Resul s in Enginee ing 21 (2024) 101954
A ailable online 29 Feb ua y 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-NC-ND license (h p://c ea i ecommons.o g/licenses/by-
nc-nd/4.0/).
Cha ac e is ics o hea anspo a ion in MHD low o chemical eac i e
mic opola nano luid wi h mo ing slip condi ions ac oss s agna ion poin s
Muhammad Jawad
a
,
**
, Hamiden Abd El-Wahed Khali a
b
,
c
, Abee A. Shaaban
d
,
e
, Ali Akgül
,
g
,
h
,
Muhammad Bilal Riaz
i
,
j
,
*
, Naeem Sadiq
k
a
Facul y o Sciences, The Supe io Uni e si y, Laho e, 54000, Pakis an
b
Depa men o Ma hema ics, College o Science, Qassim Uni e si y, Bu aydah, 51452, Saudi A abia
c
Depa men o Ope a ions and Managemen Resea ch, Facul y o G adua e S udies o S a is ical Resea ch, Cai o Uni e si y, Giza, 12613, Egyp
d
Depa men o Ma hema ics, Facul y o Educa ion, Ain Shams Uni e si y, Heliopolis, Cai o, Egyp
e
Depa men o Managemen In o ma ion Sys ems and P oduc ion Managemen , College o Business and Economics, Qassim Uni e si y, Bu aydah, Saudi A abia
Depa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Bei u , Lebanon
g
Sii Uni e si y, A and Science Facul y, Depa men o Ma hema ics, 56100, Sii , Tu key
h
Nea Eas Uni e si y, Ma hema ics Resea ch Cen e , Depa men o Ma hehama ics, Nea Eas Boule a d, PC: 99138, Nicosia /Me sin 10, Tu key
i
IT4Inno a ions, VSB – Technical Uni e si y o Os a a, Os a a, Czech Republic
j
Depa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Byblos, Lebanon
k
Depa men o Ma hema ics, Go e nmen College Uni e si y Faisalabad, Faisalabad, 38000, Pakis an
ARTICLE INFO
Keywo ds:
Mic opola luid
Con ec i ely hea ed
Nonlinea he mal adia ion
S agna ion poin
Nanopa icle
ABSTRACT
In cu en s udy, he s eady elec ically conduc ing low o mic opola nano luid pas a po ous s e ched su ace
ac oss s agna ion poin s is in es iga ed. Fo mo i a ion o p oblem, he imp essions o he nonlinea he mal
adia ion and con ec i ely hea ed ha e been analyzed. In addi ions, he in luence o he mopho esis and hea
ans e a e he pa o his s udy. The go e ning i m o PDEs a e con e ed o a sys em o nonlinea and coupled
ODEs using he simila i y app oach. Mo eo e , he esul ing p oblem is nume ically in eg a ed wi h he aid o
shoo ing app oach by u ilizing he b p4c p og amme o MATLAB. The nume ical ou comes a e calcula ed using
a ious physical pa ame e alues and con as ed wi h p e iously published esul s. Nume ical alues o he
physical quan i ies like eloci y, mic opola o a ion, empe a u e and concen a ion p o ile o in ol ing pa-
ame e s such as P and l numbe P , adia ion R, slip pa ame e
α
, he mopho esis N , suc ion/injec ion eloci y
w, B ownian mo ion Nb, magne ic pa ame e M and Lewis numbe Le a e also compu ed and delibe a ed in his
conside a ion.
1. In oduc ion
O e he pas ew decades, luid dynamics esea ch has ocused
hea ily on hea and mass ansmission. Hea and mass ansmission ha e
nume ous indus ial and enginee ing uses. In essence, mass ans e is a
key componen o e e y e inemen p ocess. Hea and mass ans e
phenomena ha e coun less applica ions in en iched oil eco e y, he -
mal insula ion, d ying o po ous solids, unde g ound species anspo
and geo he mal ese oi , snack oods, poul y p ocessing, ood indus y
and chemical manu ac u ing o consignmen essel. Nume ous in-
s iga o s including Rashad e al. [1] add essed he in e change o hea
and mass and ela ed applica ions. Majeed e al. [2] in es iga ed he
elec ical conduc ing nano luid low wi h hea ans e induced by
s e chable cylinde . Jawad e al. [3] explo ed he in luence o hea
ans e on magne ized low o maxwel luid induced by con ec i e
bounda y and A henius ene gy using Da cy-Fo chheime low heo y.
Chamka e al. [4] s udied he low o ime in depended luids wi h hea
gene a ion owing o o a ing cone in he mani es a ion o hea ans e
heo y. In high empe a u e ope a ions and space echnologies, he
impac o he mal ays is c ucial. The linea he mal adia ion p oduces
a no able inaccu acy when he empe a u e a ia ion is a i s highes .
Nonlinea he mal adia ion is aken in o accoun o co ec o such
inaccu acies. K ishna e al. [5] p o ided an explana ion o he imp es-
sions o he mal adia ion chemical eac ion on he low ac oss s e ched
shee wi h ou side hea sou ce. Jawad e al. [6] s udied he in luence o
* Co esponding au ho . IT4Inno a ions, VSB – Technical Uni e si y o Os a a, Os a a, Czech Republic.
** Co esponding au ho .
E-mail add esses: [email p o ec ed] (M. Jawad), [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.2024.101954
Recei ed 8 Janua y 2024; Recei ed in e ised o m 14 Feb ua y 2024; Accep ed 26 Feb ua y 2024
Resul s in Enginee ing 21 (2024) 101954
2
he mal adia ion and chemical eac ion on magne ized low o nano-
luid due o ex ending shee ia simila i y solu ion. Jan e al. [7]
analyzed he impac o magne ic ield and con ec i e bounda y on
e na y nano luids wi h he mal adia ion and a iable he mal con-
duc i i y induced by s e chable shee . Nuhash e al. [8] delibe a ed
hea ans e analysis on hyb id nano luid due o la pla e in he p es-
ence o he mal adia ion. Islam e al. [9] examined he impac o
he mal adia ion on wa e based magne ized hyb id nano luids in he
p esence o s e chable shee . To emphasise he signi icance o hea a-
dia ion, schola s and in en e s ha e conduc ed a numbe o s udies
[10–13]. Di e en base luids can change he luid s eam and hea
ans e when nanopa icles a e p esen . The e m nano luids (liquid and
gas) e e s o hese luid suspensions o iny pa icles. Due o i s wide
ange o uses in ae ospace ilogy, mic o luid con eyance de ices,
in elligen building design, polyme co e ing, uel cell echnology, ood,
anspo a ion and a omic eac o s, nano luids ha e d awn conside able
a en ion. Addi ionally, nano luids ind applica ions in enhanced hea
ans e , medical imaging, and ad anced ma e ials due o hei unique
he mal p ope ies. The size o nanopa icles is be ween 1 and 100 nm.
Choi [14] i s p oposed he undamen al idea o adding nanopa icles o
he base luid o imp o e hea conduc i i y in 1995. Makinde e al. [15]
sc u inized he MHD low o nano luid o e a s e ched plane e ealed
o a con ec i e bounda y condi ion. The uses o nano luids along wi h
di e en geome y we e co e ed by Wong [16]. Jawad e al. [17]
explo ed he magne ized low o nano luid induced by pe meable shee
wi h ac i a ion ene gy and he mal adia ion ia simila i y solu ion.
Hosseinzadeh e al. [18] examined he in luence o magne ic lux on
s e ched low o second g ade nano luid wi h hea ans e . To accen-
ua e he implica ion o nano luid, schola s and in en e s ha e con-
duc ed a numbe o analyses [19,20]. Nowadays, mic opola luids a e
he ocus o mos esea ch due o hei e sa ili y in applica ions such as
chemical suspensions, liquid c ys als and polyme ic luids.
Mic o-cons i uen s ha can o a e a e ound in mic opola luids, and
hei appea ance can ha e a no iceable non-New onian e ec on he
s eam’s hyd odynamics. In e e yday li e, a ious non-ideal luids can
be ound, including shampoo, ca chup, sy up, blood, honey, oo hpas e
and mac omolecules, and mo e. Jawad e al. [21] sc u inized he in-
luence o B ownian mo ion and he mal adia ion on magne ized low
o mic opola luid ia mo ing s e chable su ace. The opic o cha -
ac e is ics o mic opola luids and i s applica ions has been co e ed in
Re s. [22,23]. The doo s o non-ideal luids we e opened by E ingen’s
[24] g oundb eaking wo k in mic opola luids. In a o a ional me hod,
Nadeem e al. [25] in es iga ed he imp essions o hyd odynamic low
o a mic opola nano- luid be ween pa allel pla es. Magne ohyd ody-
namics is he e iew o he magne ic ea u es o elec ically conduc ing
luids. Examples o MHD include elec oly es, wa e , sal , me als, liq-
uids, and magne ic luids. Magne ohyd odynamics inds applica ions in
powe gene a ion, ae ospace p opulsion, me al p ocessing, and as o-
physics by le e aging magne ic ields o con ol luid low and
conduc i e media. The mass exchange and MHD low o an uppe con-
ec ed Maxwell luid wi h s e ched pla e we e made clea by Haya
e al. [26]. Sheikholeslami e al. [27] disco e ed MHD o a ional low o
wa e -based nano- luids ouching in pa allel su ace. Ib ahim e al. [28]
in es iga ed he inspi a ion o iscous dissipa ion and Joule on MHD
low o hea ing Ma angoni low ac oss a ho izon al su ace. El-hakiem
e al. [29] discussed he impac s o Joule hea ing and sola adia ion
on MHD low o a mic opola luid. Jawad e al. [30] inspec ed he in-
luence o magne ic ield and chemical di usion on s e ched low o
Maxwell luids ia simila i y solu ion. Fo mo e applica ion o magne ic
lux, in e es ed may ead [31–33]. Acco ding o he a o emen ioned
su ey o li e a u e, no a emp has been made o analyses has been
pe o med ye . This in es iga ion in elligence he pu poses o Mic o-
pola nano luid low ac oss s agna ion poin in he exis ence o he mal
adia ion and hea sou ce. The mophe osis and chemical di usion a e
he pa o s udied. The go e ning couple o PDEs o angen hype bolic
Nano luid is dis o ed in o coupled se o ODEs. Nume ical esul s a e
achie ed by using b p4c sol e package o Ma lab. In o de o alida e
he indings, we compa ed hem o ea lie published esea ch and
disco e ed a high le el o ag eemen . The s udy’s applica ion lies in
Nomencla u e
a dimensional cons an
C∞ ambien concen a ion
C concen a ion a wall
q adia i e hea lux
C nanopa icles concen a ion
qm su ace hea lux
B0 magne ic ield s eng h
P P and l numbe
qw su ace mass lux
Re Reynolds numbe
Cp speci ic hea
C x skin ic ion in x−di ec ion
T ho luid empe a u e
T luid empe a u e
DT coe icien o he mopho esis di usion
T∞ ambien empe a u e
Ec Ecke numbe
R adia ion pa ame e
Shx She wood numbe
α
slip pa ame e
(u, ) eloci y ec o s
dimensional eloci y componen
γ spin g adien iscosi y
h dimensionless ans e se eloci y
γ1 he mal Bio numbe
G mic o- o a ion o angula eloci y
w suc ion/injec ion eloci y
DB coe icien o B ownian di usion
γ2 solu al Bio numbe
(x,y)axial and no mal coo dina es
Table 1
Compa ison o P on −φ(
η
)when M and K bo h ze o.
P Ishak e al. [34] P esen ou comes
0.71 0.7641 0.7641
1 0.8708 0.8708
7 1.7225 1.7225
Table 2
Nume ical ou comes o Skin- ic ion o di e en alue o
α
, w, K and M.
M K w
α
Skin- ic ion
0.2 0.3 0.2 1.4 −0.3709264
0.3 −0.3705195
0.5 −0.3711154
0.2 −0.3718267
0.3 −0.3748274
0.5 −0.3776894
0.2 −0.3815360
0.3 −0.3913272
0.5 −0.4011963
1.4 −0.3562091
1.6 −0.3419061
1.8 −0.3378470
M. Jawad e al.
Resul s in Enginee ing 21 (2024) 101954
3
unde s anding hea anspo in magne ohyd odynamic lows o eac-
i e mic opola nano luids, especially wi h mo ing slip condi ions,
c ucial o ad anced hea ans e sys ems (see Tables 1–4).
Ma hema ical Analysis.
A s eady elec ically conduc ing low o mic opola nano luid nea a
s agna ion poin pas a po ous s e ched shee is conside ed. The im-
p essions o he nonlinea he mal adia ion and con ec i ely hea ed
ha e been analyzed o incen i e o p oblem. The p ime assump ions
used o sen imen alize he go e ning model a e desc ibed as.
•Flow is wo-dimensional and incomp essible.
•The ole o he mopho esis and hea ans e is conside ed.
•The app op ia e simila i y ans o ms ha e been used o ob ain he
model p oblem.
•A iden ical magne ic ield o s eng h B0 is supposed
•The esul s a e analyzed nume ically and g aphically.
The go e ning model is designa ed as
∂
u
∂
x+
∂
∂
y=0(1)
u
∂
u
∂
x+
∂
u
∂
y=Ud U
dx +(
μ
+k
ρ
)
∂
2u
∂
y2+
σ
B2
ρ
(U−u)− k
ρ
∂
N
∂
y,(2)
ρ
j(u
∂
N
∂
x+
∂
N
∂
y)=γ
∂
2N
∂
y2−κ(2N+
∂
u
∂
y),(3)
u
∂
T
∂
x+
∂
T
∂
y=k1
ρ
Cp
∂
2T
∂
y2+(uB0−E0)2
σ
ρ
Cp +
τ
[DB
∂
T
∂
y
∂
C
∂
y+DB
T∞(
∂
T
∂
y)2]
+1
ρ
Cp
16
σ
∗
3k∗T∗
∞
∂
2T
∂
y2+(
μ
+k
ρ
Cp )(
∂μ
∂
y)2(4)
u
∂
C
∂
x+
∂
C
∂
y=DB
∂
2C
∂
y2+DB
T∞
∂
2T
∂
y2−k0(C−C∞).(5)
The associa ed equa ions Subjec ed o ollowing BCs.
u=ax +
σ
∗[(
μ
+κ)
∂
u
∂
y+κN,] = w,
N= − n
∂
u
∂
y,−κ(
∂
T
∂
y)=h [Tw−T],
−DB
∂
C
∂
y=hm(Cw−C),a y =0,
u→ 0,N→ 0,T→T∞,C→C∞,as y →∞.

(6)
In o de o ans o m Eqs. (2)–(5) in o coupled ODEs, he simila i y
ans o ma ion and dimension less pa ame e s a e desc ibed as
(
η
) =
ψ
x
a
υ
√,N=ax 
a
υ
√h(
η
),
η
=
a
υ
√y,u=
∂ψ
∂
y, = −
∂ψ
∂
x,
h(
η
) = N
a
a
υ
√x
,φ(
η
) = C−C∞
Cw−C∞
,θ(
η
) = T−T∞
Tw−T∞
,
N =(
ρ
c)pDT(T −T∞)
(
ρ
c) T∞
,M=
σ
B2
0
ρ
a,P =
μ
Cp
k,K=k
μ
,
(
ρ
c)pDB(Cw−C∞)
(
ρ
c) ,
α
=a∗
μ

a
υ
√, w= − (a
υ
)−1
2
υ
w,
γ=k0
a,Le =
α
DB
,R=4
σ
∗T3
∞
k∗k1
.

(7)
equa ion (1) sa is ied iden ically and equ. (3–5) ans o med in o
(8–11) by using equa ion (7).
′
+(1+K)
‴
+1−
′
2+Kh
′
+M(1−
′
)(8)
(1+K
2)h
″
+ h
′
−K(2k+
″
)−
′
h
′
=0(9)
θ
″
[1+4
3Rd((Qw−1)θ+1)3]+P θ
′
+M2EcP [
′
2+E2−2E
′
]+
4Rd((θw−1)θ+1)2((θw−1)θ
′
2)+(1+K)EcP
″
2
+P (Nbθ
′
φ
′
+N θ
′
2)=0(10)
φ
″
+P Le φ
′
−Leγφ +N
Nbθ
″
=0(11)
The ans o med BCs a e designa ed as
(
η
) = w,
′
(
η
) = 1+
α
(1+k(1−n)
″
(
η
)),
h(
η
) = − n
″
(
η
),θ
′
(
η
) = − (1−θ(
η
)),
φ
′
(
η
) = − 2(1−φ(
η
)) a
η
=0,
′
(
η
) = 0,h(
η
) = 0,θ(
η
) = 0,φ(
η
) = 0a
η
→∞.

(12)
The physical quan i y o in e es a e desc ibed as
C x =2
τ
w
ρ
(ax)2,Nux =xqw
k(T −T∞),Shx=xqm
DB(Cw−C∞),(13)
Whe e
Table 3
Nume ical esul o Nussel numbe o di e en alue o Ec, Nb, P , R and Qw.
Qw R P Nb Ec Nussel numbe
1.3 0.2 1.7 0.4 0.1 0.1059563
1.5 0.1066479
1.6 0.1073487
0.2 0.1158682
0.4 0.1262158
0.5 0.1363276
1.1 0.1059692
1.5 0.1065849
1.8 0.1071552
1.1 0.0988183
1.5 0.1051001
1.6 0.1049276
0.2 0.1071593
0.4 0.1031154
0.5 0.1009690
Table 4
Nume ical esul s o Nussel numbe and Shewood numbe o di e en alue o
Le, γ1 and γ2.
γ1 γ2 Le Shewood numbe
0.1 0.1 0.2 0.0866890
0.3 0.0862373
0.5 0.0859179
0.1 0.1564274
0.3 0.2124592
0.5 0.2588001
1.2 0.0879486
1.3 0.0885011
1.4 0.0889751
M. Jawad e al.
Resul s in Enginee ing 21 (2024) 101954
4
Tw=((u+k)
∂
u
∂
y+KN)y=0
,
qw= − k1(
∂
T
∂
y)y=0
,
qm= − DB(
∂
C
∂
y)y=0
,

(14)
The dimensionless o m is de ine as
C 
R
√ex=2(1+(1−n)K)
″
(0)),
Re−1/2
xNux= − 1
3[3+4Rd((Qw−1)θ(0)+1)3]θ
′
(0),
Re−1/2
xShx= − φ
′
(0).

(15)
Whe e Rex=xU
ν
pos ula es he local Reynolds numbe .
2. Me hodology
The bounda y alue issue (8)–(11) does no ha e an analy ical so-
lu ion since hese equa ions a e linked and nonlinea . As a esul , we
employ a nume ical app oach called he shoo ing app oach wi h b p4c
in ma lab. The sys em o ODEs (Eq. (8) o (11) mus be wo ked ou in o
i s o de ODEs. We mus i s upda e he bounda y condi ions (12),
using he shoo ing me hod. Tu ning hese exp essions in o a se o ODEs
o he i s o de . The shoo ing s eps a e speci ied in Fig (1).
Fig. 1. S eps o shoo ing App oach.
Fig. 2. Imp in o M on
′
(
η
).
M. Jawad e al.
Resul s in Enginee ing 21 (2024) 101954
5
3. Resul s and discussions
This sec ion sc u inize he in luence o eme ging pa ame e s, Tem-
pe a u e p opo ion a io Qw, P and l numbe P , The mopho esis N ,
Radia ion R, Lewis numbe Le, ma e ial pa ame e K, B ownian mo e-
men Nb, Ha man numbe M, slip pa ame e
α
and Ecke numbe Ec on
non-dimensional eloci y
′
(
η
), non-dimensional ene gy θ(
η
), non-
dimensional mic o o a ion h(
η
)and non-dimensional concen a ion
φ(
η
)dis ibu ion. Figu e (2) and (3) depic di e ences in he eloci y
dis ibu ion and ene gy dis ibu ion, o a ious alue o magne ic
pa ame e M. The empe a u e ield and he mal bounda y laye
hickening a e all snowballing unc ions o magne ic pa ame e M,
howe e he eloci y ield alls as Ha man numbe M inc eases i s due
o opposing o ce. The magne ic pa ame e in luences he eloci y
p o ile by al e ing luid dynamics, a ec ing low pa e ns, and modi-
ying bounda y laye cha ac e is ics due o Lo en z o ces. The e ec s o
he ma e ial pa ame e K on he eloci y ield
′
and angula eloci y h
a e depic ed in Figu es (4) and (5) espec i ely.As K is aised, eloci y
unc ion dec eases, in he lowe hal o he su ace, and in he op hal , i
inc eases. Veloci y ini ially dec eases as K, he ma e ial pa ame e , in-
c eases in alue. Howe e , he angula eloci y h and he eloci y dis-
ibu ion bo h ise o
η
>2.2 and
η
>0.6, espec i ely. Fu he ,
ma e ial pa ame e signi ican ly impac eloci y dis ibu ion in luid
dynamics. Viscosi y a ec s in e nal ic ion, while densi y al e s mass-
Fig. 3. Imp in o M on θ(
η
).
Fig. 4. Imp in o K on
′
.
M. Jawad e al.

Resul s in Enginee ing 21 (2024) 101954
6
ela ed e ec s, shaping he o e all low pa e n. The in luence o he
empe a u e a io pa ame e Qw on he uni less ene gy ield θ is depic ed
in Figu e (6). The he mal condi ion o he luid imp o es as he em-
pe a u e a io pa ame e Qw ises, esul ing in an expansion o ene gy
dis ibu ion θ. Physically, empe a u e a io pa ame e s g ea ly in lu-
ence empe a u e dis ibu ions. Al e ing his pa ame e a ec s hea
ans e , con ibu ing o a ia ions in empe a u e p o iles and he mal
beha io in di e se sys ems. The luc ua ions o he slip pa ame e
α
on
he olume ic concen a ion ield is ep esen in Figu es (7). I should
be no iced ha he e is a di ec ela ionship be ween slip
α
and φ.
E iden ly, as moun s, he la e al su ace begins o mo e in he y-
di ec ion, inc easing he concen a ion unc ion φ. Physically, he slip
pa ame e in luences concen a ion p o iles by modi ying luid-solid
in e ac ions, impac ing mass anspo and di usion, leading o
dis inc i e concen a ion dis ibu ions in di e se sys ems. The inspi a-
ion o Ecke numbe Ec on uni less empe a u e ield is seen in
Figu e (8). The luid empe a u e is seen o g ow close o he pla e.
Howe e , as he alue o he Ecke numbe is aised, i becomes less
signi ican a he om he su ace. Physically, he Ecke numbe in-
luences empe a u e p o iles by cha ac e izing he a io o kine ic en-
e gy o en halpy, impac ing hea ans e and empe a u e dis ibu ion
in physical sys ems. The in luence o R on uni less empe a u e ield is
Fig. 5. Imp in o K on h.
Fig. 6. Imp in o Qw on θ(
η
).
M. Jawad e al.
Resul s in Enginee ing 21 (2024) 101954
7
seen in Figu e (9). The hea ays pa ame e R a ec s he mal bounda y
laye hickening. I is obse ed ha a ise in R causes an expansion o he
he mal bounda y laye . The mal adia ion signi ican ly impac s em-
pe a u e p o iles by con ibu ing hea ans e , in luencing empe a u e
dis ibu ion and he mal beha io in physical sys ems. The empe a u e
ield is edac ed when he P and l numbe P inc eases, as shown in
Figu e (10). As a esul o he low ange empe a u e, a highe P and l
numbe P has less he mal di usi i y. This deno es a dec ease in he
abili y o exchange ene gy, which ul ima ely esul s in a educ ion in he
he mal bounda y su ace. Physically, P and l numbe a ec s hea
ans e , shaping empe a u e p o iles in physical sys ems. The e ec o
N on he uni less concen a ion ield is shown in Figu e (11). Because
he concen a ion unc ion φ is an inc easing unc ion o he he mo-
pho e ic pa ame e N , he mopho esis is a ac o ha pushes iny ma-
e ials om he ho laye o he coole end. The consequences o Lewis
numbe Le on he uni less concen a ion ield a e shown in Figu e (12).
When we inc ease he alues o Le, he concen a ion p o ile dec eases.
Physically, Lewis numbe in luences concen a ion p o iles by indica ing
he ela i e a es o hea and mass ans e , a ec ing di usion in sys-
ems. Table. (1) displays a good compa ison o he p esen esul s o −
φ(0)wi h p e ious esul s. Compu a ional esul s o
″
(0),θ
′
(0)and φ(0)
o di e en alue o eme ging pa ame e s a e p esen ed in able (2–4)
Fig. 7. Imp in o
α
on φ.
Fig. 8. Imp in o Ec on θ.
M. Jawad e al.
Resul s in Enginee ing 21 (2024) 101954
8
4. Conclusion
In his wo k, he low o mic opola nano luid ac oss s agna ion poin
is explo ed. The imi a ions o he nonlinea he mal adia ion and con-
ec i ely hea ed ha e been sc u inized. The uni less eloci y, uni less
ene gy, uni less olume ic concen a ion and uni less mic o- o a ion
a e examined and discussed in he o m o g aphical ends and ables.
In special case, We p esen he Shx,C x and Nux in he o m o ables. We
concluded ha Highe alues o M esul in an inc ease in he empe -
a u e ield θ and concen a ion ield φ, whe eas he eloci y unc ion
′
has he e e se e ec . Wi h an inc ease in Nb, he empe a u e ield
inc eases bu he uni less concen a ion ield dec eases. Enla ging Ec
esul s in a highe ene gy dis ibu ion θ. Concen a ion and ene gy bo h
dis abu ions a e enhanced o inc easing alue o slip
α
. Bo h he ene gy
dis ibu ion exhibi a declining end o highe alues o P . By making
K la ge , he eloci y dis ibu ion ge s la e . Enla ging Qw and R
b oadens he empe a u e dis ibu ion θ. Le ises as he concen a ion
dis ibu ion φ dec eases.
CRediT au ho ship con ibu ion s a emen
Muhammad Jawad: W i ing – o iginal d a , Valida ion, So wa e.
Hamiden Abd El-Wahed Khali a: Valida ion, W i ing – e iew &
edi ing. Abee A. Shaaban: Visualiza ion, W i ing – e iew & edi ing.
Ali Akgül: Resou ces, Valida ion. Muhammad Bilal Riaz: Concep u-
aliza ion, Resou ces, Fo mal analysis. Naeem Sadiq: So wa e,
Fig. 9. Imp in o R on θ.
Fig. 10. Imp in o P on θ.
M. Jawad e al.
Resul s in Enginee ing 21 (2024) 101954
9
Supe ision, W i ing – e iew & edi ing.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inancial
in e es s o pe sonal ela ionships ha could ha e appea ed o in luence
he wo k epo ed in his pape .
Da a a ailabili y
Da a will be made a ailable on eques .
Acknowledgemen
The Au ho Muhammad Bilal Riaz is highly obliged o Minis y o
Educa ion, You h and Spo s o he Czech Republic o hei suppo
h ough he e-INFRA CZ (ID:90254).
Re e ences
[1] M. Rashad, A.J. Chamkha, S.M.M. El-Kabei , E ec o chemical eac ion on hea
and mass ans e by mixed con ec ion low abou a sphe e in a sa u a ed po ous
media, In . J. Nume . Me hods Hea Fluid Flow 21 (4) (2011) 418–433.
[2] A. Majeed, A. Zeeshan, M. Jawad, Double s a i ica ion impac on adia i e MHD
low o nano luid owa d a s e chable cylinde unde he mopho esis and
b ownian mo ion wi h mul iple slip, In . J. Mod. Phys. B (2023) 2350232.
Fig. 11. Inspi a ion o N on φ.
Fig. 12. Imp in o Le on φ.
M. Jawad e al.