Citation: Ashraf, A.; Zhang, Z.;
Saeed, T.; Zeb, H.; Munir, T.
Convective Heat Transfer Analysis
for Aluminum Oxide (Al
2
O
3
)- and
Ferro (Fe
3
O
4
)-Based Nano-Fluid
over a Curved Stretching Sheet.
Nanomaterials 2022, 12, 1152.
https://doi.org/10.3390/
nano12071152
Academic Editors: Ki-Hyun Kim and
Deepak Kukkar
Received: 24 February 2022
Accepted: 23 March 2022
Published: 30 March 2022
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Article
Convective Heat Transfer Analysis for Aluminum Oxide
(Al
2
O
3
)- and Ferro (Fe
3
O
4
)-Based Nano-Fluid over a Curved
Stretching Sheet
Asifa Ashraf
1
, Zhiyue Zhang
1
, Tareq Saeed
2,
* , Hussan Zeb
3
and Taj Munir
4
1
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University,
Nanjing 210023, China; asifa.ashraf70@yahoo.com (A.A.); zhangzhiyue@njnu.edu.cn (Z.Z.)
2
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics,
Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3
Department of Mathematics & Statistics, Hazara University, Mansehra 21120, Pakistan;
hussan_maths@hu.edu.pk
4
Abdus Salam School of Mathematical Sciences, Government College University Lahore,
Lahore 54600, Pakistan; taj.munir@sms.edu.pk
* Correspondence: tsalmalki@kau.edu.sa
Abstract:
In this work, the combined effects of velocity slip and convective heat boundary conditions
on a hybrid nano-fluid over a nonlinear curved stretching surface were considered. Two kinds of
fluids, namely, hybrid nano-fluid and aluminum oxide (
Al
2
O
3
)- and iron oxide (
Fe
3
O
4
)-based nano-
fluid, were also taken into account. We transformed the governing model into a nonlinear system of
ordinary differential equations (ODEs). For this we used the similarity transformation method. The
solution of the transformed ODE system was computed via a higher-order numerical approximation
scheme known as the shooting method with the Runge–Kutta method of order four (RK-4). It is
noticed that the fluid velocity was reduced for the magnetic parameter, curvature parameter, and slip
parameters, while the temperature declined with higher values of the magnetic parameter, Prandtl
number, and convective heat transfer. Furthermore, the physical quantities of engineering interest,
i.e., the behavior of the skin fraction and the Nusselt number, are presented. These behaviors are
also illustrated graphically along with the numerical values in a comparison with previous work in
numerical tabular form.
Keywords: Al
2
O
3
aluminum oxide and ferro
Fe
3
O
4
nano-particles; non-linear curved sheet; convective
heat transfer; velocity slip boundary condition
1. Introduction and Literature Review
Boundary layer flows are characterized by Newtonian and non-Newtonian fluids.
Newtonian fluids are those in which the stress is linearly proportional to strain. Examples
of Newtonian fluids are mineral oil, water, gasoline, organic matter, kerosene, solvents,
glycerin, alcohol, etc. The boundary layer flows of non-Newtonian fluids have attracted
much attention owing to their tremendous applications in industry, manufacturing, and
geothermal engineering. Examples of these applications are nuclear reactors, metallurgical
processes, the spinning of fibers, casting, liquid metals space technology, crystal growth,
and many more. The models of non-Newtonian fluids cannot be expressed in a single
relationship because of their mutual aspects. Crane et al. [
1
] determined the boundary
layer flow over exponentially and linear stretching surfaces. Then, Vleggaar [
2
] explored
the laminar flow of the boundary layer on a continuous and accelerating stretching surface.
The similarity analysis for the Navier–Stokes equation over the stretching surface was
studied by Wang [
3
]. The time-dependent boundary layer flow of viscous fluid over a
stretching curved surface was investigated by Sajid et al. [4].
The time-dependent boundary layer flow over a permeable curved shrinking/stretching
Nanomaterials 2022, 12, 1152. https://doi.org/10.3390/nano12071152 https://www.mdpi.com/journal/nanomaterials
