S abiliza ion o pe iodically o ced Hele-Shaw lows by means o a non-mono onous
iscosi y p o ile
Vicen e P´e ez-Mu˜nuzu i∗
CRETUS. G oup o Nonlinea Physics, Facul y o Physics,
Uni e si y o San iago de Compos ela. E-15782 San iago de Compos ela, Spain
(Da ed: May 21, 2022)
The onse o iscous inge ing in he p esence o a iscosi y p o ile is in es iga ed heo e ically o
wo immiscible luids unde going a ime-dependen injec ion. He e, we show ha he p esence o a
posi i e iscosi y g adien a he in e ace be ween bo h luids s abilizes he in e ace acili a ing he
sp ead o he pe u ba ion. This e ec is much mo e p onounced in he case o sinusoidal injec ion
lows. The in luence o he iscosi y g adien on he dispe sion ela ion is analyzed. Nume ical
simula ions o he Na ie -S okes equa ion con i m he linea s abili y analysis.
INTRODUCTION
Sa man-Taylo ins abili y [1] may a ise when wo lu-
ids o di e en iscosi ies a e pushed by a p essu e g a-
dien h ough wo ho izon al pa allel pla es (Hele-Shaw
(HS) cell). I is well known, bo h expe imen ally and
heo e ically, ha when a less iscous New onian luid
displaces a mo e iscous one, a inge ing ins abili y de-
elops a he in e ace be ween bo h immiscible luids [2].
Fo non New onian luids, an unexpec ed p opaga ion o
ac u es may de elop in he in aded luid o speci ic
condi ions [3–5].
Due o chemical and oil eco e y applica ions, he pos-
sibili y o con ol his in e acial ins abili y has been
he subjec o nume ous s udies in he las decades [6].
Chemical eac ions occu ing a he in e ace d i e is-
cosi y changes ha modula e he hyd odynamic iscous
inge ing ins abili y, bo h o uns able [7–10] o s able
[11] on s. Fo he las case, o example, when a change
in pH is chemically induced a he in e ace, some inge s
may de elop [12, 13]. Wi hou using a chemical eac ion,
he p ocess in ol es displacing he mos iscous luid i s
wi h some polyme - hickened luid (usually an aqueous
phase con aining wa e and polyme in app op ia e p o-
po ion o ha e he desi ed p o ile) ollowed by he less
iscous one. This s a egy basically in ol es a h ee-laye
luid wi h an in e media e laye o ini e hickness con-
aining he polyme solu ion [14]. O he possibili ies o
iscous inge ing in he p esence o non-mono onic is-
cous p o iles ha e also been s udied [15, 16]. O he non-
chemical s a egies o con ol his ins abili y include, im-
posing a ime-dependen injec ion a e, ape ing he gap
be ween pla es so hey a e no longe pa allel [17, 18], e-
placing one o he pla es wi h an elas ic memb ane [19],
o a ing he HS cell while he in iscid luid is injec ed
[25], o compe i ion be ween g a i y and iscous o ces
[20], among o he s.
Expe imen s and nume ical simula ions in ol ing he
∗Email: icen e.pe ez.m[email p o ec ed]
injec ion o a less iscous luid in o a mo e iscous one
in a adial HS cell showed ha s onge injec ion a es
esul in an enhancemen o iscous inge ing. Mos o
he exis ing s udies ha e ocused on displacemen s unde
cons an injec ion. The e a e howe e nume ous p ac i-
cal p ocesses whe e he injec ion is ac ually ime depen-
den [21–26]. Mos o hese s udies showed ha be e
eco e y can be achie ed h ough ime-dependen injec-
ion schemes in compa ison wi h cons an injec ion ones
due o an a enua ion o he inge ins abili y. On he
o he hand, se e al au ho s [27] demons a ed ha o
a pe iodic injec ion on a adial HS cell he numbe o
inge s and hei s uc u es could be con olled. As op-
posi e, o he au ho s ha e shown ha inge ing des abi-
liza ion due o pe iodic injec ion a o s luid mixing in
con ined sys ems, such as in mic o luidic de ices [28].
Based on he esul s epo ed, i is clea ha injec ion
a es and pe iodici y play an impo an ole o con ol
he low beha io and in e ace ins abili y. The objec-
i e o he p esen s udy is o analyze he combina ion o
imposing a ime-dependen injec ion a e wi h he p es-
ence o non mono onic iscosi y p o iles a he in e ace.
While we obse ed ha he pe iodic injec ion leads o a
esonance-like e ec des abilizing he in e ace, he p es-
ence o a h ee-laye non-mono onic iscosi y p o ile s a-
bilize he on achie ing be e eco e y esul s o he
iscous luid. In his wo k, we p esen a linea s abili y
analysis o he onse o iscous inge ing o wo immis-
cible luids subjec o a pe iodic injec ion and a iscosi y
g adien a he in e ace. Viscous inge ing has been ob-
se ed nume ically con i ming his analysis and he de-
pendence wi h pa ame e s o he model s udied.
LINEAR STABILITY ANALYSIS
The equa ion o mo ion o an incomp essible New o-
nian luid is gi en by,
ρDui
D =−∂p
∂xi
+∂
∂xj
(2µeij),(1)
2
FIG. 1. Ske ch o he Hele-Shaw cell (a) and he piecewise
iscosi y p o ile µ(x1) (b) used he e. Ls ands o he egion
whe e a linea iscosi y g adien is conside ed be ween bo h
luids wi h cons an iscosi ies µaand µb.
∂ui
∂xi
= 0 (2)
whe e D/D is he ma e ial de i a i e, ρis he cons an
densi y, ui he eloci y ield, p he p essu e, µ he dy-
namic iscosi y, and eij = (∂ui/∂xj+∂uj/∂xi)/2 is he
s ain a e enso . Fo a non mono onic iscosi y p o ile
µ=µ(xi), Eq. (1) becomes,
ρDui
D =−∂p
∂xi
+µ∂2ui
∂xj∂xj
+ 2eij
∂µ
∂xj
(3)
We conside he case whe e a iscous luid wi h is-
cosi y µais pushing a mo e iscous one µbin he x1-
di ec ion be ween closely spaced pa allel pla es sepa a ed
some dis ance has i is shown in Fig. 1(a), subjec o
a pe u ba ion a he in e ace x1=ξ(x3, ). In be-
ween bo h luids a egion wi h a iscosi y g adien is
conside ed, Fig. 1(b). In he basic HS low, we sup-
pose a cons an nega i e p essu e g adien along he x1
axis so ha he low goes om he le (x1<0) o he
igh (x1>0) wi h eloci y ield [u0
1(x2),0,0]. Follow-
ing he s anda d decomposi ion in no mal modes, we
pe u b he sys em o equa ions (3). Thus, he pe -
u bed eloci y ield is [u0
1(x2) + ϵu1
1(xi, ),0, ϵu1
3(xi, )],
he p essu e P0+ϵP1(xi, ), and he in e ace equa ion
is x1=ξ0( ) + ϵξ1(x3, ) (ϵ≪1). Assuming a sinusoidal
pe u ba ion along he x3axis, he pe u bed quan i ies
can be w i en as,
ξ1(x3, ) = ξexp(ık3x3+ω )
u1
1(xi, ) = ua,b(x2) exp(ık3x3+ω ) exp (±k1x1)
u1
3(xi, ) = wa,b(x2) exp(ık3x3+ω ) exp (±k1x1)
P1(x1, x3, ) = Pa,b exp(ık3x3+ω ) exp (±k1x1)
whe e ±s ands o he le (a) and igh (b) luids wi h
iscosi ies µaand µb, espec i ely. ξ,u(x2), w(x2), and P
a e he ampli udes o he no mal modes o he pe u ba-
ion. The p oblem is comple ed by he no-slip bounda y
condi ion a he pla es ui= 0 o x2= 0, h. Wi hou
loss o gene ali y, om now on, we assume a piecewise
iscosi y p o ile µ=µ(x1) as depic ed in Fig. 1(b).
Rew i ing Eq. (3) a ze o o de we ob ain,
u0
1(x2) = G
2µa,b x2
2−hx2(4)
wi h G he nega i e p essu e g adien along he x1
axis. A e aging in he x2di ec ion, he Da cy’s law
U=⟨u0
1⟩=−Gh2/12µa,b is ob ained. The in e ace
be ween he wo luids is x1=ξ0( ) wi h ξ0( ) = U .
A i s o de , he ampli udes o he no mal modes ua,b
a e gi en by,
ρ(ωua,b ±Ukua,b) = ∓Pa,b −12µa,b
h2ua,b ±2µ′
a,b(x)kua,b
(5)
wi h µ′
a,b(x) = ∂µa,b/∂x, and we ha e used he incom-
p essibili y o he low o show ha k=k1=k3. A
simila equa ion can be ob ained o he mode wa,b.
Th ee condi ions mus be imposed a he in e ace,
namely he kinema ical condi ion and he condi ion o
con inui y o no mal and angen ial s esses,
u1
1(x2)a,b =∂ξ1
∂ ,
P0
a+P1
a−P0
b+P1
b=−γ∂2ξ1
∂x2
3
,(6)
µa
∂ua
∂x2
=µb
∂ub
∂x2
whe e γis he su ace ension a he in e ace, and P0is
ob ained in eg a ing he Da cy’s law.
Sol ing analy ically he abo e equa ions, we ob ain he
dispe sion ela ion,
ω(k) = Uk(µb−µa)−γh2
12 k3
µa+µb+kh2
6(µ′
a−µ′
b)(7)
No e ha he classical Sa man-Taylo dispe sion ela-
ion (µb> µa) [2] is eco e ed o µ′
a,b = 0. An im-
p o emen o he s abili y (lowe alues o ω(k)) is ob-
ained o µ′
a−µ′
b>0 compa ed o he Sa man-Taylo
case. Fig. 2 shows his imp o emen as he iscosi y g a-
dien inc eases o µ′
a= (αµb−µa)/L (α=µ(0)/µb) and
µ′
b= 0.
NUMERICAL SIMULATIONS
Two-dimensional (x1, x3) di ec nume ical simula ions
(DNS) o he Na ie -S okes equa ions (3) ha e been used
wi h bounda y condi ions ui= 0 o x3= 0, xmax
3,
and ∂1ui= 0 o x1= 0, xmax
1. A he in e ace be-
ween bo h luids, he kinema ical condi ion and he
condi ion o con inui y o no mal and angen ial s esses
hold. Eqs. (3) we e in eg a ed using an implici C ank-
Nicholson me hod o he ad ec ion and di usion e ms.
P essu e is ob ained by means o a ac ional-s ep, o
ime-spli ing scheme [29, 30]. P ocess o sol ing he mo-
men um and he p essu e Poisson equa ions is epea ed
un il he eloci y ield is di e gence ee a each ins an
o ime. The p essu e g adien is assumed o oscilla e as
3
FIG. 2. Maximum g ow h a e ωmax as a unc ion o α,µ′
a=
(αµb−µa)/L, o he uns able case (µb> µa). Dashed line
co esponds o he Sa man-Taylo case. Se o pa ame e s:
h= 10−3m, U= 0.01 m/s, wa e µa= 0.896 ·10−3Pa s,
glyce ine µb= 0.95 Pa s, γ= 29.06 ·10−3N/m, L= 0.01, and
µ′
b= 0
∂1P=G+P′sin(Ω ) wi h G he nega i e p essu e g adi-
en along he x1-axis. The p essu e bounda y condi ions
a e o Neumann ype on x3= 0, xmax
3.
The assumed ec angula compu a ional domain is dis-
c e ized on a s agge ed Ca esian g id when p essu e P
and eloci ies a e de e mined on h ee g ids shi ed ela-
i e o each o he . So, Pis loca ed in he cen e o each
cell, he x1-componen eloci y u1is on he middle poin s
o e ical aces, and he x3-componen eloci y u3is on
he middle poin s o ho izon al aces.
The con e gence o he code was i s es ed by a ying
he spa ial esolu ion om 1000 ×100 o 4000 ×400.
Acco dingly, he ime s ep was changed om 0.001 o
0.1, and nea ly he same ime e olu ion o inge s we e
ob ained. To ensu e a good nume ical con e gence and
ela i ely sho e calcula ion ime, a spa ial esolu ion o
2000 ×200 and a ime s ep ∆ = 0.01 we e adop ed o
he ange o pa ame e s examined h oughou his s udy.
Ini ially we suppose an inle on o iscosi y µapush-
ing pe iodically ano he luid wi h iscosi y µb> µaand
he in e ace be ween bo h luids has an ini ial sinusoidal
pe u ba ion (ampli ude 20 and wa enumbe π/50) and
a iscosi y g adien µ′
a.
RESULTS
I is well known ha o HS lows, unde uns able con-
di ions (µb> µa) he in e ace be ween bo h luids be-
comes uns able and some inge s de elop. This p ocess
is enhanced unde a pulsa ing inle low. Figu e 3 shows,
o wo ins an s o ime, he luids in e ace o h ee di -
FIG. 3. In e ace on s be ween luids o h ee o cing e-
quencies Ω, wi h (blue and g een dashed lines) and wi hou
pe iodic o cing ( ed do -dashed line). G een dashed in e ace
co esponds o an in e media e ime. α= 0.4, G=−1 Pa/m,
P′= 1 Pa/m and es o pa ame e s as in Fig. 2. F om le
o igh , Ω = 0.004,0.012,0.026, espec i ely.
FIG. 4. Time e olu ion o he su ace a io S o wo alues
o he a io α=µ(0)/µb. Ω = 0.012.
e en o cing equencies Ω compa ed o he case wi hou
o cing. No e ha in all cases he in e ace has ad anced
as e unde o cing, and he ne displacemen depends
on he o cing equency. Fo an in e media e alue o Ω
he on ad ances as e han o he o he wo equen-
cies. The a io o he su aces S co e ed by he low
in e ace, wi h and wi hou o cing, allows us o quan-
i y he e ec o o cing, and can be conside ed p opo -
ional o he a io be ween he linea g ow h a es o
bo h cases. Figu e 4 shows he ime e olu ion o S o
wo alues o α o he in e media e equency shown in
Fig. 3. No e ha he on accele a es wi h ime, as e
as αdiminishes.
4
Figu e 5 shows he a io S as a unc ion o he o cing
equency Ω o he same ins an o ime as in Fig. 3.
No e he signi ican enhancemen o he low ins abili y a
ce ain in e media e equencies ha was clea ly isible
in Fig. 3. Fo e y small and la ge o cing equencies,
S →S (P′= 0) = 1. Maximum S alue was a ained
always o he same esonan equency Ω es o any alue
o he a io α=µ(0)/µb, and seems o be he signa u e o
a esonan beha io be ween he cha ac e is ic ime scale
o he low, a io o he su ace ension and he iscous
(p essu e d op) o ces,
ch ≡γ
U∆P∝γ
µb−µa
(8)
and he o cing equency. A u he in es iga ion o his
phenomenon was ca ied ou o o he alues o iscos-
i y µband su ace ension γ. Changing hese pa ame e s
modi ies he s uc u e o he in e ace ( o example, in-
c easing γleads o a smoo hing o he on ), howe e ,
he phenomenon o esonance was ound in all examined
cases, al hough a change in he esonan pe iod T es was
obse ed (Fig. 6) consis en wi h ch (8). The p e ious
esul s show ha a he esonan equency, he in e ace
can be cha ac e ized by an inc eased ins abili y, which
appea s o eme ge ea lie and be s onge han o he
o he low pe iods.
The dependence o he low ins abili y enhancemen
a he esonance peak on he alues o αand he leng h
o he middle laye Lis shown in Fig. 7. No e ha in-
c easing p(L), inc eases (dec eases) he iscosi y g adi-
en µ′
a= (αµb−µa)/L a he in e ace, and S diminishes
(inc eases) as expec ed om he heo e ical dispe sion e-
la ion, Eq. (7) (Fig. 2). Inc easing he g adien leads o
s abilize he on diminishing he g ow h a e o ins a-
bili ies a he in e ace. Fo α→µa/µb he classical
wo-laye Sa man-Taylo ins abili y is ob ained bu he
isk o in e ace b eakup inc eases. Fo smalle a ios
µb/µa han hose used he e, i is possible o each ha
limi wi hou he in e ace e en ually b eaks up. No e in
Fig. 3 he o ma ion o plumes o spikes when pe iodic
o cing is applied. When no pe iodic o cing is consid-
e ed, he ini ial sinusoidal pe u ba ion e ol es in o a
single inge and no cusps we e obse ed. These plumes
smoo h ou as su ace ension inc eases. As αdiminishes,
he in e ace mo es as e , inc easing ine ial o ces com-
pa ed o su ace ension (la ge Webe numbe ), a ou ing
he o ma ion o elonga ed plumes and he in e ace may
b eak up as i has been obse ed in o he in e acial in-
s abili ies [31]. Thus, we may assume ha pe iodic o c-
ing inc eases ine ial o ces as he in e ace mo es as e
han wi hou o cing, and cusps/plumes appea be o e
he in e ace b eaks up. Once his happens ou nume ical
algo i hm is no s able. Di e en nume ical app oaches
should hen be unde aken bu hey a e beyond he scope
o his pape .
FIG. 5. Su ace a io S co e ed by he luid in e aces, wi h
and wi hou pe iodic o cing, as a unc ion o he o cing e-
quency Ω o wo alues o he a io α=µ(0)/µb. Pa ame e s
as in Fig. 2.
FIG. 6. Resonan pe iod T es as a unc ion o he iscosi y
(a) and he su ace ension γ(b). In all cases, he beha io o
T es coincides wi h he cha ac e is ic ime o he low ch ∝
γ/(µb−µa). γ= 29.06 ·10−3N/m (a) and µb= 0.2 Pa s (b).
α= 0.4 and µa= 0.89 ·10−3Pa s in all cases.
CONCLUSIONS
The onse condi ion o iscous inge ing o a luid dis-
placing a mo e iscous one in a Hele-Shaw cell unde going
ime-dependen injec ion in a homogeneous medium has
been s udied in he p esence o a non mono onic iscos-
i y p o ile in he di ec ion o mo ion. Fo wa e numbe s
abo e a c i ical one, pe u ba ions a bo h sides o he
in e ace sp ead in he same di ec ion, des abilizing he
in e ace. The sp eading eloci y is modula ed by he
e m exp (ω ) ha b oads he inge ing in he di ec ion
o mo ion. The g ow h ins abili y a e is smalle when
he iscosi y p o ile has a posi i e slope on he in e -
ace, Eq. (7). Pe iodic o cing enhances o ma ion o a
single inge displacing he mo e iscous luid, while he
p esence o an in e media e laye wi h a posi i e iscous
g adien s abilizes he in e ace p e en ing b eakage.
Fo a c i ical alue o he injec ion pe iod he in e -
ace be ween bo h luids g ows signi ican ly i compa ed
5
FIG. 7. Maximum su ace a io S as a unc ion o he a io
α=µ(0)/µb(a) and he leng h Lo he middle laye (b). In
panel (b) lines co espond o α= 0.4 (ci cles and blue solid
line), α= 0.5 (squa es and do -dashed ed line) and α= 0.8
( iangles and dashed g een line). Pa ame e s as in Fig. 2.
o o he examined pe iods. This c i ical alue o he
pe iod is suspec ed o be associa ed wi h a esonance-
like dynamics be ween he ex e nal injec ion o cing and
he cha ac e is ic ime scale o he low ( a io o he su -
ace ension and he iscous (p essu e d op) o ces). The
e ec o a iscous linea g adien a he in e ace was
s udied nume ically and he esul s quali a i ely coincide
wi h hose ob ained om he linea s abili y heo y.
These esul s open new possibili ies o expe imen s on
iscous inge ing unde pe iodic o cing in he p esence
o a non-mono onic iscosi y p o ile o immiscible luids.
Ano he pe spec i e o his wo k is o pe o m new nu-
me ical simula ions o he obse ed plumes o unde s and
he mechanism ha e en ually may lead o he in e ace
b eaks up.
ACKNOWLEDGMENTS
This esea ch has been suppo ed by he Minis e io
de Ciencia e Inno aci´on (g an no. RTI2018-097063-B-
I00) and by Xun a de Galicia (g an no. 2021-PG036).
Bo h p og ammes a e co- unded by ERDF (EU). Com-
pu a ions ook place a CESGA (Cen o de Supe com-
pu aci´on de Galicia).
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