*Corr. Author’s Address: University of Science and Technology of Oran, Faculty of Mechanical Eng., Maritime Science and Eng., Algeria, djamila.derbal@univ-usto.dz 135
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154 Received for review: 2022-06-16
© 2023 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2023-01-03
DOI:10.5545/sv-jme.2022.239 Original Scientific Paper Accepted for publication: 2023-02-14
Effect of the Curvature Angle in a Conduit  
with an Adiabatic Cylinder over a Backward Facing Step  
on the Magnetohydrodynamic Behaviour in the Presence  
of a Nanofluid
Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
Djamila Derbal
1,*
 – Mohamed Bouzit
1
 – Abderrahim Mokhefi
2 
–
 
Fayçal Bouzit
1
1 
University of Science and Technology of Oran, Faculty of Mechanical , Algeria 
2 
Bechar University, Faculty of Sciences and Technology, Algeria
A backward facing duct are present in various industrial applications especially those focused on heat transfer. The flow through a curved 
backward facing duct especially in the presence of nanofluid presents complexities compared to a straight backward facing step (BFS) duct. 
Therefore, the present numerical study deals a nanofluid flow (Fe
3
O
4
-H
2
O) forced convection in a curved backward facing duct. The objective 
of this investigation is to visualize at different curvature angles g (0°, 30°, 45°, 60°, 90°) imposed on the top wall of the duct, the effect of 
Hartmann number Ha (0, 50, 100), magnetic field inclination angle γ (0°, 60°, 90°), Reynolds number Re (10, 100, 200) and nanoparticle 
volume fraction φ (0 %, 0.05 %, 0.1 %). The dimensionless governing equations are solved using the multigrid finite element method.  The 
results showed that the heat transfer was enhanced at the curved angle g = 90° for large Hartman numbers, thus, the average Nusselt 
number increased with a ratio of 240.74 % in the case of Hartmann number (Ha = 100).
Keywords: forced convection, backward-facing step, curved conduit, fixed cylinder, ferrofluid, finite element method, magnetohydro-
dynamic
Highlights
• The discussion of magnetohydrodynamic behavior of laminar ferrofluid flow through a curved backward facing duct with a fixed 
cylinder is presented. 
• The use of the horizontal magnetic field in the case of a curved geometry provide an enhancement of the heat transfer. Indeed, 
we note an increase of about 240 % in the heat transfer rate in case of important magnetic flux density.
• In the case of a curved backward facing duct, the presence of a vertical magnetic field has no effect on heat transfer.
• The increase in Reynolds number, Hartmann number and volume fraction of the nanoparticles enhance the heat transfer 
through a curved backward facing duct.
0  INTRODUCTION
The sudden expansion geometry in a duct creates 
a separation and reattachment of the flow, as this 
expansion is a determining factor in the structure of the 
flow so the heat transfer is significantly affected. Thus, 
this geometry has a very important role in various 
engineering applications such as gas turbine engines, 
heat exchangers, combustion chambers, aircraft, and 
electronic devices and around buildings and many 
others. Indeed, it is receiving a lot of attention, 
so many studies focused on the separation and 
reattachment of the flow backward-facing step. For 
important Reynolds numbers Erturk [1] investigated a 
simulation of incompressible laminar steady flow in 
two dimensions (2D). In three-dimensional (3D), Lan 
et al. [2] presented in rectangular duct, a numerical 
study of laminar forced convection flow by means of 
a K-ε-ζ-f turbulence model. Iwai et al. [3] presented at 
little Reynolds number, the effect of the duct aspect 
ratio. An agreement of the numerical results with 
experimental results was revealed. Moreover, an 
increase of the Nusselt number near the sidewalls in 
the cases studied. Star et al. [4] developed for different 
Richardson number a reduced-order proper orthogonal 
decomposition (POD)-Galerkin simulation Reynolds-
averaged Navier–Stokes (RANS) extended to low 
Prandtl number flow. In the fact, the buoyancy marks 
its influence on flow and heat transfer. In addition 
to a more accurate prediction of the flow behavior, 
the description of the fluid parameters as well as 
the identification of the rheological consequences 
are interesting. Kahine et al. [5] demonstrated the 
stabilizing impact of the shear thinning character 
of the fluid by numerical simulation compared to 
experimental data. To represent Newtonian fluids 
and the power law relationship between viscosity 
and shear rate, Mahmood et al. [6] analyzed the 
flow of Newtonian materials and a power law fluid 
representing the characteristics of shear thinning, 
thickening, with the finite element method. The 
control of turbulent flows can be realized using either 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
136 Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
external, stable or unstable forcing, and is of basic as 
well as practical interest [7]. Benard et al. [8] and Li et 
al. [9] studied the effect of periodic perturbations with 
an imposed frequency and amplitude on the evolution 
of vortices formed downstream of a backward facing 
step (BFS). For the little Reynolds number, Koide 
et al. [10] to enhance heat transfer; they provided a 
numerical simulation based on experimental data, 
behind a backward facing step, on the flow attachment 
in the transient regime. Tihon et al. [11] evaluated the 
effects of various control parameters on the periodic 
disturbance introduced in the flow. They presented an 
experimental approach, to examine the flow behavior 
in the backward flow step by means of the fluctuating 
wall shear rate downstream of the step, as well as the 
effect of the inlet flow pulses on the overall flows 
structure. 
Heat transfer is the concern of many researches, 
so its improvement according to the process needs. A 
new method has been introduced to intensify the 
convective heat transfer, is to inject nanoparticles into 
the base fluid. In this regard, several recent researches 
have been carried out to show the effectiveness of 
using these nanoparticles in base fluids. The addition 
of nanoparticles in the base fluid marks their 
significant effect in different processes particularly 
those that concentrate on the thermal transfer [12] to 
[14]. Selimefendigil et al. [15] studied the effects of 
nanoparticles volume fraction, the inlet oscillation 
frequency and the Reynolds number on the fluid flow 
and thermal transfer features. They showed that the 
heat transfer is increased with the enhancement of the 
volume fraction of the nanoparticles, the Reynolds 
number and the frequency of the oscillation, 
experimentally, the degree of heat transfer flow  over 
a backward-facing step duct increases as the volume 
concentration of the nanoparticles increases [16]. In 
order to observe clearly the effect of expanding rate, 
Togun et al. [17] introduces a numerical simulation of 
a turbulent and laminar heat transfer flow of nanofluid 
on a backward step, and various control parameters 
assumed. The results reveal that the angle of incidence 
of the vortex generators has significant effects on the 
heat transfer increase. Ahmed et al. [18] examined the 
forced convection heat transfer of a laminar fluid flow 
on a microscale BFS, the result indicates that the heat 
transfer is enhanced with a little increase in pressure 
drop in the case of a rectangular vortex generator 
(VG) wing with an angle of attack of 60° and a 
Reynolds number of 180. Abbassi and Nassrallah [19] 
analyzed the laminar flow of a viscous incompressible 
nanofluid through a backward-facing step duct under 
the effect of a magnetic field. They discussed the 
magneto hydrodynamic behavior under the effect of 
some control parameters. Furthermore, they 
demonstrated that the magnetic field significantly 
improves the heat transfer at high Prandtl numbers. 
Kumar and Dhiman [20] investigated for a particular 
Prandtl number and at different values of Re the forced 
convection characteristics of backward laminar flow 
in a two-dimensional channel. They discussed the 
effect of cylinder position on flow and heat transfer, 
an enhancement of the Nusselt number while using a 
circular cylinder compared to the case without a 
cylinder. Hussain et al. [21] we applied the Galerkin 
finite element method to compute the forced 
convective flow of a ferrofluid within a BFS 
containing a rotating cylinder with a fixed diameter, 
The magneto hydrodynamic behavior was analyzed 
assuming a wide range of control parameters namely 
Reynolds number, Hartmann number, magnetic field 
inclination angle and nanoparticle volume fraction. 
The effect of these parameters on the thermo 
magnetohydrodynamic (MHD) structure has been 
demonstrated. In addition, Selimefendigil and Öztop 
[22] presented the impact of these parameters on the 
flow and on the heat transfer in the case of mixed 
convection through a duct without cylinder. 
Mohammed et al. [23] investigated numerically, the 
effects on hydrodynamic and heat transfer of four 
different types of shapes that block the mixed 
convective of the laminar transient nanofluid flow 
through microscale BFS placed in a horizontal duct. 
Kherbeet et al. [24] showed the heat transfer 
characteristics reported experimentally and 
numerically on a laminar convective flow using a 
nanofluid with two nanoparticles types. It has been, 
shown that the Nusselt number increases and improves 
in the case of one nanofluid over the other. Lv et al. 
[25] in two-dimensional backward step flow, based on 
the analysis of several parameters, the flow 
characteristics are studied quantitatively using particle 
image velocimetry (PIV) while changing the Reynolds 
number and the weight nanofluid fraction. The mass 
and momentum transfer in the nanofluid improved, 
which leads to an improvement of the heat transfer. 
Amiri et al. [26] added an experimental study on the 
thermo-physical properties of egg nucleoplasmin 
(EggNP). The results indicate that a greater weight 
nanofluid concentration involves a faster rate of heat 
transfer on a backward facing step. Niemann and 
Fröhlich [27] in a turbulent mixed convective flow, 
they studied the buoyancy effect on the velocity field 
and on heat transfer is by comparing two simulations, 
one with the buoyancy term removed from the 
equations, and the other with a Richardson number of 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
137 Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour...
0.338. The results contribute to the physical 
understanding of the effects of buoyancy on heat 
transfer in the considered regime. In addition, the data 
generated provides validation and improvement of 
turbulence models for turbulent heat transfer at low 
Prandtl numbers. To show the fin's impact on heat 
transfer. Boruah et al. [28] investigated the thermal-
hydrodynamic properties and entropy creation for a 
mixed convective flow across a channel BFS with 
various baffle geometries. The results indicate that the 
wedge-shaped elliptical baffle arrangement in the duct 
BFS is an ideal design from the point of view of 
thermal-hydraulic efficiency and entropy generation. 
Thermal-hydrodynamic characteristics of fluids 
flowing through downward flowing duct in the 
presence of obstacles have received relatively little 
attention in the literature, despite the important effect 
that these obstacles can have on the flow structure and 
heat transfer [29]. Selimefendigil and Öztop [30] 
studied numerically the forced convection heat 
transfer over a backward facing step duct with a baffle 
on the top wall on a pulsating laminar flow. The 
effects of several important parameters were 
discussed, compared to a steady flow without baffle. 
They showed that in the case of a lower wall 
downstream of the enlargement, the presence of a 
deflector is not beneficial for the improvement of heat 
transfer. Selimefendigil and Öztop [31] studied 
numerically the laminar forced convection of a pulsed 
nanofluid flow over a BFS. They treated the effect of 
several control parameters and different geometrical 
factors namely: length and height of the bottom wall 
surface corrugation. The results show that the rate 
heat transfer increases with the inclusion of 
nanoparticles. Moreover, the rate of enhancement 
depends on the volume fraction of nanoparticles in the 
base fluid. Boruah et al. [32] demonstrated numerically 
a mixed convection flow of an incompressible non-
Newtonian fluid across a channel BFS with a chicane. 
The results showed the impact of baffle location and 
nanoparticle diameter on the convective transfer. 
Furthermore, these results are beneficial for the design 
of a thermodynamic system capable of achieving 
maximum heat transfer with minimum irreversibility. 
Mohammed et al. [33] have numerically simulated a 
mixed convection flows of laminar and turbulent 
nanofluid on a backwards step.  They analyzed the 
effect of geometrical parameters such as height, 
position of baffles, width and number of baffles 
analyzed. Heshmati et al. [34] presented the effect of 
several control parameters as well as deflectors with 
four different geometries. In addition, the effect of 
inclination and location was included. Due to the 
importance of thermal phenomena in curved ducts, 
several researches have been conducted in the 
literature. In order to give a detailed knowledge, to 
understand the complexity of non-isothermal flows, to 
prevent the phenomena of hydrodynamic instability 
that can occur, researchers have carried out work on 
non-isothermal flows to highlight these phenomena in 
this type of duct. Yanase et al. [35] proposed a 
numerical simulation of the convective non-isothermal 
heat transfer flow through a curved rectangular duct. 
The calculations performed by the spectral method for 
different Grashof numbers and Dean numbers. Several 
satisfactory results are obtained in terms of velocity 
and temperature. In addition, the convection 
remarkably enhanced from the hot wall to the fluid. 
Mondal et al. [36] studied numerically a convective 
heat transfer flow through a curved duct. Therefore, 
they treated the effect of Grashof number and aspect 
ratio on the thermo-hydrodynamic flow behavior over 
a wide temperature difference range applied to the 
cold inside and hot outside. Li et al. [37] presented a 
combination of experimental and numerical study to 
demonstrate the three-dimensional behavior of 
laminar flow in 120° curved rectangular ducts with 
three aspect ratios (Ar) and continuous curvature 
variation (Cr). The results show the complex changes 
in the formation, evolution of Dean vortices of 
different types. Furthermore, they demonstrated the 
effect of these control parameters on the instability of 
Dean vortices in the case of turbulent flow. Choi and 
Park [38] studied numerically the mixed convection 
flows through curved concentric annular ducts with 
constant wall temperature. The simulation was 
performed for different Dean numbers, different 
Grashof numbers and different radius ratios. Indeed, 
they presented the effect of these control parameters. 
Furthermore, they showed that for the Grashof number 
critical value, the average Nusselt number showed a 
strong dependence on the Dean number and the radius 
ratio.  Chandratilleeke et al. [39] presented a numerical 
study to examine the development of recirculation 
zones and the appropriate heat transfer phenomenon 
that takes place from a laminar fluid flow through 
curved rectangular duct. In fact, they carefully 
analyzed the flow conditions leading to hydrodynamic 
instability and the generation of Dean vortices in 
curved duct, identifying the influences of the different 
control parameters. In addition, they examined the 
buoyancy force effect and secondary flow on the 
thermal process.  Choi and Zhang [40] carried out a 
numerical simulation to evaluate the forced convection 
of a laminar flow of Al
2
O
3
 nanofluid through a curved 
duct. They found that the addition of Al
3
O
2
 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
138 Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
nanoparticles at low and increasing concentrations 
increases the average Nusselt number. Ajeel et al. [41] 
showed that different control parameters such as the 
corrugated shape of the curved duct, the installation of 
baffles with different geometric ratios as well as the 
hybridization of the nanoparticles positively affect the 
heat transfer.
The analysis of various previous works has shown 
that the thermo-magneto-hydrodynamic behavior in 
curved duct has not been widely studied especially 
the one with backward-facing step and in the presence 
of nanoparticles. In this regard, the objective of the 
present study is to provide a numerical simulation of 
a forced laminar convective flow, in the presence of 
Fe
3
O
4
 nanoparticles in the base fluid (water), through 
a curved backward-facing step duct containing a 
fixed adiabatic cylinder under an inclined magnetic 
field and using a single-phase nanofluid model. The 
finite element method was used to solve the equations 
governing the considered fluid flow. The influence of 
the curvature angle on the thermal and hydrodynamic 
structures was demonstrated for different values of the 
several control parameters.
1  GEOMETRIC DESCRIPTION
The physical study presentation, consists of a curved 
backward facing duct, containing a fixed cylinder of 
diameter D, centrally located at (4H, H), see (Fig. 1). 
In addition, an angle of curvature (g) is imposed on the 
duct and an external magnetic field has been imposed 
at an angle (γ). The expansion ratio is set to 2 and the 
step height is taken as H, halfway up the duct. The 
effects of viscous dissipation and Joule heating are 
not taken in to account the energy equation modelling. 
The induced magnetic field was assumed negligible. 
The ferrofluid at the inlet was considered cold and 
maintained at a temperature T
c
, on the other hand, the 
bottom wall downstream of the step is maintained at a 
high temperature T
h 
and the other walls are maintained 
thermally adiabatic. The water and iron oxide 
thermo physical properties are presented in Table 1. 
The physical situation with boundary conditions is 
presented in Fig. 1.
Table 1.  Thermophysical properties of water (H
2
O) and iron oxide 
nanoparticles [21]
Physical properties
Water 
(H
2
O)
Nanoparticles 
(Fe
3
O
4
)
Density [kg·m
–3
] 997.1 5,200.0
Specific heat [J·kg
–1
·K
–1
] 4,179 670.0
Thermal conductivity [W·m
–1
·K
–1
] 0.613 6.0000
Thermal expansion [K
–1
] (×10
–5
) 0.613 1.1800
Electrical conductivity [S·m
–1
] 0.050 25,000
2  MATHEMATICAL MODEL
2.1 Governing Equations
 






u
x
v
y
0, (1)
 


ff ff
ff
u
u
x
v
u
y
p
x
u
u
x
v
u
y
B



























2
2
2
2
0
2
v vu sinc os sin,   

2
 (2)
Fig. 1.  The physical configuration with the boundary conditions

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
139 Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour...


ff ff
ff
u
v
x
v
v
y
p
y
u
v
x
v
v
y
B



























2
2
2
2
0
2
u uv sinc os cos,   

2
 (3)
 u
T
x
v
T
y
T
x
T
y
ff


















2
2
2
2
. (4)
The following equation present the stream 
function:
 







x
v
y
u ,. (5)
2.2  Boundary conditions
For the numerical calculation of this study situation, 
the imposed boundary conditions are expressed as 
follows:
• Input condition: a parabolic velocity profile in 
the direction of the x-axis and a uniform cold 
temperature:
 uu yv TT
c
   ,, . 0  =  (6)
• At the bottom wall downstream of the expansion: 
The conditions of no-slip velocity and uniform 
hot temperature:
 uv TT
h
= = = 0, . (7)
• Outlet condition: The gradient of velocity and 
temperature in the normal direction (normal 
direction coincide with x-axis):
 









u
n
v
n
T
n
00 ,. (8)
• The no-slip boundary conditions for the velocity 
are applied on the duct walls and on the cylinder 
contour and the adiabatic conditions for the 
temperature are imposed except for the bottom 
hot wall downstream of the expansion:
 uv
T
n



 00 ,. (9)
2.3  Dimensionless Governing Equations
To transform the set of partial differential equations 
into a dimensionless form, the following variables 
given in Table 2 were used. The boundary conditions 
used in the dimensionless form are summarized 
in Table 3. The thermo-physical properties of the 
ferrofluid are depend on the properties of the base 
fluid and the magnetic nanoparticles. Hence, the 
effective thermo-physical properties of the nanofluid 
are given above in Table 4.
Table 2.  The variables used to transform the system of partial 
differential equations in to dimensionless form
X
x
H
Y
y
H
=
=
, U
u
u
V
v
u
=
=
,
P
p
u
ff


2
 


TT
TT
c
hc
Pr
Re


v
Hu
v
f
f
f

,
Ha BH
u
ff
ff

0

The dimensionless equations governing the flow 
summarized as follows:
 






U
X
V
X
0,
 (10)
U
U
X
V
U
Y
P
X
u
v
U
X
U
Y
ff
ff f
f
ff
ff





















1
2
2
2
2
Re 




f f
Ha
VU
2
2
Re
(s in coss in ),    (11)
U
V
X
V
V
Y
P
Y
u
v
V
X
V
Y
ff
ff f
f
ff
ff





















1
2
2
2
2
Re 




f f
Ha
UV
2
2
Re
(s in cosc os ),    (12)
 U
X
V
Y XY
ff
f




















 1
2
2
2
2
Re Pr
. (13)
The dimensionless equations stream function 
given as follows:
 







X
V
Y
U ,. (14)
2.4 The Local and Nu
ave
 Number 
A convective heat transfer occurred due to the 
difference temperature between the flowing nanofluid 
and the hot wall downstream of the expansion. This 
heat flow can be estimated using the Nusselt number. 
This local dimensionless number is defined as a 
thermal gradient over the heated surface. In effect, it 
gives a measure of the convective heat transfer that 
occurs on that surface.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
140 Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
 Table 3.  Boundary conditions introduced in dimensionless form
At the channel 
inlet:
UU yV
Y
 


 () ,,, 00 1 

At the 
downstream 
bottom wall:
UV
X



 0010 ,, , 

At the channel 
outlet:












U
X
V
XX Y
U
S

0,

On the channel 
walls (except the 
bottom wall):
UV
n





00 , 
On the boundary 
of cylinder:
UV
n



 00 0 ,,


Table 4.  The effective thermo-physical characteristics of the 
nanofluid
Density:
  
ff fs
   1
Thermal 
diffusivity:


ff
ff
p
ff
K
c


Electrical 
conductivity:


 



ff f
s
f









 1
31
21
()
() ()
,
Specific 
heat:
   cc c
p
ff
p
f
p
s
      1
Thermal 
expansion 
coefficient:
         
ff fs
1
Thermal 
conductivity:
k
k
kk kk
kk kk
ff
f
sf fp
sf fp

  
  
22
2


Dynamic 
viscosity:



ff
f

  1
25 .
The local Nusselt number on the heated wall is 
defined as follows:
 Nu
hH
k
ff
f
= , (15)
where h
ff
 is the heat transfer coefficient represented as 
follows:
 h
q
TT
ff
w
hc


, (16)
where q
w
 stands the heat flux on the heated wall given 
as follows:
 qk
TT
Hn
wf f
hc
L

 



1

. (17)
We substitute Eq. (17) into Eq. (16) and Eq. (16) 
into Eq. (15); we will have the following result:
 Nu
k
kn
ff
f




. (18)
The Nusselt number given by integrating the local 
Nusselt along the length of the hot wall as follows:
 Nu
Ll
Nu dx Nu
ave
L










1
1 00
1
  dl
l
, (19)
where L
1
 represent the horizontal length part hot wall, 
l
 
represent the inclined length part hot wall.
3  VALIDATION CODE AND GRID INDEPENDENCE TEST
The general solving procedure focuses on numerical 
methods. The discretization of the governing 
equations consists in transforming the differential 
form of the governing systems Eqs. (10) to (14) into 
a discrete algebraic form that defines all unknowns at 
each point of a used grid. The finite element method 
based on Galerkin discretization is implemented to 
solve the present problems because of its flexibility 
for complex geometries through different types of 
meshes on the one hand, and the ease of introducing 
boundary conditions in the form of flows on the other 
hand. The application of the present method requires 
a rewriting of the governing equations in integral 
form. The weak formulation is used to include the 
boundary conditions. The numerical solution goes 
through the following steps: introduction of the mesh, 
approximation of the dependent functions, assembly 
and application of the boundary conditions and finally 
the solution of the global system of equations. The 
mesh adopted for the present study, shown in Fig. 2, 
has in its entirety a triangular shape. It is refined near 
the duct walls and near the cylinder circumference. 
However, an unstructured triangular mesh has 
been introduced in the rest of the duct.  It should be 
noted that near the duct walls and near the cylinder 
circumference, a rectangular boundary layer mesh 
was used to ensure that the grid presentation is well 
aligned with the flow. In each node of the defined 
grid, to complete the computation in terms of iteration 
number, the convergence criterion used consists in 
stopping the calculation when the absolute difference 
between the old and the new values of the dependent 
variables, namely U, V, P and θ becomes less than  
10
-6
. If Ψ and n present respectively a dependent 
variable and iteration order, this criterion is established 
using the following formula:

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
141 Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour...
 
YY
Y
nn
n




1
1
6
10  . (20)
The results obtained from the computational 
code should be independent of the grid form and 
not be modified by increasing the mesh elements 
number. Therefore, different grids were verified. 
Table 5 gives the evolution of the average Nusselt 
number, the dimensionless mean temperature 
and the dimensionless temperature for different 
numbers of mesh elements that were considered in 
the present simulation. The verification of the mesh 
was carried out for the case g = 90° at Pr = 6.2, Re 
= 300, Ha = 0, γ = 0°, φ = 0.05, We have chosen to 
use a mesh corresponding to a number of elements 
equal to 41401 beyond which there was no change 
in terms of average Nusselt number, mean and local 
dimensionless temperature. To confirm the accuracy 
of these numerical results, validation with other 
previous work is necessary. The numerical results of 
this work were compared with the numerical results 
of Hussain and Ahmed [18]. They numerically studied 
the forced convection of a laminar flow of a ferrofluid 
around a cylinder inside a backward facing step. 
Therefore, the isotherm forms obtained in the present 
work were compared with the results of the reference 
mentioned above, see Fig. 4, this comparison was 
done in the case of Ha = 0 and Ha = 100 for values of 
Pr = 6.2, Re = 100, γ = 0° and φ = 0.05. In a further 
step, a comparison of the average Nusselt number 
as a function of the Hartmann number has also been 
plotted in Fig. 3 under the same conditions mentioned 
above. It is clear from these figures that the isotherms 
and the average Nusselt number are almost similar.  
Therefore, the results of this work show very good 
agreement with the reference Hussain and Ahmed 
[18].
Fig. 2.  Mesh: a) inside the computational domain,  
b) zoom near the cylinder, and c) zoom near the curvature
Fig. 3.  Comparison of the Nusselt average as a function of Ha of 
the present study with the reference Hussain and Ahmed [21]
Table 5. The average number of Nusselt and temperature for several numbers of elements
Number of elements 17040 23713 31528 40401 50293 60970
Time of simulation [s] 17 25 30 50 55 78
Nu
ave
4.88585 4.89666 4.90630 4.92318 4.92170 4.92185
θ
ave
0.14977 0.15111 0.15257 0.15413 0.15424 0.15461
θ 0.05355 0.05361 0.05462 0.05530 0.05537 0.05534
Ha = 0
Ha = 100
Hussain and Ahmed [21]                                           Present study
Fig. 4.  Comparison of the isotherms of the present study with the reference the reference Hussain and Ahmed [21]

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
142 Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
the Hartmann number, the angle of inclination of 
the magnetic field and the volume fraction of the 
nanoparticles respectively to Re = 300, Ha = 50, γ = 
0° and φ = 0.05 when the influence of one of them is 
highlighted.
4.1  Effect of Hartmann number
Fig. 5 gives the influence of the Hartmann number on 
the streamlines at different curvature angles. In the 
absence of the magnetic field for a straight geometry 
(g = 90°), an equilibrium between the pressure gradient 
and the centrifugal force is established downstream 
of the cylinder, so the action of the centrifugal and 
viscous forces simultaneously leads to the appearance 
of two counter-rotating vortices in the downstream 
part of the cylinder. Furthermore, by increasing 
4  RESULTS AND DISCUSSIONS
In this section, the results of the numerical simulation 
of the nanofluid flow in a downward curved duct 
are shown.  The isotherms, streamlines and average 
Nusselt number are presented for different values of 
Hartmann number Ha (0, 50, 100), Reynolds number 
Re (10, 100, 200), nanoparticle volume fraction 
φ (0 %, 5 %, 1 %) and magnetic field inclination 
angle γ (0°, 60°, 90°). The simulation was based on 
demonstrating for each value of the curvature angle 
g (0°, 30°, 45, 60°, 90°) the effect of these control 
parameters on the thermo hydrodynamic nanofluid 
behavior. In this study, water is the basic fluid in 
the formation of ferrofluid with Pr = 6.2. On the 
other hand, the presentation of the different profiles 
and graphs consists in setting the Reynolds number, 
Fig. 5.  Streamlines at different angles of curvature (g) for a) Ha = 0, b Ha = 50, and c) Ha = 100, when Pr = 6.2.

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143 Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour...
duct curvature  angle g (30°, 45°, 60°) again in the 
absence of the magnetic field Ha = 0,  we observe 
the formation of recirculation zones which are in this 
case dead zones, especially in the immediate vicinity 
of the curved hot wall, these vortices tend to widen 
by increasing the angle of curvature. As the magnetic 
field intensity increases from Ha = 0 to Ha = 100, 
these vortices disappear, favoring the contact of the 
ferrofluid with the hot wall. The increase in magnetic 
field intensity in the case of a straight duct (g = 0°) 
leads to an opposite impact to that of a curved duct. 
The mobility of nanoparticles in the duct decreases in 
terms of current function and velocity magnitude with 
increasing magnetic field in the case of a straight duct 
geometry (g = 0°), in contrast to a curved duct the flow 
intensifies with increasing Hartmann number. This 
behaviour is due to the Lorentz force, which obstructs 
the ferrofluid in the first case and drives it in the 
second case. The magneto hydrodynamic behaviour 
has an effect on the temperature contours as shown 
in Fig. 6. In fact, the intensity of forced convection 
increases as the Hartmann number increases from Ha 
= 0 (without magnetic field) to Ha = 100 in curved 
duct case g (30°, 45°, 60°, 90°). A reduction of the 
thermal boundary layer has been noted by virtue of 
the good heat transfer. Therefore the Nusselt number 
is better in straight curvature angle case g = 90° on the 
one hand. On the other hand, in a straight duct case, 
the convection decreases by increasing the Hartmann 
number, the lines of the isotherms start to move 
towards the adiabatic wall at the top so the thermal 
boundary increases. Fig. 7 illustrates the effect of the 
curvature angle on the local Nusselt number along the 
hot wall in the case of Hartmann number values Ha 
= 0 and Ha = 100 respectively. It was found that in 
the absence of the magnetic field (Ha = 0), the local 
Nusselt number marks an increase in the initial part 
of the hot wall (just below the cylinder), so with the 
increase of the curvature angle it is found that the 
local Nusselt number decreases. Moreover in the 
case of a strong magnetic field (Ha = 100) the local 
Nusselt number shows an increase in the initial part of 
Fig. 6.  Isotherms at different angles of curvature (g) for a) Ha = 0, b Ha = 50, and c) Ha = 100

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144 Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
a)           b) 
Fig. 7.  The effect of the curvature angle around the hot wall on local Nusselt: a) Ha = 0, and b) Ha = 100
Fig. 8.  Streamlines at different angles of curvature (g) for: a) γ = 0°, b) γ = 60° , and c) γ = 90°

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145 Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour...
the curvature hot wall part. Indeed the increase of the 
curvature angle leads to an improvement of the local 
Nusselt number in the curvature part. 
This behavior is due to the Lorentz force that 
drives the flow in the curved part.
4.2 The Effect of the Inclination Angle of the Magnetic 
Field
Fig. 8 shows the hydrodynamic behavior of the 
nanofluid for different magnetic field inclination 
angles. It has been observed that in the case of straight 
duct (g = 0°), the ferrofluid flow is slow at the bottom 
of the cylinder at horizontal magnetic field (γ = 0°). 
Therefore by increasing the magnetic field inclination, 
the mobility of the nanoparticles increases to its 
maximum at the bottom in the vicinity of the cylinder 
near the hot wall at a vertical magnetic field (γ = 90°). 
Moreover, a mobility was noted in terms of current 
function in the curved part by increasing the duct 
curvature angle of g (30°, 45°, 90°) in the presence 
of a horizontal magnetic field. In contrast, in the 
presence of a vertical magnetic field and increasing 
the curvature angle, the current function lines are 
almost identical except in the case of curvature angle 
g = 90°, where the nanofluid flow marks a significant 
recirculation region at zero velocities in the immediate 
vicinity of the adiabatic wall. Physically this 
phenomenon is by virtue of the Lorenz force, which 
is driving as we increase the angle of inclination of 
the magnetic field from 0°, 60° to 90° where we note 
a strong flow in the case of a straight duct. On the 
other hand, the increase in the angle of inclination of 
the magnetic field directs this force in direction that 
obstructs the flow in the case of a curved for all angles 
of curvature. The isotherms presented in Fig. 9 show 
the thermal behavior, which is strongly influenced 
by the hydrodynamic state of the nanofluid under the 
Fig. 9.  Isotherms at different angles of curvature (g) for: a) γ = 0°, b) γ = 60°, and c) γ = 90°

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146 Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
a)          b) 
Fig. 10.  The effect of the curvature angle around the hot wall on local Nusselt number:  
a) at horizontal magnetic field (γ = 0°), and b) at a vertical magnetic field (γ = 90°)
Fig. 11.  Streamlines at different angles of curvature (g) for: a) Re = 10, b) Re = 100, and c) Re = 300

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147 Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour...
effect of the magnetic field inclination angle. In the 
case of a horizontal magnetic field, a decrease of the 
thermal layer is underlined in the case of a straight 
duct as the magnetic field inclination angle increases. 
Indeed, a reduced thermal layer corresponding to a 
good thermal convection is observed for a vertical 
magnetic field (γ = 90°). On the other hand, for a 
horizontal field by increasing the curvature angle, 
the thermal layer is reduced in the curved part due to 
the good thermal transfer. Moreover, by increasing 
the angle of curvature, the increase of the angle of 
inclination of the magnetic field has no effect on the 
heat transfer. Fig. 10 shows the variation of the local 
Nusselt number along the hot wall for the two values 
of inclination angle respectively γ = 0° and γ = 90°. 
We observe an improved local Nusselt number in 
the curved part in the case of a horizontal inclination 
angle (γ = 0°). On the other hand in the case of a 
vertical inclination angle (γ = 90°) the local Nusselt 
number in the curved part is practically the same by 
increasing the curvature angle and in the straight part, 
it increases relatively.
4.3  The Effect of Reynolds Number
Fig. 11 shows the influence of Reynolds number 
(or inertia) on the streamlines of the Fe
2
O
3
-water 
nanofluid flow in the downward curved duct. A 
significant increase in Reynolds number from a value 
of 10 to a value of 300, leads to significant changes in 
the hydrodynamic behavior of the nanofluid flow. In 
fact, in the case of a straight duct, the flow becomes 
more intensity as the Reynolds number increases, thus 
favoring the attachment of the nanofluid due to the 
enlarged duct type used. Furthermore, by increasing 
the curvature angle of the duct as well as the Reynolds 
number, the flow of the nanofluid becomes stronger 
where it marks its intensity at a curvature angle (g 
Fig. 12.  Isotherms at different angles of curvature (g) for: a) Re = 10, b) Re = 100, and c) Re = 300

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
148 Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
= 90°) thus favoring the attachment at a Reynolds 
number (Re = 300). This phenomenon is due on the 
one hand to the inertial forces which predominate the 
viscous forces as well as the Lorenz force when the 
Reynolds number increases in the case of a horizontal 
curvature angle (g = 0°). Moreover, by increasing the 
angle of curvature, the viscous forces and the Lorenz 
force carry away the flow next to the inertia forces, 
which increase with the increase of the Reynolds 
number. The thermal behavior is strongly influenced 
by the hydrodynamic state of the nanofluid under 
the effect of the inertia as shown in Fig. 12. Indeed 
the thermal band marks a decrease with the increase 
of the inertia in the case of a straight duct. On the 
other hand, we note a decrease of the thermal layer, 
which extends towards the hot wall increasingly by 
raising simultaneously the angle of curvature and the 
Reynolds number, reflecting an improved thermal 
transfer. Moreover, a significantly reduced thermal 
layer is mentioned in the case of a curvature angle (g 
= 90°) for a Reynolds number (Re = 300), by virtue 
of a good convection that took place in this situation.
4.4  Effect of Nanoparticle Concentration
Fig. 13 shows the streamline profiles for different 
curvature angles at different average nanoparticle 
concentrations φ (0, 0.05, 0.1). 
It can be seen from this figure that there is 
no change in the streamline pattern by adding 
nanoparticles at different volume fractions to the base 
fluid. This is due on the one hand to the use of particles 
Fig. 13.  Streamlines at different angles of curvature (g) for: a) φ = 0, b) φ = 0.05, and c) φ = 0.1.

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149 Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour...
of nanometric diameters at low concentrations. 
On the other hand, the injection of nanoparticles 
in applications related to heat transfer. In fact, the 
absence of buoyancy force based on the conditions 
of this study (forced convection) does not show the 
influence of the temperature formerly affected by 
the presence of nanoparticles on the hydrodynamic 
behavior. Indeed a weak influence of the viscosity and 
density on this behavior. From Fig. 14, it appears that 
the isotherms profiles show similarly no change. With 
the increase of the concentration of nanoparticles, the 
temperature increases in the middle of the duct in the 
straight part as well as the curved part. Indeed, a good 
increase is marked in the case of a curvature angle (g 
= 90°) and a nanoparticle concentration φ = 0.1, see 
Fig. 15. Effectively the increase of the temperature 
of the cold medium leads to a reduction in terms of 
thermal gradient.
4.5 Effect of Hartmann Number on Nusselt Number
The profiles of the average Nusselt number for 
different Hartmann numbers are shown in Fig. 16a. 
Indeed at an angle of a horizontal magnetic field (γ 
= 0°), the Lorenz force obstructs the flow in the case 
of a straight duct (g = 0°) due to the perpendicularity 
between them. 
Moreover, the increase of the magnetic field 
disfavors the forced convection, which explains the 
reduction of the mean Nusselt number by increasing 
the Hartmann number. In contrast, for a curved duct 
g (30°, 45°, 60°, 90°) the increase of the Hartmann 
Fig. 14.  Isotherms at different angles of curvature (g) for: a) φ = 0, b) φ = 0.05, and c) φ = 0.1

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
150 Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
number as well as the curvature angle marks an 
increase of the average Nusselt number which 
achieves its maximum in the case of a curvature angle 
g = 90°. This behavior is in virtue of the magnetic 
force that drives the ferrofluid flow at latter case. 
Thus, a decrease of the thermal boundary layer was 
noted in this case. Overall the increase in the average 
heat transfer rate throughout the system reaches about 
240.74 % when the Hartmann number increases from 
0 to 100 at curvature angle g = 90°.
4.6  Effect of the Magnetic Field Inclination Angle on the 
Nusselt Number
Fig. 16b shows the average Nusselt number profiles 
at different curvature angles. It can be seen that the 
direction of the magnetic field significantly affects the 
average Nusselt number by increasing the curvature 
angle. Thus, an improvement of the average Nusselt 
number was noted with the increase of the magnetic 
field angle and the curvature angle except in the 
case of a vertical magnetic field (γ = 90°) where the 
increase  of the curvature angle does not show any 
effect. This is in contrast to the case of a straight duct 
where the increase in the inclination of the magnetic 
field promotes heat transfer. Moreover, in the cases 
of curvature angles g (30°, 45°, 60°, 90°), the vertical 
magnetic field increases the average Nusselt number 
in the duct straight part and decreases it in its curved 
part. 
Physically, this behavior is due to the Lorenz 
force that drives the flow in the first part. In the 
a)         b) 
Fig. 15.  Dimensionless temperature component for different volume fractions (φ) at curvature angle (g = 90°) for:  
a) along the vertical line X = 5, Y = -1, and b) along the horizontal line X = 6.55, Y = -3
a)         b) 
Fig. 16.   Effect on average Nusselt number at different curvature angle (g) of:  
a) Hartman number (Ha), and b) the magnetic field inclination (γ)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
151 Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour...
second part, i.e. the curved part, as the curvature angle 
increases, the Lorenz force acts so that the ferrofluid 
moves away from the hot wall, thus disfavoring the 
thermal transfer. Indeed, on the one hand by increasing 
the  curvature angle in the case of horizontal magnetic 
field, the heat transfer rate increase of 191.95 % has 
been noted at a curvature angle (g = 90°). On the other 
hand in  the case of a vertical magnetic field the heat 
transfer is not significantly affected by  the curvature 
angle increase, were we note the heat transfer rate 
increase of 0.31 % at an angle of curvature (g = 90°).
4.7  Effect of Reynolds NUMBER on Nusselt Number
Fig. 17a shows the variation of the average Nusselt 
number with increasing Reynolds number.  In the 
case of a straight duct the average Nusselt number 
increases with increasing Reynolds number. Indeed 
the forced convection is improved as the Reynolds 
number increases. This behavior is explained by 
the predominance of inertial forces over viscosity 
forces. Moreover, the convection is better at high 
Reynolds numbers in the case of a curved duct. This 
phenomenon is due to the viscosity forces, which are 
damped as the curvature angle increases. As a result, 
the average Nusselt number increases to its maximum 
for a Reynolds number value Re = 300 in the case of a 
right curvature angle (g = 90°). 
Thus, the heat transfer is improved. Indeed, it 
was recorded an increase of the Nusselt number from 
the value of 0.90 to the value of 2.85 with an increase 
in the thermal rate of almost 214.62 % following an 
increase of the Reynolds number from 10 to 300 in the 
case of a straight duct. Furthermore, by increasing the 
curvature angle, it was found that the average Nusselt 
number increased from 1.80 to 8.32 with an increase 
in the heat transfer rate of 362.77 % at a curvature 
angle (g = 90°).
4.8  Effect of Nanoparticle Concentration on Nusselt 
Number
The variation of the heat exchange rate with the 
nanofluid use is an objective consideration for all 
applications including this type of heat transfer 
fluid. The average Nusselt number distribution 
shown in Fig. 17b obviously shows that the heat rate 
increases with increasing nanoparticle concentration 
and with increasing curvature angle.  Moreover, an 
improvement in heat transfer at a curvature angle (g 
= 90°) of up to 12.5 % is recorded with the use of 
nanofluid with a concentration of 0.1 compared to the 
pure water.
5  CONCLUSIONS
In the present numerical study, the 
magnetohydrodynamic behavior of forced convection 
of a laminar Fe
3
O
4
-H
2
O flow in a backward facing 
curved duct comprising a fixed cylinder has been 
investigated. The governing equations solved by the 
finite element method and wide control parameters 
ranges have been considered. Hence, the effects of 
Hartmann number, Reynolds number, magnetic field 
inclination angle and nanoparticle concentration on 
the thermo-magnetohydrodynamic nanofluid behavior 
a)          b)  
Fig. 17.  Effect on average Nusselt number at different curvature angle (g) of:  
a) the Reynolds number (Re), and b) the nanoparticle volume fraction (φ)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)3-4, 135-154
152 Derbal, D. – Bouzit, M. – Mokhefi, A. – Bouzit, F.
have been demonstrated. Through this study, the 
following essential points have been demonstrated:  
• The increase in the Hartmann number slows down 
the ferrofluid flow within the straight duct with 
a backward facing step. However, this increase 
in the Hartmann number decreases the average 
Nusselt number in the straight duct case. On the 
other hand, the increase of the curvature angle as 
well as the magnetic field intensity favors the heat 
transfer. Therefore, the average Nusselt number 
improves by reaching its maximum value at the 
right curvature angle. 
• The increase in the inclination angle degree of the 
magnetic field affects positively the heat transfer 
in the case of a straight duct with a backward 
facing step. Indeed this increase improves the 
average Nusselt number in this case. Moreover, 
the rise of the curvature angle as well as the 
magnetic field angle increases the convective 
heat transfer except for the straight magnetic field 
where the curvature angle shows no effect on the 
heat transfer.
• The increase in the Reynolds number leads to a 
significant enhancement of the convective heat 
transfer. In addition, with the increase of the 
curvature angle dampens the effect of viscous 
forces, thus, promoting heat transfer even more.
• The addition of Fe
3
O
4
 nanoparticles to pure 
water improves the heat transfer. In this case, the 
value of the average Nusselt number increases 
by almost 12.5 % compared to the pure water 
used. However, due to the low concentrations 
introduced, the nanofluid hydrodynamic was not 
largely affected due to low concentrations used.
7  NOMENCLATURE
Cp  specific heat [J·kg
-1
·K
-1
]
K  thermal conductivity [W·m
-1
·K
-1
]
Ha  Hartmann number
q  heat flux density [W·m
-2
]
Re  Reynolds number
Nu
ave 
average Nusselt number
Nu  local Nusselt number
Pr  Prandtl number
P  pressure [Pa]
P
atm  
atmospheric pressure [Pa]
x, y  Cartesian coordinates [m]
X, Y  dimensionless coordinates
g  curvature angle [°]
H  step height
T  temperature [K]
T
c  
cold temperature [K]
T
h  
hot temperature [K]
u  velocity in x direction [m·s
-1
]
v  velocity in y direction [m·s
-1
]
V, U Dimensionless velocity  
u  Vertical wall velocity [m·s
-1
]
Greek symbols
α  thermal diffusivity [m
2
·s
-1
]
β  thermal expansion coefficient [K
-1
]
θ  dimensionless local temperature
μ  dynamic viscosity [kg·m
-1
·s
-1
]
ρ  density [kg·m
-3
]
σ  electrical conductivity [Ω
-1
·m
-1
]
γ  magnetic field inclination angle
φ  volume fraction 
θ
ave  
dimensionless average temperature
Subscripts:
s  solid
c  cold
f  base fluid
h  hot
ff  ferro-fluid
p  nanoparticles
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