This research will investigate heat transfer dynamics in thinners, applying transformer model learning in analyzing Casson fluid flow with heat transport influenced by Hall currents and Darcy–Forchheimer effects. The objective is to enhance predictions of heat transfer rates in complex non-Newtonian fluid flows, where conventional models often fall short. This work is focused on the proper modeling of dissipative Casson fluids, with applications in engineering, particularly in biomedical and industrial settings where certain effects, such as Hall currents and Darcy–Forchheimer, are present. The work simplifies the complicated governing partial differential equations through the use of similarity transformations into ordinary differential equations. These equations are subsequently transformed into an optimization problem, which is solved using an innovative approach that combines transformer model learning with mean square error and the Adam optimization algorithm. Graphical results of this new modeling approach that depicted unusual characteristics for Casson fluids under examined conditions were obtained. This research highlights the integration of advanced numerical methods and transformer model learning to refine fluid behavior modeling, significantly improving the accuracy of heat transfer predictions. These advancements have substantial practical implications, offering potential enhancements in the optimization of processes within industrial and biomedical fields.

1.
K.
Bhattacharyya
and
G. C.
Layek
, “
Effects of suction/blowing on steady boundary layer stagnation-point flow and heat transfer towards a shrinking sheet with thermal radiation
,”
Int. J. Heat Mass Transfer
54
(
1–3
),
302
307
(
2011
).
2.
G. G.
Pereira
, “
Effect of variable slip boundary conditions on flows of pressure driven non-Newtonian fluids
,”
J. Non-Newtonian Fluid Mech.
157
(
3
),
197
206
(
2009
).
3.
F.
Khani
,
A.
Farmany
,
M. A.
Raji
,
A.
Aziz
, and
F.
Samadi
, “
Analytic solution for heat transfer of a third-grade viscoelastic fluid in non-Darcy porous mediawith thermophysical effects
,”
Commun. Nonlinear Sci. Numer. Simul.
14
(
11
),
3867
3878
(
2009
).
4.
L. J.
Sademaki
,
M. D.
Shamshuddin
,
B. P.
Reddy
, and
S. O.
Salawu
, “
Unsteady Casson hydromagnetic convective porous media flow with reacting species and heat source: Thermo-diffusion and diffusion-thermo of tiny particles
,”
Partial Differ. Equations Appl. Math
11
,
100867
(
2024
).
5.
M.
Mujahid
,
Z.
Abbas
, and
M. Y.
Rafiq
, “
A study on the pressure-driven flow of magnetized non-Newtonian Casson fluid between two corrugated curved walls of an arbitrary phase difference
,”
Heat Transfer
53
(
8
),
4510
4527
(
2024
).
6.
M. F.
Khan
,
M.
Sulaiman
,
A. N.
Ali
,
G.
Laouini
,
F. S.
Alshammari
, and
M.
Khalid
, “
A computational study of magneto-convective heat transfer over inclined surfaces with thermodiffusion
,”
IEEE Access
11
,
57046
57070
(
2023
).
7.
K.
Nonlaopon
,
M. F.
Khan
,
M.
Sulaiman
,
F. S.
Alshammari
, and
G.
Laouini
, “
Analysis of MHD Falkner–Skan boundary layer flow and heat transfer due to symmetric dynamic wedge: A numerical study via the SCA-SQP-ANN technique
,”
Symmetry
14
(
10
),
2180
(
2022
).
8.
E. A.
El-Sayed
,
F. A.
Alwawi
,
F.
Aljuaydi
, and
M. Z.
Swalmeh
, “
Computational insights into shape effects and heat transport enhancement in MHD-free convection of polar ternary hybrid nanofluid around a radiant sphere
,”
Sci. Rep.
14
(
1
),
1225
(
2024
).
9.
A.
Moghtadaei
, “
Numerical simulation of heat transfer in a solar flat plate collector using nano-fluids
,”
Adv. Bioeng. Biomed. Sci. Res.
7
(
2
),
01
06
(
2024
).
10.
S.
Al Omari
,
A. M.
Ghazal
,
M.
Syam
,
H.
El Sayed
,
R.
Al Najjar
, and
M. Y.
Selim
, “
An investigation on the thermal degradation performance of crude glycerol and date seeds blends using thermogravimetric analysis (TGA)
,” in
5th International Conference on Renewable Energy: Generation and Applications (ICREGA)
(
IEEE
,
2018
), pp.
102
106
.
11.
X. Q.
Wang
and
A. S.
Mujumdar
, “
A review on nanofluids—Part I: Theoretical and numerical investigations
,”
Braz J. Chem. Eng.
25
,
613
630
(
2008
).
12.
A. V.
Kuznetsov
and
D. A.
Nield
, “
Natural convective boundary-layer flow of a nanofluid past a vertical plate
,”
Int. J. Therm. Sci.
49
(
2
),
243
247
(
2010
).
13.
J.
Buongiorno
, “
Convective transport in nanofluids
,”
J. Heat Transfer
128
(
3
),
240
250
(
2006
);
N.
Casson
, “
Flow equation for pigment-oil suspensions of the printing ink-type
,” in
Rheology of Disperse Systems
(
Pergamon Press
,
Oxford
,
1959
), pp.
84
104
.
14.
T.
Hayat
,
G.
Bashir
,
M.
Waqas
, and
A.
Alsaedi
, “
MHD flow of Jeffrey liquid due to a nonlinear radially stretched sheet in presence of Newtonian heating
,”
Results Phys.
6
,
817
823
(
2016
).
15.
M.
Sheikholeslami
,
Application of Control Volume Based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer
(
Elsevier
,
2018
).
16.
M.
Vinodkumar Reddy
,
K.
Vajravelu
,
M.
Ajithkumar
,
G.
Sucharitha
, and
P.
Lakshminarayana
, “
Analysis of entropy generation and activation energy on a convective MHD Carreau–Yasuda nanofluid flow over a sheet
,”
Mod. Phys. Lett. B
38
(
28
),
2450266
(
2024
).
17.
M.
Ajithkumar
et al, “
A numerical simulation of the magneto-micropolar nanofluid flow configured by the stimulus energies and chemical interaction
,”
J. Comput. Theor. Transp.
53
(
7
),
469
489
(
2024
).
18.
S. K.
Das
,
S. U.
Choi
, and
H. E.
Patel
, “
Heat transfer in nanofluids—A review
,”
Heat Transfer Eng.
27
(
10
),
3
19
(
2006
).
19.
K.
Govindarajulu
,
A.
SubramanyamReddy
,
D.
Rajkumar
,
T.
Thamizharasan
,
M.
Dinesh Kumar
, and
K. R.
Sekhar
, “
Numerical investigation on MHD non-Newtonian pulsating Fe3O4-blood nanofluid flow through vertical channel with nonlinear thermal radiation, entropy generation, and Joule heating
,”
Numer. Heat Transfer, Part A
45
,
1
20
(
2024
).
20.
M.
Ajithkumar
,
P.
Lakshminarayana
, and
K.
Vajravelu
, “
Diffusion effects on mixed convective peristaltic flow of a bi-viscous Bingham nanofluid through a porous medium with convective boundary conditions
,”
Phys. Fluids
35
(
3
),
032008
(
2023
).
21.
M. V.
Reddy
et al, “
Numerical treatment of entropy generation in convective MHD Williamson nanofluid flow with Cattaneo–Christov heat flux and suction/injection
,”
Int. J. Model. Simul.
45
,
1
18
(
2024
).
22.
M.
Ajithkumar
,
K.
Vajravelu
,
G.
Sucharitha
, and
P.
Lakshminarayana
, “
Peristaltic flow of a bioconvective sutterby nanofluid in a flexible microchannel with compliant walls: Application to hemodynamic instability
,”
Phys. Fluids
35
(
12
),
122005
(
2023
).
23.
H.
Tahir
,
U.
Khan
,
A.
Din
,
Y. M.
Chu
, and
N.
Muhammad
, “
Heat transfer in a ferromagnetic chemically reactive species
,”
J. Thermophys. Heat Transfer
35
(
2
),
402
410
(
2021
).
24.
B. A.
Bhanvase
,
D. P.
Barai
,
S. H.
Sonawane
,
N.
Kumar
, and
S. S.
Sonawane
, “
Intensified heat transfer rate with the use of nanofluids
,” in
Handbook of Nanomaterials for Industrial Applications
(
Elsevier
,
2018
), pp.
739
750
.
25.
M.
Gupta
,
V.
Singh
,
R.
Kumar
, and
Z.
Said
, “
A review on thermophysical properties of nanofluids and heat transfer applications
,”
Renew Sustain Energy Rev
74
,
638
670
(
2017
).
26.
N. S.
Khashi'ie
,
N. M.
Arifin
,
I.
Pop
, and
N. S.
Wahid
, “
Flow and heat transfer of hybrid nanofluid over a permeable shrinking cylinder with Joule heating: A comparative analysis
,”
Alexandria Eng. J.
59
(
3
),
1787
1798
(
2020
).
27.
B.
Mehta
,
D.
Subhedar
,
H.
Panchal
, and
Z.
Said
, “
Synthesis, stability, thermophysical properties and heat transfer applications of nanofluid—A review
,”
J. Mol. Liq.
364
,
120034
(
2022
).
28.
A.
Manan
,
S. U.
Rehman
,
N.
Fatima
,
M.
Imran
,
B.
Ali
,
N. A.
Shah
, and
J. D.
Chung
, “
Dynamics of Eyring-Powell nanofluids when bioconvection and Lorentz forces are significant: The case of a slender elastic sheet of variable thickness with porous medium
,”
Mathematics
10
(
17
),
3039
(
2022
).
29.
M.
Ishaq
,
S. U.
Rehman
,
M. B.
Riaz
, and
M.
Zahid
, “
Hydrodynamical study of couple stress fluid flow in a linearly permeable rectangular channel subject to Darcy porous medium and no-slip boundary conditions
,”
Alexandria Eng. J.
91
,
50
69
(
2024
).
30.
N.
Fatima
,
S. U.
Rehman
, and
B.
Ali
, “
Significance of stratification and Lorentz force on the transport phenomena of gyrotactic microorganisms in tangent hyperbolic nanofluid with Darcy-Forchheimer law
,”
J. Therm. Anal. Calorim.
149
(
4
),
1477
1493
(
2024
).
31.
S.
Ur Rehman
,
N.
Fatima
,
B.
Ali
, and
A.
Shafiq
, “
Significance of mono and hybrid nanoparticles on the dynamics of Prandtl fluid subject to Darcy Forchiemmer law, Lorentz and Coriolis forces: The case of 3D stretched surface
,”
Waves Random Complex Medium
43
,
1
17
(
2022
).
32.
B.
Ali
,
S. U.
Rehman
,
M.
Fiaz
,
M. B.
Riaz
, and
M.
Zahid
, “
Significance of quadratic density variation on the heat transport phenomena in Careau dusty fluid subject to Lorentz force via stretching surface
,”
Int. J. Thermofluids
22
,
100703
(
2024
).
33.
M.
Ali
,
R.
Nasrin
, and
M. A.
Alim
, “
Axisymmetric boundary layer slip flow with heat transfer over an exponentially stretching bullet-shaped object: A numerical assessment
,”
Heliyon
9
(
3
),
e13671
(
2023
).
34.
A.
Vaswani
et al., “
Attention is all you need
,” in
Advances in Neural Information Processing Systems
(NIPS 2017),
2017
, Vol.
30
.
You do not currently have access to this content.