Impact of Reynolds and Prandtl Numbers Coupled with Viscous
Dissipation on Mixed Convection in A Square Cavity
EMMANUEL O. SANGOTAYO1,*, NIKOS MASTORAKIS2
1Department of Mechanical Engineering,
Ladoke Akintola University of Technology, Ogbomoso,
NIGERIA
2Technical University of Sofia,
Sofia,
BULGARIA
*Corresponding Author
Abstract: - Efficient thermal management in industrial manufacturing and electronic cooling systems can be
achieved by comprehending characteristics of Reynolds and Prandtl numbers in mixed convection scenarios,
which aids in the optimization of heat transfer systems through the utilization of forced and natural convection
effects. The impact of Reynolds and Prandtl numbers on mixed convection in a square cavity connected to a
moving heated horizontal plate is investigated in this research paper. The convective behavior under different
conditions was evaluated by discretizing the flow governing equations, which included the momentum and
energy equations, using the finite difference method. The research assessed various fluids, including air
(Pr = 0.7), liquid metal (Pr = 0.01), and oil (Pr = 10). The Reynolds number ranged from 0.001 to 100, the
Eckert number ranged from 0.01 to 40, and a constant Richardson number (Ri = 1) was maintained throughout.
The results revealed that the Reynolds number substantially impacts the velocity and temperature
characteristics, especially when coupled with a Prandtl number over one and when viscous dissipation remains
constant. The utmost velocity that can be attained within the cavity is significantly diminished as the Reynolds
number rises, underscoring the critical importance of dynamic fluid properties in determining heat transfer
efficiency and fluid flow characteristics. The research unveiled the critical importance of Reynolds and Prandtl
numbers in the field of fluid dynamics concerning enhancing heat transfer attributes for engineering purposes,
thereby guaranteeing the effectiveness of thermal systems.
Key-Words: - Reynolds number, Prandtl number, moving plate, finite difference method, heat transfer, fluid
dynamics.
Received: January 16, 2024. Revised: August 11, 2024. Accepted: September 7, 2024. Published: October 21, 2024.
.
1 Introduction
The study of mixed convection flow and heat
transfer in lid-driven enclosures has received
significant interest in academic literature.
Conjugate mixed convection heat transfer is
applied in a wide range of engineering and natural
processes. These include cooling electronic
components, lubrication technologies, drying
processes, food production, flow and heat transfer
in solar ponds, and the thermal-hydraulics of
nuclear reactors. Flow and heat transfer occur
frequently in obstructed enclosures in various
technical applications, such as improving heat
transfer efficiency in microelectronic devices, flat-
plate solar collectors, and flat-plate condensers in
refrigeration systems. These specific matters have
primarily been investigated about natural
convection occurring within enclosed spaces. [1],
performed a study on the properties of natural
convective flow and heat transfer around a heated
cylinder placed inside a square area with different
thermal boundary conditions. In addition, [2],
conducted a detailed investigation of the combined
effects of natural convection and conduction within
a complex enclosure. The findings revealed a clear
correlation between thermal conductivity in the
solid region and the improvement of flow and heat
transfer. The total flow and heat transfer dynamics
were shown to be considerably affected by
geometric forms and Rayleigh numbers.
[3], conducted a study on the natural
convection processes occurring in a closed cavity
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of a refrigerator. Similarly, [4], used numerical
techniques to investigate the continuous flow of
heat caused by natural convection in a square
container filled with a fixed amount of solid
material that conducts heat that contains circular or
square barriers. The investigation showed a minor
variation in the average Nusselt number between
cylindrical rods and square rods. [5], conducted a
separate investigation on natural convection in a
horizontal fluid layer that contained a conducting
body. They utilized a precise and efficient
Chebyshev spectral collocation approach and [6]
furthered their research by examining spontaneous
convection in a horizontal fluid layer that contained
a heat-generating conducting body. In addition, [7],
examined the phenomenon of natural convection
around a heated square cylinder that was positioned
inside an enclosure, with the Rayleigh number
varying between 103 and 106. The work focused on
analyzing the intricate flow and heat transfer
properties under different thermal boundary
circumstances, revealing precise differences
between heating with uniform wall temperature and
heating with uniform wall heat flux. The use of
enclosures with movable lids is of utmost
significance in the field of heat transfer
mechanisms, especially in applications such as
cooling electronic chips, harnessing solar energy,
and the food industry. [8], investigated the impact
of the Prandtl number on both the flow patterns and
the mechanisms of heat transmission in a square
container. The results revealed that the influence of
buoyancy becomes more noticeable as the Prandtl
number increases. Additionally, the authors derived
a correlation between the average Nusselt number
and the Prandtl number, Reynolds number, and
Richardson number. A numerical analysis of heat
transport through mixed convection in a two-
dimensional square cavity with an aspect ratio of 1
was conducted by [9].
2 Physical Domain
The flow of mixed convection around a heated
bluff body with a square cross-section is a
fundamental engineering problem that is relevant in
several practical scenarios. These scenarios include
heat exchangers, chemical industries, electronic
cooling, and the flow around buildings, among
others. The flow around unheated square barriers is
characterized by the interplay of a free shear layer,
a boundary layer that forms on the surfaces of the
obstacle. Heating a cylinder causes the added
buoyancy to greatly complicate the flow. The
quantification of the buoyancy effect is done using
a non-dimensional metric called the Richardson
number. This number indicates the ratio of the
buoyancy force to the inertial force. The flow field
surrounding a hot object is mainly influenced by
the Reynolds number (Re), Richardson number
(Ri), and Prandtl number (Pr).
Several empirical and computational
investigations have been carried out to analyze the
influence of these parameters on fluid dynamics.
The impact of Reynolds number (Re) on the flow
over a square cylinder has been recorded by several
studies, [10], [11]. When a cylinder is placed in a
free stream, the flow remains constant for Reynolds
numbers (Re) below 40 and becomes irregular for
Re values beyond 50, [12]. At low Reynolds (Re)
values, there is no separation of flow at the leading
and trailing edges of the cylinder, resulting in the
top and bottom surfaces behaving similarly to a flat
plate. As a result, the transfer of heat is highest at
the surfaces close to the front corners and decreases
towards the rear corners, [13]. Many scholars have
extensively studied the influence of buoyancy on
the flow field surrounding a square cylinder and
vortex shedding occurs in a cross-buoyancy flow
scenario for all Ri values in the unstable flow
regime, as shown by [14]. However, in the steady
flow regime, [15] found that vortex shedding
begins after a threshold Ri value. The reduction of
vortex shedding is observed after attaining a
threshold Ri value, which helps to enhance
buoyancy, [16]
According to [17], the requirement for a certain
Re value to initiate flow separation becomes more
important as the Ri value increases. The increase in
Ri was associated with a greater length of vortex
formation, resulting in the suppression of vortex
shedding, [18]. The Prandtl number (Pr) is a crucial
factor that affects the flow field. It is calculated by
dividing the momentum diffusivity by the thermal
diffusivity in the fluid. The range of values varies
significantly, ranging from approximately 0.001 for
liquid metals to 1025 for the Earth's mantle. Fluids
with varying Prandtl numbers are widely used in
chemical industries and nuclear reactors. However,
there have been few studies that have examined the
influence of Pr on the flow around a square
cylinder.
[19], investigated the impact of Prandtl
number (Pr) on the restricted cross buoyancy flow
around a square cylinder positioned in a channel
under constant flow circumstances, with Pr values
ranging from 0.7 to 100. The behavior of
streamlines, isotherms, and drag and lift
coefficients for different Pr values was
demonstrated. [20], examined the impact of
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the Prandtl number (Pr) on the phenomenon of
unstable forced convection around a square
cylinder. It was found that the Nusselt number,
which represents the convective heat transfer over
the surface of the cylinder, rose as the Prandtl
number grew. [21], examined the movement of
fluid around two square cylinders placed one after
the other, with a constant and controlled flow, for a
range of Prandtl numbers from 0.7 to 1000. It was
observed that when the Pr values were low, the
isotherms were wider, and when the Pr values were
high, the isotherms became narrower as a result of
decreased thermal diffusion. [22], investigated the
movement of fluid with different densities passing
between two square cylinders arranged in a line.
They conducted their study for a range of Prandtl
numbers (Pr) between 0.7 and 100, while
maintaining a constant flow rate. As the Pr values
increased, they saw a decrease in the asymmetry
caused by buoyancy in the flow field. Additionally,
at higher Pr values, there was a less pronounced
change in the lift coefficient with Ri. The current
investigation examined the flow of a heated
rectangular cavity under specific conditions: 0.001
< Prandtl number (Pr) < 10, 0.001 < Reynolds
number (Re) = 100, 0.01 < Eckert number (Ec) =
40, with a Richardson number (Ri) of 1. This study
investigated the impact of changing the Reynold
number and Prandtl number, along with the
viscous-energy dissipation function, on the flow
patterns, energy distribution, and heat transfer rate
within a rectangular cavity.
2.1 The Physical and the Mathematical
Models
Figure 1 illustrates the continuous motion of a
horizontal plate emerging from a slot at a velocity
Uw and temperature Tw into a quiescent fluid
within a rectangular enclosure. This plate serves as
the upper boundary of the enclosure, which is also
defined by a fixed horizontal isothermal wall at the
bottom, a fixed isothermal vertical wall on the left,
and an adiabatic vertical wall on the right. The
upper horizontal wall's temperature Tw is higher
than the lower horizontal wall (i.e. Tw > T∞),
leading to free convective motion within the
enclosure. The flow is considered steady,
incompressible, laminar, and two-dimensional,
with the fluid being Newtonian. Negligible heat
transfer by radiation and internal heat generation is
assumed while accounting for the viscous-energy
dissipation function effect. The fluid properties are
assumed to be temperature-independent except for
the buoyancy term in the momentum equation, for
which the Boussinesq approximation is utilized.
The extrusion die wall is both stationary and
impermeable, imposing non-slip boundary
conditions.
The governing equations for the flow at each
point in the continuum consist of mass, momentum,
and energy conservation expressions, including the
viscous dissipation term, these equations for a two-
dimensional rectangular domain are as presented in
equations (1-6):
Continuity equation:
0
y
v
x
u
(1)
The Navier-Stokes equations in the x- and y-
directions as presented in equations (2 and 3):
2
2
2
2
1
- y
u
x
u
x
p
y
u
v
x
u
u
(2)
TTβg
y
v
x
v
y
p
y
v
v
x
v
u
y
v
v
x
v
u
-
1
-
2
2
2
2
(3)
where
T-Tg
is the body force per unit
volume in the y-direction.
x
y
Continuously moving plate
c c
T/ x = 0; = = 0u v
L
H
Fig. 1: Schematic representation of the physical
model with the boundary constraints and the
coordinate axes
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The thermal energy transport equation is
expressed in equation (4):
μ
2
2
2
2
y
T
x
T
k
y
T
v
x
T
ucp
(4)
where
represents the viscous-energy-dissipation
function, defined by equation (5):
222
2
y
u
x
v
y
v
x
u
(5)
The consideration of this function becomes
crucial in cases of high fluid viscosity or flow
velocities.
The specified boundary conditions for
velocities and temperature are:
; L x 0 H, y atT T 0, v ,U u ww
; L x 0 0, y atT T 0, v 0, u
; H y 0 0, xat0 T 0, v 0, u
. H y 0 L, xat0
x
T
x
v
x
u
(6)
3 Procedure of Analysis and the
Solution Techniques
The Navier-Stokes equations represent a group of
partial differential equations that can be categorized
as elliptic, parabolic, or hyperbolic based on the
specific problem being addressed. When
considering these equations in their incompressible
form, one can opt to solve them using either the
vorticity-stream function approach or in their
primitive-variable form. For this study, the former
method is employed resulting in equations (2) and
(3) being simplified into a vorticity transport
equation by removing the pressure gradient terms
through the use of the continuity equation (1),
along with the scalar value of the vorticity, w, in a
two-dimensional Cartesian coordinate system
defined in equation (7):
y
u
x
v
. (7)
This derived expression manifests as the
dimensional vorticity transport equation (8):
2
2
2
2
yx
x
T
g
y
v
x
u
(8)
The velocity components within a two-
dimensional Cartesian coordinate system are
delineated as derivatives of the stream function, as
indicated in eqn(9):
x
v
y
u
,
(9)
Upon substitution into equation (7), the Poisson
equation for the stream function is obtained in eqn
(10):
- 2
2
2
2
yx
(10)
The energy equations developed alongside the
prescribed boundary conditions were transformed
into non-dimensional form to allow for
generalization across various physical scenarios
using L, (Tw - T),
W
U
,
LUW
and
LUW
respectively for length, temperature, velocity,
stream function, and vorticity as presented in
equation (11) :
,,
,,
ww U
v
V
U
u
U
L
y
Y
L
x
X
,
,,
LU
LUTT
TT
w
ww
(11)
The normalized versions of the X- and Y-
velocity components, stream function, vorticity,
and energy transport equations are presented in
equations (12-15):
X
V
Y
U
,
(12)
2
2
2
2
YX
(13)
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1
22
2
2
2
XRe
Gr
YXRe
Y
V
X
U
(14)
2
Y
θ
X
θ
Y
θ
X
θ
222
2
2
2
2
Y
U
X
V
Y
V
X
U
Re
Ec
ReC
k
VU
p
(15)
Within the equations (14-15), Ec represents the
Eckert number, k denotes thermal conductivity, μ
signifies dynamic viscosity, Cp stands for specific
heat capacity, Re represents the Reynolds number,
and Gr represents the Grashof number. The Eckert
number serves to relate the flow viscous-dissipation
term to energy distributions. This number acts as a
criterion for determining the inclusion of the
viscous-energy dissipation effect in heat transfer
analysis. The Prandtl number establishes a link
between the rates of heat and momentum diffusion.
The Grashof number serves as a dimensionless
parameter reflecting the ratio of buoyancy force to
viscous force in free-convection flow issues, it
signifies whether the flow is laminar or turbulent,
and which dynamic process holds dominance.
The boundary conditions, when expressed in
non-dimensional form, are as follows:
; 1 X 0 1; Yat
1 U0; V 0; 0;
; X ; at Y
θ V U ; Ψ Ω
100
00
100
00
; Y ; at X
θ VU ; Ψ Ω
1 Y 0 1; Xat
0
X
X
V
X
U
; 0
. (16)
The vorticity and energy transport equations
(14) and (15) exhibit non-linear characteristics.
Currently, there are no universally accepted
analytical solutions available for these
interconnected equations. Among the most
effective methods for solving equations (12) – (15)
is the finite difference technique, where each term
within the differential equations is approximated by
their corresponding differential quotient. The
resulting linear equations are subsequently
addressed concurrently by employing the relaxation
technique.
The convective heat transfer inside the
enclosure is calculated based on the Nusselt
number, a dimensionless quantity that characterizes
the proportion of heat transfer via convection and
conduction across the fluid layer. The temperature
gradient resulting from the exchange of heat
between the fluid and the wall can be associated
with the local Nusselt number, Nux, through
Equation (17):
1
Y
x
xYk
xh
Nu
(17)
The mean Nusselt number is derived through
the integration of the local Nusselt number along
the entire length of the heated wall as shown in
Equation (18):
1
010
dX
Y
Q
Q
uN
orY
cond
conv
(18)
The attainment of a stable flow state was
determined by monitoring the convergence of
temperature and vortex field, utilizing the
prescribed criterion given by Equation (19):
-
2
1
2
2
n
ij
1n
ij
2
M
j
n
ij
N
i
M
j
N
i
(19)
The parameter δ represents α, θ, or ω, with n
indicating the number of iterations until the results
converge. The literature reports variations in the
value of δ ranging from 10-3 to 10-8, [23].
4 Discussion of Numerically
Generated Results
An investigation was conducted to assess the
influence of the convergence criterion on the
numerical results. This was accomplished by
calculating the mean Nusselt number for various
values of the convergence parameter, which
ranged from 10-1 to 10-8. The results, shown in
Figure 2, indicate that a value of 10-4 was enough
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to achieve convergence. To verify the code used in
this study and the precision of the simulations, the
Nusselt number was calculated for a convective
flow situation with a Prandtl number of 0.7 and a
Rayleigh number of 1000. The Nusselt number
calculated using the program was Nu = 1.1210,
which closely matches the value of Nu = 1.132
given by [24] for the same Prandtl and Rayleigh
values, with a discrepancy of around 2%.
Additional validation was conducted by calculating
the Nusselt number for a Rayleigh number (Ra) of
105 and a Prandtl number (Pr) of 0.7., [24],
documented 4.6201, and the current simulation
produced Nu = 4.7438, indicating consistent
findings across the three investigations.
10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
Average Nusselt number, Nu
Convergence parameter,
Fig. 2: Plot of average Nusselt number, Nu versus
the convergence parameter,
The numerical results were rigorously validated
to ensure grid independence. This was done by
obtaining solutions using progressively larger grid
sizes until a point was reached where a significant
change in the solutions occurred with further
increases in the number of nodes. This is shown in
Figure 3 for Re = 100, Pr = 0.7, and Ra = 1000,
represented by a dotted line. The accuracy of the
computed numerical results was found to be highly
dependent on the number of nodal points. The
numerical results closely matched several well-
established benchmarks using a grid structure
consisting of 41 × 41 nodal points. The grid
independence tests demonstrated that a grid system
with dimensions of 41 × 41 was sufficient in terms
of numerical stability, field resolution, and
accuracy, which aligns with the conclusions of a
previous study conducted by [23], [25].
Figure 4.2 Average Nusselt number, Nu versus log(Grids size)
with different Reynolds number, Re , for Pr = 0.7, Ra = 1000
3
4
5
6
7
8
9
10
2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 4.1
log(Grids size)
Average Nusselt number, Nu
Re=0.001
Re=10
Re=100
Fig. 3: Average Nusselt number, Nu versus log
(Grid sizes) with different Reynolds numbers, Re,
for Pr=0.7, Ra=1000
Figure 4 shows the non-dimensional
temperature distribution at Y = 0.5 for various
Eckert numbers (Ec), with Ra = 1000, Pr = 0.7, and
Re = 100. The data illustrates that an elevation in
the Eckert number results in an intensified
temperature gradient. The Eckert number quantifies
the relationship between the dynamic temperature
resulting from fluid motion and the optimum
temperature gradient of the fluid flow. The study
concludes that the Eckert number has a
considerable impact on the temperature gradient,
especially at higher altitudes where there are
matching strong temperature gradients. This
discovery supports the research conducted by [23],
which emphasizes the significant influence of
viscous energy dissipation in flows with high-
temperature gradients.
Fig. 4: Temperature distributions at Y = 0.5, with
varying Eckert numbers, for given values of Ra =
1000, Pr = 0.7, and Re = 100
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Figure 5 displays temperature profiles that have
been scaled to remove units of measurement for
different Prandtl numbers, Pr, at a specific location
Y = 0.5. The values of Ra, Re, and Ec are fixed at
1000, 100, and 0.4, respectively. The diagram
illustrates the impact of the Prandtl number on
thermal patterns. The relationship between greater
Prandtl numbers and enhanced temperature
gradients in various fluids is apparent; it was
supported by [8]. On the other hand, when the
Prandtl number (Pr) is less than 1, there are only
small variations in temperature gradient. This is
due to either weak convection (low momentum
diffusivity) or strong thermal diffusivity. This
figure highlights the influence of the Prandtl
number on temperature profiles, demonstrating a
significant reduction in the thickness of the thermal
boundary layer as the surface temperature
differential increases.
Figure 4.7 Temperature fields with different Prandtl numbers, Pr.
at Y = 0.5, for Ra = 1000, Re = 100, Ec = 0.4.
-0.05
0.15
0.35
0.55
0.75
0.95
1.15
1.35
1.55
00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
X axis
Dimensionless temperature
Pr = 0.001
Pr = 0.01
Pr = 0.1
Pr = 0.7
Pr = 10
Fig. 5: Temperature distributions with varying
Prandtl numbers, at Y = 0.5, for given values of
Ra = 1000, Ec = 0.4, and Re = 100
Figure 6 depicts the dimensionless temperature
distribution for various Reynolds numbers, labeled
as Re, at a specific location Y = 0.5. The
parameters Pr = 0.7, Ec = 0.4, and Ra = 1000 are
also considered. The results suggest that as the
Reynolds number increases, there is a proportional
increase in the thermal fields, as evidenced by the
temperature gradient. The figure demonstrates that
when the Reynolds number (Re) is much smaller
than 1, the thermal fields exhibit minor alterations.
Nevertheless, when the Reynolds (Re) values reach
50, 70, 80, 90, and 100, the fields experience a
sudden rise and converge at a shared point
positioned at x = 0.4. Beyond this point, any further
alterations in the fields become negligible. This
pattern is caused by the prevalence of inertia forces
over viscous forces in the fluid flow.
Figure 4.8 Temperature fields with different Reynolds numbers, Re.
at Y = 0.5, for Ra = 1000, Ec = 0.4, Pr = 0.7
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
00.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
X axis
Dimensionless temperature
Re = 0.001
Re = 0.1
Re = 1
Re = 10
Re = 20
Re = 50
Re = 70
Re = 80
Re = 90
Re = 100
Fig. 6: Dimensionless temperature distribution for
various Reynolds numbers, Re, at Y = 0.5 for Pr =
0.7, Ec = 0.4, and Ra = 1000
Figure 7 depicts the non-dimensional
maximum stream function profile for different
Eckert numbers while keeping the values of Ra =
1000, Pr = 0.7, and Re = 100 constant. The
illustrated diagram demonstrates that the Eckert
number has a negligible impact on the flow fields.
More precisely, an increase in the Eckert number
does not cause major changes in the flow patterns.
Figure 4.9 Effect of Eckert number on convective flow vigours
as related by the maximum stream function, for Re = 100, Ra = 1000 and Pr = 0.7
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0 2 4 6 8 10 12 14 16 18 20
Eckert number, Ec
Maximum Stream function
Fig. 7: Dimensionless maximum stream function
profile for different Eckert numbers, at Ra = 1000,
Pr = 0.7, and Re = 100
Figure 8 displays non-dimensional streamline
profiles corresponding to different Prandtl
numbers, Pr, while keeping the circumstances fixed
at Ra = 1000, Re = 100, and Ec = 0.4. The figure's
representation demonstrates that an increase in the
Prandtl number results in a decrease in the flow
fields. Convection has a significantly reduced
impact on the fields in the case of liquid metal. The
Prandtl number is a measure of the material
characteristics of a fluid, which might vary among
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various fluids. A Prandtl number below 1 indicates
either low momentum diffusivity (indicating weak
convection) or high thermal diffusivity. This
number is used to quantify the correlation between
the rates at which heat and momentum diffuse,
which is crucial in calculating the thickness of
boundary layers in a specific external flow field.
Increased momentum or thermal diffusivity
indicates that the effects of viscosity or temperature
spread over a larger area inside the flow field.
Figure 4.13 Flow fields with different Prandtl numbers, Pr.
at Y = 0.5, for Ra = 1000, Re = 100, Ec = 0.4.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
00.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
X axis
Dimensionless stream function
Pr = 0.001
Pr = 0.01
Pr = 0.1
Pr = 0.7
Pr = 10
Fig. 8: Dimensionless streamline profiles
corresponding to different Prandtl numbers, Pr, at
mid-plane, for Ra = 1000, Re = 100, and Ec = 0.4
Figure 9 displays the non-dimensional
streamline profile for different Reynolds numbers,
Re while keeping Ra = 1000, Pr = 0.7, and Ec = 0.4
constant. The findings indicate that as the Reynolds
number rises, there is a corresponding increase in
the flow fields. This discovery emphasizes the
dominant impact of inertia force compared to
viscous force. When the Reynolds number (Re) is
much less than 1, the flow fields experience little
changes because the dominant force in the fluid
flow is viscosity, rather than inertia. In this
situation, diffusion is less significant compared to
the inertial and buoyant forces within the enclosure.
Figure 4.14 Flow fields with different Reynolds numbers, Re.
at Y = 0.5, for Ra = 1000, Ec = 0.4, Pr = 0.7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
00.05 0.1 0.15 0.2 0.25 0.3
x axis
Dimensionless stream function
Re = 0.001
Re = 0.1
Re = 1
Re = 10
Re = 20
Re = 50
Re = 70
Re = 80
Re = 90
Re = 100
Fig. 9: Dimensionless streamlines profile for
different Reynolds numbers, Re, for Ra = 1000,
Pr = 0.7, and Ec = 0.4
Figure 10 illustrates the relationship between
the Nusselt number (Nu) and the Prandtl number
(Pr) for different values of the Eckert number (Ec).
The graph shows the Nu-Pr relationship for Ra =
1000, Ec = 0.4, and 0.0, with Re fixed at 100. The
findings demonstrate that a higher Prandtl number
corresponds to an increase in the Nusselt number.
Figure 4.15 Nuselt number, Nu versus Prandtl numbers, Pr.
for Ra = 1000, Re = 100,
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
0 1 2 3 4 5 6 7 8 9 10
Prandtl numbers, Pr.
Nusselt number, Nu
Ec = 0.4
Ec = 0.0
Fig. 10: plot of Nu against Pr for Ra = 1000,
Ec = 0.4, and 0.0, with Re fixed at 100.
Figure 11 depicts the graph of the Nusselt
number, Nu, against the Reynolds number, Re, for
Eckert numbers, Ec = 0.4, Ra = 1000, and Pr = 0.7.
The obtained data demonstrates that when the
Reynolds number increases, the Nusselt number
displays harmonic patterns that are associated with
energy-viscous-dissipation. The graphical
representation suggests that the Reynolds number
has a notable influence on the heat transfer
properties within this particular region.
Furthermore, at lower Reynolds numbers, the
influence on the rate of heat transfer by different
convection modes is insignificant because the
diffusion effects decrease in both the inertial and
buoyancy forces near the extrusion slot.
Fig. 11: Nusselt number, Nu versus Reynolds
number, Re, for Pr = 0.7, Ra = 1000, Ec = 0.4
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2024.19.27
Emmanuel O. Sangotayo, Nikos Mastorakis
E-ISSN: 2224-347X
303
Volume 19, 2024
5 Conclusions
The study examined the heat transfer
characteristics, flow patterns, and thermal
distributions on a constantly moving horizontal
sheet of extruded material in a stable fluid
environment. This analysis was conducted both
near and far from the extrusion slot and the
numerical model was discretized using a central
finite difference method. The study investigated the
influence of the Prandtl number and Reynolds
numbers at various values of the Eckert number on
the thermal distributions, flow patterns, and heat
transfer rates. The following findings were derived
based on the analyzed flow conditions and
parameter ranges:
The Eckert number has a substantial effect on
the distribution of energy and the transfer of heat. It
improves the distribution of energy without having
a noticeable impact on the patterns of flow. An
increase in Prandtl numbers results in enhanced
thermal distributions and heat transfer rates,
although it leads to diminished flow patterns. At
lower Reynolds numbers, the heat transfer rates
from distinct convection modes show negligible
changes. However, as the Reynolds numbers
increase, the thermal distributions, flow patterns,
and heat transfer rates become more pronounced.
The findings of this research underscore the
considerable significance of fluid dynamics in
optimizing heat transfer for the benefit of
engineering. Additional research that expand the
range of Reynolds and Prandtl numbers so that the
effects in various industrial and technological
settings were thoroughly elucidated and ultimately
contributed to the enhancement of thermal system
design and performance.
Declaration of Generative AI and AI-assisted
Technologies in the Writing Process
During the preparation of this work the authors
used Grammarly for language editing. After using
this service, the authors reviewed and edited the
content as needed and take full responsibility for
the content of the publication
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E-ISSN: 2224-347X
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problem to the final findings and solution.
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Conflict of Interest
The authors have no conflicts of interest to declare.
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WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2024.19.27
Emmanuel O. Sangotayo, Nikos Mastorakis
E-ISSN: 2224-347X
305
Volume 19, 2024
