Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 3, pp. 519-530, Warsaw 2015
DOI: 10.15632/jtam-pl.53.3.519

INCLINATION ANGLE IMPLICATIONS FOR FLUID FLOW AND MIXED

CONVECTION IN COMPLEX GEOMETRY ENCLOSURE-MESHLESS

NUMERICAL ANALYSES

Mohammad Najafi

Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran,

Iran; e-mail: najafi@srbiau.ac.ir

Mehdi Nikfar

Department of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran

Ali Arefmanesh

Department of Mechanical Engineering, University of Kashan, Kashan, Iran

The meshless local Petrov-Galerkin (MLPG) method is extended to analyze the mixed co-
nvection and fluid flow in an inclined two-dimensional lid-driven cavity. The enclosure con-
sidered comprises two insulated vertical walls and a wavy bottom wall which is subjected
to a higher constant temperature than its top counterpart, the sliding lid. For the proposed
scheme, the stream function formulationwith aweighting function of unity is employed.The
simulation results reveal that the local Nusselt number increases with a clockwise increase
in the inclination angle. Also, a decrease in the aspect ratio results in an increase in the hot
wavy wall averageNusselt number.

Keywords: meshless, Petrov-Galerkin,mixed convection, rectangular cavity, wavy wall

1. Introduction

In the past two decades, a number of mesh-free numerical methods were proposed for so-
lving partial differential equations. The principal objective of developing these methods was
to get around the difficulties associated with generating a mesh in conventional numerical
techniques such as the finite difference and the finite element methods. In the earlier mesh-
free techniques, an auxiliary grid had been required to evaluate the integrals resulting from
applying the Galerkin method to the differential equations. Hence, these methods could not
be considered totally meshless. Subsequently, two truly meshless techniques, the meshless lo-
cal boundary equation (MLBE) method and the meshless local Petrov-Galerkin (MLPG) me-
thod were proposed by Zhu et al. (2004), and Atluri and Zhu (1998), respectively. To ob-
tain the discretized equations in the latter scheme, instead of a mesh, an arbitrary point-
-distribution, without any pre-specified relationship among the points, is generated in the com-
putational domain. Subsequently, the shape functions from the moving least squares (MLS)
interpolations are employed in a local weak form of the differential equations over a local sub-
domain. The arbitrariness of the point-distribution as well as the possibility of adding or remo-
ving points to or from pre-existing point-distributions without any considerable computational
effort make the MLPG method quite appealing for domains with complex geometries such as
enclosures having wavy walls.
As far as application of the MLPG method on the fluid flow and heat transfer analysis,

a number of recent studies may be mentioned. Lin and Atluri (2001,2001) applied the newly-
-developed MLPG method to the solutions of the convection-diffusion and the Navier-Stokes
(N-S) equations. Theymodified the local weak forms to overcome the so-called Babuska-Brezzi



520 M. Najafi et al.

conditions while solving the primitive variables form of the N-S equations. Moreover, they pre-
sented a different upwinding scheme in order to obtain stabilized solutions underhighPeclet and
Reynolds (Re) numbers.Arefmanesh et al. (2005) applied a variation of theMLPGmethodwith
unity as the test function to the convection-diffusion and the potential flow equations. Compari-
sons of their resultswith the analytical solution for the convection-diffusion equation in a square
domain demonstrated high accuracy of their proposed method. More recently, Arefmanesh et
al. (2008) used a variation of theMLPG method with the unit test function to solve a number
of non-isothermal fluid flow problems. They employed the stream function-vorticity formulation
to solve different test cases such as a non-isothermal lid-driven cavity flow with an inlet and
an outlet. In another study, Haji Mohammadi (2008) applied the MLPG method to simulate
the incompressible viscous fluid flow. He formulated his considered cases in terms of the stream
function-vorticity and employed the radial basis function interpolations in his approach.

Mixed convection flow and heat transfer in an enclosure involving a moving lid has drawn
extensive attention due to its broad applications in engineering,modern technology, and natural
situations. Among these applications, there is cooling of electronic devices, float glass produc-
tion, food processing, and fluid flow and heat transfer in solar ponds. Flow and heat transfer
from irregular surfaces are often encountered in many engineering applications to enhance heat
transfer such as micro-electronic devices, solar collectors, electrical machinery, etc. Moreover,
roughened and wavy surfaces could be used in the cooling of electrical and nuclear components
to enhance the heat transfer.On the other hand, the studyof natural andmixed convection heat
transfer in inclined enclosures is relevant in geophysical situations, honeycomb solar collector
plates, heat exchangers and crystal growth processes from melts, etc. Determining the optimal
inclination angle is of paramount importance in such cases. As far as fundamental research is
concerned, tilting the enclosure allows one to study different instabilities and their correspon-
ding interactions through making a suitable choice of the values of the external parameters.
Many recent studies regarding the applications of the conventional numericalmethods onmixed
convection are available in the literature. Chamkha (2002) conducted a numerical simulation
of unsteady and laminar mixed convection heat transfer in a square cavity in the presence of
internal heat generation or absorption and a magnetic field using the finite volume method.
Mixed convection heat transfer in a two-dimensional rectangular cavity consisting of isothermal
side walls moving vertically and a partially-heated bottomwall was studied numerically byGuo
and Sharif (2004) using the finite volume method. In another study in the same year, Oztop
and Dagtekin (2004) numerically simulated a steady state two-dimensional mixed convection
problem in a cavity having two differentially-heated and vertically driven side walls using the
finite volumemethod and the SIMPLE algorithm. The study showed that both the Richardson
number and the direction of the moving walls affect the fluid flow and heat transfer inside the
cavity.More recently, a finite element analysis of unsteady laminarmixed convection heat trans-
fer in a cavity with sinusoidally oscillating horizontal top-lid was carried out by Khanafer et
al. (2007). The laminar mixed convection heat transfer in two-dimensional shallow rectangular
driven cavities with an aspect ratio of 10 was studied numerically by Sharif (2007) using the
FLUENT version 6 commercial code. In another study, Al-Amiri et al. (2007) investigated the
mixed convection heat transfer and fluid flow in a lid-driven cavity being heated from a sinuso-
idalwavy bottom surface using the finite elementmethod.Their results showed that the average
Nusselt number increased with an increase in both the amplitude of the wavy surface and the
Reynolds number. Very recently, Nasrin (2012) conducted a finite element simulation of the
mixed convection in an enclosure with a wavy bottomwall maintained at a higher temperature
than those of the lid and two insulated sidewalls. For Ra=104, Re=100, andPr=0.7, Nasrin
investigated the effects of the cavity aspect ratio, amplitude of the wavy wall and the number
of undulations on the heat transfer inside the enclosure. It was concluded that increasing the
wavy surface amplitude and cavity aspect ratio improved the heat transfer.



Inclination angle implications for fluid flow and mixed convection... 521

In view of the above literature survey, there is not any type ofmeshless numerical treatment
in the literature which would be applied to the mixed convection heat transfer in an irregular
geometry enclosure with a wavy bottom wall within which the flow is induced by a shear force
resulting from motion of the cavity lid combined with a buoyancy force resulting from the
differentially-heated cavity walls. Moreover, a comprehensive study of the mixed convection in
such an enclosure for a wide range of the Grashof and Richardson numbers as well as different
inclination angles remains to be carried out. In the present study, therefore, the meshless local
Petrov-Galerkin method is extended to analyze the mixed convection heat transfer in a lid-
-driven cavity with awavy bottomwall and the effects of the inclination angle, Grashof number,
Richardson number and the aspect ratio of the cavity on the fluid flow and heat transfer are
investigated.

2. Problem formulation

A mixed convection fluid flow and heat transfer within a lid-driven rectangular cavity with a
wavy bottomwall is simulated numerically using theMLPGmethod. As depicted in Fig. 1, the
height and thewidth of the cavity are denoted by H and L, respectively, and the aspect ratio of
the cavity is definedasA = H/L.The left and the rightwalls of the cavity are insulatedwhile the
bottomwall of the enclosure, which is a wavywall, is maintained at a constant temperature Th,
here referred to as “hot” temperature. The enclosure’s top wall, which moves in its own plane
from left to right with a constant speedul, is kept at a constant temperature Tc, here referred to
as “cold” temperature, where Th > Tc. The cavity is filledwith a fluidwithPr=1. The thermo-
-physical properties of the fluid are considered to be constant with the exception of density in
the buoyancy term which, as will be noticed, varies according to the Bousinesq approximation
(Bejan, 2004).

Fig. 1. Lid-driven differentially heated cavity, domain, boundary conditions and a typical control volume

The steady-state fluid flow and heat transfer in the cavity are governed by the continuity,
momentumandenergy equations.Thenatural convection term is incorporated in themomentum
equation by employing the Bousinesq approximation (Bejan, 2004). In this study, the two-
-dimensional fluid flow problem is formulated in terms of the stream function and vorticity.

To cast the governing equations into a dimensionless form, the following dimensionless va-
riables are introduced



522 M. Najafi et al.

X =
x

H
Y =

y

H

U =
u

ul
V =

v

ul
θ =

T −Tc
Th−Tc

Ψ =
ψ

Hul
Ω =

ωH

ul

(2.1)

where u and v are the velocity components in the x and y-directions, respectively, ψ is the
stream function and ω is the vorticity. Substituting the above dimensionless variables into the
stream function, vorticity and energy equations results in the following dimensionless form of
the governing equations

∂2Ψ

∂X2
+
∂2Ψ

∂Y 2
=−Ω

U
∂Ω

∂X
+V

∂Ω

∂Y
=
1

Re

(∂2Ω
∂X2
+
∂2Ω

∂Y 2

)
+Ri

(
cosγ

∂θ

∂X
− sinγ

∂θ

∂Y

)

U
∂θ

∂X
+V

∂θ

∂Y
=
1

RePr

( ∂2θ
∂X2

+
∂2θ

∂Y 2

)

(2.2)

where γ is the inclination angle of the cavity (Fig. 1). The Reynolds number Re and the Ri-
chardson number Ri are defined as

Re=
ulH

ν
Ri=

Gr

Re2
(2.3)

where

Gr=
gβ(Th−Tc)H

3

ν2
(2.4)

where ν is the kinematic viscosity of the fluid, Gr is the Grashof number, and β is the thermal
expansion coefficient. The rate of heat transfer in the enclosure is obtained from the Nusselt
number calculations. On the hot and the cold walls of the cavity, the local Nusselt number (Nu)
is obtained from the following relation

Nu=−
∂θ

∂n
(2.5)

The Grashof numbers used in this study are 104 and 105. Two configurations for tilting the
horizontal cavity, namely, clockwise and counter-clockwise about the origin, point O (Fig. 1),
are considered in this work. For each configuration, the inclination angle γ is taken as 0, 30◦,
60◦, and 90◦. Also, for γ =0, the simulation is carried out for three aspect ratios of A =0.5, 1,
and 2.

3. Numerical procedure

In the present study, the governing equations are solved numerically by the MLPG method.
To implement the method, an arbitrary collection of points is selected in the computational
domain (Fig. 1). Subsequently, a control volume is generated around each of the points. A
typical rectangular control volume ΩI generated around point I is shown in Fig. 1.
As the first step in developing the discretized equations for the control volume ΩI, the

weak forms of the governing equations are to be obtained. This is accomplished bymultiplying
the stream function, vorticity and energy equations by the test function WI. Subsequently, the
resulting equations are integrated over ΩI. After performing the integration by parts, the weak



Inclination angle implications for fluid flow and mixed convection... 523

forms of the stream function, vorticity and energy equations for the control volume ΩI are
expressed respectively

−

∫

ΩI

∇WI ·∇Ψ dΩ +

∫

ΓI−ΓI∩Γh

WI
∂Ψ

∂n
dΓ +

∫

ΓI∩Γh

WI
∂Ψ

∂n
dΓ =−

∫

ΩI

WIΩ dΩ

∫

ΩI

WI
(
U
∂Ω

∂X
+V

∂Ω

∂Y

)
dΩ =−

1

Re

∫

ΩI

∇WI ·∇Ω dΩ +
1

Re

∫

ΓI−ΓI∩Γh

WI
∂Ω

∂n
dΓ

+
1

Re

∫

ΓI∩Γh

WI
∂Ω

∂n
dΓ +Ri

∫

ΩI

WI
( ∂θ
∂X
cosγ −

∂θ

∂Y
sinγ

)
dΩ

∫

ΩI)

WI
(
U
∂θ

∂X
+V

∂θ

∂Y

)
dΩ =−

1

RePr

∫

ΩI

∇WI ·∇θ dΩ +
1

RePr

∫

ΓI−ΓI∩Γh

WI
∂θ

∂n
dΓ

+
1

RePr

∫

ΓI∩Γh

WI
∂θ

∂n
dΓ

(3.1)

whereΓh is the portion of the domain boundary forwhich anatural boundary condition is enfor-
ced. Here, ΓI is the boundary of the control volume ΩI, and ΓI ∩Γh represents the intersection
of ΓI with Γh.

Next, the integrals in Eqs. (3.1) are to be evaluated. The Gaussian quadrature is employed
for this purpose. To perform the numerical integrations, the control volume ΩI is divided into a
number of sub-partitions (Fig. 1). The proper number of the Gauss points for the surface and
the contour integrations in each sub-domain are selected. Subsequently, the field variables are
approximated at each of the Gauss points using theMLS interpolations. Considering Ωk as the
interpolation domain for a typical Gauss point k, and Sj,j =1(1)nk, as a collection of nk nodal
points with coordinates xj ∈ Ωk (whose influence domains cover the considered Gauss point),
the unknown stream function, vorticity and temperature fields are approximated within Ωk,
respectively (Onate et al., 1996),

Ψ
(k)
(x)=

nk∑

j=1

N
(k)
j (x)Ψ̂j Ω

(k)
(x) =

nk∑

j=1

N
(k)
j (x)Ω̂j

θ
(k)
(x)=

nk∑

j=1

N
(k)
j (x)θ̂j

(3.2)

where Ψ̂j, Ω̂j, and θ̂j are fictitious nodal values for the stream function, vorticity, and tem-

perature, respectively. In Eqs. (3.2), N
(k)
j (x), j = 1(1)nk, are the MLS interpolation functions

(Onate et al., 1996).

Having obtained the weak formulations of the governing equations and the moving least-
-squares approximations of the field variables, the final step in the discretization process is to
present the fully-discretized equations for a typical control volume ΩI with the boundary ΓI.
For this purpose, the control volume ΩI is divided into a number of sub-domains. The proper
number of the Gauss points is selected in each of the sub-domains. Subsequently, the MLS
interpolations for the stream function, vorticity and temperature (Eqs. (3.2)) at a typical Gauss
point k are substituted into Eqs. (3.1). The resulting discretized equations are given as



524 M. Najafi et al.

nk∑

j=1

( ∫

ΓI−ΓI∩Γh

WIN
(k)
j,n(x) dΓ

)

Ψ̂j =−
nk∑

j=1

(∫

ΩI

WIN
(k)
j (x) dΩ

)

Ω̂j

−

∫

ΓI∩Γh

WI
∂Ψ

∂n
dΓ +

nk∑

j=1

(∫

ΩI

(
WI,XN

(k)
j,X(x)+WI,YN

(k)
j,Y (x)

)
dΩ

)

Ψ̂j

nk∑

j=1

(∫

ΩI

WI
(
UN

(k)
j,X(x)+V N

(k)
j,Y (x)

)
dΩ −

1

Re

∫

ΓI−ΓI∩Γh

WIN
(k)
j,n(x) dΓ

)

Ω̂j

+
1

Re

nk∑

j=1

(∫

ΩI

(
WI,XN

(k)
j,X(x)+WI,YN

(k)
j,Y (x)

)
dΩ

)

Ω̂j =
1

Re

∫

ΓI∩Γh

WI
∂Ω

∂n
dΓ

+Ri
nk∑

j=1

(∫

ΩI

WI
(
N
(k)
j,X(x)cosγ −N

(k)
j,Y (x)sinγ

)
dΩ

)

θ̂j

nk∑

j=1

(∫

ΩI

WI
(
UN

(k)
j,X(x)+V N

(k)
j,Y (x)

)
dΩ −

1

RePr

∫

ΓI−ΓI∩Γh

WIN
(k)
j,n(x) dΓ

)

θ̂j

+
1

RePr

nk∑

j=1

(∫

ΩI

(
WI,XN

(k)
j,X(x)+WI,YN

(k)
j,Y (x)

)
dΩ

)

θ̂j =
1

RePr

∫

ΓI∩Γh

WI
∂θ

∂n
dΓ

(3.3)

The weighting function WI is taken to be unity (WI =1) in the present study.

Similar discretized equations are obtained for all the Gauss points within the control volu-
me ΩI and, in turn, for all the control volumes of the domain. Assembling these equations for
the control volumes and enforcing the essential boundary conditions by the direct interpolation
method yield a set of algebraic equations for the stream function, vorticity and temperature.
Solving this system of algebraic equations using a band solver and the Gaussian elimination
process yields the unknown values of the field variables at the points.

4. Benchmarking of the code

In order to validate the numerical procedure, the fluid flow and heat transfer inside a square
cavity is solved using the present method and the obtained results are compared with those of
Moallemi and Jang (1992) who used the FVM. The left and the right walls of the considered
cavity are insulated and the enclosure’s top and bottom walls are maintained at constant tem-
peratures Tc and Th, respectively, with Th > Tc. The top wall of the cavity moves in its own
plane from the left to the right with a constant speed Ul. The simulations are performed for
Pr = 1. Figure 2 shows the streamlines and the isotherms obtained by the MLPG simulation
for Ri = 1 (Gr = 106, Re = 1000), and Pr = 1. The results presented by Moallemi and Jang
(1992) using the FVM are also shown in this figure. As it can be seen from Fig. 2, very good
agreements exist between the streamlines and the isotherms obtained by theMLPG simulation
and those by Moallemi and Jang (1992). Comparisons between the local Nusselt numbers for
the cold and the hot walls of the cavity obtained by the proposed MLPG method for Pr = 1,
and the results ofMoallemi and Jang (1992) are presented in Fig. 3. As it is observed from this
figure, excellent agreements exist between the local Nusselt numbers for the two methods.



Inclination angle implications for fluid flow and mixed convection... 525

Fig. 2. Streamlines and isotherms, comparison between theMLPGmethod and the results of Moallemi
and Jang (1992) for Ri= 1, (Gr=106, Re= 1000), Pr=1; (a) streamlines, (b) isotherms

Fig. 3. Local Nusselt numbers, comparisons between theMLPGmethod and the results of Moallemi
and Jang (1992) for Ri=1, (Gr=106, Re= 1000), and Pr=1

5. Results and discussions

Having validated the numerical scheme, the proposed code is employed to investigate the cha-
racteristics of the mixed convection heat transfer in rectangular cavities having wavy bot-
tom walls (Fig. 1). The geometry of the wavy surface analyzed in this study is described by
Y =0.05(1−cos(6πX)), where X and Y are the dimensionless Cartesian coordinates as shown
in Fig. 1. The calculations are performed for a fluid with Pr = 1, for Gr = 104 and 105, and
Ri=1.



526 M. Najafi et al.

To analyze the effects of the inclination angle of the cavity γ on the fluid flow and heat
transfer, the simulations are performed for two configurations of tilting the cavity clockwise and
counter-clockwise directions about point O in Fig. 1. For each of the two configurations, the
inclination angles of 0◦, 30◦, 60◦, and 90◦ are considered. The results discussed here are for
A =1, Ri= 1 and 0.01, and Gr=104.
Figure 4 shows the streamlines and the isotherms inside the cavity forRi= 1and for different

inclination angles obtained by tilting the cavity counter-clockwise about point O (Fig. 1).The
natural convection from the wavy hot wall intensifies with the increasing inclination angle.
This, in turn, enhances the downstream eddy (DSE), and results in a relatively large counter-
-clockwise rotating vortex taking place in the lower portion of the cavity in the vicinity of the
wavy wall with its illuminating eye coming closer to the cavity center away from the hot wavy
wall. Consequently, the primary recirculation vortex weakens, and diminishes in size (Fig. 4a).
The effect of the natural convection opposing the direction of the driven lid ismore prominent in
thevicinity of thewavywall right edge, therefore, as it canbe seen fromFig. 4b, the temperature
gradients along the cold and a significant portion of the hot wall reduce substantially with the
increasing inclination angle.

Fig. 4. Streamlines and isotherms for different counter-clockwise inclination angles, Ri=1, Gr=104,
and A =1; (a) streamlines, (b) isotherms

Fig. 5. Local Nusselt number distributions for different counter-clockwise inclination angles, Ri= 1,
Gr=104, and A =1; (a) along the hot wall, (b) along the cold wall

Figures 5a and 5b show the variations of the local Nusselt number along the hot and the cold
walls of the cavity for Ri= 1 and for different counter-clockwise inclination angles, respectively.



Inclination angle implications for fluid flow and mixed convection... 527

Moving from the left to the right along the hot wavy wall, it is noticed that the local Nusselt
number increases in the vicinity of the first hill by increasing the inclination angle (Fig. 5a). This
increase in the local Nusselt number is attributed to the enhancement of the natural convection,
andthe resultingboundary layer formedalong thehotwavywall.Along the remainingportionsof
thewavywall, the localNusseltnumberdecreases substantiallywithan increase in the inclination
angle (Fig. 5a).The reductionof theNusseltnumber isdue to the fact that thenatural convection
along the wavy wall in these regions opposes the primary recirculation vortex. Moreover, as it
can be observed from Fig. 5b, the local Nusselt number along the cold wall decreases with an
increase in the inclination angle which is attributed to the shrinkage of the primary vortex.

The streamlines and the isotherms inside the cavity for different inclination angles obtained
by tilting the cavity clockwise are displayed in Figs. 6a and 6b, respectively. The results in
these figures are obtained for Ri = 1, Gr = 104, and A = 1. By tilting the cavity clockwise
about point O (Fig. 1), the natural convection along the hot wavy wall, unlike the counter-
-clockwise case, assists the primary recirculation vortex (Fig. 6a). This, in turn, would result
in the enhancement of temperature gradients along the cold and the hot walls of the cavity
(Fig. 6b). Therefore, the local Nusselt numbers along the hot and the cold cavity walls, which
are depicted in Figs. 7a and 7b, respectively, increase with an increase in the inclination angle.

Fig. 6. Streamlines and isotherms for different clockwise inclination angles, Ri=1, Gr=104,
and A =1; (a) streamlines, (b) isotherms

Fig. 7. Local Nusselt number distributions for different clockwise inclination angles, Ri= 1, Gr=104,
and A =1; (a) along the hot wall, (b) along the cold wall



528 M. Najafi et al.

Figures 8a and 8b show the variations of the average Nusselt number with the inclination
angle for tilting the cavity counter-clockwise and clockwise about point O (Fig. 1), respectively.
The results in these figures are for Ri = 1, Gr= 104, and A =1. It is observed in Fig. 8a that
the optimal angle which provides the highest average Nusselt numberwhile tilting the enclosure
counter-clockwise is γ = 0◦, i.e. the horizontal cavity. However, the optimal angle while tilting
the cavity clockwise is about γ =60◦ (Fig. 8b).

Fig. 8. Variation of the averageNusselt number with the inclination angle for Ri=1, Gr=104,
and A =1; (a) tilting counter-clockwise, (b) tilting clockwise

As far as the effects of the aspect ratio of the cavity on the fluid flow and heat transfer
behavior are concerned, Figures 9a and 9b show the streamlines and the isotherms inside the
cavity, respectively. The figures are obtained for the aspect ratios of 0.5, 1, and 2 while keeping
Ri= 1 andGr=105. As it can be observed fromFig. 9a, theDSE enhances with the increasing
aspect ratio of the cavity, anda relatively large counter-clockwise eddy is generated in this region
(Fig. 9a). In fact, for the aspect ratio equal to two, the effect of the forced convection does not
reach the bottom of the cavity. Therefore, the temperature gradient decreases in the vicinity
of the hot wavy wall with increasing the aspect ratio of the cavity (Fig. 9b). As a result, the
average Nusselt number of the hot wall of the cavity decreases from 6.3855 (for the aspect ratio
of 0.5) to 4.5281 for the aspect ratio of 1, and to 2.6750 for the aspect ratio of 2.

Fig. 9. Streamlines (a) and isotherms (b) inside the cavity for different aspect ratios, γ =0, Ri= 1,
andGr=105



Inclination angle implications for fluid flow and mixed convection... 529

6. Conclusions

TheMLPGmethod is extended tonumerically investigate the incompressible steady laminarflow
of a fluid in an irregular geometry domain sustainingmixed convection heat transfer. The forced
convective flow within the cavity is attained by a sliding top lid, while the natural convective
effect is obtained by subjecting the bottom wavy wall to a higher temperature than its top
sliding counterpart. The attention on this present study stems from furthering the applicability
of the MLPG method by focusing on, perhaps, the most important mode of the heat transfer,
the mixed convection for which the broadest spectra of engineering, technological and natural
applications exist.Theresults showthat theclockwise inclinationangle tiltingassists theprimary
recirculation vortex, causing the localNusselt number for both thehot and coldwalls to increase.
The optimal inclination angle to achieve themaximumheat transfer is about 60◦. Also, for zero
inclination angle, the results show that a decrease in the aspect ratio enhances the hot wall
average Nusselt number as the effect of the forced convection on the hot wall increases for the
shorter cavity.

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Manuscript received June 7, 2014; accepted for print September 5, 2014