EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 833-846
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Numerical Simulation of Convection and Conduction
Heat Transfer for a Fluid in a Porous Medium

Ghanim M. Algwauish1,∗, Alaa A. Hammodat1, Hamsa D. Saleem1

1 Department of Mathematics, College of Education for Pure Sciences, University of Mosul,
Mosul, 41001, Iraq

Abstract. Building a computational formula which is expressed by a system of nonlinear partial
differential equations in two components was done in this job. The matter of energy transfer by
free convection of a dissipative liquid running in a horizontal channel with a porous medium was
examined in this study, and we describe the behavior of fluid flow through the channel and the
temperature distribution within it. Numerical methods were used as the ODE45 method, one
of the Rang-Kutta methods for processing the resulting differential equations. The effect of the
Prandtl number, the Schmidt multitude, and the Gratshof amount were also studied.

2020 Mathematics Subject Classifications: 65Lxx, 65Mxx, 80Mxx

Key Words and Phrases: Heat Transfer, Prandtl number, Schmidt number, Gratshof number,
ODE45 Method.

1. Introduction

The flow of reactive viscous fluids via porous media is a theoretically tricky issue in
many scientific, technological, and engineering fields. Food drying, extraction of geother-
mal energy, removal of nuclear waste, thermal and liquid flow inside of internal tissue, con-
structing insulation, water supply flow, the output of oil and gas, oxidizer astrophysical,
Metal extraction and exfoliation, magneto hydrodynamic (MHD) pumps and turbines, air-
craft thrusters and ship propellers, and vehicle destroy devices are all examples of extreme
flow systems. Such flow systems naturally contain exothermic reactions. As a result, we
examine the thermal effects and stability requirements for reactive varying non-Newtonian
fluids flowing instability through saturated porous substrates . Since they make up the
majority of chemical fluids, this study focuses on non-Newtonian fluids. Differential fluids,
such as third-grade fluid, will be given particular care. [13]. Several researchers have made

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4711

Email addresses: g.m.abdullah@uomosul.edu.iq (Ghanim M. Algwauish),
alaahammodat@uomosul.edu.iq (Alaa A. Hammodat),
hamsa-dawood@uomosul.edu.iqedu.iq (Hamsa D. Saleem)

https://www.ejpam.com 833 © 2023 EJPAM All rights reserved.



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 834

contributions to this field.

Wright et al. [17] looked at the effect of on specified surface temperature flux by con-
vective on a nanofluid flow in a horizontal sheet . A study of the effect of a continuous
horizontal mass modulus on the thermal conductivity of a conical duct with porous walls
by Yih [18]. Gupta and Gupta [6] explored the movement of mass and heat along a sliding
plate with suction or blow acts. Andersson et al. [3] then generalized Wang’s problem to
the context of radiant heat. Abel et al. [1] have used the magnetic properties effect to
examine the influence of variable viscosity on unsettled radiant heat and flow in a fluid
flick above a boundary layer flow. Noor and Hashim [12] looked into the effects of magneti-
zation on an unreliable mutable stretching plate in a tinny ocean segment. Hasanuzzaman
et al. [9] explored the Dufour and thermal permeability impacts on the occasional free
magnetic characteristics of heat flow passing via an unbounded vertical permeable layer.
Infrared energy’s affects on irregular energy transmission and stream in a thin liquid film
across a straining slab in a saturated medium were examined by Khader and Megahed [10].

Mehta et al. [15] explored cyclical heat and flow transmission through a plastic struc-
ture between steel plates, as well as heat exchange flux and boiler, in the proximity of
oblique ferrites. Hammodat A. et al.,[7], deal with this problem of convection and radia-
tion heat transfer of a fluid in a porous channel under influence of a magnetic field using
the opposite direction implicitly technique. We observed that shifting the temperature of
the fluid within the tube had an important influence on the measurements. In the same
year, Hammodat A. et al. [8] analyse the flow of fluids in a bridge under the influence of
an electromagnetic force (EMF) and actually fix the partial differential equations that de-
scribe the situation using the strategy of lines. Besides that, we aspire to demonstrate the
behavior of temperature within the cross-section as well as the effects of physical quantities.

Tarammim et al.[14] examenid instable MHD free convection stream over a perpendic-
ular dish in a two-dimensional coordinate system and examined the stability conditions
using an explicit finite difference method. They examined the governing equations of un-
stable free convection heat transfer flow in the presence of radiant energy in 2021 (see
[16]). Nagarju [11] studied the behavior of fluid velocity and temperature by utilizing the
finite difference approach, and they evaluated the impact of misaligned magnetic force
and temperature radiation on unstable MHD flow on a sloped plate. Bordoloi [4] They
tested the free convection of a fluid flowing in a narrow, almost infinite porous plate. The
results showed the effect of temperature and velocity in the matter and under the pres-
ence of thermal radiation and chemical interaction. The goal of this study is to examine
numerically transient combined mass and heat exchange by mixed convection flow in a
moveable parallel rectangular layer. The basic equations of the issue are a group of partial
differential equations that are transformed into nondimensional equations. To solve the
problem numerically, the ODE45 method was used. The results of this investigation will
be presented and graphically displayed for well-known parameters with varying parame-
ters.



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 835

Figure 1: Physical geometry.

2. Basic Equations and Problem Formulation

In a power generation system, the simultaneous mass and heat transmission in mixed
convection rivers on a rising horizontal porous material with thermal diffusion is regarded.
Let the X − axis be chosen along the porous plate in the flow path and the Y − axis
is vertical to the channel. Consider the temperatures at the two horizontal walls to be
constant and T̄1 represents the temperature range of the horizontally, T̄0 unchanged where
T̄1 > T̄0.

As well as let U, V be the components of velocity in the channel’s bottom (top) walls
given X and Y directions, respectively, and that all components in the Z direction have
disappeared. Figure 1 depicts a drawing of the structure and coordinate system.

According to the aforementioned premises, the governing equations for continuity,
momentum, energy, and concentration can be written as:

∂U

∂X
+

∂V

∂Y
= 0, (1)



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 836

∂

∂T

[
∂V

∂X
− ∂U

∂Y

]
+

[
∂

∂X

(
U
∂V

∂X
+ V

∂V

∂Y

)
− ∂

∂Y

(
U
∂U

∂X
+ V

∂U

∂Y

)]
= υ∇2

[
∂V

∂X
− ∂U

∂Y

]
+

υ

κ

[
∂V

∂X
− ∂U

∂Y

]
, (2)

∂T̄

∂T
+ U

∂T̄

∂X
+ V

∂T̄

∂Y
=

κ∗

ρCp

[
∂2T̄

∂X2
+

∂2T̄

∂Y 2

]
+

υ

Cp

[(
∂U

∂Y

)2
]
, (3)

∂C̄

∂T
+ U

∂C̄

∂X
+ V

∂C̄

∂Y
= D

[
∂2C̄

∂X2
+

∂2C̄

∂Y 2

]
, (4)

T is the occurrence and T̄ , C̄, ρ, υ, p, k, k∗, Cp are the aspects to evaluate which are
temperature, concentrations, density, kinematic viscosity, pressures, media porosity, ther-
mal transfer, and specific heat at constant pressure. The incremental boundary conditions
accompanying (1)-(4) are:

U = V = 0,

T̄ = T̄0, T̄1,

C̄ = C̄0, C̄1,

 at Y = 0, d. (5)

In order to remove dimensions from the governing equations under the initial and boundary
conditions, the following variables are introduced [5]

x̃ = X
d , ỹ = Y

d , ũ = Ud
υ
√
Gr

, ṽ = V d
υ
√
Gr

,

t̃ = Tυ
√
Gr

d2
, θ = T̄−T̄0

T̄1−T̄0
, ϕ̃ = C̄−C̄0

C̄1−C̄0
,

 . (6)

Dimensionless quantities involved:

α = κ∗

ρCp
, N = υ

d2∆T
, Sc = υ

d , P r = υ
α ,

Gr = gβd3(T̄1−T̄0)
υ2 , ϵ = υ

Cp
,

 , (7)

where α is referred to as the thermal diffusion, N is a newly discovered physical quantity,
Sc Schmidt’s number, Pr that Prandtl number, Gr is the Gratshof number for heat
transmission, ϵ is the parameter for dispersion. By employing the above variables and
parameters and aftar the necessary anayisis, the equations (1)-(5) can reduce to;
Continuity equation

∂ũ

∂x̃
+

∂ṽ

∂ỹ
= 0. (8)

Momentum equation

∂

∂t̃

[
∂ṽ

∂x̃
− ∂ũ

∂ỹ

]
+

[
∂

∂x̃

(
ũ
∂ṽ

∂x̃
+ ṽ

∂ṽ

∂ỹ

)
− ∂

∂ỹ

(
ũ
∂ũ

∂x̃
+ ṽ

∂ũ

∂ỹ

)]



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 837

=
1√
Gr

∇2

[
∂ṽ

∂x̃
− ∂ũ

∂ỹ

]
+

1√
Gr

[
∂ṽ

∂x̃
− ∂ũ

∂ỹ

]
. (9)

But ũ = ∂Ψ
∂ỹ and ṽ = ∂Ψ

∂x̃ is stream function[12]. Put ξ = −∇Ψ, then equation (9) become:

∂ξ

∂t̃
=

1√
Gr

∇2ξ +
1√
Gr

ξ, (10)

Energy equation

∂θ

∂t̃
+ ũ

∂θ

∂x̃
+ ṽ

∂θ

∂ỹ
=

1

Pr

1√
Gr

[
∂2θ

∂x̃2

]
+ ϵN

√
Gr

(
∂ũ

∂ỹ

)2

. (11)

Diffusion equation

∂ϕ̃

∂t̃
+ ũ

∂ϕ̃

∂x̃
+ ṽ

∂ϕ̃

∂ỹ
=

1

Sc

1√
Gr

[
∂2ϕ̃

∂x̃2
+

∂2ϕ̃

∂ỹ2

]
. (12)

The boundary conditions (5) take on the non-dimensional form as follows:

ũ = ṽ = 0,

θ = 0, 1

ϕ̃ = 0, 1,

 at ỹ = 0, 1. (13)

3. Results: Solution Approach

The concept behind this approach is based on keeping the time derivative continuous
and discretizing the other terms using the ODE45 solver (Runge-Kutta formula) with
recurrent iterations in time. To solve the two-dimensional motion equation (10) and
energy equation (11) diffusion equation (12) and the boundary conditions (13), we try
to use the numerical simulation for −L ≤ x̃ ≤ L, and −H ≤ ỹ ≤ H, where L,H are
the arbitrary length of the computational domain between x̃ and ỹ-direaction. The main
equation of the motion, energy, and diffusion equations (11-13) are solved as mentioned
above using the ODE45 solver. To discretize the domain above in to N + 1 points in
X-direction and M + 1 in Y -direction, we use x̃i = −L + (i − 1)∆x̃, i = 1, . . . , N + 1
and ỹi = −H + (i − 1)∆ỹ, j = 1, . . . ,M + 1, Where ∆x̃ = 2L

N , and ∆ỹ = 2H
M . Consider

the forward and backward finite difference methods which are used to discretize the main
equation in this paper [6].

ξx̃,ij =
ξi+1j − ξij

∆x̃
, ξx̃,ij =

ξij − ξi−1j

∆x̃
, (14)

ξt̃,ij =
1√
Gr

[[(
ξi+1j − 2ξij + ξi−1j

∆x̃2

)
+

(
ξij+1 − 2ξij + ξij−1

∆ỹ2

)]
− ξij

]
,



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 838

i = 2, ..., N, j = 2, ...,M, (15)

θt̃,ij = −
(
ũij

θi+1j − θi−1j

2∆x̃
+ ṽij

θij+1 − θij−1

2∆ỹ

)
+

1

Pr

1√
Gr

(
θi+1j − 2θij + θi−1j

∆x̃2

)
+

ϵN
√
Gr

(
ũij+1 − ui,j−1

2∆ỹ

)2

, i = 2, ..., N, j = 2, ...,M. (16)

The below is the diffusion equation:

ϕ̃t̃,ij =

(
ũij

ϕ̃i+1j − ϕi−1j

2∆x̃
+ ṽij

ϕ̃ij+1 − ϕ̃ij−1

2∆ỹ

)

+
1

Sc

1√
Gr

[(
ϕ̃i+1j − 2ϕ̃ij + ϕ̃i−1j

∆x̃2

)
+

(
ϕ̃ij+1 − 2ϕij + ϕij−1

∆ỹ2

)]
, i = 2, ..., N, j = 2, ...,M.(17)

4. Discussion and Results

In this study, the system of coupled dimensionless value problem equations for velocity,
diffusion, and temperature with the boundary conditions have been solved numerically and
carried out for various parameters of Schmidt number (Sc), Gratshof number (Gr), and
Prandtl number (Pr). The topic of heat transmission by free convection of a dissipative
fluid moving in a horizontal tube with porous walls was looked into in this study. The final
dimensionless PDEs with the boundary conditions are solved using the ODE45 MATLAB
software [2]. In Fig. 2, we introduce a cartesian coordinate system (X,Y, Z), with the
X,Y plane describing the location of points on the mesh, i.e., the spots in the T∗-matrix
resulting from the energy equation simulation results, and the Z-axis normal to the plane
indicating temperature readings at any of those sites.

Figure 2: Temperature Behavior inside the Channel



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 839

4.1. Temperature variations for various parameter/number ratios

Figs. 3, 4, and 5 analyze the effects of temperature using a variety of Schmidt values,
the Prandtl number, and the Gratshof number, respectively. In Figure 3, the temperature
transfer is plotted for different values of the (Sc) number, it is clear that the change in
temperature profile is not noticeable when increasing the Schmidt number, otherwise in
figs. 4 and 5 the effect of the (Pr) number and (Gr) number on temperature is significant
respectively. it is observed that when increasing Gr gives a rise in temperature and goes
beyond the stability. Furthermore, figure 4 displays that for small values of (Gr), the
temperature field falls down and gives rise to higher values. However, the temperature
profile increase for any value of (Pr) sees fig. 5.

Figure 3: Temperature Distribution for Schimdt number (Sc).



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 840

Figure 4: Computation temperature distribution against Gratshof number (Gr).

Figure 5: Computation temperature distribution against Prandtl number (Pr).



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 841

4.2. Diffusion variations for various parameter/number ratios

Figures 6 and 7 show the effect of (Sc) number as well as (Gr) number on the diffusion
equation; these numbers have no discernible effect, and the diffusion profile remains stable.
However, for higher values of these parameters, the diffusion profile takes a long time to
reach stability.

Figure 6: Impact of Schmidt number (Sc) diffusion distribution.



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 842

Figure 7: Impact of Gratshof number (Gr) on Diffusion distribution.

4.3. Velocity variations for various parameter/number ratios

For the velocity case Figs. 8, 9, the same stability behaviour (see Fig. 6) is illustrated
for various values of the (Gr) number in the motion equations. It is worth noticing that
the velocity profile increase with rising values of (Gr) for a very short time and declines
with time increasing. The motion distribution is plotted in the Fig. 9 for various (Pr)
numbers. we find that the influence of increasing the (Pr) number is unremarkable.



G. M. Algwauish, A. A. Hammodat, H. D. Saleem / Eur. J. Pure Appl. Math, 16 (2) (2023), 833-846 843

Figure 8: Distribution of the motion equation for Gartshof number (Gr).

Figure 9: Effect of Prandtl number (Pr) on Motion distribution.



REFERENCES 844

5. Conclusion

The influence of mass and heat transfer on mixed convection fluid flow past a horizontal
channel are investigated in the present study. The data are plotted to show when velocity,
temperature, and concentration change as a function of various parameters. The following
are some of the investigation’s key findings.

(i) The influence of temperature profile for different values of Schmidt number, Prandtl
number, and Gratshof number is shown in Fig. 3 and Fig. 4. It is seen that
increasing the Schmidt number and Prandtl number leads to stability;

(ii) As shown in Fig. 5, increasing the Gratshof number causes a shift away from stability.

(iii) The effect of Schmidt’s number and Gratshof number in the diffusion equation is
revealed in Figs. 6, 7, where it is noticed that for higher values of these parameters,
the results take a long time to be stability.

(iv) In Figs. 8, 9 the effect of the Gratshof number and Prandtl number on the diffusion
equation is depicted, where it is noticed that the higher the Gratshof number, the
diffusion profile takes a long time to be stable and the influence of increasing the
Prandtl numberin Fig. 9 is not remarked.

Acknowledgements

The authors gratefully acknowledge the services and support from the University of
Mosul / College of Education for Pure Science improving the quality of this paper. We
would also like to thank the reviewers whose constructive comments and suggestions helped
to improve this paper.

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