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Resea ch A icle
T ansien Bounda y-Laye Dynamics o An Elas ico-Viscous Medium
Ac oss a De o mable Shee Subjec ed o Chemical Reac i i y
Bikash Koli Saha *
Independen Resea che , Ma hema ics, Guwaha i, Assam, India
Co esponding Au ho : * Bikash Koli Saha DOI: h ps://doi.o g/10.5281/zenodo.17532511
Abs ac
Manusc ip In o ma ion
The ansien bounda y-laye e olu ion o an elas ico- iscous medium along a pe sis en ly
de o ming shee unde going a i s -o de chemical eac ion is in es iga ed. The heological
beha iou o he elas ico- iscous luid is cha ac e ised h ough Wal e s’ liquid model (B′
o mula ion). By in oking simila i y ans o ma ions, he con olling pa ial di e en ial
equa ions a e ecas in o a coupled sys em o nonlinea sel -simila o dina y di e en ial
equa ions subjec o pe inen bounda y cons ain s. These nonlinea ela ions a e subsequen ly
educed o a se o i s -o de di e en ial equa ions accompanied by hei associa ed bounda y
speci ica ions. The esul ing sys em is ackled nume ically ia MATLAB’s in insic bounda y-
alue sol e b p4c. The nume ical solu ions hus ob ained a e employed o cons uc eloci y
and concen a ion dis ibu ions o a ying magni udes o he go e ning pa ame e s. C i ical
examina ion o he gene a ed p o iles demons a es ha bo h he hyd odynamic ield and mass
anspo phenomena a e p o oundly modula ed by he in luence o he con olling low
pa ame e s.
▪ ISSN No: 2583-7397
▪ Recei ed: 08-09-2025
▪ Accep ed: 30-10-2025
▪ Published: 05-11-2025
▪ IJCRM:4(6); 2025: 11-18
▪ ©2025, All Righ s Rese ed
▪ Plagia ism Checked: Yes
▪ Pee Re iew P ocess: Yes
How o Ci e his A icle
Saha. B. K. T ansien Bounda y-
Laye Dynamics o An Elas ico-
Viscous Medium Ac oss a
De o mable Shee Subjec ed o
Chemical Reac i i y. In J Con emp
Res Mul idiscip. 2025;4(6):11-18.
Access his A icle Online
www.mul ia iclesjou nal.com
KEYWORDS: Uns eady Flow; Elas ico-Viscous Fluid; Bounda y Laye ; De o ming Su ace; Chemical Reac ion.
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1. INTRODUCTION
The powe -law model is commonly applied o desc ibe luids
whose iscosi y a ies wi h shea a e. Howe e , i ails o
accoun o elas ic e ec s. In con as , second and hi d-g ade
luid models can cap u e elas ici y, bu hei iscosi y does no
depend on shea a e (Haya e al. [1]. Addi ionally, hese
models canno ep esen s ess elaxa ion phenomena.
Fu he mo e, hese models ail o cap u e s ess elaxa ion
e ec s. Con e sely, he Maxwell cons i u i e amewo k,
classi ied wi hin he amily o a e- ype iscoelas ic luids,
possesses he inhe en capabili y o cap u e s ess- elaxa ion
phenomena, he eby secu ing b oade ecogni ion and
applicabili y (Abel e al. [2]). Unlike o mula ions in ol ing
shea -dependen iscosi y, his model ci cum en s such
analy ical in icacies in bounda y-laye in es iga ions, he eby
enabling he analysis wi h pa icula emphasis on he ole o
luid elas ici y in go e ning bounda y-laye dynamics (Heyha
and Khabazi [3]).
The bu geoning in e es in magne o-hyd odynamic (MHD)
phenomena is p incipally a ibu ed o i s ex ensi e spec um o
p ac ical applica ions, encompassing domains such as
pe oleum and na u al gas ex ac ion, geophysical luid
dynamics, and inno a ions in ag o-enginee ing sys ems. The
modula ion o elec ically conduc ing luids by applied
magne ics lux exe s a decisi e in luence on he ope a ional
e iciency o di e se indus ial de ices, including
hyd odynamic bea ings, MHD-based gene a o s, and
elec omagne ic pumps. Beyond he indus ial ealm, MHD has
demons a ed e sa ili y wi hin biomedical enginee ing, whe e
i unde pins echniques o umou abla ion, accele a ion o
wound healing, gas ic he apies, and s e ilisa ion o su gical
ins umen s (Shehzad and Heyha [4–5]). In es iga ions o
MHD bounda y-laye dynamics unde he mal s a i ica ion
ha e elucida ed ha in ensi ica ion o he magne ic in e ac ion
pa ame e supp esses he eloci y ield, while simul aneously
inducing a ansien in ensi ica ion o he skin- ic ion
coe icien . Analogous obse a ions we e epo ed by Tian e al.
[6], who analysed he coupled e ec s o adia i e op ical
cha ac e is ics and Lo en z body o ces on MHD bounda y-
laye low pas a de o mable su ace. Complemen a y s udies
ha e also b oadened he scope o inqui y: one examined he
exis ence o dual simila i y solu ions in he con ex o MHD
anspo o e a nonlinea po ous sh inking shee imme sed in a
iscous medium, whe eas ano he , conduc ed by Jusoh e al.
[7], explo ed MHD-d i en o a ing bounda y laye s in he
p esence o a pe meable s e ching/sh inking shee .
The analysis o luid low and he mal anspo a ains
heigh ened complexi y when examined wi hin po ous domains
exhibi ing spa ially a ying pe meabili y. In such sys ems, he
low beha iou and he mal anspo cha ac e is ics a e
signi ican ly in luenced by spa ial a ia ions in pe meabili y
(Ullah e al. [8]). The he e ogeneous s uc u e o he po ous
ma e ial c ea es in ica e in e ac ions be ween he luid and he
ma ix, leading o luc ua ions in eloci y, p essu e, and
empe a u e. These a ia ions can educe he e iciency o hea
ans e and may igge low ins abili ies, including o ex
o ma ion and low eci cula ion zones (San os-Mo eno e al.
[9]). Wi hin po ous s uc u es, bo h con ec ion and conduc ion
con ibu e o hea ans e , while changes in po osi y in oduce
he mal non-uni o mi y. Consequen ly, he he mal ield wi hin
a po ous subs a e unde goes modula ion as he luid a e ses’
zones o spa ially he e ogeneous pe meabili y (Ullah [10]). The
de elopmen o obus p edic i e amewo ks and high- ideli y
models o a wide spec um o enginee ing and en i onmen al
applica ions— anging om subsu ace hyd ological p ocesses
o enewable ene gy sys ems—necessi a es an in e disciplina y
syn hesis o luid dynamics, hea - ans e heo y, and po ous-
media physics. This equi emen o igina es om he highly
nonlinea in e ac ion be ween luid momen um ans e and
hea anspo p ocesses embedded in spa ially he e ogeneous
po ous amewo ks, a opic ha has ecen ly a ac ed
conside able esea ch a en ion ac oss mul iple dimensions
(Sowmiya and Kuma , Nabwey e al., Reddy e al., Rehman
and Salleh [11–14]).
Mo eo e , hyb id nano luids—enginee ed h ough he
suspension o mul iple dis inc nanopa icle species wi hin a
base ca ie luid—ha e been employed o augmen he he mo-
physical pe o mance beyond ha achie able wi h con en ional
mono-nanopa icle o mula ions, as he combined physical
cha ac e is ics o di e en nanopa icles p omo e enhanced
ene gy ans e (Gangadha e al. [15]). Magne ically
in luenced, yield-s ess-based hyb id nano luid (wi h sodium
algina e as a ca ie ) beha es when subjec ed o squeezing
o ces, while also conside ing he coupled he mal and mass
di usion e ec s (So e and Du ou ) by Noo e al. [16].
Se e al esea che s ([17–18]) ha e in es iga ed uns eady
s e ching su ace p oblems unde a ious condi ions,
employing simila i y ans o ma ions o educe he go e ning
uns eady bounda y-laye model is mapped in o a sys em o
o dina y di e en ial equa ions. Mukhopadhyay and
Bha acha yya [19] conduc ed an assessmen o he ansien
low dynamics o Maxwell luid along a s e ching su ace
subjec ed o a chemical eac ion.
2. Go e ning Equa ions and Fo mula ion:
A bidi ec ional lamina bounda y-laye anspo accompanied
by mass anspo o an incomp essible non-New onian luid
o e an uns eady s e ching shee is explo ed. The solu e
concen a ions p esc ibed a he shee and in he a - ield a e
ep esen ed by 𝐶𝑊 and 𝐶∞, espec i ely. The di using species
is assumed o pa icipa e in a i s -o de homogeneous chemical
eac ion go e ned by a ime-dependen a e cons an . 𝑍1.
Fo <0Bo h luid and mass lows a e pe sis en , while
uns eadiness a ises a = 0. The shee issues om a sli a he
o igin (x = 0, y = 0) and s e ches wi h a eloci y U(x, ) = 𝑏𝑥
1−𝛼𝑡
whe e b and 𝛼 a e posi i e cons an s o dimension (𝑡𝑖𝑚𝑒)−1.
In his o mula ion, he pa ame e b cha ac e ises he p ima y
s e ching a e, whe eas he e m 𝑏
1−𝛼𝑡 signi ies he
ins an aneous e ec i e s e ching a e, which p og essi ely
ampli ies as ime ad ances. Wi hin he amewo k o polyme
ex usion, he empo al e olu ion induces a ia ions in he
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physico-mechanical a ibu es o he eme ging shee . The
co esponding sys em o go e ning equa ions is exp essed as
ollows:
𝜕𝑢
𝜕𝑥+𝜕𝑣
𝜕𝑦=0 (2.1)
𝜕𝑢
𝜕𝑡+𝑢𝜕𝑢
𝜕𝑥+𝑣𝜕𝑢
𝜕𝑦=𝜐𝜕2𝑢
𝜕𝑦2−𝑘0
𝜌[𝜕
𝜕𝑡(𝜕2𝑢
𝜕𝑦2) + 𝑢 𝜕3𝑢
𝜕𝑥𝜕𝑦2+𝑣𝜕3𝑦
𝜕𝑦3−𝜕𝑢
𝜕𝑦𝜕2𝑢
𝜕𝑥𝜕𝑦−𝜕𝑣
𝜕𝑦𝜕2𝑢
𝜕𝑦2] (2.2)
𝜕𝑐
𝜕𝑡+𝑢𝜕𝑐
𝜕𝑥+𝑣𝜕𝑐
𝜕𝑦= 𝐷𝜕2𝑐
𝜕𝑦2− 𝑧1(𝑐−𝑐∞) (2.3)
Fig. 1 .1. Geome ic Con igu a ion o he Flow Field
In his con ex , u and co espond o he espec i e eloci y
componen s aligned wi h he Ca esian axes; x and y designa e
he kinema ic iscosi y o he wo king luid; c signi ies he
scala ield desc ibing species concen a ion wi hin he
medium; D deno es he molecula di usi i y o he solu e
dispe sed in he luid. The eac ion kine ics, exp essed as a
unc ion o empo al a ia ion, a e gi en by 𝑧1(𝑡)= 𝑧0
1−𝛼𝑡
whe e 𝑧0 I is a cons an . A posi i e alue o 𝑧1> 0
co esponds o a des uc i e eac ion, while a nega i e alue
𝑧1<0 Indica es a cons uc i e eac ion.
The o mula ion o he p oblem adhe es o he subsequen se o
bounda y cons ain s:
𝑢=𝑈(𝑥,𝑡),𝑣=0, 𝑐= 𝑐𝑤(𝑥,𝑡) 𝑎𝑡 𝑦=0 (2.4)
𝑢 →0, 𝑐 → ∞ 𝑎𝑠 𝑦 → ∞ (2.5)
The shee ’s su ace concen a ion a ies wi h bo h posi ion and ime, de ined as
𝑐𝑤(𝑥,𝑡) = 𝑐∞+𝑏𝑥(1−𝛼𝑡)−2, whe e 𝑐∞ Is he uni o m ee-s eam concen a ion. Fo (b > 0) 𝑐𝑤 inc eases wi h x; o ( b < 0 ), i
dec eases, wi h he a ia ion ampli ying o e ime. The ela ions o 𝑈(𝑥,𝑡), 𝑐𝑤(𝑥,𝑡) and 𝑧1(𝑡) a e alid o 𝛼−1.
We de ine u and h ough he ollowing exp essions:
𝑢= 𝜕𝜓
𝜕𝑦,𝑣= −𝜕𝜓
𝜕𝑥 𝑎𝑛𝑑 𝜙= 𝑐−𝑐∞
𝑐𝑤−𝑐∞ (2.6)
Simila i y-based ans o ma ion
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𝜂=𝑦√𝑏
𝜐(1−𝛼𝑡), 𝜓= √𝜐𝑏
1−𝛼𝑡𝑥𝑓(𝜂), 𝑐= 𝑐∞+𝑏𝑥(1−𝛼𝑡)−2𝜙(𝜂) (2.7)
By applying ela ions (2.6) and (2.7) o equa ions (2.2) and (2.3), he go e ning equa ion educes o
𝑀(𝜂
2𝑓′′+𝑓′) + (𝑓′)2−𝑓𝑓′′− 𝑓′′′+ 𝑘1[ 𝑀𝜂
2𝑓′𝑣+(2𝑀+1)𝑓′′′+ 𝑓𝑓′𝑣+𝑓′𝑓′′′− (𝑓′′)2= 0 (2.8)
𝑀(𝜂
2𝜙′+2𝜙)+𝑓′𝜙−𝑓𝜙′− 1
𝑆𝑐𝜙′′− 𝛾𝜙= 0 (2.9)
He e, M = 𝛼
𝑏 Deno es he uns eadiness pa ame e , k1 = 𝑘0𝑏
𝜇(1−𝛼𝑡) ep esen s he elas ico- iscous pa ame e , Sc = 𝜐
𝐷 Is he
Schmid numbe , and 𝛾= 𝑧0
𝑏 co esponds o he eac ion a e pa ame e
Unde he ans o ma ion, he bounda y condi ions become
𝑓′(𝜂)=1,𝑓(𝜂)=0,𝜙(𝜂)=1 𝑎𝑡 η =0 (2.10)
𝑓′(𝜂) →0, 𝜙(𝜂) →0 𝑎𝑠 η →∞ (2.11)
3. Solu ion Scheme
Equa ions (2.8) and (2.9), ep esen ing he sel -simila nonlinea o m, a e ans o med in o i s -o de di e en ial ela ions
speci ied by:
𝑓=𝑧1,𝑓′=𝑧2, 𝑓′′=𝑧3,𝑓′′′=𝑧4 ,𝜙=𝑧5,𝜙′=𝑧6 (3.1)
F om ela ion (3.1), i ollows ha .
𝑧1′=𝑧2,𝑧2′=𝑧3,𝑧3′=𝑧4,𝑧5′=𝑧6 (3.2)
Employing ela ions (3.1) and (3.2), equa ions (2.8) and (2.9) may be exp essed as:
𝑧4′= 1
𝑀𝜂
2+ 𝑧1 [𝑧32−𝑧2𝑧4−(2𝑀+1)𝑧4+ 1
𝑘1 {𝑧1𝑧3+ 𝑧4−𝑧22− 𝑀 ( 𝜂
2𝑧3+ 𝑧2)}] (3.3)
𝑧6′=𝑠𝑐 { 𝑧2𝑧5−𝑧1𝑧6− 𝛾𝑧5+ 𝑀 ( 𝜂
2𝑧6+2𝑧5) (3.4)
While he bounda y condi ions (2.10) and (2.11) a e ans o med as ollows:
𝑧1(0)=0,𝑧2(0)=1 𝑎𝑛𝑑 𝑧3(0)=0 ,𝑧5(0)=1 (3.5)
𝑧2(∞)=0,𝑧5(∞)=0 (3.6)
Equa ions (3.3) – (3.4), subjec o bounda y cons ain s (3.5) – (3.6), a e nume ically esol ed ia MATLAB’s b p4c scheme
o a ange o low-pa ame e alues examined in his s udy.
4. Resul s and Discussion:
The MATLAB sol e b p4c is used o compu e eloci y and
concen a ion p o iles, demons a ing he e ec o go e ning
pa ame e s on low beha iou (Figu es 2–8). Valida ion is
ca ied ou by compa ing he compu ed skin ic ion coe icien
𝑓′′(0) Wi h es ablished esul s showing close ag eemen (Table
4.1) and con i ming accu acy.
Table 4.1: The alues o 𝑓′′(0) co esponding o di e en uns eadiness pa ame e s M when 𝑘1=0
M
Sha idan e al. [17]
Chamakha e . al. [18]
Mukhopadhyay and Bha acha yya [19]
P esen S udy
0.8
-1.261042
-1.261512
-1.261479
-1.261450
1.2
-1.377722
-1.378052
-1.377850
-1.377845
Fig.4.1. The eloci y dis ibu ion co esponding o di e en
alues o he elas ico- iscous coe icien 𝑘1 A e p esen ed. The
eloci y exhibi s an ini ial a enua ion, ollowed by a
p og essi e augmen a ion wi h inc easing elas ico- iscous
e ec s, and ul ima ely unde goes a downs eam decay o a
p esc ibed uns eadiness pa ame e M.
Fig.4.2. Depic s he implica ion o he uns eadiness pa ame e
M on he eloci y dis ibu ion. 𝑓′(𝜂). Wi h inc easing M, he
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eloci y along he shee exhibi s a ma ked educ ion, while
a he om he su ace, i e eals non-uni o m luc ua ions. In
he ini ial egion, he momen um di usion-laye hickness, bo h
adjacen o and away om he shee , enla ges wi h highe M.
The special case M = 0 signi ies he s eady-s a e egime.
Fig.4.3 p esen s he concen a ion dis ibu ion. ∅(𝜂) Unde
a ying magni udes o he uns eadiness pa ame e M. The
indings e eal ha concen a ion exhibi s an ini ial ise,
ollowed by a g adual decay as M inc eases. The mass ans e
a e, hough signi ican nea he onse , diminishes p og essi ely
wi h highe M. Mo eo e , a any ixed spa ial loca ion,
concen a ion declines sha ply wi h inc easing M. Since luid
mo ion is induced exclusi ely by he s e ching shee —whe e
su ace concen a ion exceeds he ambien ee-s eam le el—
he concen a ion p o ile consis en ly dec eases wi h 𝜂 Unde
he in luence o he elas ico- iscous pa ame e .
Fig.4.4. Depic s he implica ion o he Schmid numbe Sc on
he concen a ion dis ibu ion. ∅(𝜂). An in iguing beha iou
eme ges whe ein concen a ion i s exhibi s a ise bu
subsequen ly diminishes wi h escala ing Sc. Fu he mo e, he
solu e di usion-laye hickness con ac s as Sc g ows and
ul ima ely app oaches a s abilised s a e a a ini e loca ion.
Fig.4.5. Depic s he consequence o he gene a i e eac ion
coe icien . 𝛾 (< 0) on he concen a ion dis ibu ion ∅(𝜂). Wi h
inc easing 𝛾 The concen a ion diminishes signi ican ly,
esul ing in a lowe ed mass di usion a e. Fo cons uc i e
chemical eac ions 𝛾 (< 0), he concen a ion ield unde goes a
ma ked educ ion unde he go e ning in luence o he elas ico-
iscous e ec s. The p o ile i s dec eases, hen ises a e a
ce ain dis ance as he eac ion a e pa ame e dec eases, and
e en ually s abilises a a poin along he shee . Addi ionally, he
di usion laye hickness is ound o dec ease p og essi ely.
Fig.4.6. Po ays he in luence o he des uc i e eac ion
coe icien . 𝛾 (> 0) on he concen a ion dis ibu ion ∅(𝜂). The
concen a ion exhibi s a p onounced decline wi h inc easing. 𝛾,
ye beyond 𝜂=2 i ises, he eby augmen ing he mass di usion
a e om he luid domain owa d he su ace. Owing o he
elas ico- iscous pa ame e , concen a ion ini ially d ops
apidly, hen dec eases mo e g adually wi h ising eac ion a e
pa ame e , and e en ually s abilises a he shee . The di usion
laye hickness ini ially g ows quickly due o he combined
elas ic and iscous e ec s, bu wi h u he inc ease in he
eac ion a e pa ame e , i g adually becomes hinne .
Fig.4.7. As he elas ico- iscous pa ame e escala es, he
concen a ion p o ile ∅(𝜼) d ops sha ply a 𝜂=4, hen ises
quickly and con e ges a a ce ain poin . This end indica es
ha highe elas ico- iscous pa ame e alues cause a signi ican
educ ion in he mass di usion a e. In gene al, he
concen a ion dis ibu ion diminishes wi h ising elas ico-
iscous pa ame e , whe eas he hickness o he di usion laye
co espondingly expands.
Fig. 4.1. Implica ion o 𝑘1 on Veloci y P o ile 𝑓′(𝜂)
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Fig.4.2. Implica ion o M on Veloci y P o ile 𝑓′(𝜂)
Fig.4.3. Implica ion o M on Concen a ion P o ile ∅(𝜂)
Fig.4.4. Implica ion o Sc on Concen a ion P o ile ∅(𝜂)
In . J . o Con emp. Res. in Mul i. PEER-REVIEWED JOURNAL Volume 4 Issue 6 [No - Dec] Yea 2025
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Fig4.5. Implica ion o 𝛾 (𝛾 < 0) on Concen a ion P o ile ∅(𝜂)
Fig.4.6. Implica ion o 𝛾 (𝛾 > 0) on Concen a ion P o ile ∅(𝜂)
Fig.4.7. Implica ion o 𝑘1 On Concen a ion P o ile ∅(𝜂)
In . J . o Con emp. Res. in Mul i. PEER-REVIEWED JOURNAL Volume 4 Issue 6 [No - Dec] Yea 2025
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CONCLUSION
The key indings o his esea ch a e summa ised below:
(i) The eloci y p o ile climbs down and oscilla es o he
g ow h o he elas ic- iscous pa ame e , and oscilla es
a e a e sing a ew dis ances on he shee wi h he
g ow h o he uns eadiness pa ame e .
(ii) The concen a ion p o ile dec eases wi h he g ow h o
he luid low pa ame e in ol ed in his esea ch.
(iii) The shee su ace expe iences a dec ease in mass
ans e a e as g ow h o bo h uns eadiness and
elas ico- iscous pa ame e .
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Abou he Co esponding Au ho
D . Bikash Koli Saha is an Independen Resea che
and Educa o in Ma hema ics, specialising in luid
dynamics, bounda y laye heo y, and ma hema ical
physics. Fo me ly an Assis an P o esso a Don
Bosco College, Tu a, he emains dedica ed o
ad ancing esea ch in luid mechanics, hea ans e ,
and applied di e en ial equa ions h ough analy ical
p ecision and inno a ion.
