Case S udies in The mal Enginee ing 54 (2024) 104023
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Mono and hyb id nano luid analysis o e sh inking su ace wi h
he mal adia ion: A nume ical app oach
S. Saleema, Bilal Ahmadb, Az a Naseemb, Muhammad Bilal Riazc,d,
Tasawa Abbasb,*
aDepa men o Ma hema ics, College o Science, King Khalid Uni e si y, Abha 61413, Saudi A abia
bDepa men o Ma hema ics, Uni e si y o Wah, Wah Can , 47040 Pakis an
cIT4Inno a ions, VSB –Technical Uni e si y o Os a a, Os a a, Czech Republic
dDepa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Byblos, Lebanon
ARTICLE INFO
Handling Edi o : Huihe Qiu
Keywo ds:
Hyb id nano luid
Sh inking su ace
MHD
The mal adia ion
Viscous dissipa ion
ABSTRACT
The s udy o magne ohyd odynamics (MHD) incomp essible low o a luid ha ing hyb id
nanopa icles making he colloidal combina ion wi h base luid is p esen ed in his esea ch.
Compa a i e analysis is ca ied ou o he nano luids Al2O3/Ke osene and ZnO/Ke osene oil
wi h he hyb id nano luid Al2O3–ZnO/Ke osene oil. The subjec lows a e in luenced wi h he -
mal adia ion and iscous dissipa ion. A nume ical echnique Kelle box is employed o examine
he en ision ma hema ical model. Fo compu a ional p ocedu e MATLAB so wa e will be used.
Tabula ed and g aphical ou comes o di e en e ec s a e p esen ed o he app aisal o eloci y
and empe a u e dis ibu ions. I is comp ehend ha he he mal adia ion and iscous dissipa-
ion pa ame e s possess up su ging ends o he hyb id and nano luid empe a u e p o ile bu
opposi e end has been obse ed o di e en olume ac ions o nanopa icles.
1. In oduc ion
The bounda y laye low caused by a sh inking o s e ching shee has been in es iga ed and discussed by a ious esea che s be-
cause o i s widesp ead uses in indus ies. The a e a which he su ace con ac s o s e ches and he a e o cooling (exchange o
hea ) du ing his p ocess de e mine he su ace's ac ual beha iou . The low o luid b ough abou by a sh inking shee was i s s ud-
ied and a nume ical solu ion was gi en by Mikla cic and Wang [1]. Fo sh inking shee hey ound dual solu ion. Wang [2] examined
he luid low caused by sh inking o a shee nea he s agna ion poin egion and as a esul ob ained dual solu ions. Shouka Ahmed,
Ma yam Ahmed [3] ca ied ou he s udy on mixed con ec i e MHD low and concluded ha he mal low beha iou expands wi h
ising adia ion ac o , Ecke numbe , adia ion and s eng h o hea sou ce. The analy ical solu ion o he low o bounda y laye
b ough on by he s e ching o a shee was p o ided by M. Hassani [4]. Lok e al. [5] conside ed he hyd omagne ic low o liquid on
a sh inking shee close o s agna ion poin egion. Dual solu ion low cases in a ea o s agna ion poin h ough a sh inking/s e ching
shee we e s udied by No i ah Bachok, Anau Ishak [6]. The slip condi ion's impac on s agna ion poin low was s udied by Bha -
acha yya e al. [7]. By employing shoo ing me hod o he solu ion o sel -simila equa ions hey ob ained dual solu ions. Begawada
and Nandeppana a [8] examined he impac o he mal adia ion on mic opola luid low h ough e ical po ous medium. The e-
sea che also conside ed he slip condi ion and ob ained nume ical solu ion by he use o Runge-Ku a-Fehlbe g (RKF) me hod. On a
sh inking/s e ching po ous su ace s agna ion poin low was s udied by Bachok [9]. Fo a sh inking shee dual solu ions we e ob-
* Co esponding au ho .
E-mail add ess: asawa [email protected] (T. Abbas).
h ps://doi.o g/10.1016/j.csi e.2024.104023
Recei ed 29 Sep embe 2023; Recei ed in e ised o m 10 Janua y 2024; Accep ed 12 Janua y 2024
Case S udies in The mal Enginee ing 54 (2024) 104023
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S. Saleem e al.
Table 1
The mal and physical p ope ies o base luid and nanoma e ials [29].
Ma e ials
ρ(kg/m3)
𝜎((Ω.m))−1
KCp
Ke osene oil 783 6 × 10−10 0.15 2090
ZnO 5700 10–1 × 10−325 523
Al2O33970 1 × 10−10 40 765
Table 2
P ope ies o nano luid and hyb id nano luid [29].
P ope ies Nano luid Hyb id Nano luid
Hea Capaci y (ρCp)n = (ρCp) (1-
∅1)
+(ρCp)s1∅1. (ρCp)hn = (ρCp)n (1-
∅2)
+(ρCp)s2∅2.
The mal Conduc i i y
kn =k 2k +ks1−2k −ks1∅1
2k +ks1+k −ks1∅1
khn =kn 2kn +ks2−2kn −ks2∅2
2kn +ks2+kn −ks2∅2
Elec ical Conduc i i y
𝜎n =𝜎 (𝜎s1(1+2∅1)+2(1−∅1)𝜎
𝜎s1(1−∅1)+(2+∅1)𝜎
𝜎hn =𝜎n (2𝜎 (1−∅2)+(2∅2+1)𝜎s2
𝜎 (2+∅2)+(1−∅2)𝜎s2
Densi y ρn = (1 − ∅1)ρ +ρs1∅1ρhn = (1 − ∅2)ρn +ρs2∅2
Dynamic Viscosi y
𝜇n =𝜇
(1−∅1)2.5
𝜇hn =𝜇n
(1−∅2)2.5
ained and in case o s e ching shee unique solu ion was ob ained by him. By using di e en physical condi ions some mo e e-
sea che s also ob ained he dual solu ions. Maxwell [10] examined he impac on he he mal conduc i i y o luid using a ious ma-
e ials wi h be e conduc i i y. SP Sam a and MG Reddy [11] in es iga ed he magne ohyd odynamic ee con ec i e low along he
uppe egion o a pa aboloid o e olu ion while keeping in check he e ec s o B ownian mo ion and he mopho esis. Nano luid was
in oduced by Choi & Eas man [12] o in ensi y he conduc i i y o a luid. They obse ed ha when nanopa icles a e mixed in a base
luid hen nano luid is ob ained. Wi h he ad ancemen in he ield o hea ans e by means o nano echnology, nano luid is cha ac-
e ized as a mix u e o nanopa icles in a base luid. The inse ion o nanopa icles imp o es luids he mal conduc i i y and abili y o
cooling becomes limi ed [13–15]. Some nanopa icles used a e ca bon me al oxides and me als. Nano luids a e widely used in indus-
ies he e o e esea che s a e in e es ed in he s udy o hese luids.
The es ablishmen o hyb id nano luid which is ob ained by blending nanopa icles in a base luid has imp o ed hea ans e and
o he ea u es o he luid [16–20]. The in oduc ion o his luid a ac ed many esea ches owa ds he ex ension wo k. Amal aj and
Michael [21] p o ed he hyb id nano luid Al2O3/CuO as a be e coolan o he sola panel. The mal conduc i i y o wo nano luids
CuO/Wa e and Al2O3/Wa e and hyb id nano luid Al2O3–CuO/Wa e was in es iga ed by S. Sen hil aja [22]. The inc ease in he -
mal conduc i i y was 8%, 6.1% and 9% espec i ely. The indings demons a ed ha in compa ison o he o he wo nano luids he
conduc i i y a e o hyb id nano luid was high. Flow o hyb id nano luid Al2O3–CuO/wa e o e a s e ching su ace in h ee dimen-
sions unde he e ec o Lo en z o ce was s udied by De i and De i [23] and p o ed ha by using di e en nanopa icles hea ans-
e a e can be maximized. E ec o nonlinea adia ion on MHD low o Casson hyb id nano luid caused by a cu ed s e ching shee
was in es iga ed by N Sandeep e al. [24]. SP Sam a e al. [25] examined he hea ans e and low cha ac e is ics o MHD low o
dus y nano and dus y hyb id nanoliquids caused by a s e ching su ace. Recen ly Khan e al. [26] in es iga ed biocon ec i e ca lized
Casson hyb id nano luid o e e ical cone. Di e en s udies on hyb id nano luid's low o e a sh inking o s e ching su ace by em-
ploying di e en physical condi ions a e [27,28].
Based on abo e li e a u e e iew i is indica ed ha a s udy which includes compa a i e analysis o di e en ypes o nano luids
and hyb id nano luid keeping he same e ec s is missing. This s udy indica es he Idiosync a ic beha iou o mono- and hyb id
nano luids along wi h hei applica ions in a ious he mal sys ems including sola he mal sys ems, au omo i e cooling sys ems,
hea sinks, o he mal ene gy s o age. He e we conside h ee di e en ypes o luids iz., Al2O3/Ke osene oil and ZnO/Ke osene oil
nano luid and hei mix u e Al2O3–ZnO/Ke osene oil named as Hyb id nano luid o e a sh inking shee . The impo an objec i es o
his wo k a e.
•To analyze he low beha iou o mono and hyb id nano luids agains he magne ic ield.
•To in es iga e he he mal aspec s o luid in he p esence o he mal adia ion and iscous dissipa ion
•Jus i ica ion o he an icipa ed solu ion o he hea ans e phenomena by compa ing esul s wi h p e ious s udies.
The model includes he nonlinea Pa ial di e en ial equa ions which a e ans o med in o o dina y di e en ial equa ions by ap-
plica ion o sui able simila i y ans o ma ion. Kelle box me hodology will be used o adop nume ical solu ions. The impac o pa a-
me e s in ol ed in he modeled equa ions will be s udied on he beha iou o eloci y and empe a u e p o iles o he unde s udy lu-
ids and he indings will be shown g aphically.
2. Ma hema ical model and o mula ion
He e wo dimensional bounda y laye low o h ee di e en luids nea he egion o s agna ion poin is in es iga ed. The low is
de eloped by S e ching/

S
h inking su ace. Ke osene oil is aken as base luid while nanopa icles used a e aluminum oxide and zinc
Case S udies in The mal Enginee ing 54 (2024) 104023
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S. Saleem e al.
Fig. 1. P oblem's geome y.
Fig. 2. G aph o eloci y p o ile o a ious alues o M.
oxide. The mophysical p ope ies o hese nanopa icles a e gi en in Tables 1 and 2. The di ec ion o su ace is along he ho izon al
axis (
x
) while e ical axis (
y
) is pe pendicula o i . The a e a which su ace is s e ched o sh unk is gi en by
uw(x)=ax
, whe e a
is nega i e o su aces ha sh ink and posi i e o su aces ha s e ch. The eloci y o he o hogonal low o s agna ion poin is
gi en by
ue(x)=bx
whe e b is posi i e and gi es he s eng h o s agna ion low. A magne ic ield o s eng h B0is also applied no -
mal o he su ace as shown in Fig. 1.
Wi h he addi ional e ec o iscous dissipa ion and con ec i e bounda y he s eady s a e con inui y, momen um and ene gy
equa ions o he modeled p oblem a e [29].
𝜕u
𝜕x
+𝜕
𝜕y
=0,
(1)
u𝜕u
𝜕x+ 𝜕u
𝜕y= − 1
𝜌hn
ue
due
dx+
𝜇hn
𝜌hn
𝜕2u
𝜕y2+
𝜎hn
𝜌hn
B0
2(ue−u),
(2)
u𝜕T
𝜕x+ 𝜕T
𝜕y=k
𝜌Cphn
𝜕2T
𝜕y2−1
𝜌Cphn
𝜕q
𝜕y+
𝜇hn
𝜌Cphn 𝜕u
𝜕y2
(3)
Wi h bounda y condi ions
Fo y=0∶u=uw, =0,−khn 𝜕T
𝜕y=h1T −T,
(4)
Fo y→∞ ∶ u→ue, →0,T→T∞,
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Fig. 3. G aph o empe a u e dis ibu ion o a ying alues o P and l numbe .
Fig. 4. Impac o a ying alues o R on empe a u e dis ibu ion o h ee luids.
He e

is componen o eloci y pe pendicula o su ace whe eas
u
is eloci y componen along he su ace,
ue
is ee s eam eloci y
and
uw
is eloci y a wall. T,
Tw
,T
∞
ep esen s luid empe a u e, empe a u e a wall and ee s eam empe a u e espec i ely.
Also μ,σ,k,ρ,q ,Cp, ep esen s he dynamic iscosi y, elec ical conduc i i y, he mal conduc i i y, densi y o luid, adia i e hea
lux and speci ic hea a cons an p essu e espec i ely. Subsc ip s s1 and s2 ep esen s he solid pa icles o aluminum oxide and
zinc oxide. Subsc ip s hn , and n s ands o hyb id nano luid, luid and nano luid espec i ely.
Using Rosseland app oxima ion [30]q akes he o m
q = −
4𝜎1
3k1
𝜕T4
𝜕y,
(5)
he e k1 ep esen s he abso p ion coe icien and σ1s ands o S e an-Bol zmann cons an . I is supposed ha a ia ion o empe a u e
in luid is such ha T4can be w i en as a unc ion (linea ) o T. By applying Taylo se ies expansion o T4abou he poin T
∞
we ge
T4∼
=4TT3
∞−3T4
∞,
(6)
he e ms in ol ing highe powe s o T
∞
a e igno ed.
By i ue o (5) and (6) equa ion (3) becomes
u𝜕T
𝜕x+ 𝜕T
𝜕y=k
𝜌Cphn
𝜕2T
𝜕y2−1
𝜌Cphn −16𝜎1T∞
3
3k1
𝜕2T
𝜕y2+
𝜇hn
𝜌Cphn 𝜕u
𝜕y2
,
(7)
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S. Saleem e al.
Fig. 5. Impac o ising alues o Econ empe a u e dis ibu ion o h ee luids.
Fig. 6. Impac o a ying alues o Bion empe a u e dis ibu ion o h ee luids.
Using he simila i y ans o ma ions
u= ′(6)b, = − (6)√b𝜇
𝜌
, 𝜃 (6)=T−T∞
Tw−T∞
,η = √b𝜌
𝜇
y,
(8)
Whe e p ime is used o show di e en ia ion wi h espec o ƞ.
By i ue o equa ion (8) he dimensionless o m o equa ions (2) and (7) is
A1
A2
′′ +1− ′2+MA3
A2
(1− ′)=0,
(9)
(A4+R)𝜃′′ = −A5P 𝜃′−EcA1P ′′2,
(10)
Toge he wi h bounda y condi ions
A η = 0 = 0 : (η) = 0, (η) = ʎ, (η) = − (1 − 𝜃(η))𝑓′
𝜃
′
𝐵
𝑖
Fo 6→∞ ′→1, 𝜃 →0,
(11)
whe e P , R, M,
Ec,Bi
and ʎ s ands o P and l numbe , adia ion pa ame e , magne ic pa ame e , Ecke numbe , Bio numbe and
eloci y a io pa ame e espec i ely and a e gi en by

Case S udies in The mal Enginee ing 54 (2024) 104023
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Fig. 7. Impac o enhancing alues o ∅1,∅2on eloci y o luids.
Fig. 8. Impac o enhancing alues o ∅1,∅2on empe a u e o luids.
M=
𝜎 B0
2
b𝜌
,P =
𝜇 (𝜌Cp)
𝜌 k
,R=
16𝜎1T∞
3
3k1k
,
(12)
In addi ion he a ios A1,A2,
A3
,A4and
A5
a e gi en as
A1=
𝜇hn
𝜇
,A2=
𝜌hn
𝜌
,A3=
𝜎hn
𝜎
,A4=
khn
k
,A5=
(𝜌Cp)hn
(𝜌Cp)
,
(13)
The wo quan i ies he skin ic ion coe icien (C ) and local Nussel numbe (Nux)a e o p ime impo ance om enginee ing poin
o iew, which a e gi en by [31,32]
C =𝜏w
𝜌 ue
2,Nux=
xqw
k (Tw−T∞),
(14)
Whe e qwis hea lux om he pla e and τwis su ace shea s ess along he pla e. They a e compu ed by
𝜏w=𝜇hn (𝜕u
𝜕y)y=0
,qw= −khn (𝜕T
𝜕y)y=0
(15)
By using equa ion (8) we ob ain
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Table 3
Hyb id nano luid Al2O3–ZnO/Ke osene oil: Following ables depic he in luence o di e en pa ame e s on − ″(0), −θ′(0) o ∅1,
= 0.1, 𝑀=𝑅= 0.5, = 1, ʎ= −1.2,∅2𝑃𝑟𝐸𝑐
= 0.7.
∅1∅2M R P ʎEc−θ′(0) - ″(0)
0.1 0.1 0.5 0.5 1 −1.2 0.7 0.99424 2.96009
1 1.09229 3.37927
1.5 1.17353 4.36705
0.5 0.99424
1.5 1.0406
2.5 1.08421
0.8 0.88424
1.7 1.57612
2.5 2.3008
−1.2 0.99424 2.96009
−1.3 1.0758 2.73136
−1.4 1.1574 2.37629
0.1 0.1 0.99424 2.96009
0.4 0.4 1.1796 4.53701
0.7 0.7 2.5452 7.0732
0.7 0.99424
0.9 1.25503
1.2 1.64621
Table 4
Nano luid Al2O3/Ke osene oil.
<! − − Col Coun ∶9− − >∅1
∅2M R P ʎEc−θ′(0) − ″(0)
0.1 0 0.5 0.5 1 −1.2 0.7 0.71743 2.21103
1 0.78355 2.80108
1.5 0.83863 3.27592
0.5 0.7174
1.5 0.7503
2.5 0.7831
0.8 0.66743
1.7 1.1586
2.5 1.7258
−1.2 0.78498 2.21103
−1.3 0.85704 2.04995
−1.4 0.85704 1.08262
0.1 0.71743 2.21103
0.4 0.9903 2.60597
0.7 1.2846 3.12959
0.7 0.71743
0.9 0.90499
1.2 1.18634
Rex
1∕2C =A1 ′′ (0),Rex−1∕2Nux= −A4𝜃′(0).
(16)
He e
Rex
ep esen s he local Reynolds numbe .
3. Compu a ional me hod
The sys em o equa ions (9) and (10) which is a se o nonlinea pa ial di e en ial equa ions wi h i s bounda y condi ions (11) a e
compu ed nume ically by employing Kelle Box [33–35] echnique. Le
′=a,
(17)
a′=b,
(18)
𝜃′= ,
(20)
Equa ions (9) and (10) becomes
A1
A2
b′+1−a2+ b +MA3
A2
(1−a)=0,
(21)
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Table 5
Nano luid ZnO/Ke osene oil.
<! − − Col Coun ∶9− − >∅1
∅2M R P ʎEc−θ′(0) − ″(0)
0 0.1 0.5 0.5 1 −1.2 0.7 0.71150 2.58499
1 0.78768 3.3252
1.5 0.85012 3.91741
0.5 0.71150
1.5 0.75155
2.5 0.78019
0.8 0.67150
1.7 1.10111
2.5 1.5842
−1.2 0.71150 2.58499
−1.3 0.77148 2.45382
−1.4 0.83528 2.25918
0.1 0.71150 2.58499
0.4 0.9219 3.4617
0.7 1.1220 4.6560
0.7 0.71150
0.9 0.88785
1.2 1.15236
(A4+R) ′+A5P +EcA1P b2=0,
(22)
And bounda y condi ions ake he o m
ƞ= 0 𝑎=ʎ,𝑓= 0, 𝑡= − (1 − 𝜃)
𝐵
𝑖
6→∞a→1, 𝜃 →0,
By applying ini e di e ence me hod
j− j−1−
hj
2(aj−aj−1)=0,
(23)
aj−aj−1−
hj
2(bj−bj−1)=0,
(24)
𝜃j−𝜃j−1−
hj
2( j− j−1)=0,
(25)
Using (23) o (25) in (21) and (22), we ha e
A1
A2bj−bj−1+1−hjaj+aj−1
22
+ j+ j−1
2bj+bj−1
2+MA3
A21−
aj+aj−1
2=0,
(26)
A4+R j− j−1+hjA5P  j+ j−1
2 j+ j−1
2+EcA1P bj+bj−1
22=0,
(27)
New on's me hod o linea iza ion
jk+1= jk+𝛿 jk,
(28)
ajk+1=ajk+𝛿ajk,
(29)
bjk+1=bjk+𝛿bjk,
(30)
𝜃jk+1=𝜃jk+𝛿𝜃jk,
(31)
jk+1= jk+𝛿 jk,
(32)
So equa ions (23)–(25) becomes
𝛿 j−𝛿 j−1−
hj
2{𝛿aj−𝛿aj−1}=(Q1)j,
(33)
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S. Saleem e al.
Table: 6
Compa ison o he Skin ic ion coe icien wi h R=M=EC =∅1=∅2= 0.
ʎ[29] P esen s udy
1 0 0
0.5 0.71330 0.71330
0 1.23258 1.23257
−0.25 1.40224 1.40220
−0.5 1.49567 1.49569
−0.75 1.48930 1.48925
−1 1.32880 1.32881
−1.15 1.08220 1.08220
𝛿aj−𝛿aj−1−
hj
2{𝛿bj−𝛿bj−1}=(Q2)j,
(34)
𝛿𝜃j−𝛿𝜃j−1−
hj
2{𝛿 j−𝛿 j−1}=(Q3)j,
(35)
Whe e
(Q1)j= j−1− j+hjaj−1
2
,(Q2)j=aj−1−aj+hjbj−1
2
and (Q3)j=𝜃j−1−𝜃j+hj j−1
2
,
So equa ions (26) and (27) becomes
C1𝛿aj+C2𝛿aj−1+C3𝛿 j+C4𝛿 j−1+C5𝛿bj+C6𝛿bj−1=(R1)j,
(36)
D1𝛿 j+D2𝛿 j−1+D3𝛿 j+D4𝛿 j−1+D5𝛿bj+D6𝛿bj−1=(R2)j,
(37)
Whe e