Ci a ion: Khan, M.S.; Mei, S.;
Shabnam; Fe nandez-Gamiz, U.;
Noeiaghdam, S.; Khan, A.; Shah, S.A.
Elec o iscous E ec o Wa e -Base
Nano luid Flow be ween Two
Pa allel Disks wi h Suc ion/Injec ion
E ec . Ma hema ics 2022,10, 956.
h ps://doi.o g/10.3390/ma h10060956
Academic Edi o s: Mos a a Sa da i
Shadloo, Mohammad Mehdi Rashidi
and Alessio Alexiadis
Recei ed: 14 Decembe 2021
Accep ed: 14 Ma ch 2022
Published: 17 Ma ch 2022
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ma hema ics
A icle
Elec o iscous E ec o Wa e -Base Nano luid Flow be ween
Two Pa allel Disks wi h Suc ion/Injec ion E ec
Muhammad Sohail Khan 1, Sun Mei 1,*, Shabnam 1, Unai Fe nandez-Gamiz 2, Samad Noeiaghdam 3,4 ,
Aami Khan 5,* and Said Anwa Shah 6
1School o Ma hema ical Sciences, Jiangsu Uni e si y, Zhenjiang 212013, China;
[email p o ec ed] (M.S.K.); [email p o ec ed] (S.)
2Nuclea Enginee ing and Fluid Mechanics Depa men , Uni e si y o he Basque Coun y UPV/EHU,
Nie es Cano 12, 01006 Vi o ia-Gas eiz, Spain; [email p o ec ed]
3Indus ial Ma hema ics Labo a o y, Baikal School o BRICS, I ku sk Na ional Resea ch Technical Uni e si y,
664074 I ku sk, Russia; [email p o ec ed]
4Depa men o Applied Ma hema ics and P og amming, Sou h U al S a e Uni e si y, Lenin P ospec 76,
454080 Chelyabinsk, Russia
5Depa men o Pu e and Applied Ma hema ics, Uni e si y o Ha ipu , Ha ipu 22620, Pakis an
6Depa men o Basic Sciences and Islamia , Uni e si y o Enginee ing and Technology Peshawa ,
Peshawa 25000, Pakis an; anwa shah@ue peshawa .edu.pk
*Co espondence: [email p o ec ed] (S.M.); aami [email p o ec ed] (A.K.)
Abs ac :
This a icle, in es iga es he beha iou o an ionized nanoliquid mo ion ega ding hea
ansmission be ween wo pa allel discs. In he p oposed model, he squeezing low o Cu-wa e
nano luid wi h elec ical po en ial o ce is analysed o s udying he low p ope ies and an uni o m
magne ic ield is applied o ha luid, by aking he su ace o he bo om disc po ous. We ha e
also s udied he e ec s o di e en nanoma e ials on he ansmission o hea h ough nano luids.
Fu he mo e, he in luence o a ious physical pa ame e s in he p oposed model o nano luids low
like olume ac ion o nanoma e ials, squeezing numbe , Ha mann numbe , Ecke numbe , and
P and l numbe a e analysed and discussed quan i a i ely h ough a ious ables and g aphs. The
sys em o nonlinea pa ial di e en ial equa ions (PDE’s) has been used o o mula e he p oposed
low model and la e con e ed o a se o nonlinea ODE’s by mean simila i y ans o ma ion.
Fu he , he educed o m o ODEs has been sol ed by Pa ame ic Con inua ion Me hod (PCM),
which is a s able nume ical scheme. The ou comes ob ained om he p oposed model could also
be used o analyse nano luid low in se e al ields, such as polyme p ocessing, powe ans e and
hyd aulic li s.
Keywo ds:
nano luid; elec o- iscous luid; Lo en z o ce; pa ame ic con inua ion me hod and BVP4C
MSC: 76N17; 76N25
1. In oduc ion
The squeezing low o nanoliquid in he gap o wo pa allel disks is used in se e al
indus ial p ocesses such as d illing ools, sola de ices, upda ing punching equipmen
and cooling de ices. Nanoliquids di e om o dina y liquids, especially in hei he -
mophysical p ope ies like he mal di usi i y, iscosi y, and he mal conduc i i y. I is
cu en ly being used in many indus ial p ocesses such as anspo a ion, pha maceu icals,
nuclea eac o s and p ocess in ol ing hea and mass ans e . Many esea che s ha e
begun o explo e he ba ie s o using simple luids. I came o he discussion a e he
in oduc ion o nanoliquids. Tha is why many esea ch is being done on nano luids in
e ms o hea ans e .
Analysis o nanoliquid low be ween wo pa allel squeezing discs wi h he same
geome y can be ound in he ollowing esea ch a icles. In he li e a u e, he idea o
Ma hema ics 2022,10, 956. h ps://doi.o g/10.3390/ma h10060956 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 956 2 o 15
squeezing low has been in oduced o he i s ime by S e an [
1
] . He did his esea ch
using a mode a e lub ica ion me hod and analyzed New onian luid using an Adhoc
asymp o ic app oach. Von Ka man [
2
] in oduced he well-known simila i y ans o ma ion
echnique ha educes he sys em o PDEs o a sys em o ODEs. Such sys ems o ODEs
can be sol ed h ough a ious analy ical and nume ical me hods. Engmann e al. [
3
]
has p o ided he undamen al heo y o squeezing low along wi h i s de elopmen s. I
was e iewed ha he squeezing low is empo a y in na u e. Empi ical s udies ha e
ailed o p o ide a gene al co ela ion be ween he basic heological quan i ies o luid
low phenomena. Bu bidge e al. [
4
] desc ibed he luid low o hin lub ica ing ilms.
The luid low is ca ego ised in o h ee main ypes, poo ly lub ica ed egime, slip egime
and supe lub ica ed egime. The e is a s ong conside a ion be ween heo e ical and
expe imen al analysis.
Ali and Bu [
5
]s udied he in luence o en opy gene a ion in he o a ing disk and
concluded ha he a e o squeezing luid mo ion is in e sely p opo ional o he e ec s o
local en opy gene a ion a e. Buongio no [
6
]in es iga ed ha he low o nano luid ia
slippe y e ec s be ween nanoma e ials and base luid molecules. The analysis concludes
ha signi ican slippe y e ec s a e due o he mopho esis and B ownian mo ion. Fo he
c i ical s udy o nano luid low, he slippe y and homogenous models we e used simul a-
neously. La e on, Haya e al. [
7
] s udied he axisymme ic low e ec s on MHD unde
he e ec o con ec i e bounda y condi ions o hi d-g ade luid low. In hei p oposed
model, hey ha e analysed he luid low be ween wo pa allel discs, one ixed and he
o he squeezing. Nanoma e ials, whe he non-me allic o me allic wi h a diame e o less
han 100 nm, a e impo an elemen s o nano luids. Expe imen al s udies ha e shown
ha nano- luids ha e maximum he mal conduc i i y bu low emissions, hus showing
signi ican s abili y when mixed wi h o he pa icles. Because o hese ea u es, nano- luids
a e e y use ul in adia o s, hea exchange s and cooling elec onic sys ems. Choi [
8
]
concluded ha blending me allic nanoma e ials in o base luids inc emen s he he mal
conduc i i y o nano luids. A e his, many scien is s ha e u ned o nano- luids. Rashdi
e al. [
9
] s udied MHD e ec s on he low ia blending alumina and coppe oxide in o a
base luid wi h a uni o m magne ic ield be ween wo pe meable channels.
In addi ion, Mus a a e al. [
10
] in es iga ed he squeezing low o iscous liquid
be ween wo pa allel pla es ega ding mass and hea ans e , and he low model was
sol ed in MATHEMATICA by HAM. They no iced ha he Nussel numbe inc emen s
due o he inc emen in he alues o
P
and
Ec
as he Nussel numbe in ol ed bo h
Ec
and
P
. Pou meh an e al. [
11
] nume ically sol ed he squeezing low model o incomp ess-
ible nanoliquids by colloca ion and leas squa e me hod o s udy hea and mass ans e
phenomena. In hei analysis, hey used wo pa allel pla es ho izon ally and s udied he
e ec s o a ious pa ame e s such as skin ic ion and Nussel numbe . Khilap e al. [
12
]
in es iga ed he mo ion o an incomp essible liquid be ween wo pa allel pla es o see he
eloci y, empe a u e and magne ic ield p o ile. The p oposed model o luid low has
been sol ed using (RK-4, RK-5) nume ical scheme in he shoo ing me hod. The nume ical
ou comes o he model solu ion we e used o desc ibe a ious low p ope ies and pa am-
e e s such as eloci y and empe a u e p o ile, nanoma e ials olume ac ion, Ha man
numbe and Schmid numbe . Siddiqui e al. [
13
] examined he hyd o-magne ic e ec o
iscous luid low in he gap o wo ho izon al pla es. The hea ans e phenomenon o
nano-liquid along wi h magne ic ield was analysed by Ha ami e al. [14].
Acha ya e al. [
15
] in es iga ed he beha iou o squeezing nano- luid low o (Cu-
wa e , Cu-ke osene) unde he in luence o a a iable magne ic ield. The p oposed low
model has been sol ed nume ically by RK-4 nume ical scheme. Hussain e al. [
16
] analysed
he low o nano- luid o he e ec o mic oci cula ion and concluded ha mic oci cula ion
has an inc easing e ec on skin ic ion as well as hea ans e . I has been obse ed
ha Ag-wa e nanoliquid has a lowe hea ans e a e han Ag-ke osene oil nanoliq-
uid. Nanoma e ials exis in many shapes and sizes. Ag and Cu a e sphe ical-shaped
me al nanopa icles, while single and mul i-walled ca bon nano ubes a e ube-shaped.
Ma hema ics 2022,10, 956 3 o 15
A compa a i e s udy o nanopa icles in di e en shapes and sizes has been done by Timo-
ee a e al. [
17
] o analyse hei in luence on he he mal conduc i i y in a iscous nano luid.
They ha e concluded ha he blade-shaped nano-pa icles ha e mo e he mal conduc i i y.
Vaj a elu e al. [
18
] analysed he low beha iou o Ag-wa e and Cu-wa e nano luid wi h
espec o hea ans e and concluded ha he hickness o he he mal bounda y laye
o Ag-wa e nano luid inc eases mo e as compa ed o Cu-wa e nano luid. In he las
cen u y, nume ous s udies [
19
] ha e been conduc ed on nano luids in di e en geome ies
and discussed i s uses in se e al ields such as ex usion, coolan s, packaging p ocesses
and hea exchange s. Sheikholeslami e al. [
20
] explained asymme ical nano- luid low
be ween squeezing bounda ies. I has been obse ed ha he alues o skin ic ion and
Nussel numbe o sil e a e highe han o he nanoma e ials. In addi ion, he Nussel
numbe elies hea ily on nanoma e ials olume ic concen a ion.
Elec oosmo ic luid low and elec ical po en ial ene gy a e usually modelled by he
Poisson equa ion. The Poisson–Bol zmann equa ion has been alida ed by he modynamic
equilib ium, whe e he dis ibu ion o ions does no apply o luid low. Fu he mo e, i
is conside ed o be a s eady elec oosmo ic luid h ough he mic o-channel and is one o
he mos impo an poin s ha ing an unusual e ec on he con ec i e anspo o ions.
This o mula es he physical p ocess wi h an in e nal elec ic ield, so he Ne ns –Planck
equa ion can be used as a subs i u e o he Poisson–Bol zmann equa ion. Hu e al. [
21
]
examined he elec o-kine ic luid low in T-shaped squeezing pla es. The nonlinea Poisson–
Bol zmann equa ion o he elec okine ic su ace is nume ically sol ed o ob ain he elec ic
po en ial. I is used in he luid low whe e he po en ial o ce is applied a he end o
he channel. The impac s o elec okine ics on he mo emen o luids in a mic o-channel
be ween wo pa allel pla es ha e been in es iga ed by Mala e al. [
22
]. Yang e al. [
23
]
ound ha he ini ial low e ec is due o he use o elec ic po en ial h ough mic ochannels.
The low model has been sol ed nume ically o look a he beha iou o ion dis ibu ion
and Ne ns –Planck equa ions.
Da idson e al. [
24
] e iewed he e o es ima es in high- equency RDF schemes
wi h espec o in e ace e ec s and in oduced a me hod o analysing scale in e ace and
wa e b eak mul imedia scenes. Pa k e al. [
25
] analysed he phenomena o elec ically
conduc i e luids and he powe o elec oosmo ic ia Poisson–Bol zmann equa ions whe e
he dis ibu ion o ions is no a ec ed by he low o luids. Al hough his is no accep able
o elec oosmo ic luids lowing be ween wo pla es, he e a e se e al issues, which c i ically
a ec he empe a u e o conc e e ions. The wo equa ions, Ne ns –Planck and Poisson–
Bol zmann, ha e been compa ed o he elec o-osmo ic luid low be ween channels, whe e
he ideal ioniza ion o ions has no been speci ied. Rojas e al. [
26
] analysed he beha iou
o low agg ega e ze a po en ial and sol ed he low model analy ically, while a nume ical
solu ion was a ailable. The ou comes o he model disclose ha he slope o he wall
su ace and olume ic low a e in he mic ochannel a e inc easing. Thiyaga ajan e al. [
27
]
sc u inized he pleu al e usion p ocess in he lung wall as obs uc ion o he pleu al ca i y.
The e e sal p ocess o he lung and ches wall causes he accumula ion o pleu al luid
in he pleu al space. Pa ie al lympha ic dila ion is caused by an inc ease in pleu al liquid.
This app oach has been in oduced o acqui e new ou comes o espi a o y ac in ec ions,
and has been injec ed in o an uns able na u al and o ced con ec ion anspo low o
neu al pleu al luid in wo ypes o e ical po ous spaces, which was la e esea ched.
M. K. Alam e al. [
28
] examined he impac s o mass and hea ans e a he ansien
squeezing low o iscid liquid in he p esence o a a iable magne ic ield. Bilal e al. [
29
]
and Khosh oi [
30
] esea ched ad ances in a ious echnologies like powe enginee ing
and mic oelec onics based on he de elopmen o e icien cooling sys ems. This p ocess
in ol es he use o ins o conside ably a iable geome y wi hin ca i ies o inc emen
he hea dissipa ion om he hea gene a ion p ocess. Since ins a e hough o play an
e ec i e ole in enhancing hea ans e , he aim o he ongoing esea ch is o examine he
e ec s o di e en pa ame e s on ene gy ans e as well as he ene gy ansmission in ins
embedded in he ca i ies.
Ma hema ics 2022,10, 956 4 o 15
Rizwan e al. [
31
] in es iga ed he beha iou o hea ans e in he squeezing low o
nano luid using wa e -based coppe nanoma e ials in he gap o wo pa allel discs wi h
injec ion/suc ion e ec s. As me allic componen s a e g ea ly in luenced by he magne ic
ield due o he in ol emen o MHD e ec s, i is applied o hogonally o he su ace,
and he bo om disc usually aken is po ous. Khan e al. [
32
,
33
] examined he Poisson-
Bol zmann model, which de i es om he hypo hesis o he modynamic equilib ium on he
condi ion ha he dis ibu ion o ions will no be a ec ed by he low o liquid. Ne e heless,
i is conside ed a easonable hypo hesis o he s able low o elec oosmo ic luid h ough
s aigh mic o-channels, he e a e some key si ua ions whe e he con ec i e anspo o
ions has ex ao dina y e ec s. In all hese si ua ions, he Ne ns -Planck equa ion mus
be used in spi e o he Poisson-Bol zmann equa ion o o mula e he elec ic ield in he
domain. Khan e al. [
34
–
37
] analysed he in luence o a iable magne ic ields in he low
o hyb id nanoliquids o see he imp o emen in he hea ans e a e. The pu pose o hei
s udy was o see he in luence o nano luid (Cu-H
2
O) be ween wo pa allel discs unde he
in luence o a a iable magne ic ield.
F om he abo e li e a u e e iew, I is no iced ha he in es iga ion o nano luids by
dissol ing coppe nanoma e ial wi h a changeable magne ic ield in he gap o wo pa allel
po ous discs so a has no been conside ed. In addi ion, he impac o a iable magne ic
ields on he mass and hea ans e in such low o nano luid is a no el y in cu en esea ch.
This ype o nano- luid low is e y impo an in many indus ial and enginee ing p ocesses.
In his a icle, we a e going o analyse he luid low in he p esence o ions. Veloci y p o ile,
empe a u e p o ile, Nussel numbe and skin ic ions a e calcula ed, which explain he
low p ope ies o he p oposed low model. In addi ion, he impac s o a iable magne ic
ields in he nano luid low o coppe nanoma e ials a e analysed o see he enhancemen in
hea ans e a e. The go e ning equa ions o he p oposed hyb id nano luid a e modelled
unde ce ain assump ions and sol ed nume ically by (pa ame ic con inua ion me hod)
in MATLAB. The nume ical ou comes o se e al eme ging pa ame e s like skin ic ions,
Nussel numbe , e c., a e discussed using a ious ables and g aphs Figu e 1.
Figu e 1. Geome y o he p oblem.
2. Fo mula ion
An uns eady, incomp essible, elec o- iscous nano luid is conside ed be ween he
ci cula space o wo squeezing disks. The wo discs apa om one ano he by
h( ) =
lp1−β
, whe e,
l
deno es he leng h, he wo discs will be pa allel i
=
0. Subsequen ly,
we suppose ha he luid has symme ic posi i e
(+)
and nega i e
(−)
ions along wi h
Ma hema ics 2022,10, 956 5 o 15
alencies o
z+=−z−=z=
1 and
no
in he bulk ions concen a ion o ionic species.
Elec okine ic luid lows ha ing ionic species a e explained by he Na ie –S okes equa ions
h ough he inclusion o uni o m magne ic ield and elec ical body o ce e ms. The model
in ol ing hese equa ions is u he upda ed h ough he inclusion o Poisson equa ion,
Ne ns –Planck equa ion and cha ge dis ibu ion o he con e sion o e e y ion species.
The ma hema ical modelling o he p oposed nano luid low as ollow [27–29]:
Con inui y equa ion:
∇.~
U=0, (1)
The upda ed momen um equa ion h ough elec o iscous and magne ic e ec [
25
,
31
]:
∂~
U
∂ + (~
U.∇)~
U=−1
ρn
∇P+µn
ρn
∇2~
U−µ2
n
ρ2
n
BK2(n+−n−)∇V−σ
ρn µe
(~
H×~
U)×~
H(2)
The Poisson equa ion [25]:
∇2V=−1
2K2(n+−n−)(3)
The Ne ns –Planck equa ions [25]:
∂n+
∂ +∇.(~
Un+) = µn
ρn Sc
(∇2n++∇.(n+∇V)) (4)
∂n−
∂ +∇.(~
Un−) = µn
ρn Sc
(∇2n−+∇.(n−∇V)) (5)
and he Equa ions o ene gy [31]:
∂T
∂ +~
U.∇T=kn
(ρCp)n
∇2T+1
(ρCp)n
a(τ.L)(6)
whe e
H
he magne ic ield,
U
eloci y o he luid,
P
luid p essu e,
ρn
nano-liquid
densi y,
σ
luid elec ical conduc i i y,
Sc
Schmid numbe ,
T
luid empe a u e p o ile,
κn
nanoliqui he mal conduc i i y,
V
o al local elec ical po en ial induced,
K2
is he in e se
Debye cons an ,
(ρCp)n
speci ic hea o he nano luid,
(ρCp)
speci ic hea o he base
luid,
µn
nanoliquid kinema ic iscosi y,
n+
,
n−
a e he anions and ca ions,
τ=µ A1
Shea s ess, µ he dynamic iscosi y o he low, A1=L+LTand L=∆U, espec i ely.
Nano luids a e de ined as [31]:
ρn =ρ (1−φ+φρs
ρ
),µn = (1−φ)−2.5µ ,
κn
κ
=κs+2k −2φ(κ −κs)
κs+2k +φ(κ −κs)and (ρCp)n
(ρCp)
=1−φ+φ(ρCp)s
(ρCp)
,
(7)
whe e
κn
and
κs
a e he base luid and solid ac ion he mal conduc i i ies, espec i ely,
and φis he olume ac ion o he solid nanoma e ials.
Bounda y Condi ions
The bounda y condi ions o he p oposed model a e aken as ollow:
u=0, w=−dh
d ,V= 2
2l(1−β ),n−=0, n+=0, T=Tua z =h( )
u=0, w=−wo
p1−β ,V=0, T=Tl,n+=β
1−β ,n−=β
1−β ,a z =0
(8)
Ma hema ics 2022,10, 956 6 o 15
Fo con e ing PDEs in o ODEs, he ollowing simila i y a iables [
31
] ha e been
used,
u=β
2(1−β ) 0(η),w=−βl
p1−β (η),n+=βµ m(η)
ρ (1−β ),n−=βµ n(η)
ρ (1−β ),
θ(η) = T−Tu
Tl−Tu,V= 2P(η)
l2(1−β ), whe e η=z
lp1−β
(9)
So, Equa ion
(
1
)
is sa is ying au oma ically and he emaining Equa ions (2)–(6) akes
he ollowing o m.
i −ρn
µn
S(η 000 +3 00 − 000)−1
µn
M 00
+µn
ρn
2BK2(P0m+Pm0−Pn0−P0n) = 0,
(10)
P00 +4P−SK1K2(n−m) = 0, (11)
m00 −ρn
µn
SSc(ηm0+2m−2m0 ) + 2mP +δP0m0−(4Pm −SK2K1(m2−mn)) = 0, (12)
n00 −ρn
µn
SSc(2n+ηn0−2n0 )−2nP −δP0n0+ (4Pn +SK2K1(mn −n2)) = 0, (13)
θ00 −κn
κ
P ((ρCp)n
(ρCp)
S(ηθ0−θ0 )−µn Ec(6 02+ 002)) = 0. (14)
The bounda y condi ions in ans o m o m as ollow:
0(0) = 0, (0) = A,P(0) = 0, m(0) = 1, n(0) = 1, θ(0) = 1,
(1) = 0.5, 0(1) = 0, P(1) = 1, m(1) = 0, n(1) = 0, θ(1) = 0, (15)
whe e
Sc =ν
D
Schmid numbe ,
S=βl2
2ν
squeeze numbe ,
P =ν (ρCp)
κ
P and l numbe ,
Ec =1
(Cp) (T0−Th)(β
2(1−β ))2
Ecke numbe ,
B=ρk2T2ε0ε
2z2e2µ2
is ixed a a speci ied empe a u e,
K2
1=2z2e2l2n0
ε0εkbT
he dimensionless in e se o Debye leng h and
M= l2σB2
0
ρ µe
Magne ic
Pa ame e . The pa ame e
A=wo
βl>
0 co esponds o suc ion and
A=wo
βl<
0 co esponds
o injuc ion o luid om he lowe disk.
Nussel numbe and skin ic ion a e he desi ed physical quan i ies, which can be
w i en as ollow:
C =µn
ρn (β
2(1−β ))2(∂u
∂z)z=h( ),Nu=−κn (∂T
∂z)z=h( )
k (T0−Th), (16)
In case o Equa ion (16), we ge
p1−β SC =µn
ρn
00(1),−κn
κ
θ0(0) = Nulp1−β . (17)
Ma hema ics 2022,10, 956 7 o 15
3. Nume ical Solu ion by PCM
This sec ion explains he p ocedu e o he selec ion o an op imal alue o he con inu-
a ion pa ame e s along wi h he p ac ical implemen a ion o PCM [
32
,
33
], which is used o
he solu ion o non-linea ODEs in (10)–(14) wi h p ede e mined bounda y condi ions (15).
•Fi s o de o ODE
We conside he ollowing o educing Equa ions (10)–(14) in o i s -o de ODEs:
=Y1, 0=Y2, 00 =Y3, 000 =Y4
P=Y5,P0=Y6,m=Y7,m0=Y8
n=Y9,n0=Y10,θ=Y11,θ0=Y12
(18)
By using hese ans o ma ions in Equa ions (10)–(14), we ge ,
Y0
4=ρn
µn
S(3Y3+ηY4−Y1Y4) + 1
µn
MY3
−µn
ρn
2BK2(Y5Y8+Y6Y7−Y6Y9−Y5Y10),
(19)
Y0
6=−4Y5−SK2K1(Y7−Y9), (20)
Y0
8=ρn
µn
ScS(ηY8+2Y7−2Y1Y8)−2Y5Y7−δY8Y8
+δ(4Y5Y7−SK2K1(Y2
7−Y7Y9)),
(21)
Y0
10 =ρn
µn
ScS(ηY10 +2Y9−2Y1Y10) + 2Y5Y9+δY6Y10
−δ(4Y5Y9+SK2K1(Y7Y9−Y2
9)),
(22)
Y0
12 =(ρCp)n
(ρCp)
κ
κn
SP (ηY12 −Y1Y12)−κ
κn
µn P Ec(6Y2
2+δY2
3),(23)
and he bounda y condi ions becomes
Y2(0) = 0, Y1(0) = A,Y2(1) = 0, Y1(1) = 1
2,Y5(0) = 0, Y5(1) = 1,
Y7(0) = 1, Y7(1) = 0, Y9(0) = 1, Y9(1) = 0, Y11(0) = 1, Y11(1) = 0,
(24)
•In oduc ion o he q-Pa ame e
The ODEs in he q-pa ame e g oup is explained h ough he in oduc ion q-pa ame e
in Equa ions (19)–(23) and we ha e,
Y0
4=ρn
µn
S(3Y3+ηY4−Y1(Y4−1)q) + 1
µn
MY3
−µn
ρn
2BK2(Y5Y8+Y6Y7−Y6Y9−Y5Y10),
(25)
Y0
6=−4Y5−SK2K1(Y7−Y9+Y6−(Y6−1)q), (26)
Y0
8=ρn
µn
ScS(ηY8+2Y7−2Y1(Y8−1)q)−2Y5Y7−δY8Y8
+δ(4Y5Y7−SK2K1(Y2
7−Y7Y9)),
(27)
Y0
10 =ρn
µn
ScS(ηY10 +2Y9−2Y1(Y10 −1)q) + 2Y5Y9+δY6Y10
−δ(4Y5Y9+SK2K1(Y7Y9−Y2
9)),
(28)
Ma hema ics 2022,10, 956 8 o 15
Y0
12 =(ρCp)n
ρCp)
(κ
κn
SP (ηY12 −Y1(Y12 −1)q)−κ
κn
µn P Ec(6Y2
2+δY2
3).(29)
•Di e en ia ion by q, eaches a he ollowing sys em w. . he sensi i i ies o he
pa ame e -q
Di e en ia ing he Equa ions (25)–(29) w. . by q
D0
1=H1D1+E1(30)
whe e H1is he coe icien ma ix, E1is he emainde and D1=dYi
dτ, 1 ≤i≤12.
•Cauchy P oblem
D1=P1+A1V1, (31)
whe e P1and V1 a e ec o alue unc ions.
E1+H1(A1V1+P1) = (A1V1+P1)0(32)
and le he bounda y condi ions.
•Using by Nume ical Solu ion
An absolu e scheme is used o sol e he p oblem,
V1i+1−V1i
4η=H1V1i+1(33)
Pi+1−Pi
4η=H1Pi+1+E1(34)
•Taking o he co esponding coe icien s
Fo he solu ion o he ODEs, gi en bounda ies a e commonly used o
qi
, whe e
1≤i≤12, bu i is needed o apply D2=0, which looks in ma ix o m as gi en,
L1.D1=0 o L1.(A1V1+P1) = 0 (35)
whe e A1=−L1.P1
L1.V1.
4. Resul s and Discussions
To in es iga e he low o nano luid ega ding hea and mass ans e unde he
e ec s o ions dis ibu ion, we ha e displayed he nume ical ou comes o di e en low
p ope ies g aphically like adial eloci y
0(η)
, axial eloci y
(η)
, posi i e ions
m(η)
,
nega i e ions
n(η)
, Poisson a iable
P(η)
, and hea ans e
θ(η)
o he di e en alues
o low pa ame e s such as squeezing numbe , Ha man numbe , he olume ac ion
o nanoma e ials and suc ion/injec ion pa ame e
A
. Mos o he low p ope ies o he
p oposed model a e desc ibed h ough he a ious g aphs in he p esc ibed domain 0
≤
η≤
1. Figu es 2–10 a e plo ed o analyse he impac o squeezing nano luids low wi h
a iable magne ic ield in he p esence o ions dis ibu ion.
Ma hema ics 2022,10, 956 9 o 15
(a) (b)
Figu e 2.
Impac o (
a
)
(η)
and (
b
)
0(η)
o squeeze pa ame e and ixed alues o
M= 0.41, K= 0.20, B=1.41, K1=0.50, δ=1, Sc =1.50, P =1.50, Ec =0.50.
(a) (b)
(c)
Figu e 3.
Impac o (
a
)
m(η)
, (
b
)
n(η)
and (
c
)
P(η)
o squeeze pa ame e and ixed alues o
B=1.41, M= 0.41, K=2.20, Sc =1.5, K1=0.50, δ=1.0, P =1.50, Ec =0.50.
