50

.0

0.04

0.03

0.02

0.01

0.005

0

.001

0.005

0.01

0.02

0.03

0.04

0.05

0.08

0.1

0 0.5 10

0.5

1

.1

0.09

0.0

80

.07

0.0

40

.01

0.04

0.1

0.15

0.2

0.25

0.28

0 0.5 10

0.5

1

0.30.20.12

0

.01

0.15

0.30

.5

0.70.8

1

0 0.5 10

0.5

1

Fig. 11. Streamfunction (w), temperature (h) and heatfunction (P) contours for case 4 with Re = 1,Pr= 7.2,Gr= 105 and (a) Da= 105, (b) Da= 104 and (c)Da = 103.



14/22

found that Nub forDa = 104 is less compared to Da = 105 along

0.356X6 0.65 of the bottom wall due to less intense heatlines

corresponding to jPj = 00.05 in this region (seeFig. 4(b)). Larger

distribution in Nub is observed for Da = 103 compared to lower

Da due to largely intense heatlines along the bottom wall corre-

sponding to 06 jPj6 0.5. Smaller gradients in heatfunction withdistance are observed along X= 0.40.6, towards the center of the

bottom wall for Da= 103. Thus Nub starts to decrease faster at

X= 0.40.6 and reaches a minima towards the middle portion of

the bottom wall (seeFig. 12(a)). It may be noted that larger gradi-

ents in heatfunctions corresponding to 0.46 jPj 6 0.6 are observed

to the corner of the bottom wall for Da = 103 whereas 0.26 jPj

6 0.25 occur for Da = 104 andjPj = 0.080.1 for Da = 105. Thus

larger values of Nub are observed at corners of the bottom wall

for higherDa.

The distribution in Nub for Da = 105 with Re = 100 (see

Fig. 12(a), upper panel, dotted line) is similar to the distribution

with Re = 1. This is due to the conduction dominant heat transfer

and less magnitudes in heatfunctions (jPj = 00.001) (Fig. 5(a)) sig-

nifying smaller heat transfer rates. However, non-symmetric dis-tribution in Nub is observed at higher Da(Da = 10

4 103). It is

observed that Nub reaches a minima at X= 0.3 withNub smaller

than that withDa = 105. Lower gradients in jPj corresponding to

0.0023 6 jPj6 0.0024 are observed for Da = 104 whereas larger

gradients in jPj(jPj = 0.00030.0006) were observed in this zone

for Da = 105 (Fig. 5(a) and (b)). Thus heat transfer rates in this

zone for Da = 104

are lower than those for Da = 105

. Similar toRe = 1,Nub attains a maxima at the corners of the bottom wall.

It is interesting to note that similar to Da = 104, Nub first

reaches a minima at X= 0.1 andNub is less compared to lower Da

for Da= 103, due to lesser gradients in heatlines corresponding

to 0.0056 jPj6 0.006 in this zone. Further Nub increases and

reaches a maxima at X= 1 for Da= 103. It is also observed that

the maxima in Nub atX= 1 is larger compared to that atX = 0.1,

whereas larger values ofNub is observed for Da = 104 atX = 0.1,

compared to X= 1. This is based on the fact that larger gradients

in heatfunctions corresponding to 06 jPj6 0.0032 are observed

at X= 1 whereas that varies within jPj = 0.00550.006 at X= 0.1

forDa = 103 (seeFig. 5(c)).

The upper panel plot of Fig. 12(b) illustrates the local heat

transfer rates along the right wall (Nur) for case 1. A maxima inNur for smaller Da is observed at Y= 0, thereafter that decreases

0

3

6

9

12

15 Case 1

Bottom Wall(a)

Da=10-3

Da=10-4

Da=10-5

0.2 0.4 0.6 0.8

Distance, X

0

3

6

9

12

LocalNusseltNumber,

Nub

Case 2 Da=10-3

Da=10-4

Da=10-5

0

3

6

9 Case 1 Da=10-3

Da=10-4 Da=10

-5

Right Wall(b)

0.2 0.4 0.6 0.8

Distance, Y

0

3

6

9

Nur

Case 2

Da=10-3

Da=10-4Da=10

-5

0

3

6

9 Case 1

Da=10-3

Da=10-4

Da=10-5

Left Wall(c)

0.2 0.4 0.6 0.8

Distance, Y

0

3

6

9

Nul

Case 2Da=10

-3 Da=10-4

Da=10-5

LocalNusseltNumber,

LocalNusseltNumber,

Fig. 12. Variation of local Nusselt number with distance at (a) bottomwall (b)right wall (c) left wall for Pr= 7.2, Gr= 105, Re=1 () and 100 (). In each plot, upper panel

corresponds to case 1 and lower panel corresponds to case 2.



15/22

linearly to attain a minima atY= 1. Larger values ofNurdue to lar-

ger gradients in heatfunctions (jPj = 0.050.1) are observed to the

bottom portion of the right wall for Da = 105. On the other hand,

smaller gradients in heatfunctions (jPj = 00.04) are observed

along 0.36 Y6 1 ofthe right wall (Fig. 5(a)) and thus Nur decreases

along the length. Similar qualitative trend is observed for Da= 104

but lower Nur compared to Da = 105 is observed. Further, Nur

slowly increases to a maxima atY= 1. It may be noted that jPj var-

ies within 0.20.22 along Y= 0 0.2 whereas larger gradients in

jPj (jPj= 0.020.2) are observed along 0.2 6 Y6 0.9 of the right

wall for Da = 104. Thus Nur first reaches a minima at Y= 0.15

and thereafter increases monotonically along the length. It is inter-

esting to observe that Nur increases linearly along the right wall

and that reaches maxima at Y = 1, for Da = 103. This is based on

the fact that large gradients in heatfunctions are observed towards

the top portion of the right wall as largely intense heatlines corre-

sponding to 06 jPj6 0.2 are observed towards the top portion of

the right wall (Fig. 5(c)).

The distribution in Nurfor Re = 100 is qualitatively similar dis-

tribution to that ofRe = 1 forDa = 105

and 104

. It may be notedthat smaller values in Nurare observed due to less intense heat-

lines corresponding to jPj6 0.001 along a large portion on the

right wall for lower Da (Fig. 5(a), (b)). However, larger gradients

in heatfunctions (jPj = 00.0035) are observed along the right wall

forDa = 103. Thus larger distribution in Nuris observed compared

to lower Da. Also, it may be noted that dense heatlines (jPj = 0

0.002) are observed to the top portion of the right wall signifyinglarger Nur in this zone for Da = 10

3. Thus Nurreaches a sudden

maxima atY= 1.

The upper panel plot ofFig. 12(c) illustrates the local heat trans-

fer rates along the left wall (Nul). Due to the symmetry in heatline

distribution irrespective ofDa, the variation in Nulis similar to that

ofNurfor Re = 1 and may be explained in a similar manner.

It may be noted that larger values ofNulare observed compared

toNurirrespective ofDa, due to larger gradients in heatfunctions

for Re = 100. It is observed that jPj varies within 00.003 along

the left wall whereas that varies within 00.0008 along the right

wall for Da = 105 (Fig. 5(a)). On the other hand, smaller magni-

tudes in heatfunctions corresponding to jPj6 0.0001 are observed

to the top portion of the left wall signifying lowerNulin this zone.

Similar qualitative trend is observed with Da = 104

. It is interest-ing to observe thatNulfirst reaches a maxima at Y= 0.2 and there-

0

3

6

9Case 3

Bottom Wall(a)

Da=10-3

Da=10-4

Da=10-5

0.2 0.4 0.6 0.8

Distance, X

0

3

6

9

12

15

LocalNusseltNumber,

Nub

Case 4

Da=10-3

Da=10-4

Da=10-5

-2

0

2

4

6 Case 3

Da=10-3

Da=10-4

Da=10-5

Right Wall(b)

0.2 0.4 0.6 0.8

Distance, Y

0

3

6

9

Nur

Case 4Da=10-3

Da=10-4

Da=10-5

-2

0

2

4

6 Case 3

Da=10-3Da=10-4

Da=10-5

Left Wall(c)

0.2 0.4 0.6 0.8

Distance, Y

0

3

6

9

Nul

Case 4

Da=10-3

Da=10-4

Da=10-5

LocalNusseltNumber,

LocalNusseltNumber,

Fig. 13. Variation of local Nusselt numberwith distance at (a)bottom wall (b)right wall (c)left wall for Pr= 7.2, Gr= 105, Re = 1 () and 100 (). In eachplot, upper panel

corresponds to case 3 and lower panel corresponds to case 4.



16/22

after, that reaches a minima at Y= 0.5 and then monotonically in-

creases to reach a maxima at Y= 1 forDa = 103. Larger gradients

in heatfunctions with jPj = 0.0050.006 are observed at Y = 0.15

0.2 whereas smaller gradients corresponding to 0.0036 jPj6

0.005 are observed over a large zone near the central portion of

the left wall. Thus Nul first reaches a maxima that further decreases

to a minima at Y= 0. 5 (Fig. 5(c)). Dense heatlines from the center of

bottom wall withjPj

= 00.001 are observed atY

= 1, illustrating a

maxima inNur(upper panel ofFig. 12(c)).

3.6.2. Case 2: Non-uniform heating of bottom wall

The lower panel plots ofFig. 12(a)(c) represent the distribution

of local heat transfer rates along the bottom and side walls for case

2. It is observed that Nub for non-uniform heating is less than that

for uniform heating case at the edges irrespective of allDaas there

is no discontinuity in temperature due to sinusoidal type of heat-

ing. ThusNub is zero near the junction of hot and cold walls.

Symmetric distribution in Nubis observed for Re = 1 dueto sym-

metric distribution in sinusoidal heating of the bottom wall and

also due to symmetric distribution in heatlines irrespective ofDa.

It is observed that Nub increases towards the middle portion of

the bottom wall and a maxima is observed at X= 0.5 for Da=

105 andRe = 1. This is based on the fact that larger gradients in

heatfunctions corresponding to largely intense heatlines (jPj =

00.01) is observed at the center of bottom wall (figure not

shown). Also, sinusoidal heating provides maximum temperature

atX= 0.5 and the temperature decreases towards the end. The var-

iation ofNubwithin 0.36X6 0.7 is qualitatively similar to that of

case 1 forDa = 104. Heatlines corresponding to 0.16 jPj6 0.4 are

observed along X = 0.20.3 and 0.70.8, signifying larger Nub in

that zone. Thus a maxima inNubis observed in this zone. Qualita-

tively similar variation inNubis observed forDa = 103. Sinusoidal

variation in heat transfer rates (Nub) are also observed with a max-

ima atX= 0.30.4 and 0.60.7 and local minima atX= 0, 0.5 and 1.

Largely intense heatlines corresponding to jPj = 0.10.4 are ob-

served signifying a maxima in these zones and less intense heat-

lines with jPj = 00.05 (figure not shown) at the centerillustrating a minima inNub. Although lower values ofNub are ob-

served at the edges compared to case 1, but larger values ofNubcompared to case 1 are observed towards the center of the bottom

wall as larger heating effects are observed atX= 0.5. This is alsoex-

plained by the larger heatfunction gradients in case 2.

The distributions in Nubfor Da = 105 and Re = 100 are similar to

that of Da = 105 and Re= 1, due to conduction dominant heat

transfer and very low magnitudes in heatfunctions. Non-symmet-

ric sinusoidal variation ofNub forDa= 104 and 103 is observed

at higher Re. A local minima of Nub at X= 0.4 and a maximum

Nub atX = 0.60.7 (Nub= 5.8) is observed for Da = 104. Heatlines

corresponding to jPj = 0.00150.0016 are observed along X= 0.4

signifying lesser heat transfer rates in this regime. Largely intense

heatlines corresponding to jPj = 00.0008 are observed withinX= 0.60.7 (figure not shown) illustrating larger Nubin this regime.

High values of Nub are also observed for Da = 103 compared to

those for lower Da atRe = 100. Sinusoidal variation is clearly ob-

served from the heat transfer rates with a maxima at X = 0.60.7

and a local minima at X= 0.250.3 atDa = 103. Less intense heat-

lines corresponding to jPj = 0.00220.0024 are observed along

06X6 0.3 of the bottom wall whereas dense heatlines with jPj

varying within 00.0015 at X= 0.60.65 illustrate higher heat

transfer rates in this zone (figure not shown).

The lower panel ofFig. 12(b) illustrates the local heat transfer

rates for cold right wall for various governing parameters. The spa-

tial distribution ofNuris almost similar to that of case 1 except that

a minima is observed to the bottom corner of right wall, due to

non-uniform heating. An overall decreasing trend in Nur is ob-served for Da = 105 similar to case 1. At higher Da(103), Nur is

found to increase linearly towards the top end, due to dense heat-

lines corresponding to 06 jPj6 0.4 in that zone. The distribution

is qualitatively similar to that of case 1 along a large zone on the

right wall and a similar explanation based on heatlines follows.

Lower distributions ofNurare observed for Re = 100 compared

to Re = 1, similar to case 1. Lower magnitudes of heatlines with

jPj6 0.0001 occur at the top portion of right wall even at higher

Dafor

Pr= 7.2 and

Gr= 105. Overall, the distribution in

Nuris qual-

itatively similar to that of case 1 and a similar explanation follows.

It is observed that jPj varies within 06 jPj6 0.003 along

0.66 Y6 1 of the right wall for case 1 whereas that varies within

06 jPj6 0.002 in the same region on the right wall for case 2 at

higher Da. Thus larger values ofNur is observed with case 1 com-

pared to case 2.

The lower panels ofFig. 12(c) illustrate the local heat transfer

rates for cold left wall for various governing parameters. Due to

the symmetry in heatline distribution irrespective ofDa, the vari-

ation inNul is similar to that ofNurfor Re = 1 and the trend may

be explained in a similar manner. Also, the variation in Nul for

Re = 100 (lower panel) is qualitatively similar to that of case 1. Sim-

ilar toNur, lowerNul values are observed compared to case 1 and

that may be explained based on similar arguments for case 1. It

may be noted that larger gradients in jPj are observed along the

left wall with jPj = 00.004 whereas smaller gradients in heatfunc-

tions corresponding to 06 jPj6 0.0023 are observed along

0.26 Y6 1 on the right wall at higherDa (Da = 103). Thus larger

values ofNul is observed compared to those ofNur.

3.6.3. Case 3: Linearly heated side walls with uniformly heated bottom

wall

The solid line in upper panel plots ofFig. 13(ac) illustrate the

local Nusselt number vs. distance along the bottom and side walls

for case 3. Due to linearly heated side walls, the heat transfer rate,

Nub, is 1 at the edges of the bottom wall. It may also be noted that

due to symmetry in the temperature field and conduction domi-

nant heat transport, the heat transfer rate is symmetric along thebottom wall and is almost constant atDa = 105 andRe = 1 (figure

not shown). Also, parallel and uniformly distributed heatlines with

jPj = 00.04 are observed along the bottom wall. Lower values in

Nub are also observed for Da= 104 due to uniform distribution

of heatlines corresponding to 06 jPj6 0.05 along the bottom wall.

On the other hand, it is observed that Nub has a maxima atX= 0.25

and 0.65 and a minima is observed at X= 0.5, similar to case 2 for

Da = 103. This trend is observed due to multiple heatline circula-

tion cells within the cavity. Smaller gradients in jPj corresponding

to 06 jPj6 0.01 are observed at X = 0.5, illustrating a minima in

this zone. Also, largely intense heatlines corresponding to

0.056 jPj6 0.1 are observed at X= 0.25 and 0.65 signifying larger

Nub in this zone.

It may be noted that constant and low values in Nub are ob-served for Da= 105 due to less magnitudes in heatlines corre-

sponding to jPj = 00.0008 along the bottom wall (seeFig. 10(a))

even at highRe(Re = 100). It is interesting to observe that variation

inNubis symmetric to the bottom wall forDa= 104. This is based

on the fact that uniform distribution in heatfunctions correspond-

ing to jPj = 00.001 is observed along the bottom wall. Note that,

smaller gradients in jPj corresponding to 0.00126 jPj6 0.0013

are observed alongX= 00.15 (Fig. 10(b)). Thus smallerNub is ob-

served in this regime. Similar trend is observed near the right cor-

ner of the bottom wall. However, it is interesting that non-

symmetric distribution inNubis observed for Da = 103. Note that

Nub first reaches a maxima at X= 0.2 and further, that decreases

linearly to a minima atX= 1. Largely intense heatlines correspond-

ing to jPj = 0.00120.0023 are observed atX= 0.150.25 illustrat-ing a maxima in Nub in this zone (Fig. 10(c)). In contrast, lower



17/22

magnitudes in heatlines with jPj6 0.001 are observed towards the

right corner of the bottom wall, signifying lower Nub.

The solid line in upper panel ofFig. 13(b) illustrates the heat

transfer rates along the right wall forRe = 1. The heat transfer rate

at the bottom edge is zero due to linearly heated side wall. It is

interesting to observe that Nurhas a maxima at the top edge due

to larger gradients in heatfunctions, irrespective of Da. Uniform

and low values inNur

are observed forDa

= 105, due to conduc-

tion dominant heat transfer illustrated by the uniform variation

in heatfunctions on the right wall (figure not shown). It may be

noted that Nur is negative up to Y6 0.6 and thereafter that in-

creases to a maxima at Y= 1 forDa = 104. Negative values ofNurare observed due to change in the sign of the gradient of heatfunc-

tion as described by the definition of heatfunction along the right

wall (Eq.(16)). This is also based on the fact that heatlines corre-

sponding to jPj = 0.050.07 start from 06 Y6 0.6 of the right wall

and these end towards the top portion of the right wall signifying

largerNuratY= 1. It is interesting to observe that the heat transfer

rate (Nur) shows an oscillatory trend with a maxima and minima

attaining alternatively for Da = 103 andRe = 1. Also,Nuris found

to be positive up to Y= 0.4, thereafter, that becomes negative up

to Y= 0.7 and finally that increases monotonically with distance.

This is based on the fact that largely dense heatlines are observed

along the multiple heatline circulation cells near Y= 0.2. Thus a

maxima is observed. It is also observed that dense heatlines corre-

sponding to jPj = 00.1, that emanate from 0.46 Y6 0.7 of the

right wall end towards 0.76 Y6 1 of the right wall. Thus large neg-

ativeNuris observed atY= 0.6.

Negative heat transfer rates are observed along the right wall

except at the top corner, due to less intense heatlines with

jPj6 0.0002 and negative gradients of jPj signifying less heat

transfer, occurring along a large zone on the right wall for

Da = 105 at Re= 100 (seeFig. 10(a)). Similar qualit

fem solution

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