The Effect of Variable Magnetic Field on Viscous Fluid between 3-D Rotatory Vertical Squeezing Plates: A Computational Investigation



Ci a ion: Alam, M.K.; Bibi, K.; Khan,
A.; Fe nandez-Gamiz, U.;
Noeiaghdam, S. The E ec o
Va iable Magne ic Field on Viscous
Fluid be ween 3-D Ro a o y Ve ical
Squeezing Pla es: A Compu a ional
In es iga ion. Ene gies 2022,15, 2473.
h ps://doi.o g/10.3390/en15072473
Academic Edi o : Dmi y Eskin
Recei ed: 3 Feb ua y 2022
Accep ed: 22 Ma ch 2022
Published: 28 Ma ch 2022
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ene gies
A icle
The E ec o Va iable Magne ic Field on Viscous Fluid
be ween 3-D Ro a o y Ve ical Squeezing Pla es:
A Compu a ional In es iga ion
Muhammad Kam an Alam 1,*, Khadija Bibi 1, Aami Khan 1, Unai Fe nandez-Gamiz 2and
Samad Noeiaghdam 3,4
1Depa men o Ma hema ics & S a is ics, The Uni e si y o Ha ipu , Ha ipu 22620, Pakis an;
[email p o ec ed] (K.B.); aami [email p o ec ed] (A.K.)
2
Nuclea 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] o [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
*Co espondence: [email p o ec ed]
Abs ac :
In his pape , he 3-D squeezing low o iscous incomp essible luid be ween wo pa allel
pla es o a ing a he same a e is in es iga ed. The low is obse ed unde he in luence o he
a ying magne ic ield. The low phenomena a e modeled by u ilizing he basic go e ning equa ions,
i.e., equa ion o con inui y, coupled Na ie S okes, and Magne ic Field equa ions. Using app op ia e
simila i y ans o ma ions, he esul an pa ial di e en ial equa ions a e hen ans o med in o
a sys em o o dina y di e en ial equa ions. The compu a ional echnique is de eloped ia he
Homo opy Analysis Me hod (HAM) o ob ain he solu ion o ans o med sys ems o o dina y
di e en ial equa ions. The in luence o se e al enginee ing luid pa ame e s, such as squeeze
Reynolds numbe , magne ic ield s eng h pa ame e , and magne ic Reynolds numbe , on eloci y
and magne ic ield componen s, a e obse ed om di e en g aphs. I has been in es iga ed ha by
inc easing he squeeze Reynolds numbe , luid eloci y in he yand zdi ec ions will be inc eased
as well. On he magne ic ield componen along he y-axis, an inc easing in luence o squeezing
Reynolds numbe is also no iced. Simila ly, aising he magne ic Reynolds numbe inc eases he
eloci y along he y-axis, whe eas he in e se ela ionship is ound o magne ic ield componen s.
Fu he mo e, o each low phenomenon, an e o analysis is also p esen ed.
Keywo ds:
MHD; Homo opy Analysis Me hod; hea and mass ans e ; ime dependen squeeze
phenomenon; a iable magne ic iled; iscous luid
1. In oduc ion
When a luid is squeezed be ween wo pa allel pla es app oaching one o he , i is
called a squeeze low. The uns eady squeezing low be ween wo pla es o a ing a di e en
angula eloci ies is ega ded as one o he mos impo an s udy subjec s due o i s ex en-
si e applica ions in science and echnology. Among hese a e hyd odynamic lub ica ion,
polyme echnology, biomechanics, he pe oleum sec o , and ae odynamic hea ing. The
in e ac ion o conduc ing luids wi h elec omagne ic ields is widely known as Magen o-
Hyd o Dynamics (MHD). The use o an MHD luid as a lub ican in indus ial applica ions
is appealing because i p e en s he unan icipa ed a ia ion o lub ican iscosi y wi h
empe a u e unde such high wo king condi ions. Many expe s a e showing in e es in his
ield; o example, he uns eady squeezing low be ween pa allel pla es was conside ed o
iscous MHD luid by Siddiqui e al. [
1
]. Fu he , E ik Swee [
2
] in es iga ed he analy ical
solu ion o a iscous luid low be ween mo ing pa allel pla es in an uns able MHD low.
Ene gies 2022,15, 2473. h ps://doi.o g/10.3390/en15072473 h ps://www.mdpi.com/jou nal/ene gies
Ene gies 2022,15, 2473 2 o 21
They used he Homo opy Analysis Me hod o ind he solu ion, which indica ed ha he
magne ic ield’s s eng h has a signi ican impac on he low. La e on, Mu y e al. [
3
]
obse ed he elec ically conduc ing luid in a wo-phase MHD con ec i e low unde
he ac ion o a cons an ans e se magne ic ield h ough an inclined channel in a o a -
ing sys em. Onyango e al. [
4
] expe imen ed on an uns eady MHD low o iscous luid
be ween wo pa allel pla es unde a cons an p essu e g adien . Khan e al. [
5
] obse ed
he low o a iscous luid be ween comp essing pa allel pla es unde he in luence o a
a ying magne ic ield. They in es iga ed he en opy gene a ion due o magne ic ields,
luid ic ion, and hea ans e in a wo-dimensional low p oblem.
Muhammad e al. [6]
discussed he squeezing MHD low be ween wo pa allel pla es using Je ey luid. MHD
luid low be ween wo pa allel pla es was in es iga ed by Ve ma e al. [
7
]. La e on, Haya
e al. [
8
] analy ically ea ed he squeeze low o MHD nano luids be ween wo pa allel
pla es. Fu he mo e, Linga Raju [
9
] discussed he MHD wo- luid low o ionized gases
and in es iga ed he e ec o hall cu en on empe a u e dis ibu ion. The e ec o mag-
ne ohyd odynamics on a luid ilm was hen obse ed by Hamza [
10
], who s udied he
squeezed low be ween wo su aces while o a ion was added o he su aces. Uns eady
Coue e low was hen s udied by Das e al. [
11
] whe e he low was uns eady, and he
MHD e ec was added. The low was obse ed in a o a ing sys em.
A iscous luid low be ween o a ing pa allel pla es wi h a ying bu cons an angula
eloci ies was in es iga ed by Pa e e al. [
12
]. In addi ion, [
13
] also added ema ks on he
low when he iscous luid is lowing be ween wo pa allel o a ing pla es. Fu he on,
Rajagopal [
14
] also s udied second o de ed luid lowing in a o a ing sys em. La e on, he
MHD double-di usi e low o nano luids was s udied by T ipa hi e al. [
15
], he low was
obse ed in a o a ing channel wi h iscous dissipa ion and hall e ec .
The MHD low o iscous luids in a o a ing ame was also s udied in cylind ical coo -
dina es, as was discussed by Hughes e al. [
16
]. They examined he lub ica ion low o such
iscous luids be ween o a ing pa allel disks. In addi ion, Elshekh e al. [
17
] alked abou
he ilm o a luid squeezed be ween o a ing pa allel disks whe e an ex e nal magne ic ield
was applied. The in luence o a changing magne ic ield on he uns eady squeezing low o
iscous luids be ween o a ing discs was also examined by Shah e al. [
18
]. The squeezing
uns eady low o MHD luid be ween wo disks was also discussed by
Ganji e al. [19]
,
hey obse ed he low wi h suc ion o injec ion in ol ed. Be ween squeezing discs mo -
ing a a ious eloci ies, he e ec s o MFD iscosi y and magne ic ield-based (MFD)
he mosolu al con ec ion o he luid dynamics we e examined by Khan e al. [20].
The uns eady squeeze low o iscous luids is also obse ed in h ee-dimensional
o a ing sys ems. Recen ly, Munawa e al. [
21
] s udied he squeeze low o iscous luids
in a h ee-dimensional o a ing sys em. The low was conside ed be ween pa allel pla es
wi h he lowe s e ching pla e kep po ous. Fu he on, Alzah ani e al. [
22
] nume ically
ea ed he squeezed low o iscous luid be ween o a ing pa allel pla es in a h ee-
dimensional sys em and examined he e ec o Du ou and So e numbe . Simila wo k
has been done on hi d-g ade nano luids in a h ee-dimensional o a ing sys em, whe e he
he mopho esis e ec and B ownian mo ion we e obse ed by Shah e al. [
23
]. In addi ion,
he hin- ilm low o Da cy Fo chheime hyd omagne ic nano luid be ween o a ing pa allel
disks in a h ee-dimensional sys em was discussed by Riasa e al. [
24
]. They examined he
impo ance o he Magne ic Reynolds numbe in such a sys em. Mo eo e , Fiza e al. [
25
]
examined he low o Je ey luid in a h ee-dimensional o a ing sys em. Ve y ecen ly, o
di e en luid low phenomena, he well-known HAM me hod was u ilized by di e en
au ho s [
26
–
30
]. They u ilized he HAM me hod o examine he beha io o hei s udy
and p edic he beha io o di e en p oblems.
The abo e exis ing li e a u e wi nessed ha no s udy in pas has been conduc ed so a
on he 3-dimensional squeeze low o iscous luids be ween wo pa allel pla es unde he
in luence o he a iable magne ic ield, while bo h he pla es ha e some angula eloci y.
Hence, he sugges ed wo k is he bes app oach owa d such p oblems and is a way o
Ene gies 2022,15, 2473 3 o 21
mo i a ion o esea che s b inging a new idea o s udying he low be ween uns eady
o a ing pa allel pla es.
2. Modeling and Fo mula ion o he Physical P oblem
The incomp essible iscous luid low be ween wo ho izon al squeezing pla es sepa-
a ed by a dis ance
D( ) = l(1−β )1/2
(Please see Figu e 1), whe e
l
is he spacing be ween
pla es a ime
=
0. The uppe pla e o a es wi h an angula eloci y o
Ωu
, whe eas he
lowe pla e mo es wi h angula eloci y
Ωl
. The e ec o a iable magne ic ield
M
is added
ex e nally, which p oduces he induced magne ic ield
B
wi h he ollowing componen s,
Bx,By, and Bz.
Figu e 1. Geome y o he low p oblem
The sys em o coo dina es selec ed, is Ca esian coo dina es. The o igin is placed in he
lowe pla e’s cen e , in which he x-axis is aken along he ho izon al axis and he z-axis is
a a igh angle o bo h he pla es (along he e ical axis). The o a ion o pla es is along he
y-axis. The low be ween he pla es occu s due o he mo ion o pla es owa ds each o he ,
i.e., he squeezing e ec . The e ec o g a i y on he luid is negligible. Now we obse e
he eloci y p o ile o he gi en luid, and he e ec o he magne ic ield on he eloci y
o luid o hese iscous luids in a h ee-dimensional sys em. Since he coo dina es o
he low a e in such a way ha he
x
-componen is along he di ec ion o he luid and he
z
-componen is no mal o he di ec ion o low; hus, he componen o m o equa ion o
con inui y, Na ie –S okes, and magne ic ield equa ion a e,
Con inui y equa ion:
∂u
∂x+∂
∂y+∂w
∂z=0, (1)
Na ie –S okes equa ion x-componen :
ρ∂u
∂ +u∂u
∂x+ ∂u
∂y+w∂u
∂z=−∂P
∂x+µ∂2u
∂x2+∂2u
∂y2+∂2u
∂z2+
1
µ2Bz∂Bx
∂z−Bz∂Bz
∂x−By
∂By
∂x+By∂Bx
∂y.
(2)
Na ie –S okes equa ion y-componen :
ρ∂
∂ +u∂
∂x+ ∂
∂y+w∂
∂z=−∂p
∂y+µ∂2
∂x2+∂2
∂y2+∂2
∂z2+
1
µ2Bx
∂By
∂x−Bx∂Bx
∂y−Bz∂Bz
∂y+Bz
∂By
∂z.
(3)
Ene gies 2022,15, 2473 4 o 21
Na ie –S okes equa ion z-componen :
ρ∂w
∂ +u∂w
∂x+ ∂w
∂y+w∂w
∂z=−∂p
∂z+µ∂2w
∂x2+∂2w
∂y2+∂2w
∂z2+
1
µ2By∂Bz
∂y−By
∂By
∂z−Bx∂Bx
∂z+Bx∂Bz
∂x.
(4)
Magne ic ield equa ion x-componen :
∂Bx
∂ =u∂By
∂y+By∂u
∂y− ∂Bx
∂y−Bx∂
∂y−w∂Bx
∂z−Bx∂w
∂z+u∂Bz
∂z+Bz∂u
∂z+
1
δµ2∂2Bx
∂x2+∂2Bx
∂y2+∂2Bx
∂z2.
(5)
Magne ic ield equa ion y-componen :
∂By
∂ = ∂Bz
∂z+Bz∂
∂z−w∂By
∂z−By∂w
∂z−u∂By
∂x−By∂u
∂x+ ∂Bx
∂x+Bx∂
∂x+
1
δµ2"∂2By
∂x2+∂2By
∂y2+∂2By
∂z2#.
(6)
Magne ic ield equa ion z-componen :
∂Bz
∂ =w∂Bx
∂x+Bx∂w
∂x−u∂Bz
∂x−Bz∂u
∂x− ∂Bz
∂y−Bz∂
∂y+w∂By
∂y+By∂w
∂y+
1
δµ2∂2Bz
∂x2+∂2Bz
∂y2+∂2Bz
∂z2.
(7)
3. Bounda y Condi ions
The bounda y condi ions o he abo e luid low a e gi en as:
u=0, =Ωlx
1−β ,w=0, Bx=By=Bz=0, a z=0.
u=0, =Ωux
1−β ,w=dD( )
d ,Bx=0, By=xN0
1−β ,Bz=−βM0
(1−β )1/2 ,
a z=D( )whe e D( ) = l(−β )1/2.
He e,
ρ
is he densi y o luid,
P
is p essu e, and
B
is he induced magne ic ield. Now,
using he ollowing ans o ma ion o con e he abo e pa ial di e en ial equa ions o
o dina y di e en ial equa ions:
u=βx
(1−β ) 0(η), =Ωlx
(1−β )g(η),w=−βl
(1−β )1/2 (η),Bx=βxM0
l(1−β )m0(η),
By=xN0
(1−β )n(η),Bz=−βM0
(1−β )1/2 m(η),η=z
l(1−β )1/2 .
A e non-dimensionlizing he abo e equa ions and gi en bounda y condi ions will
be con e ed o he ollowing O.D.E.’s:
0000 =Szh3 00 +(η−2 ) 000 −2 0 00i+
2SzM2
xh2Rmmm0+ηmm00 − mm00 +m2 00−m0m00i,(8)
g00 =Szh2g+ηg0+2 0g−2g0 i−2SzMxMyhm0n−n0mi, (9)
m00 =Rmhm+ηm0−2m0 +2 0mi, (10)
Ene gies 2022,15, 2473 5 o 21
n00 =Rm2n+ηn0−2n0 +2 0n−2Mx
Myg0m−m0g,(11)
whe e
Sz=βl2
2ν
, deno es he Squeezing Reynolds numbe ,
Mx=M0
l√ρµ2
, ep esen s he
Magne ic ield s eng h along he x-axis ,
My=N0
Ωl√ρµ2
is he Magne ic ield s eng h along
x-axis and
Rm=SzB
which is gi en by,
Rm=βl2
2ν(νσµ2)
is he Magne ic Reynolds
numbe , and he bounda y condi ions become o he o m:
(0) = 0, 0(0) = 0, g(0) = 1, m(0) = 0, n(0) = 0.
(1) = 1
2, 0(1) = 0, g(1) = Ωu
Ωl
=S,m(1) = 1, n(1) = 1.
4. Me hod o Solu ion
An analy ical echnique was used o ind he solu ion o Equa ions (8)–(11), known as
he Homo opy Analysis Me hod. We exp ess he unc ions
,
g
,
m
, and
n
(whe e
,
g
,
m
,
and na e he unc ions o η,ηK,K≥0) as a se o base unc ions:
n=
∞
∑
K=0
aKη (12)
gn=
∞
∑
K=0
bKη (13)
mn=
∞
∑
K=0
cKη (14)
nn=
∞
∑
K=0
dKη (15)
whe e he cons an co-e icien s aK,bK,cK, and dka e o be de e mined. Ini ial app oxima-
ions a e chosen as ollows:
0=1.5 ∗η2−η3; (16)
g0= (S−1)∗η+1; (17)
m0=η; (18)
n0=η(19)
now o choose he auxilia y ope a o s:
£ =∂4
∂η4,£g=∂2
∂η2,£m=∂2
∂η2,£n=∂2
∂η2(20)
wi h he ollowing p ope ies
£ (k1∗η3+K2∗η2+K3∗η+K4∗) = 0 (21)
£g(K5∗η+K6∗) = 0 (22)
£m(K7∗η+K8∗) = 0 (23)
£n(K9∗η+K10∗) = 0 (24)
whe e K1∗,K2∗,K3∗,K4∗,K5∗,K6∗,K7∗,K8∗,K9∗, and K10∗a e a bi a y cons an s.

Ene gies 2022,15, 2473 6 o 21
We can ob ain he Ze o h o de de o ma ion as:
(1−s)£ [ (η;s)− 0(η)] = s¯h ℵ [ (η;s),m(η;s)] (25)
(1−s)£g[g(η;s)−g0(η)] = s¯hgℵg[ (η;s),g(η;s),m(η;s),n(η;s)] (26)
(1−s)£m[m(η;s)−m0(η)] = s¯hmℵm[ (η;s),m(η;s)] (27)
(1−s)£n[n(η;s)−n0(η)] = s¯hnℵn[ (η;s),g(η;s),m(η;s),n(η;s)] (28)
F om Equa ions (14)–(17), he nonlinea ope a o s a e de ine as:
ℵ [ (η;s),m(η;s)] = ∂4 (η;s)
∂η4−Sz3∂2 (η;s)
∂η2+(η−2 )∂3 (η;s)
∂η3
−2∂ (η;s)
∂η
∂2 (η;s)
∂η2−2SzM2
x
M∂3M(η;s)
∂η3−∂M(η;s)
∂η
∂2M(η;s)
∂η2
(29)
ℵg[ (η;s),g(η;s),m(η;s),n(η;s)] = ∂2g(η;s)
∂η2−Sz2g+η∂g(η;s)
∂η +2∂ (η;s)
∂η g
−2∂g(η;s)
∂η −2SzMxMy
∂m(η;s)
∂η n−m∂n(η;s)
∂η 
(30)
ℵm[ (η;s),m(η;s)] = ∂2m(η;s)
∂η2−Rmm+η∂m(η;s)
∂η −2
 ∂m(η;s)
∂η −m∂ (η;s)
∂η  (31)
ℵn[ (η;s),g(η;s),m(η;s),n(η;s)] = ∂2n(η;s)
∂η2−R2n+η∂n(η;s)
∂η
−2∂n(η;s)
∂η −n∂ (η;s)
∂η +2∂Mx
∂My
∂g(η;s)
∂η m−g∂m(η;s)
∂η 
(32)
whe e
s
is a ixed pa ame e , nonlinea pa ame e s a e
ℵ
,
ℵg
,
ℵm
and
ℵn
, while
¯h
,
¯hg
,
¯hm
,
and ¯hna e he nonze o auxilia y pa ame e s.
Fo s=0 and s=1, we ha e:
(η, 0) = o, (η, 1) = (η)
g(η, 0) = go,g(η, 1) = g(η)
m(η, 0) = mo,m(η, 1) = m(η)
n(η, 0) = no,n(η, 1) = n(η)
(33)
as
s
a ies om 0 o 1, exac solu ions o
(η)
,
g(η)
,
n(η)
, and
n(η)
can be ob ained om
ini ial guesses o 0,g0,m0, and n0, espec i ely.
Fo hese unc ions, he Taylo ’s se ies a e gi en by:
(η;s) = 0+
∞
∑
n=1
qn n(η)(34)
Ene gies 2022,15, 2473 7 o 21
g(η;s) = g0+
∞
∑
n=1
qngn(η)(35)
m(η;s) = m0+
∞
∑
n=1
qnmn(η)(36)
n(η;s) = n0+
∞
∑
n=1
qnnn(η)(37)
n(η) = 1
n!
∂n (η;s)
∂ηns=0
,gn(η) = 1
n!
∂ng(η;s)
∂ηns=0
mn(η) = 1
n!
∂nm(η;s)
∂ηns=0
,nn(η) = 1
n!
∂nn(η;s)
∂ηns=0
(38)
I can be no ed ha in he abo e se ies con e gence s ongly depends upon
¯h
,
¯hg
,
¯hm
,
and ¯hn.
Assuming ha hese nonze o auxilia y pa ame e s a e chosen so ha he equa ions
con e ge a s=1, one can ob ain:
(η) = 0+
∞
∑
n=1
n(η)(39)
g(η) = g0+
∞
∑
n=1
gn(η)(40)
m(η) = m0+
∞
∑
n=1
mn(η)(41)
n(η) = n0+
∞
∑
n=1
nn(η)(42)
Di e en ia ing he Equa ions (28)–(31) n- imes wi h espec o
s
and pu ing
s=
0,
we ha e:
£ [ n(η)−χn n−1(η)] = ¯h R ,n(η)(43)
£g[gn(η)−χngn−1(η)] = ¯hgRg,n(η)(44)
£m[mn(η)−χnmn−1(η)] = ¯hmRm,n(η)(45)
£n[nn(η)−χnnn−1(η)] = ¯hnRn,n(η)(46)
wi h he gi en bounda y condi ions,
n(0) = 0, 0
n(0) = 0, gn(0) = 1, mn(0) = 0, nn(0) = 0
n(1) = 0.5, 0
n(1) = 0, gn(1) = S,mn(1) = 1, nn(1) = 1(47)
R ,n(η) = 0000
n−1(η)−Sz3 00
n−1(η) + (η) 000
n−1(η)−2 0
n−1(η) 00
n−1(η)
−2
n−1
∑
j=0
j(η) 000
n−j−1(η)+2SzM2
x2Rm
n−1
∑
j=0
mj(η)m0
n−j−1(η)
+ηm00
n−j−1(η) + mn−j−1(η) 00
n−j−1(η)−m0
n−1(η)m00
n−1(η)
(48)
Ene gies 2022,15, 2473 8 o 21
Rg,n(η) = g00
n−1(η)−Sz2gn−1(η) + (η)g0
n−1(η) + 2
n−1
∑
j=0
gj(η) 0
n−j−1(η)− j(η)g0
n−j−1(η)+2SzMxMy
n−1
∑
j=0nj(η)m0
n−j−1(η)−nj(η)m0
n−j−1(η)
(49)
Rm,n(η) = m00
n−1(η)−Rmmn−1(η) + (η)m0
n−1(η)+
2
n−1
∑
j=0mj(η) 0
n−j−1(η)− j(η)m0
n−j−1(η) (50)
Rn,n(η) = n00
n−1(η)−Rm2nn−1(η) + (η)n0
n−1(η)+
2
n−1
∑
j=0nj(η) 0
n−j−1(η)− j(η)n0
n−j−1(η)
−2Mx
Mymj(η)g0
n−j−1(η)−gj(η)m0
n−j−1(η)
(51)
and χn=1, i n >1, and 0, i n =1.
Finally, he gene al solu ion o (41–43) can be w i en as:
n(η) = Zη
0Zη
0Zη
0Zη
0¯h R ,n(z)dzdzdzdz +χn n−1+K1∗η3+K2∗η2+K3∗η+K4∗(52)
gn(η) = Zη
0Zη
0¯hgRg,n(z)dzdz +χngn−1+K5∗η+K6∗(53)
mn(η) = Zη
0Zη
0¯hmRm,n(z)dzdz +χnmn−1+K7∗η+K8∗(54)
nn(η) = Zη
0Zη
0¯hnRn,n(z)dzdz +χnnn−1+K9∗η+K10∗(55)
and so o (η),g(η),m(η), and n(η), he exac solu ion becomes:
(η)≈
n
∑
m=0
m(η)
g(η)≈
n
∑
m=0
gm(η)
m(η)≈
n
∑
m=0
mm(η)
n(η)≈
n
∑
m=0
nm(η).
(56)
5. Op imal Con e gence Con ol Pa ame e s
I sholud be no ed ha he nonze o auxilia y pa ame e s
¯h
,
¯hg
,
¯hm
, and
¯hn
con ained
in he se ies solu ions (41–43), h ough which he a e o he homo opy se ies solu ions and
con e gence egion can be de e mined. A e age esidual e o s we e used o ob ain he
op imal alues o ¯h , ¯hg, ¯hm, and ¯hn:
ε
n=1
K+1
K
∑
j=0ℵ n
∑
i=0
(η),
n
∑
i=0
m(η)m=jδm2
dη(57)
Ene gies 2022,15, 2473 9 o 21
εg
n=1
K+1
K
∑
j=0ℵgn
∑
i=0
(η),
n
∑
i=0
g(η),
n
∑
i=0
m(η),
n
∑
i=0
n(η)m=jδm2
dη(58)
εm
n=1
K+1
K
∑
j=0ℵmn
∑
i=0
(η),
n
∑
i=0
m(η)m=jδm2
dη(59)
εn
n=1
K+1
K
∑
j=0ℵnn
∑
i=0
(η),
n
∑
i=0
g(η),
n
∑
i=0
m(η),
n
∑
i=0
n(η)m=jδm2
dη(60)
Fu he mo e,
ε
n=ε
n+εg
n+εm
n+εn
n(61)
whe e he o al squa ed esidual e o is
ε
n
. We can minimize he o al a e age squa ed
esidual e o by applying he Ma hema ica package BVPh 2.0. To acqui e he local op imal
con e gence con ol pa ame e s, he command Minimize was used.
6. E o Analysis
Taking 10
−40
as a maximum esidual e o , he p oblem was sol ed wi h he HAM
BVPh 2.0 package. An in es iga ion was made using 40 h-o de app oxima ions. The
p o ision o e o analysis suppo s he au hen ica ion o esul s o many ele an physical
pa ame e s in Figu e 2and om esul s gi en in Table 1.
Figu e 2. To al esidual e o , wi h Sz=−0.1, Mx=0.1, My=0.3, Rm=0.01, and S=1.
Table 2is p o ided o de e mine he equa ions’ inaccu acy om he Na ie –S okes
and magne ic ield equa ions. An inc ease in he o de o app oxima ion can be seen, he
solu ion ob ained om hese equa ions con e ges o he exac analysis.
Table 1.
Es ima ing he o al esidual e o wi h ixed alues o
Sz=−
0.1,
Mx=
0.1,
My=
0.3,
Rm=0.01, and S=1, o di e en o de s o app oxima ions.
mε
mεg
mεm
mεn
mCPU Time
1 5.56818 ×10−71.50838 ×10−52.75325 ×10−83.42075 ×10−80.42 s
5 1.04968 ×10−24 4.31742 ×10−19 7.60909 ×10−23 1.89839 ×10−18 4.19 s
10 1.48326 ×10−30 9.10580 ×10−33 1.64090 ×10−35 4.81645 ×10−34 18.5 s
15 1.47350 ×10−30 9.86847 ×10−33 1.86358 ×10−35 1.38805 ×10−35 33.5 s
20 1.48626 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.50240 ×10−35 61.6 s
25 1.47036 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.48434 ×10−35 106.08 s
30 1.47036 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.48434 ×10−35 163.13 s
35 1.47036 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.48434 ×10−35 240.95 s
40 1.47036 ×10−30 9.78372 ×10−33 1.79738 ×10−35 1.48434 ×10−35 456.57 s
Ene gies 2022,15, 2473 16 o 21
Figu e 9.
Impac o magne ic ield s eng h
Mx
on eloci y componen
0
, keeping
Sz=−
2,
My=
3,
Rm=1, and S=1.
Figu e 10.
Impac o Magne ic ield s eng h
Mx
on eloci y componen
g
and magne ic ield
componen n, Keeping Sz=−0.25, My=3, Rm=0.5 and S=1.

Ene gies 2022,15, 2473 17 o 21
Figu e 11.
Impac o magne ic ield s eng h
Mx
on magne ic ield componen
m
, keeping
Sz=−
1.75,
My=3, Rm=1, and S=1.
Figu e 12.
Impac o magne ic ield s eng h
My
on eloci y componen
g
and magne ic ield compo-
nen
n
, keeping
Sz=−
0.5,
Mx=−
0.5( o g),
Mx=−
1.5 ( o n),
Rm=−
2 ( o g),
Rm=−
1 ( o n),
and S=1.
Ene gies 2022,15, 2473 18 o 21
Figu e 13.
Impac o magne ic Reynolds numbe
Rm
on eloci y componen
and
0
, keeping
Sz=−1.5, Mx=−1.5, My=3, and S=1.
Figu e 14.
Impac o magne ic Reynolds numbe
Rm
on eloci y componen
g
, keeping
Sz=−
1.5,
Mx=−0.75, My=1, and S=1.
Ene gies 2022,15, 2473 19 o 21
Figu e 15.
Impac o magne ic Reynolds numbe
Rm
on magne ic ield componen s
m
and
n
, keeping
Sz=−1.5, Mx=−0.75, My=1, and S=1.
8. Conclusions
In his pape , he 3-D squeezing MHD low o iscous luids was conside ed be ween
wo pa allel pla es, whe e bo h he pla es a e o a ing wi h he same angula eloci ies.
The e ec o he a iable magne ic ield was applied and he phenomenon was modeled
using he coupled go e ning equa ions, i.e., con inui y, Na ie –S okes, and he magne ic
ield equa ion. Fu he mo e, by using he sui able simila i y ans o ma ion, he modeled
equa ions o he low phenomena we e ans o med o he o dina y di e en ial Equa-
ions (8)–(11) and we e sol ed by using he analy ical echnique (HAM) using Ma hema ica
package BVPh 2.0. The e o analysis was ca ied ou up o 10
−40
h o de , and he e ec o
di e en pa ame e s on he eloci y and magne ic ield we e obse ed h ough g aphs and
ables. Gi en below a e some conclusions made om he abo e analysis:
•
I was obse ed ha inc easing he squeeze e ec on he uppe pla e causes an inc ease
in he low eloci y along he
y
- and
z
-di ec ion, while along he
x
-di ec ion, he
eloci y inc ease ini ially, bu a dec ease in he eloci y has been obse ed in he uppe
domain (η→1).
•
I was also in es iga ed and concluded ha , by inc easing he squeeze Reynolds
numbe , he magne ic ield componen dec eased he e ec o he magne ic ield along
he z-componen , whe eas he e ec inc eased along he y-componen .
•
Fu he mo e, om he abo e p oblem, i was obse ed ha inc easing he magne ic
ield s eng h pa ame e
Mx
, which is he s eng h o he magne ic ield along he
x
-axis, inc eases he luid eloci y along he
z
-axis; howe e , eloci y along he
y
-axis
showed a g adual dec ease by inc easing
Mx
. Mo eo e , along he
x
-axis, i s an
inc ease in he eloci y componen was obse ed, bu as
η→
1 he eloci y s a ed
dec easing.
•
An in e se ela ion was obse ed be ween he magne ic ield s eng h pa ame e
Mx
and he magne ic ield componen along he
z
-axis, i.e., inc easing he alue o
Mx
showed a dec easing e ec in he alue o he magne ic ield along he
z
-componen ,
and a di ec ela ion could be seen along he y-axis.
•
Fu he mo e, i was seen ha an inc easing alue o he magne ic ield s eng h
pa ame e along he
y
-componen caused a dec ease in he eloci y o he luid along
he y-axis and he e ec o he magne ic ield along he y-axis.
Ene gies 2022,15, 2473 20 o 21
•
I is concluded ha o he magne ic Reynolds numbe
Rm
, a dec ease in he low
eloci y along he
z
-axis was obse ed wi h inc easing
Rm
. On he o he hand, eloci y
along he
y
-axis showed an inc easing e ec by inc easing he
Rm
low eloci y along
he
x
-axis; his showed a dec easing pa e n ini ially, bu as
η→
1, an inc easing e ec
was obse ed.
•
I was also obse ed ha o he magne ic ield, inc easing he magne ic Reynolds
numbe showed a dec ease in he alue o he magne ic ield along bo h he
y
- and
z-axes.
Au ho Con ibu ions:
Concep ualiza ion, M.K.A.; Da a cu a ion, M.K.A.; Fo mal analysis, M.K.A.
and U.F.G.; Funding acquisi ion, S.N.; In es iga ion, K.B.; Me hodology, M.K.A. and U.F.-G.; Re-
sou ces, U.F.-G.; W i ing— e iew & edi ing, A.K. and U.F.-G. All au ho s ha e ead and ag eed o he
published e sion o he manusc ip .
Funding:
The wo k o U.F.-G. was suppo ed by he go e nmen o he Basque Coun y o he ELKA-
RTEK21/10 KK-2021/00014 and ELKARTEK20/78 KK-2020/00114 esea ch p og ams, espec i ely.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Re e ences
1.
Siddiqui, A.M.; I um, S.; Ansa i, A.R. Uns eady squeezing low o a iscous MHD luid be ween pa allel pla es, a solu ion using
he homo opy pe u ba ion me hod. Ma h. Model. Anal. 2008,13, 565–576. [C ossRe ]
2.
Swee , E.; Vaj a elu, K.; Van Go de , R.A.; Pop, I. Analy ical solu ion o he uns eady MHD low o a iscous luid be ween
mo ing pa allel pla es. Commun. Nonlinea Sci. Nume . Simul. 2011,16, 266–273. [C ossRe ]
3.
S i Ramachand a Mu y, P.; Balaji P akash, G. MHD Two-Fluid Flow and Hea T ans e be ween Two Inclined Pa allel Pla es in a
Ro a ing Sys em. In . Sch. Res. No . 2014,2014, 256898. [C ossRe ]
4.
Onyango, E.R.; Kinyanjui, M.; Kima hi, M. Uns eady Hyd omagne ic Flow be ween Pa allel Pla es Bo h Mo ing in he P esence
o a Cons an P essu e G adien . 2017. A ailable online: h p://www.ijesi .com/Volume%206/Issue%201/IJESIT201701_11.pd
(accessed on 1 Ma ch 2022).
5.
Khan, M.S.; Shah, R.A.; Khan, A. E ec o a iable magne ic ield on he low be ween wo squeezing pla es. Eu . Phys. J. Plus.
2019,134, 219. [C ossRe ]
6.
Muhammad, T.; Haya , T.; Alsaedi, A.; Qayyum, A. Hyd omagne ic uns eady squeezing low o Je ey luid be ween wo pa allel
pla es. Chin. J. Phys.2017,55, 1511–1522. [C ossRe ]
7.
Ve ma, P.D.; Ma hu , A. Magene ohyd odynamic Flow be ween Two Pa allel Pla es, One in Uni o m Mo ion and he O he a
Res wi h Uni o m Suc ion a he S a iona y Pla e. P oc. Indian Na l. Sci. Acad. 1969,35, 507–517.
8.
Haya , T.; Sajjad, R.; Alsaedi, A.; Muhammad, T.; Ellahi, R. On squeezed low o couple s ess nano luid be ween wo pa allel
pla es. Resul s Phys. 2017,7, 553–561. [C ossRe ]
9.
Linga Raju, T. MHD hea ans e wo-ionized luids low be ween pa allel pla es wi h Hall cu en s. Resul s Eng.
2019
,4, 100043.
[C ossRe ]
10.
Hamza, E.A. The magne ohyd odynamic e ec s on a luid ilm squeezed be ween wo o a ing su aces. J. Phys. Appl. Phys.
1991
,
24, 547–554. [C ossRe ]
11.
Das, S.; Maji, S.L.; Gu ia, M.; Jana, R.N. Uns eady MHD Coue e low in a o a ing sys em. Ma h. Compu . Model.
2009
,50,
1211–1217. [C ossRe ]
12.
Pa e , S.V.; Rajagopal, K.R. Swi ling Flow be ween Ro a ing Pla es. In The B ead h and Dep h o Con inuum Mechanics; Sp inge :
Be lin/Heidelbe g, Ge many, 1986; ISBN 978-3-642-61634-1.
13.
Rajagopal, K.R.; Pa e , S.V. Rema ks on he low be ween wo pa allel o a ing pla es. Tech. Summ. Rep. Wis. Uni .
1984
. A ailable
online: h ps://ui.adsabs.ha a d.edu/abs/1984wisc. ep Q....P/abs ac (accessed on 1 Ma ch 2022).
14.
Rajagopal, K.R. The low o a second o de luid be ween o a ing pa allel pla es. J. Non-New on. Fluid Mech.
1981
,9, 185–190.
[C ossRe ]
15.
T ipa hi, R.; Se h, G.; Manoj, M. Double di usi e low o a hyd omagne ic nano luid in a o a ing channel wi h Hall e ec and
iscous dissipa ion: Ac i e and passi e con ol o nanopa icles. Ad . Powde Technol. 2017,28, 2630–2641. [C ossRe ]
16.
Hughes, W.; Elco, R. Magne ohyd odynamic lub ica ion low be ween pa allel o a ing disks. J. Fluid Mech.
1962
,13, 21–32.
[C ossRe ]
Ene gies 2022,15, 2473 21 o 21
17.
Elshekh, S.S.; Abd Elhady, M.K.; Ib ahim, F.N. Fluid ilm squeezed be ween wo o a ing disks in he p esence o a magne ic
ield. In . J. Eng. Sci. 1996,34, 1183–1195. [C ossRe ]
18.
Shah, R.A.; Khan, A.; Shuaib, M. On he s udy o low be ween uns eady squeezing o a ing discs wi h c oss di usion e ec s
unde he in luence o a iable magne ic ield. Heliyon 2018,4, e00925. [C ossRe ]
19.
Ganji, D.D.; Abbasi, M.; Rahimi, J.; Gholami, M.; Rahimipe oudi, I. On he MHD squeeze low be ween wo pa allel disks wi h
suc ion o injec ion ia HAM and HPM. F on . Mech. Eng. 2014,9, 270–280. [C ossRe ]
20.
Khan, A.; Shah, R.A.; Shuaib, M.; Ali, A. Fluid dynamics o he magne ic ield dependen he mosolu al con ec ion and iscosi y
be ween coaxial con ac ing discs. Resul s Phys. 2018,9, 923–938. [C ossRe ]
21.
Munawa , S.; Mehmood, A.; Ali, A. Th ee-dimensional squeezing low in a o a ing channel o lowe s e ching po ous
wall. Compu . Ma h. Appl. 2012,64, 1575–1586. [C ossRe ]
22.
Alzah ani Abdullah, K.; Zaka, U.M.; Tasee , M. Nume ical T ea men o 3D Squeezed Flow in a Ro a ing Channel Wi h So e and
Du ou E ec s. F on . Phys. 2020,8, 201. [C ossRe ]
23.
Shah, Z.; Gul, T.; Islam, S.; Khan, M.A.; Bonyah, E.; Hussain, F.; Mukh a , S.; Ullah, M. Th ee dimensional hi d g ade nano luid
low in a o a ing sys em be ween pa allel pla es wi h B ownian mo ion and he mopho esis e ec s. Resul s Phys.
2018
,10, 36–45.
[C ossRe ]
24.
Riasa , S.; Ramzan, M.; Kad y, S.; Chu, Y.-M. Signi icance o magne ic Reynolds numbe in a h ee-dimensional squeezing
Da cy-Fo chheime hyd omagne ic nano luid hin- ilm low be ween wo o a ing disks. Sci. Rep. 2020,10, 17208. [C ossRe ]
25.
Fiza, M.; Alsubie, A.; Ullah, H.; Hamadneh, N.N.; Islam, S.; Khan, I. Th ee-Dimensional Ro a ing Flow o MHD Je ey Fluid Flow
be ween Two Pa allel Pla es wi h Impac o Hall Cu en . Ma h. P obl. Eng. 2021,2021, 6626411. [C ossRe ]
26.
Kam an Alam, M.; Bibi, K.; Khan, A.; Noeiaghdam, S. Du ou and So e E ec on Viscous Fluid Flow be ween Squeezing Pla es
unde he In luence o Va iable Magne ic Field. Ma hema ics 2021,9, 2404. [C ossRe ]
27.
Waqas, H.; Fa ooq, U.; Ib ahim, A.; Kam an Alam, M.; Shah, Z.; Kumam, P. Nume ical simula ion o biocon ec ional low o
bu ge nano luid wi h e ec s o ac i a ion ene gy and exponen ial hea sou ce/sink o e an inclined wall unde he swimming
mic oo ganisms. Sci. Rep. 2021,11, 14305. [C ossRe ]
28.
Khan, A.; Shah, R.A.; Kam an Alam, M.; Ahmed, H.; Shahzad, M.; Rehman, S.; Ahmed, S.; Sohail Khan, M.; Abdel-A y, A.-H.;
Zaka y, M. Compu a ional in es iga ion o an uns eady non-New onian and non-iso he mal luid be ween coaxial con ac ing
channels: A PCM app oach. Resul s Phys. 2021,28, 104570. [C ossRe ]
29.
Shah, Z.; Khan, A.; Alam, M.K.; Khan, W.; Islam, S.; Kumam, P.; Thoun hong, P. Mic opola Gold Blood Nano luid low and
Radioac i e Hea T ans e be ween Pe meable Channels. Compu . Me hods P og ams Biomed. 2020,186, 105197. [C ossRe ]
30.
Bu gan, H.I.; Aksoy, H. Daily low du a ion cu e model o ungauged in e mi en subbasins o gauged i e s. J. Hyd ol.
2022
,
604, 127429. [C ossRe ]