Resul s in Enginee ing 22 (2024) 102157
A ailable online 21 Ap il 2024
2590-1230/© 2024 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/).
E alua ing ene gy ansmission cha ac e is ics o Non-New onian luid low
in s a i ied and non-s a i ied egimes: A compa a i e s udy
S. Bilal
d
, Asadullah
a
,
*
, Muhammad Bilal Riaz
b
,
c
a
Depa men o Ma hema ics, Ai Uni e si y, Sec o E-9, P.A.F Complex, P.O. 44000, Islamabad, Pakis an
b
IT4Inno a ions, VSB – Technical Uni e si y o Os a a, Os a a, Czech Republic
c
Depa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Byblos, Lebanon
d
Dep . o Mechanical Enginee ing, College o Enginee ing, P ince Mohammad Bin Fahd Uni e si y, Al Khoba 31952, Kingdom o Saudi A abia
ARTICLE INFO
Keywo ds:
MHD
The mosolu al s a i ica ion
Williamson luid
Le enbe g-ma qua d and bayesian
egula iza ion schemes
Con ec i e inclined su ace
Chemical eac ion
ABSTRACT
Conside ing he na u al and indus ial impo ance o low cha ac e iza ion in s a i ied media, he cu en s udy
is a icula ed. This wo k highligh s he in luence o linea s a i ica ion as well as con ec i e su aces in bo h
he mal and solu al ields on he heological a ibu es o Williamson luid low h ough an inclined su ace.
No el physical aspec s o a uni o mly p o ided magne ic ield o s eng h B and chemically eac i e species a e
also included. The conce ned anspo equa ions a e de i ed om he associa ed conse a ion laws in dimen-
sional o ms. Modi ica ion in he de eloped couple sys em is achie ed by using a se o simila a iables.
Le enbe g-Ma qua d Scheme (LMS) and Bayesian Regula iza ion Scheme (BRS) a e u ilized in compa a i e
manne o analyze ini ial da a accessed o quan i ies o in e es . The da a used in he gene a ion o MLP was 80
pe cen o model aining and 20 pe cen o es ing and alida ion. E o his og ams, pe o mance plo s, i ness
cu es, and eg ession plo s o aining, es ing, and alida ion a e p esen ed. Da a in he o m o ables and
g aphs a e p esen ed, which exp ess an excellen ma ch be ween he ANN-p edic ed and a ge ed alues. I is
e ealed ha an a i icial neu al ne wo k app oach can p o ide highly e icien o ecas ing o such p oblems by
p o iding accu a e da a o quan i ies o in e es . I is no iced ha Nussel numbe and She wood numbe en-
hances up o 33 % and 29 % e sus espec i e s a i ica ion pa ame e s. Veloci y p o ile declines agains mag-
ne ic ield pa ame e (M) whe eas, skin ic ion coe icien inc emen s up o 25 %. Appliance o con ec i e
bounda y cons ain s a he su ace o inclined shee ends o enhance he empe a u e and concen a ion ields.
1. In oduc ion
O e he yea s, Na ie S okes equa ion has been u ilized o p edic
he beha io o luids agains appliance o shea s esses. Expe s ha e
expe imen ally e i ied ha owing o he dis inc i e esponse o liquids
o ex e nal o ces, hey a e ca ego ized in o iscoelas ic and iscoine-
las ic. Th ough heological cha ac e iza ion o he men ioned sub-
classes, ex ensi e dissimila i ies ha e been ound in hei physiological
a ibu es. Viscoinelas ic luids ha e been conside ed mo e ealis ic and
applicable o eal-wo ld applica ions owing o he p edic ion o low
beha io a ze o shea a es. Some mesme izing applica ions o isco-
elas ic luids include ood manu ac u ing, oil and gas e inemen p o-
cesses, powe gene a ion sys ems, p o ec i e sys ems, medicinal
de ices, elec onic de ices, and c ys al g ow h. On he basis o such
exclusi e signi icance, a ious low models ha e been p oposed o
u ilize i in di e en si ua ions; ew models among hem a e powe law,
Casson, and Williamson luids. Among hese amewo ks, he i es is
he Williamson luid because i alls in o he luid ca ego y ha possesses
a pseudo-plas ic na u e. This ype o luid is u ilized o many indus ial
and enginee ing pu poses, such as blood cells, pho og aphic ilms, and
ood p ocessing. A ew ecen s udies depic ing he low beha io o
iscoinelas ic ma e ials unde di e en physical cons ain s ha e been
conduc ed. Fo ins ance, he low beha io o iscoinelas ic luids o e
ex e nal su aces employing he simila i y app oach was in es iga ed by
Da ji and Timol [1]. In an in es iga ion, pe is al ic low o Williamson
luid in he small in es ine wi h he inse ion o an endoscope was
mani es ed by Nadeem e al. [2]. The low was con ined be ween wo
concen ic ubes. They obse ed he pa e n o o med s eamlines and
a ained ha apping occu s, and he size o he apped bolus a ies by
a ying physical pa ame e s. Hea ans e analysis o a s eady 2D
Williamson luid low h ough a pe meable exponen ially enla geable
* Co esponding au ho
E-mail add esses: [email p o ec ed] (Asadullah), [email p o ec ed] (M.B. Riaz).
Con en s lis s a ailable a ScienceDi ec
Resul s in Enginee ing
jou nal homepage: www.sciencedi ec .com/jou nal/ esul s-in-enginee ing
h ps://doi.o g/10.1016/j. ineng.2024.102157
Recei ed 11 No embe 2023; Recei ed in e ised o m 6 Ap il 2024; Accep ed 18 Ap il 2024
Resul s in Enginee ing 22 (2024) 102157
2
shee was demons a ed by Nadeem and Hussain [3]. In his s udy, hey
conside ed wo egimes o hea ans e , ha is, he p esc ibed expo-
nen ial o de hea lux (PEHF) case and he p esc ibed exponen ial o de
su ace empe a u e (PEST) case. They concluded ha , o bo h cases,
he he mal bounda y laye dep ecia es by lou ishing he P and l
numbe . A nume ical in es iga ion o Williamson luid low o e an
ex endable cylinde wi h a iable he mal conduc i i y and hea gen-
e a ion/abso p ion was mani es ed by Malik e al. [4]. They con i med
ha he empe a u e p o ile was enhanced by ampli ying he magni ude
o he he mal conduc i i y pa ame e , while he opposi e end was
a ained agains he P and l numbe . Kuma an e al. [5] illus a ed he
low o Williamson luid inco po a ed wi h he impac s o magne ohy-
d odynamic and iscous dissipa ion h ough an uppe pa aboloid o
e olu ion. They obse ed ha he eloci y dis ibu ion declined wi h
inc easing Ha mann numbe . Shah e al. [6] discussed Williamson luid
low o e ime dependen ex endable su ace inco po a ed in pe meable
media wi h hea ansmission and he mal adia ion. The hea and mass
ans e analysis o magne ized Williamson luid h ough a cu ed su -
ace o uns eady and s eady low egimes was add essed by Kuma e al.
[7]. In addi ion, he low was also cha ac e ized by con ec i e bounda y
cons ain s, chemical eac ion and he mal adia ion. Raju e al. [8]
explo ed Williamson and Casson luid low o e s e chable su ace
along wi h ans e o mass and hea . In addi ion, he low is also
cha ac e ized by homogeneous–he e ogeneous eac ions. Magne ized
Williamson luid low o e an ex endable su ace by aking he impac s
o he mal adia ion, chemical eac ion and iscous dissipa ion was
demons a ed by Kuma e al. [9]. The low cha ac e is ics o Wil-
liamson luid o e a nonlinea ex endable su ace embedded in a po ous
medium we e disclosed by Abbas e al. [10]. Some ecen de elopmen s
conce ning he desc ibed non-New onian model in a ious physical
cons ain s a e encapsula ed in Re s. [11–14].
He e ogenei y among luid laye s is gene a ed owing o densi y and
iscosi y di e ences caused by empe a u e and concen a ion a ia-
ions. The in e play be ween luid he e ogenei y and g a i y esul s in a
s iking phenomenon known as s a i ica ion. Comp ehension o he
mo emen o objec s in a s a i ied en i onmen in ol es pe asi e
en i onmen al, geophysical, ecological, and indus ial p ocedu es. Fo
ins ance, con e sion o ai pollu an s (dus , ae osols, wea he balloons,
ma e ) esides in he lowe a mosphe ic egion whe e hey encoun e
low empe a u e g adien s, and by p o iding s a i ica ion, hey will
sca e in he uppe zone, which o ms clouds and imp o es he quali y
o ai . In addi ion, he emo al o accumula ed oxican s in oceanic
laye s is e icien ly e adica ed by p oducing s a i ica ion. O he han
ecological u iliza ions, s a i ied en i onmen s a e also immensely
applicable in ene gy con e sion and s o ing sys ems. In iew o such
splendid implica ions, he phenomenon o s a i ica ion h ough bo h
he mal and solu al exchanges has accoun ed o he pe asi e in en o
esea che s’ a e ni y. Rishabh and A dekani [15] delinea ed sedi-
men a ion o an isola ed objec in luenced by he mal s a i ica ion in
shallow wa e and discussed hei hyd odynamic mechanism. The
applica ion o a s a i ied low egime in gas pipelines, by in es iga ing
he hyd o he mal a ibu es o iscoelas ic liquid low o e an ex ending
su ace subjec ed o he mally a ian s a i ica ion, was di ulged by
Lone e al. [16]. The ole o a iable magne ic lux in gene a ing s a -
i ied low o a non-New onian luid o e an exponen ial su ace by
employing a nume ical scheme was explica ed by Singh e al. [17].
Coue e low o s ably s a i ied dus y Wal e liquid h ough a pe meable
medium and unde physical insigh o a uni o mly applied ans e se
magne ic ield h ough an analy ical app oach was con empla ed by Dey
[18]. Hyd o he mal a ibu es o s a i ied Ey ing-Powel luid wi h
physical aspec s o a iable he mo physical cha ac e is ics (conduc-
i i y and iscosi y) o e a non-linea ly ex endable su ace we e de e -
mined by Wahab e al. [19]. Densi y a ia ions p oduced in he s a i ied
low o mic opola luid due o he non-pe sis en impac o buoyancy
o ces o e a pla e we e sc u inized by Waqas e al. [20]. The s a i ied
low o Powell-Ey ing luid o e an ex endable pe meable su ace wi h
dissipa ion and adia i e hea lux was in es iga ed by Abbas and
Megahed [21]. Some ele an s udies disclosing he e o s made in he
di ec ion o s a i ied lows a e p esen ed in Re s. [22–26].
In he cu en cen u y, ANN has played an impo an ole in sol ing
complex p oblems in di e en domains, such as science and enginee -
ing. The ANN model comp ises in e connec ed laye s o neu ons ha
wo k based on he wo king mechanism o he human b ain. I is due o
he ac ha neu al ne wo k ob ained in o ma ion om inpu da a; hey
ecognize inpu da a and o ganize he da a o p edic he ou comes.
Cu en ly, ANN a e used ex ensi ely in he ield o non-New onian
luids. ANN based ime dependen s udy o Williamson luid low
h ough a pe meable ex endable su ace wi h Le enbe g-Ma qua d
back p opaga ion scheme was ca ied ou by Sha iq e al. [27]. Shayya
e al. [28] de eloped ANN model o he es ima ion o ic ion ac o o
He schel-Bulkley liquids unde u bulen and lamina low si ua ion
h ough closed pipes. Machine lea ning analysis o biocon ec i e low o
Williamson luid o e an enla geable shee wi h B ownian mo ion and
he mophe o ic impac s was mani es ed by P iyadha shini e al. [29].
Shoaib e al. [30] analyzed he low ea u es o magne ized hi d-g ade
luid h ough an enla geable shee wi h ac i a ion ene gy and con ec-
i e bounda y cons ain s by u ilizing a machine lea ning algo i hm
based on he Le enbe g–Ma qua d scheme. Hussain e al. [31] de el-
oped an ANN model based on he Le enbe g–Ma qua d algo i hm o
sc u inize he ea u es o Casson luid low o e a nonlinea slan ed
su ace. Fo he sc u iniza ion o low p ope ies o mixed con ec i e
non-New onian luid low o e s e ching media, an in elligence algo-
i hm based on he Bayesian egula iza ion app oach was o mula ed by
Shah e al. [32].
The phenomenon o s a i ica ion has nume ous applica ions in
a ious ields such as geology and ea h science, ecology and en i on-
men al science, ma ke segmen a ion, a mosphe ic science, and many
mo e. In iew o such splendid implica ions, he phenomenon o s a -
i ica ion h ough bo h he mal and solu al exchanges has been accoun-
ed. In addi ion, as ange o applica ions conce ned wi h s a i ied
lows a e ound in mul iple scien i ic and enginee ing p ocesses, such as
Nomencla u e
U1,V1 Veloci y componen s
υ
Kinema ic iscosi y
Γ Time cons an
α
The mal di usi i y
cp Speci ic hea cons an βT1 The mal expansion
coe icien
β1 Inclina ion angle K1 Chemical eac ion coe icien
B Uni o m magne ic ield βC1 Solu al expansion
coe icien
A1 Williamson luid pa ame e D Di usion species
coe icien
T1∞ Ambien empe a u e k The mal conduc i i y
ρ
Densi y o luid C1 Fluid Concen a ion
g G a i a ional accele a ion T1 Fluid Tempe a u e
h Hea ans e coe icien C1∞ Ambien concen a ion
M Magne ic ield pa ame e hc Mass ans e coe icien
λ The mal buoyancy pa ame e λ1 Solu al buoyancy
pa ame e
S1 The mal s a i ica ion pa ame e S2 Solu al
s a i ica ion pa ame e
Sc Schmid numbe k Chemical eac ion pa ame e
P P and l numbe γ1 The mal Bio numbe
C x Skin ic ion coe icien γ2 Solu al Bio numbe
τ
w Wall sha e s ess Nux Nussel numbe
qw Hea lux Shx She wood numbe
qm Mass lux
S. Bilal e al.
Resul s in Enginee ing 22 (2024) 102157
3
hea ans e in buildings, ae ospace enginee ing, en i onmen al engi-
nee ing, elec onic cooling, and oceanog aphy. Subsequen ly, consid-
e a ion o con ec i e bounda y cons ain s makes he p esen
communica ion mo e impac ul. F om he abo e-men ioned li e a u e
su ey, i is wo h men ioning ha no con ibu ion has ye been made o
explo e he ea u es o s a i ied Williamson luid low wi h magne o-
hyd odynamic (MHD) and chemical eac ion e ec s. The p incipal aim
o he p esen s udy is o sc u inize he aspec s o s a i ied Williamson
luid low o e an enla geable su ace wi h con ec i e bounda y con-
di ions in he exis ence o magne ohyd odynamic and chemical eac ion
e ec s. Mo eo e , a machine lea ning algo i hm based on he Le enbe g
Ma qua d and Bayesian Regula iza ion Schemes is also employed in a
compa a i e sense o check he i ness o he a ained da a.
The inno a i e con ibu ions o he s udy a e as below.
1. The go e ning equa ions a e cons i u ed o s a i ied Williamson
luid low o e a con ec i e inclined enla geable shee .
2. Da a se s a e ob ained nume ically by u ilizing shoo ing me hod in
combina ion wi h he RK-4 scheme, and a compa a i e analysis o
he da a se is pe o med o ain h ough LMS and BRS.
3. Valida ion o he a ained ou comes is shown by e o his og am,
mean squa e e o , eg ession, and i ness plo s.
4. The impac s o signi ican low pa ame e s on he empe a u e,
concen a ion and eloci y dis ibu ions, d ag coe icien , and hea
and mass luxes a e obse ed nume ically and g aphically.
2. P oblem o mula ion
In his sec ion, a b ie desc ip ion o he implemen ed me hodologies
o he p oposed go e ning p oblem, ha is, s a i ied Williamson luid
low o e a con ec i e su ace wi h magne ic ield and chemical eac-
ion aspec s, is p esen ed. Quan i ies o enginee ing in e es such as he
skin ic ion coe icien (SFC), hea lux coe icien (HFC), and mass lux
coe icien (MFC) a e calcula ed. Fo his pu pose, he ini ial da ase
men ioned quan i ies a e p oduced h ough he shoo ing me hod (SM)
and hen di ided in o aining, es ing, and alida ion. Subsequen ly,
a i icial back-p opaga ed neu al ne wo king (ABPNN) based on
enowned schemes (LMS) and (BRS) is employed. A g aphical isuali-
za ion o he s udy is depic ed in Fig. 1.
By u ilizing a o emen ioned supposi ions, he go e ning con inui y,
momen um, ene gy and concen a ion equa ions ake ollowing o m.
Con inui y equa ion [33].
∂
U1
∂
X1+
∂
V1
∂
Y1=0,(1)
Momen um equa ion o non-New onian Williamson luid model is
ep esen ed as unde [33].
U1
∂
U1
∂
X1+V1
∂
U1
∂
Y1=ϑ
∂
2U1
∂
Y1
2+
2
√ϑΓ
∂
U1
∂
Y1
∂
2U1
∂
Y1
2−
σ
B2
ρ
U1
+(βT1(T1−T1∞)+βC1(C1−C1∞))gsinβ1,(2)
In he momen um equa ion, he las wo e ms we e o mula ed by
employing he Boussinesq app oxima ion, which exp esses he ea u es
o con ec ion a ising om empe a u e and concen a ion di e ences
caused by linea s a i ica ion [34].
Ene gy equa ion [34].
U1
∂
T1
∂
X1+V1
∂
T1
∂
Y1=k
ρ
cp
∂
2T1
∂
Y1
2,(3)
Concen a ion equa ion [34].
U1
∂
C1
∂
X1+V1
∂
C1
∂
Y1=D
∂
2C1
∂
Y1
2−K1(C1−C1∞).(4)
The ele an bounda y condi ions a e exp essed as unde [34].
U1=U1w(X1)=cX1,V1=0,−k
∂
T1
∂
Y1=h (T1 −T1),−D
∂
C1
∂
Y1
=hc(C1 −C1),a Y1=0,U1→0,T1→T1∞(x),C1→C1∞(x)as Y1→∞,
(5)
Whe e,
T1 (X1)=T10+d1X1,C1 (X1)=C10+e1X1,T1∞(X1)=T10+d2X1,C1∞(X1)=C10
+e2X1.
(6)
In abo e equa ions (2)–(6), U1 and V1 ep esen s eloci y componen
along X1- and Y1-axis, espec i ely. U1w s e ching eloci y, T1 hea ed
luid empe a u e, T1∞ a iable ambien luid empe a u e, C1 hea ed
luid concen a ion, C1∞ a iable ambien concen a ion, h and hc
ep esen s hea and mass ans e coe icien s, D is coe icien o di u-
sion species, k shows he mal conduc i i y, d1,d2,e1 and e2 highligh s
dimensional cons an s.
Dimensionless o m o go e ning equa ions is a ained by employing
ollowing ans o ma ions [34].
Fig. 1. Physical domain o he p oblem.
S. Bilal e al.
Resul s in Enginee ing 22 (2024) 102157
4
U1=cX1F
′
(ξ),V1= − 
cϑ
√F(ξ),ξ=
c
ϑ
√Y1,θ(ξ) = T1−T1∞
T1 −T10
,φ(ξ)
=C1−C1∞
C1 −C10
.(7)
A e employing simila a iables con inui y equa ion is iden ically
sa is ied and he emaining equa ions subjec o bounda y cons ain s
a e as ollows
F F
″
+F
‴
+A1F
″
F
‴
−F
′
2−MF
′
+λθ sin β1+λ1φ sin β1=0,(8)
θ
″
+P Fθ
′
−P S1F
′
−P F
′
θ=0,(9)
φ
″
+ScFφ
′
−ScF
′
φ−ScS2F
′
−Sck φ =0.(10)
Dimensionless o m o bounda y cons ain s is as unde
F
′
(0)=1,F(0)=0,θ
′
(0)= − γ1(1−S1−θ(0)),φ
′
(0)= − γ2(1−S2
−φ(0)),F
′
(∞)=0,θ(∞)=0,φ(∞)=0,
(11)
he e, A1 Williamson luid pa ame e , λ and λ1 a e he mal and solu al
buoyancy pa ame e s, espec i ely, M is magne ic ield pa ame e . P
shows P and l numbe , S2 and S1 depic s solu al and he mal s a i ied
pa ame e s, Sc highligh s Schmid numbe , k highligh s chemical e-
ac ion pa ame e , γ2 and γ1 depic s solu al and he mal Bio numbe s.
These pa ame e s a e exp essed as unde
A1=ΓX1
2c3
ϑ
√,M=
σ
B2
c
ρ
,λ=gβT1(T1 −T10)
X1c2,λ1=gβC1(C1 −C10)
X1c2,P =ϑ
α
S1=d2
d1
,S2=e2
e1
,Sc =ϑ
D,k =K1
c,γ1=h
k
c
ϑ
√,γ2=hc
D
c
ϑ
√.
(12)
The pa ame e s o physical in e es s i.e. d ag coe icien , Nussel
numbe and She wood numbe a e illus a ed by he ollowing
exp essions
C x =
τ
w
1
2
ρ
U1
2
w
,Nux=X1qw
k(T1 −T1∞),Shx=X1qm
D(C1 −C1∞),(13)
whe e, qm wall mass lux,
τ
w wall shea s ess, and qw wall hea lux a e
de ined as ollows
τ
w=
μ
0[
∂
U1
∂
Y1+Γ

2
√(
∂
U1
∂
Y1)2],qw= − k(
∂
T1
∂
Y1),qm= − D(
∂
C1
∂
Y1).(14)
A e he implemen a ion o dimensionless a iables in eq. (7), d ag
coe icien , hea lux and mass lux in dimensionless o m a e as ollows
1
2C xRe1
2
x=[F
″
(0)+A1
2(F
″
(0))2],Nux=−θ
′
(0)
Re−1
2
x(1−S1)
,Shx=−φ
′
(0)
Re−1
2
x(1−S2)
.
(15)
3. Nume ical scheme
To sol e analy ically he ansla ed nonlinea ODE’s is no easy.
App oxima e solu ion o hese nonlinea ODEs along wi h bounda y
condi ions is ob ained by implemen ing shoo ing app oach in combi-
na ion wi h Runge-Ku a me hod o o de ou , which is widely used
nume ical echnique o o igina e app oxima e solu ion o ODEs wi h
adap i e con olled s ep size. To use his echnique, highe -o de ODEs
a e ans o med in o i s o de by in oducing new a iable F= 1,
F
′
= 2,F
″
= 3,F
‴
=
′
3,θ= 4,θ
′
= 5,θ
″
=
′
5,φ= 6,φ
′
= 7,φ
″
=
′
7.
Fi s -o de ODEs a e
′
1= 2,
′
2= 3,
′
3= 2
2+M 2− 1 3−λ 4sin β1−λ1 6sin β1
1+A1 3
,
′
4= 5
′
5=P 2 4+P 2S1−P 1 5,
′
6= 7,
′
7=Sck 6+ScS2 2+Sc 2 6−Sc 1 7,
whe e ele an bounda y condi ions a e
ξ=0: 1(ξ)=0, 2(ξ)=1, 5(ξ)= − γ1(1−S1− 4(ξ)), 7(ξ)= − γ2(1−S2
− 6(ξ)),
ξ→ ∞ : 2(ξ)→ 0,y4(ξ)→ 0,y6(ξ)→0.
To a ain he nume ical solu ion o he ini ial alue p oblem i s ly
ini ial guesses a e selec ed, hen solu ion p ocess is ca ied ou o a ain
he ou comes. The key ac o o he shoo ing me hod is o selec
app op ia e alue o ξ∞.Fo his, ini ial guesses o he o pa icula
physical pa ame e s a e selec ed o ob ain
″
(0),θ
′
(0)and φ
′
(0).Modi-
ied he ini ial guesses un il he di e ence be ween wo adjacen ou -
comes o
″
(0),θ
′
(0)and φ
′
(0)is close o equi ed digi . Las compu a ion
o ξ∞ is conside ed as adequa e o he gi en da a se . A e wa ds, so-
lu ion is p ocesses u he . A e a aining he esul s o
″
(0),θ
′
(0)and
φ
′
(0)RK-4 app oach is u ilized o a ain he esul s. This p ocess
epea ed un il he equi ed deg ee o accu acy is a ained. The s eps
in ol ed in he u ilized nume ical app oach is mani es ed in he
ollowing lowcha po ayed in Fig. 2(a). Implemen a ion o u ilized
nume ical scheme on cu en ly modelled p oblem is s ep wisely shown
in Fig. 2(b), whe eas wo king mechanism o shoo ing me hod is depic ed
in Fig. 2(c). In his mechanism, we used he hi and ial echnique o
con e he bounda y alue p oblem in o an ini ial alue by choosing he
ini ial guesses h ough he hi and ial app oach.
4. A i icial neu al ne wo king (ANN)
A i icial neu al ne wo king (ANN) is a machine lea ning algo i hm
whose wo king mechanism is highly based on he wo king s uc u e o
he human b ain. A e ecei ing inpu in o ma ion, ANN is an e ec i e
ool o nonlinea s a is ical da a modeling. The s uc u e o an ANN
consis s o h ee laye s ha a e co ela ed wi h each o he . All inpu
laye s a e hidden, and a e ecei ing in o ma ion, hese laye s al e
hem om laye o laye h ough a se ies o ans o ma ions. To un-
de s and mo e complex objec s, each laye is ea ed as bo h inpu and
ou pu o he ANN. O e all, hese inne laye s a e enowned as neu on
laye s. To p edic he hyd odynamic low a ibu es o Williamson Fluid
in a s a i ied en i onmen , an ANN model de eloped o e a con ec i e
su ace is designed. The ANN model comp ises back p opaga ion (BP),
eed o wa ding (FF), and mul ilaye pe cep ion (MLP) ne wo king. The
MLP ne wo k consis s o h ee laye s: inpu , hidden, and ou pu laye s
(as shown in Fig. 3). In he inpu laye da a o SFC, HFC and MFC
agains physical pa ame e s like Ha mann numbe M,Williamson luid
pa ame e A1,P and l numbe P , he mal Bio numbe γ1,Schmid
numbe Sc and chemical eac ion pa ame e k is en e ed in o sys em o
aining pu pose. The hidden laye comp ises he numbe o neu ons o
be speci ied o aining he da ase ha p edic s da a wi h high accu-
acy. Finally, he ou pu laye in which da a a e o be es ed is a ained
S. Bilal e al.
Resul s in Enginee ing 22 (2024) 102157
5
o (SFC), (HFC) and (MFC) o a ious scena ios. Da a alida ion is
a ained o he cons uc ion o neu al ne wo king, and es ing da a is
applied o check unbiased inpu pe o mance. A wo king diag am o
mul ilaye pe cep ion (MLP) is illus a ed in Fig. 3. In he p esen s udy,
h ee quan i ies o in e es ha a e use ul in many enginee ing p oblems
a e compu ed in he ou pu and inpu laye s. A e accessing he inpu
da a ia nume ical app oaches, hey a e ca ego ized in o h ee pa s,
ha is ( aining, es ing, and alida ion) o p edic he solu ion h ough
(LMS) and (BRS). Da a is dis ibu ed in a pe cen age wise manne such
ha 80 % is assigned o ain he ne wo k and 20 % o alida ion and
es ing modules. The pe o mance o (LMS) and (BRS) on he de eloped
ANN model o Williamson luid low o e a con ec i e su ace by
es ima ing he mean squa e e o (MSE) o he aining, es ing, and
alida ion modules ela ed o he scena ios o (SFC), (HFC) and (MFC) is
Fig. 2. a. S eps in ol ed in nume ical p ocedu e. b Desc ip ion o implemen ed scheme on p esen p oblem. c Wo king mechanism o shoo ing p ocedu e.
Fig. 3. Mul ilaye pe cep ion o ANN model.
S. Bilal e al.

Resul s in Enginee ing 22 (2024) 102157
6
Fig. 4. Flow cha o de eloped ANN model [37].
Table 1
Nume ical esul s h ough LMS o he ANN model.
Pa ame e s Physical Quan i y MSE Epochs G ad Pe o mance Mu Time
T aining Tes ing Valida ion
M,A1 SFC 3.53E-8 4.66E-7 7.61E-7 218 9.02E-08 3.53E-08 1.00E-08 0:00:03
P ,γ1 HFC 4.68E-8 2.46E-7 8.39E-8 33 9.69E-08 4.68E-08 1.00E-08 0:00:00
Sc,k MFC 8.61E-9 1.60E-8 5.70E-9 19 9.39E-08 8.91E-09 1.00E-09 0:00:00
S. Bilal e al.
Resul s in Enginee ing 22 (2024) 102157
7
illus a ed in Table 1 o (LMS) and Table 2 o (BRS). Fu he mo e,
(MSE) is also e alua ed o (SFC), (HFC) and (MFC) e sus a ia ion in
(M),(A1),(P ),(γ1),(Sc)and (k ) anges om 0.1 o 2 compa a i ely
h ough (LMS) and (BRS). A educ ion in he magni ude o (MSE) is
e ealed in (BRS) as compa ed o (LMS) which signi ies ha (BRS)
p edic s mo e accu a e alues. The esul s o he low ield, he mal
g adien , concen a ion dis ibu ion, skin ic ion, and hea and mass
luxes we e also mani es ed in a nume ical and g aphical manne (see
Fig. 4).
Fig. 5(a–h) demons a es he e o his og am, aining, pe o mance
analysis, and eg ession plo s o (SFC) agains Ha mann numbe (M)
and Williamson luid pa ame e A1 o (LMS) and (BRS). The da a a e
ca ego ized in o h ee modules: aining (80 %), es ing (20 %), and
alida ion. Mul iple samples a e ob ained, he model is applied o each
sample o o ecas he ou pu , and he a ge o ac ual alues a e hen
compa ed wi h he ob ained ou comes. E o s a e calcula ed by ca e-
go izing in o 20 bins and ep esen ed h ough a his og am, as shown in
Fig. 5(a–b); all 20 bins a e displayed e sus he numbe o samples
con aining e o s, whe e e o bins a e depic ed on he X-axis and he Y-
axis ep esen s he ins ance a which e o occu s o samples ha
con ain e o in ha bin. In he ba diag am, i is no iced ha om he
aining da a, se en and i e samples ha e almos ze o e o , and om
he es da a, h ee and one samples con ain ze o e o o (LMS) and
(BRS) espec i ely. In addi ion, one sample o alida ion possesses ze o
e o , which shows ha mos o he da a ha e ze o e o s, which ce i-
ied well aining o da a h ough he Le enbe g Ma qua d Scheme
(LMS). Subsequen ly, i is also e ealed ha ange o ze o e o lies
be ween 10
−5
– 10
−3
and 10
−6
– 10
−4
which also assu es excellen
aining o da a h ough (LMS) and (BRS) espec i ely. Fig. 5(c–d) shows
a isualized g aph o he aining s a e o he ANN model h ough (LMS)
and (BRS) and p o ides c ucial in o ma ion ela ed o he ob ained
model by es ima ing he g adien , mu, and op imal epochs. (MSE) is
u ilized by aking he squa ed di e ence be ween he a ge and p e-
dic ed es ima ions o a ain con e gence o he model. The g adien
alue, which sugges s ha he simula ion model has eached he local
minimum o he objec i e unc ion a he lowes possible le el, is also
discussed. Finally, he con olling pa ame e (Mu) ha con ols he ANN
model aining mechanism is also elabo a ed, which di ec ly a ec s he
e o con e gence. Fig. 5(c) e eals ha he alue o Mu is 1.0E-08 wi h
a g adien o 9.02E-08, and Fig. 5(d) depic s ha he ou come o Mu is
0.5, wi h a g adien o 7.92E-08. I is e ealed ha as he epochs in-
c ease, he es ima ions o Mu and he g adien sh ink, and as a esul ,
he a e o con e gence will occu apidly. Fig. 5(e– ) shows alida ion
pe o mance plo s o es he model on he da ase a e each epoch o
adjus he ne wo k weigh s simul aneously h ough (LMS) and (BRS). A
he s a o he aining p ocess, MSEs a e a hei peak, wi h he passage
o ime e o s declining as he model mo es owa ds he op imal s a e.
The bes alida ion pe o mance is ep esen ed by do ed lines. F om
Fig. 5(e), i is no iced ha up o 218 epochs, he model’s alida ion
pe o mance is e y high, and i s bes alida ion pe o mance was
achie ed wi h an MSE o 7.61E-07 h ough (LMS) while om Fig. 5( ), i
is obse ed ha he bes alida ion pe o mance o he model is a ained
a 174 epochs wi h an MSE o 4.69E-08 h ough (BRS). I is deduced ha
om analysis ha (BRS) gi es mo e accu a e esul han (LMS) by
educing he e o be ween a ge and p edic ed alues. Fig. 5(g–h) di-
ulges he linea eg ession g aphs which a e a ained when he ANN
model is implemen ed on all da ase s o make p edic ion h ough (LMS)
and (BRS) and hen compa ed wi h ac ual alues. The i ness o model is
checked by de eloping linea ela ions o mula ed by Ou pu =
R* a ge +bias be ween ou pu s and a ge s. F om Fig. 5(g–h), R =1 is
obse ed in all eg ession plo s used o aining, alida ing, and es ing
da ase s, indica ing ha all o mula ed ANNs a e pe o ming lawlessly.
The aining da a we e ob ained by sol ing he go e ning physical
p oblems o (SFC), (HFC) and (MFC) agains he in ol ed physical
pa ame e s by implemen ing shoo ing and Runge-Ku a nume ical ap-
p oaches. F om he ske ch (see Fig. 5(g–h)), i is no iced ha he ou pu
is nea ly iden ical o he a ge , al hough he bias a ies in all ou cases.
O e all, i is in e ed ha ANNs p o ide excellen ou comes o eg es-
sion analysis.
Assessmen o e o s be ween p edic ed and a ge alues ia his o-
g am, pe o mance, eg ession and i plo s a e is po ayed in Fig. 6(a–j)
o HFC agains P and l numbe (P ) and he mal Bio numbe (γ1)in
compa a i e manne h ough (LMS) and (BRS). The o ange line ep e-
sen s ze o e o . Fig. 6(a–b) show he di e ence be ween he p edic ed
ou pu a ained om (LMS) and (BRS) and he ac ual alues ob ained o
(HFC) in he o m o all h ee da ase s ( aining, es ing, and alida ion)
on which sampling is execu ed. Fig. 6(a) calcula es e o s in each sample
collec ed o (HFC) h ough (LMS) ca ego ized in he o m o 20 bins
along wi h samples cons i u ing hose e o s. F om he his og am, i is
e ealed ha o he aining da a, i e samples ha e an e o o almos
ze o, whe eas o he alida ion da a, h ee samples con ain ze o e o
and one sample o es da a possesses ze o e o . This igu e shows ha
mos samples ha e ze o e o om all da ase s. In compa ison o he
p e ious diag am, Fig. 6(b) shows ha he ze o e o in he samples
collec ed o he hea lux coe icien h ough (BRS) inc eases o
aining da a up o six and es ing up o wo samples. In o ma ion abou
he aining s a e plo o e alua e he ained model om he ANN by
showing he g adien , mu, op imal epoch, and how hey each he
op imal s a e o bo h (LMS) and (BRS) is collec ed in Fig. 6(c)–(d). MSE
is u ilized o he con e gence analysis o he de eloped model by
inding a e age squa ed di e ences be ween a ge s and p edic ed es-
ima ions by ANN. The g adien alue is also discussed, which indica es
ha he local minimum o he objec i e unc ion hi s a he lowes poin .
Finally, he con olling pa ame e (Mu) o ANN model aining is dis-
cussed, which di ec ly in luences e o con e gence. Fig. 6(c) e eals
ha he alue o Mu is 1.0E-08 wi h g adien s 9.68E-08 h ough (LMS)
while; Fig. 6(d) shows ha he alue o Mu is 0.5 wi h g adien 9.63E-08
h ough (BRS). I is e ealed ha as he epochs inc ease, he es ima ions
o (Mu) and g adien dep eca e, and as a esul , he a e o con e gence
will occu apidly. The pe o mance plo o alida e model es ing a e
each epoch o adjus he ne wo k weigh s o da a ecei ed o (HFC)
h ough (LMS) and (BRS) is shown in Fig. 6(e– ). The linea eg ession
plo desc ibing he use o ained ANN da a o all aining, alida ion,
and es ing modules o p edic he hea lux coe icien (HFC) h ough
(LMS) and (BRS) along wi h a compa ison wi h ac ual alues ecei ed
om nume ical app oaches, is shown in Fig. 6(g–h). This g aph de-
e mines he goodness o he model and ollows a linea ela ionship
be ween he a ge and inpu . F om bo h igu es, i is obse ed ha R =1
in all h ee modules, which ensu es ha ANN has wo ked excellen ly. In
Fig. 6(g), o all ou da ase s, he model gi es R =1. This means ha he
ou pu is nea ly iden ical o he a ge , bu he bias a ies in each
ins ance. Fig. 6(h) also shows ha he aining da a ha e R =1, which
also e eals pe ec accu acy, and he model s ill pe o ms he bes .
Fig. 6(i–j) illus a es plo s o i ness o he de eloped ANN model o he
Table 2
Nume ical esul s h ough BRS o ANN model.
Pa ame e s Physical Quan i ies MSE Epochs E ec i e Pa ame e G ad Pe o mance Mu Sum o Squa e pa ame e Time
T aining Tes ing
M,A1 SFC 4.69E-8 1.15E-7 174 15.0 7.93E-08 4.70E-08 0.500 44.5 0:00:02
P ,γ1 HFC 2.45E-9 1.17E-8 162 14.8 9.63E-08 2.45E-09 0.500 74.0 0:00:02
Sc,k MFC 1.52E-9 7.52E-10 142 14.4 9.76E-08 1.53E-09 0.500 60.8 0:00:02
S. Bilal e al.
Resul s in Enginee ing 22 (2024) 102157
8
Fig. 5. G aphs o he o mula ed ANN model o skin ic ion h ough LMS and BRS. (a) LMS E o his og am. (b) BRS E o his og am. (c) LMS T aining s a e. (d)
BRS T aining s a e. (e) LMS Pe o mance. ( ) BRS Pe o mance. (g) LMS Reg ession. (h) BRS Reg ession.
S. Bilal e al.
Resul s in Enginee ing 22 (2024) 102157
9
Fig. 6. Plo s o he ANN model o he Nussel numbe h ough LMS and BRS. (a) LMS E o his og am. (b) BRS E o his og am. (c) LMS T aining s a e. (d) BRS
T aining s a e. (e) LMS Pe o mance. ( ) BRS Pe o mance. (g) LMS Reg ession. (h) BRS Reg ession. (i) LMS Fi ness cu e. (j) BRS Fi ness cu e.
S. Bilal e al.
Resul s in Enginee ing 22 (2024) 102157
16
sou ce/sink, in: IOP Con e ence Se ies: Ma e ials Science and Enginee ing, ol.
263, 2017.
[6] Z. Shah, E. Bonyah, S. Islam, W. Khan, M. Ishaq, Radia i e MHD hin ilm low o
Williamson luid o e an uns eady pe meable s e ching shee , Heliyon 4 (2018)
e00825.
[7] K.A. Kuma , J.R. Reddy, V. Sugunamma, N. Sandeep, Simul aneous solu ions o
mhd low o williamson luid o e a cu ed shee wi h nonuni o m hea sou ce/
sink, Hea T an. Res. 50 (2019).
[8] C.S. Raju, N. Sandeep, M.E. Ali, A.O. Nuhai , Hea and mass ans e in 3-D MHD
Williamson-Casson luids low o e a s e ching su ace wi h non-uni o m hea
sou ce/sink, The m. Sci. 23 (2019) 281–293.
[9] A. Kuma , R. T ipa hi, R. Singh, V.K. Chau asiya, Simul aneous e ec s o nonlinea
he mal adia ion and Joule hea ing on he low o Williamson nano luid wi h
en opy gene a ion, Phys. S a . Mech. Appl. 551 (2020) 123972.
[10] A. Abbas, M.B. Jeelani, A.S. Alnahdi, A. Ilyas, MHD Williamson nano luid luid
low and hea ans e pas a non-linea s e ching shee implan ed in a po ous
medium: e ec s o hea gene a ion and iscous dissipa ion, P ocesses 10 (2022)
1221.
[11] S.U. Zaman, M.N. Aslam, M.B. Riaz, A. Akgul, A. Hussan, Williamson MHD
nano luid low wi h adia ion e ec s h ough slende cylinde , Resul s in
Enginee ing (2024) 101966.
[12] T. Salahuddin, G. Fa ima, M. Awais, M. Khan, B.A. Awan, Adap a ion o nano luids
wi h magne ohyd odynamic Williamson luid o enhance he he mal and solu al
low analysis wi h iscous dissipa ion: a nume ical s udy, Resul s in Enginee ing 21
(2024) 101798.
[13] M. Saleem, M. Hussain, Imp ession o nonlinea adia ion and S e an blowing on
he magne o c oss nano-Williamson luid abo e exponen ially s e ching shee ,
Resul s in Enginee ing 17 (2023) 100864.
[14] Q. Khan, M. Fa ooq, S. Ahmad, Gene alized anspo analysis on mixed con ec ion
squeeze low o a Casson luid o e an inclined s e ching shee wi h iscous
dissipa ion and double s a i ica ion, Ain Shams Eng. J. 15 (2024) 102253.
[15] V.M. Rishabh, M.A. A dekani, Mo ion in s a i ied luids, Annu. Re . Fluid Mech.
55 (2023) 157–192.
[16] S.A. Lone, S. Anwa , A. Saeed, G. Bogn´
a , A s a i ied low o a non-New onian
Casson luid comp ising mic oo ganisms on a s e ching shee wi h ac i a ion
ene gy, Sci. Rep. 13 (2023) 11240.
[17] D.P. Singh, S.M. Chaudha y, A. Pandey, A.K. Shukla, Uns eady MHD s a i ied low
o non-New onian luid be ween pa allel pla es h ough po ous medium, Indian J.
Theo . Phys. 58 (2010).
[18] D. Dey, Non-New onian e ec s on hyd omagne ic dus y s a i ied luid low
h ough a po ous medium wi h olume ac ion, P oc. Na l. Acad. Sci., India, Sec .
A 86 (2016) 47–56.
[19] H.A. Wahab, H. Zeb, S. Bha i, M. Gulis an, S. Kad y, Y. Nam, Nume ical s udy o
he e ec s o empe a u e dependen iscosi y low o non-New onian luid wi h
double s a i ica ion, Appl. Sci. 10 (2020) 708.
[20] M. Waqas, M.I. Khan, T. Haya , A. Alsaedi, S a i ied low o an Old oyd-B
nanoliquid wi h hea gene a ion, Resul s in physics 7 (2017) 2489–2496.
[21] W. Abbas, M.A. Megahed, Powell-Ey ing luid low o e a s a i ied shee h ough
po ous medium wi h he mal adia ion and iscous dissipa ion, AIMS Ma h 6
(2021) 13464–13479.
[22] M.Y. Malik, S. Bilal, M. Bibi, U. Ali, Loga i hmic and pa abolic cu e i ing
analysis o dual s a i ied s agna ion poin MHD mixed con ec ion low o Ey ing-
Powell luid induced by an inclined cylind ical s e ching su ace, Resul s Phys. 7
(2017) 544–552.
[23] D. S ini asacha ya, O. Su ende , E ec o double s a i ica ion on mixed
con ec ion bounda y laye low o a nano luid pas a e ical pla e in a po ous
medium, Appl. Nanosci. 5 (2015) 29–38.
[24] T. Haya , M. Waqas, M.I. Khan, A. Alsaedi, Analysis o hixo opic nanoma e ial in
a doubly s a i ied medium conside ing magne ic ield e ec s, In . J. Hea Mass
T an. 102 (2016) 1123–1129.
[25] M.J. Babu, N. Sandeep, UCM low ac oss a mel ing su ace in he p esence o
double s a i ica ion and c oss-di usion e ec s, J. Mol. Liq. 232 (2017) 27–35.
[26] S. Nadeem, N. Muhammad, Impac o s a i ica ion and Ca aneo-Ch is o hea lux
in he low sa u a ed wi h po ous medium, J. Mol. Liq. 224 (2016) 423–430.
[27] A. Sha iq, A.B. Çolak, T.N. Sindhu, Q.M. Al-Mdallal, T. Abdeljawad, Es ima ion o
uns eady hyd omagne ic Williamson luid low in a adia i e su ace h ough
nume ical and a i icial neu al ne wo k modeling, Sci. Rep. 11 (2021) 14509.
[28] W.H. Shayya, S.S. Sablani, A. Campo, Explici calcula ion o he ic ion ac o o
non-New onian luids using a i icial neu al ne wo ks, De . Chem. Eng. Mine .
P ocess. 13 (2005) 5–20.
[29] P. P iyadha shini, V. Ka pagam, N.A. Shah, M.H. Alsheh i, Bio-con ec ion e ec s
o MHD williamson luid low o e a symme ically s e ching shee : machine
lea ning, Symme y 15 (2023) 1684.
[30] M. Shoaib, M.A.Z. Raja, G. Zubai , I. Fa ha , K.S. Nisa , Z. Sabi , W. Jamshed,
In elligen compu ing wi h le enbe g–ma qua d backp opaga ion neu al ne wo ks
o hi d-g ade nano luid o e a s e ched shee wi h con ec i e condi ions,
A abian J. Sci. Eng. (2021) 1–19.
[31] S. Hussain, S. Islam, M.A.Z. Raja, K.S. Nisa , M. Shoaib, In elligen compu ing
echnique o s udy hea and mass anspo o Casson nano luidic low model on a
nonlinea slan ed ex ending shee , ZAMM-Jou nal o Applied Ma hema ics and
Mechanics/Zei sch i ü Angewand e Ma hema ik und Mechanik 104 (2024)
e202300224.
[32] Z. Shah, M.A.Z. Raja, M. Shoaib, Design o Bayesian s ochas ic ne wo ks o
nume ical ea men o Williamson luid s e ching low model wi h mixed
con ec ed hea gene a ion, Nume . Hea T an., Pa B: Fundamen als (2024) 1–24.
[33] S. Nadeem, S.T. Hussain, C. Lee, Flow o a Williamson luid o e a s e ching shee ,
B az. J. Chem. Eng. 30 (2013) 619–625.
[34] I. Jabeen, M. Fa ooq, N.A. Mi , Desc ip ion o s a i ica ion phenomena in he luid
ese oi s wi h i s -o de chemical eac ion, Ad . Mech. Eng. 11 (2019)
1687814019836897.
[35] N.S. Akba , A. Ebaid, Z.H. Khan, Nume ical analysis o magne ic ield e ec s on
Ey ing-Powell luid low owa ds a s e ching shee , J. Magn. Magn Ma e . 382
(2015) 355–358.
[36] M.Y. Malik, T. Salahuddin, A. Hussain, S. Bilal, MHD low o angen hype bolic
luid o e a s e ching cylinde : using Kelle box me hod, J. Magn. Magn Ma e .
395 (2015) 271–276.
[37] P. Ka hic, S. Joseph, N. A un, S. Kuma a el, Op imiza ion o biohyd ogen
p oduc ion by En e obac e species using a i icial neu al ne wo k and esponse
su ace me hodology, J. Renew. Sus ain. Ene gy 5 (2013).
S. Bilal e al.