The present numerical work examines the effect of fractional order parameter on heat transfer and entropy generation for a thermo-magnetic convective flow of nanofluid (Cu-water) in a square porous enclosure that contains semi-circular bottom wall. The Darcy–Brinkmann–Forchheimer model is utilized to evaluate the momentum transfer in porous media, and the Caputo-time fractional derivative term is introduced in momentum as well as in the energy equation. Further, non-dimensional governing equations are simulated through the penalty finite element method, and the Caputo time derivative is approximated by L1-scheme. The study is carried out for various parameters, including Rayleigh number (Ra), Darcy number (Da), radius of the semicircle (r), fractional order (α), and Hartmann number (Ha). The comprehensive results are presented by the contour variation of isotherms, streamlines, and total entropy generation at the selected range of parameters. In addition, thermal transport and irreversibilities due to heat transfer, fluid friction, and magnetic field have been accounted through the numerical variation of mean Nusselt number ( N u m ) and Bejan number due to heat transfer ( B e h t ), fluid friction ( B e f f ), and magnetic field ( B e m f ), respectively. The key findings of the present study reveal that during the initial evolution period, the Num value increases as α 1. Additionally, time taken to achieve the steady state condition varies and depends on fractional order α. Furthermore, in the absence of Ha, the heat transfer and entropy generation intensifies with augmentation of Ra and Da for all α, while, the increasing value of Ha shows an adverse impact on the heat transfer rate.

1.
D. B.
Ingham
and
I.
Pop
,
Transport Phenomena in Porous Media III
(
Elsevier
,
2005
), Vol.
3
.
2.
N. H.
Saeid
, “
Natural convection in porous cavity with sinusoidal bottom wall temperature variation
,”
Int. Commun. Heat Mass Transfer
32
,
454
463
(
2005
).
3.
D. A.
Nield
,
A.
Bejan
et al,
Convection in Porous Media
(
Springer
,
2006
), Vol.
3
.
4.
F. A.
Zahor
,
R.
Jain
,
A. O.
Ali
, and
V. G.
Masanja
, “
Modeling entropy generation of magnetohydrodynamics flow of nanofluid in a porous medium: A review
,”
Int. J. Numer. Methods Heat Fluid Flow
33
,
751
771
(
2023
).
5.
A. I.
Alsabery
,
A. S.
Abosinnee
,
S. K.
Al-Hadraawy
,
M. A.
Ismael
,
M. A.
Fteiti
,
I.
Hashim
,
M.
Sheremet
,
M.
Ghalambaz
, and
A. J.
Chamkha
, “
Convection heat transfer in enclosures with inner bodies: A review on single and two-phase nanofluid models
,”
Renewable Sustainable Energy Rev.
183
,
113424
(
2023
).
6.
K.
Thirumalaisamy
and
S.
Ramachandran
, “
Comparative heat transfer analysis on Fe3O4–H2O and Fe3O4–Cu–H2O flow inside a tilted square porous cavity with shape effects
,”
Phys. Fluids
35
,
022007
(
2023
).
7.
P. K.
Tyagi
,
R.
Kumar
, and
P. K.
Mondal
, “
A review of the state-of-the-art nanofluid spray and jet impingement cooling
,”
Phys. Fluids
32
,
121301
(
2020
).
8.
V.
Kumar
,
S. K.
Murthy
, and
B. R.
Kumar
, “
Multi-force effect on fluid flow, heat and mass transfer, and entropy generation in a stratified fluid-saturated porous enclosure
,”
Math. Comput. Simul.
203
,
328
367
(
2023
).
9.
M.
Izadi
, “
Effects of porous material on transient natural convection heat transfer of nano-fluids inside a triangular chamber
,”
Chin. J. Chem. Eng.
28
,
1203
1213
(
2020
).
10.
A.
Abderrahmane
,
A.
Manoongam
,
A. A.
Alizadeh
,
O.
Younis
,
H.
Zekri
,
S. S. P. M.
Isa
,
S.
Baghaei
,
W.
Jamshed
, and
K.
Guedri
, “
Investigation of the free convection of nanofluid flow in a wavy porous enclosure subjected to a magnetic field using the Galerkin finite element method
,”
J. Magn. Magn. Mater.
569
,
170446
(
2023
).
11.
M.
Hamid
,
Z.
Khan
,
W.
Khan
, and
R. U.
Haq
, “
Natural convection of water-based carbon nanotubes in a partially heated rectangular fin-shaped cavity with an inner cylindrical obstacle
,”
Phys. Fluids
31
,
103607
(
2019
).
12.
V.
Kumar
,
S. K.
Murthy
, and
B. R.
Kumar
, “
Entropy generation in a chemically and thermally reinforced doubly stratified porous enclosure in a magnetic field
,”
Phys. Fluids
34
,
013307
(
2022
).
13.
S.
Kumar
,
B. R.
Kumar
,
S. V. K.
Murthy
, and
D.
Parmar
, “
Thermo-fluidic convective flow study of hybrid nanofluid in an inverted T-shaped porous enclosure under uniformly acting magnetic field
,”
J. Porous Media
26
,
75–91
(
2023
).
14.
C.-C.
Cho
, “
Natural convection of Cu-water nanofluid in enclosed cavity with porous effect and wavy surface based on energy-flux-vector visualization method
,”
Phys. Fluids
32
,
103607
(
2020
).
15.
A. I.
Alsabery
,
T.
Tayebi
,
A. J.
Chamkha
, and
I.
Hashim
, “
Effect of rotating solid cylinder on entropy generation and convective heat transfer in a wavy porous cavity heated from below
,”
Int. Commun. Heat Mass Transfer
95
,
197
209
(
2018
).
16.
A.
Fattahi
,
N.
Hajialigol
,
M.
Delpisheh
, and
N.
Karimi
, “
Lattice-Boltzmann numerical simulation of double-diffusive natural convection and entropy generation in an n-shaped partially heated storage tank
,”
Eng. Anal. Boundary Elem.
146
,
105
118
(
2023
).
17.
W.
Al-Kouz
,
A.
Abderrahmane
,
M.
Shamshuddin
,
O.
Younis
,
S.
Mohammed
,
O. A.
Bég
, and
D.
Toghraie
, “
Heat transfer and entropy generation analysis of water-Fe3O4/CNT hybrid magnetic nanofluid flow in a trapezoidal wavy enclosure containing porous media with the Galerkin finite element method
,”
Eur. Phys. J. Plus
136
,
1184
(
2021
).
18.
R.
Anandalakshmi
and
T.
Basak
, “
Numerical simulations for the analysis of entropy generation during natural convection in porous rhombic enclosures
,”
Numer. Heat Transfer, Part A
63
,
257
284
(
2013
).
19.
D. K.
Mandal
,
N.
Biswas
,
N. K.
Manna
,
R. S. R.
Gorla
, and
A. J.
Chamkha
, “
Hybrid nanofluid magnetohydrodynamic mixed convection in a novel W-shaped porous system
,”
Int. J. Numer. Methods Heat Fluid Flow
33
,
510
544
(
2023
).
20.
D. K.
Mandal
,
N.
Biswas
,
N. K.
Manna
,
D. K.
Gayen
,
R. S. R.
Gorla
, and
A. J.
Chamkha
, “
Thermo-fluidic transport process in a novel M-shaped cavity packed with non-Darcian porous medium and hybrid nanofluid: Application of artificial neural network (ANN)
,”
Phys. Fluids
34
,
033608
(
2022
).
21.
S.
Kumar
,
B. V. R.
Kumar
,
S. V. S. S. N. V. G. K.
Murthy
, and
D.
Parmar
, “
Double-diffusive convective flow of hybrid nanofluid in an inverted T-shaped porous enclosure: A numerical study
,”
Numer. Heat Transfer, Part A
(published online).
22.
M.
Sheikholeslami
,
Z.
Shah
,
A.
Shafee
,
P.
Kumam
, and
H.
Babazadeh
, “
Lorentz force impact on hybrid nanofluid within a porous tank including entropy generation
,”
Int. Commun. Heat Mass Transfer
116
,
104635
(
2020
).
23.
S.
Shehzad
,
M.
Sheikholeslami
,
T.
Ambreen
, and
A.
Shafee
, “
Convective MHD flow of hybrid-nanofluid within an elliptic porous enclosure
,”
Phys. Lett. A
384
,
126727
(
2020
).
24.
F.
Selimefendigil
and
H. F.
Öztop
, “
Magnetohydrodynamics forced convection of nanofluid in multi-layered U-shaped vented cavity with a porous region considering wall corrugation effects
,”
Int. Commun. Heat Mass Transfer
113
,
104551
(
2020
).
25.
M.
Ghalambaz
,
M.
Sabour
,
I.
Pop
, and
D.
Wen
, “
Free convection heat transfer of MgO-MWCNTs/EG hybrid nanofluid in a porous complex shaped cavity with MHD and thermal radiation effects
,”
Int. J. Numer. Methods Heat Fluid Flow
29
,
4349
4376
(
2019
).
26.
Z.
Abdelmalek
,
T.
Tayebi
,
A.
Dogonchi
,
A. J.
Chamkha
,
D.
Ganji
, and
I.
Tlili
, “
Role of various configurations of a wavy circular heater on convective heat transfer within an enclosure filled with nanofluid
,”
Int. Commun. Heat Mass Transfer
113
,
104525
(
2020
).
27.
M. S.
Asmadi
,
R.
Md Kasmani
,
Z.
Siri
, and
H.
Saleh
, “
Thermal performance analysis for moderate Rayleigh numbers of Newtonian hybrid nanofluid-filled U-shaped cavity with various thermal profiles
,”
Phys. Fluids
33
,
032006
(
2021
).
28.
S.
Kumar
,
B.
Kumar
, and
S. K.
Murthy
, “
Double diffusive convective flow study of a hybrid nanofluid in an inverted T-shaped porous enclosure under the influence of Soret and Dufour prameters
,”
J. Heat Mass Transfer
145
,
102501
(
2023
).
29.
A.
Dogonchi
,
M. A.
Ismael
,
A. J.
Chamkha
, and
D.
Ganji
, “
Numerical analysis of natural convection of Cu–water nanofluid filling triangular cavity with semicircular bottom wall
,”
J. Therm. Anal. Calorim.
135
,
3485
3497
(
2019
).
30.
I.
Chabani
,
F.
Mebarek-Oudina
,
H.
Vaidya
, and
A.
Ismail
, “
Numerical analysis of magnetic hybrid nano-fluid natural convective flow in an adjusted porous trapezoidal enclosure
,”
J. Magn. Magn. Mater.
564
,
170142
(
2022
).
31.
A.
Dogonchi
,
M.
Nayak
,
N.
Karimi
,
A. J.
Chamkha
, and
D. D.
Ganji
, “
Numerical simulation of hydrothermal features of Cu–H2O nanofluid natural convection within a porous annulus considering diverse configurations of heater
,”
J. Therm. Anal. Calorim.
141
,
2109
2125
(
2020
).
32.
A. S.
Dogonchi
,
T.
Armaghani
,
A. J.
Chamkha
, and
D.
Ganji
, “
Natural convection analysis in a cavity with an inclined elliptical heater subject to shape factor of nanoparticles and magnetic field
,”
Arabian J. Sci. Eng.
44
,
7919
7931
(
2019
).
33.
A.
Dogonchi
,
F.
Selimefendigil
, and
D.
Ganji
, “
Magneto-hydrodynamic natural convection of CuO-water nanofluid in complex shaped enclosure considering various nanoparticle shapes
,”
Int. J. Numer. Methods Heat Fluid Flow
29
,
1663
1679
(
2019
).
34.
S. M.
Seyyedi
,
A.
Dogonchi
,
D.
Ganji
, and
M.
Hashemi-Tilehnoee
, “
Entropy generation in a nanofluid-filled semi-annulus cavity by considering the shape of nanoparticles
,”
J. Therm. Anal. Calorim.
138
,
1607
1621
(
2019
).
35.
H.
Sun
,
Y.
Zhang
,
D.
Baleanu
,
W.
Chen
, and
Y.
Chen
, “
A new collection of real world applications of fractional calculus in science and engineering
,”
Commun. Nonlinear Sci. Numer. Simul.
64
,
213
231
(
2018
).
36.
H.
Karani
,
M.
Rashtbehesht
,
C.
Huber
, and
R. L.
Magin
, “
Onset of fractional-order thermal convection in porous media
,”
Phys. Rev. E
96
,
063105
(
2017
).
37.
S. E.
Ahmed
, “
Caputo fractional convective flow in an inclined wavy vented cavity filled with a porous medium using Al2O3-Cu hybrid nanofluids
,”
Int. Commun. Heat Mass Transfer
116
,
104690
(
2020
).
38.
S.
Malik
and
A. K.
Nayak
, “
MHD convection and entropy generation of nanofluid in a porous enclosure with sinusoidal heating
,”
Int. J. Heat Mass Transfer
111
,
329
345
(
2017
).
39.
R.
Nayak
,
S.
Bhattacharyya
, and
I.
Pop
, “
Numerical study on mixed convection and entropy generation of Cu–water nanofluid in a differentially heated skewed enclosure
,”
Int. J. Heat Mass Transfer
85
,
620
634
(
2015
).
40.
C.-C.
Cho
, “
Effects of porous medium and wavy surface on heat transfer and entropy generation of Cu-water nanofluid natural convection in square cavity containing partially-heated surface
,”
Int. Commun. Heat Mass Transfer
119
,
104925
(
2020
).
41.
T.
Basak
,
S.
Roy
,
T.
Paul
, and
I.
Pop
, “
Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions
,”
Int. J. Heat Mass Transfer
49
,
1430
1441
(
2006
).
42.
A.
Al-Zamily
and
M. R.
Amin
, “
Natural convection and entropy generation in a cavity filled with two horizontal layers of nanofluid and porous medium in presence of a magnetic field
,” in
Proceedings of ASME International Mechanical Engineering Congress and Exposition
(
American Society of Mechanical Engineers
,
2015
), Vol.
57502
, p.
V08BT10A029
.
43.
I.
Podlubny
,
Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications
(
Elsevier
,
1998
).
44.
M.
Stynes
, “
A survey of the L1 scheme in the discretisation of time-fractional problems
,”
Numer. Math.: Theory, Methods Appl.
15
,
1173
1192
(
2022
).
45.
Y.
Lin
and
C.
Xu
, “
Finite difference/spectral approximations for the time-fractional diffusion equation
,”
J. Comput. Phys.
225
,
1533
1552
(
2007
).
46.
M. R. S.
Ammi
,
I.
Jamiai
, and
D. F.
Torres
, “
A finite element approximation for a class of Caputo time-fractional diffusion equations
,”
Comput. Math. Appl.
78
,
1334
1344
(
2019
).
47.
P.
Nithiarasu
,
K.
Seetharamu
, and
T.
Sundararajan
, “
Natural convective heat transfer in a fluid saturated variable porosity medium
,”
Int. J. Heat Mass Transfer
40
,
3955
3967
(
1997
).
48.
J. N.
Reddy
,
An Introduction to the Finite Element Method
(
McGraw-Hill
,
New York
,
2013
), Vol.
3
.
You do not currently have access to this content.