International Communications in Heat and Mass Transfer 134 (2022) 106038
Available online 9 April 2022
0735-1933/© 2022 Elsevier Ltd. All rights reserved.
Physical specications of MHD mixed convective of Ostwald-de Waele
nanouids in a vented-cavity with inner elliptic cylinder
Wasim Jamshed
a
, Mohamed R. Eid
b
,
c
,
*
, Syed M. Hussain
d
, Aissa Abderrahmane
e
,
Rabia Safdar
f
, Obai Younis
g
, Amjad Ali Pasha
h
a
Department of Mathematics, Capital University of Science and Technology (CUST), Islamabad 44000, Pakistan
b
Department of Mathematics, Faculty of Science, New Valley University, Al-Kharga, Al-Wadi Al-Gadid, 72511, Egypt
c
Department of Mathematics, Faculty of Science, Northern Border University, Arar 1321, Saudi Arabia
d
Department of Mathematics, Faculty of Science, Islamic University of Madinah, 42351, Saudi Arabia
e
Laboratoire de Physique Quantique de la Mati`
ere et Mod´
elisation Math´
ematique (LPQ3M), University of Mascara, Algeria
f
Department of Mathematics, Lahore College for Women University, 54000 Lahore, Pakistan
g
Department of Mechanical Engineering, College of Engineering at Wadi Addwaser, Prince Sattam bin Abdul Aziz University, Wadi Addwaser 11991, KSA
h
Aerospace Engineering Department, King Abdulaziz University, Jeddah 21589, Saudi Arabia
ARTICLE INFO
Keywords:
Ostwald–De Waele nanouid
MHD
Mixed convection
Galerkin nite element method
ABSTRACT
The present work investigates heat transmission and stable MHD (magneto-hydrodynamic) mixed convective
owing in a ventilated porous enclosed space with a heated elliptic inner cylinder lled with MWCNT (multi-wall
carbon nanotube)/CMC (carboxymethylcellulose) nanouid. The enclosure is surrounded by a homogeneous
magnetic eld. By employing Galerkin nite element method (GFEM), the governing equations are conrmed.
Simulations are performed for various ranges of pertinent parameters such as power law index (between 0.8 and
1.4), Hartmann number (between 0 and 100), elliptic cylinder inclination (between 0∘ and 90∘), and Richardson
number (between 0.1 and 100). The numerical ndings are given in bounds of isotherms, Nusselt numbers, and
streamlines, which are critical governing factors for heat convective and enclosure ow. The data demonstrate
that average Nusselt numbers Nu
avg
increases when Richardson number Ri and porosity ratio, increases. Hartman
number works with the induced Lorentz force to make uidity dominates the lower-level uidity in the enclosure
which reects in isothermal displays. Even if the higher power law indexing operates against the uidity by
triggering shear force, it appears to be constructive in terms of heat transmission.
1. Introduction
The uid that consists of nano-meter sized particles as well as makes
colloidal suspension of them is called nanouid. These nanoparticles are
made of metallic oxides, carbides, nitride ceramics, carbide ceramics,
carbon-nanotubes, metal carbonitrides, or metals. Nanouids have high
abilities of heat transfer [1]. They have greater thermal conductivity as
well as rheological features than ordinary uids. Ali and Salam [2]
utilized nanouid to study the recent developments in improving heat
transfer. They also reviewed progress in preparing and increasing the
stability. Prasad et al. [3] presented a detailed procedure of preparing
nanouids, their characteristics along with their uses in energy,
biomedical and mechanical elds. Wang and Fan [4] utilized heat
conductive nanouids to examine the available techniques for handling
various difcult problems of nanouids. Dharmalingam et al. [5] gave a
detailed summary of mathematical and experimental research about
heat transfer regarding nanouids, physical as well as chemical features,
and analysis of future challenges for nanouids. Baleanu et al. [6]
considered the particle motion in a circular hollow. Karimi et al. [7]
established a thorough computational model for a tube-shaped hole
receiver that was kept at the crucial point of the system regarding
parabolic plate accumulator. Sanchugov et al. [8] presented an article
about major outcomes of the research done on a uid ow that was
passing through a cavity inlet, lled with gas. The results can be
employed for innovative procedure designing about the cleaning of
inner surfaces for hydraulic cylinders. Hazi et al. [9] focused on
behavior regarding the ow of particles as well as uid. Various
important and rare techniques and equations were employed to
* Corresponding author at: Department of Mathematics, Faculty of Science, New Valley University, Al-Kharga, Al-Wadi Al-Gadid, 72511, Egypt.
E-mail address: [email protected] (M.R. Eid).
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ichmt
https://doi.org/10.1016/j.icheatmasstransfer.2022.106038
International Communications in Heat and Mass Transfer 134 (2022) 106038
2
concentrate on particles owing in the lid-driven hollow of semi ellipse
shape.
Susmal [10] conducted a numerical analysis on forced convective
heat transmission regarding an orientated elliptical cylinder having a
stable aspect ratio. They used Al
2
O
3
nanouid based on water.
Alguboori et al. [11] utilized an Al
2
O
3
/water hybrid nanouid lled
inclined annular cavity for numerical analysis regarding natural con-
vection heat transfer. There is the cold constant temperature on the
outer side of the elliptic cavity while hot is constant on the inner side of a
circular cylinder wall. Singh and Kishore [12] provided the numerical
outcomes regarding mixed convective heat transfer happening between
shear-thinning nanouids and unconned elliptical cylinders. Bou-
zerzour et al. [13] made a computational examination about liquid
owing and natural convection heat transmission in the 2D annulus. The
annulus is made up of two confocal elliptic cylinders that are differen-
tially heated. The annulus is lled up with silver nanouids based on
water. The inner side of the elliptic cylinder is maintained at uniformly
isothermal temperature, but the outer wall is at a uniformly differenti-
ated lower temperature. Abdulkadhim [14] numerically demonstrated
natural convective heat transfer about the copper-water nanouid lying
in between inner heated circular cylinders in a cooled elliptical cavity.
Inuence of viscous dissipation and thermal radiation was studied by
Roy et al. [15] regarding heat transfer and hybrid nanouid ow passing
through the circular cylinder.
Ostwald-de Waele nanouid is also called power-law nanouid. It is
a kind of generalized Newtonian uid or time independent non-
Newtonian nanouid. Ostwald-de Waele uid model is modied for
Darcy Brinkman’s surface. It is employed for solving boundary value
problems consisting of nonlinear PDEs. Yu et al. [16] employed melting
heat transfer to highlight the ow of Ostwald-de Waele nanouid
passing through a revolving disk with variant thickness in the permeable
surface. Siddiqui [17] utilized Ostwald-de Waele uid to study the
mathematical expressions regarding its pulsatile motion in two me-
diums. Vajravelu et al. [18] made an analysis for inuences of variant
thermos-physical features on heat transfer and unsteady MHD Oatwald-
de Waele nanouid ow on extending sheet. Copper-water nanouid
lled square cavity was utilized by Guiet et al. [19] for numerical
analysis of natural convection about protruding heater inside the cavity.
Chetteti and Srivastav [20] also employed Ostwald-de Waele nanouid
ow for analysis of mixed convection. Sivaraj and Banerjee [21] pre-
sented an experimental analysis on transmission features regarding non-
Newtonian nanouids. Ellahi et al. [22] employed a compatible third-
grade non-Newtonian nanouid having thermodynamic and incom-
pressible features to examine its fully developed ow. A reader may read
articles related to non-Newtonian nanouids from [23–25].
Mittal et al. [26] presented the models regarding the physical char-
acteristics of nanouids. They focused on conducting analyses on MHD
convection regarding nanouids in many applications. Jha et al. [27]
made a theoretical analysis on an incompressible, electrically con-
ducting, and viscous nanouid ow having steady, MHD, and fully
developed natural convective features underneath the inuence of
magnetic force. Ullah et al. [28] conducted exact research on the same
uid having the same features on an inclined plate. Font [29] presented
a thorough overview of MHD and numerical hydrodynamics in general
relativity. Matthaeus et al. [30] provided an overview of turbulence that
is as same as hydrodynamics. They also included magneto-
hydrodynamics. Effects of MHD, as well as gyrotactic microorganisms,
were implemented by Shahid et al. [31] for analysis of nanouid ow
across an extending sheet. Nourbakhsh et al. [32] presented an analysis
to investigate the behavior of heat transfer regarding nanouid ow
when radiation along with MHD is employed with constant heat ux.
The effect of nanomolecules diffusion in water ow, with thermal
conductance, and mass transmit efcacy was discussed by Gowda et al.
[33]. They used KKL (Koo–Kleinstreuer and Li) scheme in the presence
of Cattaneo-Christov heat uxing and activating energy.
A review was presented by Mahian et al. [34] about computational
and theoretical analysis regarding entropy generating for heat trans-
mission and nanouids owing in different regions. Huminic and
Huminic [35] presented the entropy formation of hybrid as well as or-
dinary nanouids under different thermal systems and constraints.
Renuka et al. [36] explained the results about Joule heating impact and
Fig. 1. Physical paradigm and coordinates system.
020406080100
1
2
3
4
5
Gr=2x10
5
Nu
avg
Ha
Sheikholeslami et al[41]
Present model
Gr=2x10
4
Fig. 2. Comparison of calculated Nu
avg
of the current work (bottom) with that
of literature [41].
Table 1
Thermophysical assets of (MWCNT/CMC) [47,48].
Properties
ρ
(kg/m
3
) C
p
(J/kg. K) k(W/m. K)
σ
(S/m) β(K
−1
)
CMC 997.1 4179 0.613 5.5 ×10
−6
21 ×10
−5
MWCNT 1600 796 3000 1.9 ×10
−4
4.4 ×10
Table 2
Nu
avg
ndings for diverse grid-sizes.
Mesh size 1826 2810 3028 4084 8384 26,144
ψ
max
0.407 0.047 0.047 0.047 0.047 0.047
Nu
avg
3.912 4.047 4.065 4.131 4.218 4.335
W. Jamshed et al.
International Communications in Heat and Mass Transfer 134 (2022) 106038
3
viscous dissipative on ow and entropy analysis of nanouids. Alsabery
et al. [37] presented a computational analysis about free convective and
entropy formation analysis in a nanouid lled square enclosure. Multi-
convection regimes, as well as entropic formation owing to Marangoni
effects, side-wall motion, and double-diffusive convective, were inves-
tigated by Ahmed et al. [38] in lid-driven enclosures lled with a
penetrable material. Selimefendigil et al. [39] highlighted The numeri-
cal investigation of the mixed convective of a CuO–H
2
O nanoliquid ll
Fig. 3. Rotating angle effect on the uidity and isotherms when Ri =0.1, 1, 10.
W. Jamshed et al.
International Communications in Heat and Mass Transfer 134 (2022) 106038
4
lid driven chamber with upper and lower triangle regions under the
impact of inclined electromagnetic eld. Yaseen and Ismael [40] probed
the mixed convective heat transmission non-Newtonian uid-structure
interacting (FSI) in an open trapezoidal hollow. They discovered that the
uttering phenomena of the n occur at the largest Ri and Re values with
pseudoplastic liquid as well as Newtonian liquid. In addition to the
numerous and innite uses of uids and nanoparticles in the engineering
industries, the selection of this basic uid is based on its practical uses in
a variety of disciplines. Theses articles [41–43] presented the recent
researchs on the formation of thermodynamics and entropy in
buoyancy-induced owing in cavities and channels.
In the literature review and authors’ understandings, there is no
already published work in which the effect of power law uid and
magnetic eld on mixed convection in vented-cavity with the inner
elliptic cylinder. Due to practical application in engineering, this com-
bination needs to be explored more. Therefore, the effect of various
control parameters is examined in detail.
2. Mathematical formulation
Fig. 1 depicts the basic eld of a square chamber with a restricted
heater and elliptic cylinder, as well as the boundary conditions. The
cavity’s height is H, and the heater’s length is h. The oor wall is pre-
served at a steady cold temperature of T
c
, while the upper wall is kept at
a hot temperature of T
H
, and the perpendicular walls are believed to be
adiabatic. Under the inuence of the inclined magnetic eld, the hollow
is lled with a nanoliquid. The Boussinesq approximation is utilized to
establish the buoyancy force in the momentum equation. The ow of
this model is laminar, steady, and incompressible. Darcy-Brinkman-
Forchheimer porous model is applied. The porous medium is isotropic
and homogeneous. Viscous dissipation and radiation are neglected (See
Fig. 2.).
Fig. 3. (continued).
W. Jamshed et al.
International Communications in Heat and Mass Transfer 134 (2022) 106038
5
2.1. The governing equations
The following are the continuity, impetus, and energy formulas for a
time-independent, two-dimensional ow [17,41,44]:
∂
u
∂
x+
∂
v
∂
y=0,(1)
Fig. 4. Power law index effect on the uidity and isotherms when Ri =0.1, 1, 10.
W. Jamshed et al.
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