Jou nal o Compu a ional Physics 442 (2021) 110480
Con en s lis s a ailable a ScienceDi ec
Jou nal o Compu a ional Physics
www.else ie .com/loca e/jcp
Th ee dimensional modeling o liquid d ople sp eading on
solid su ace: An en iched fini e elemen /le el-se app oach
Mohammad R. Hashemi a,b,∗, Pa el B. Ryzhako a,b, Ricca do Rossi a,b
aCen e In e nacional de Mè odes Numè ics en Enginye ia (CIMNE), 08034 Ba celona, Spain
bUni e si a Poli ècnica de Ca alunya (UPC), 08034 Ba celona, Spain
a i c l e i n o a b s a c
A icle his o y:
A ailable online 2 June 2021
Keywo ds:
Two-phase flow
Su ace ension
We ing
Mic ofluidics
D ople s
Con ac -line
A physically consis en app oach is in oduced o simula e dynamics o d ople s in con ac
wi h solid subs a es. The nume ical me hod is de eloped by in oducing he molecula –
kine ic model wi hin he amewo k o he le el-se /en iched fini e elemen me hod and
including he heo e ically esol ed sub-elemen al hyd odynamics. The le el-se me hod
is cus omized o comply ully wi h he model acqui ed o he mo ing con ac -line. The
consis ency o he p oposed me hod is e ified by compa ing he simula ion esul s wi h
he heo e ical p edic ions. In o de o u he alida e he me hod, he sp eading o
a d ople is nume ically modeled and compa ed igo ously wi h he expe imen al da a
epo ed in he li e a u e. The p oposed me hod is also employed o cap u e he e olu ion
o a d ople apped in a conical po e. All es -cases a e simula ed on h ee-dimensional
compu a ional domains.
©2021 The Au ho (s). Published by Else ie Inc. This is an open access a icle unde he
CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
1. In oduc ion
Accu a e modeling o liquid sp eading on a solid su ace [1]is o a undamen al impo ance in he analysis o mul i-
phase flows in mic o-channels [2,3]as well as po ous [4] and fib ous [5]media, which a e encoun e ed in a wide ange
o indus ial applica ions. One such applica ion, ha mo i a ed he de elopmen s o he p esen wo k, is he wa e -ai
anspo in he gas channels and fib ous di usion laye o polyme elec oly e memb ane uel cells (PEMFCs) [6,7] ha is
an essen ial ac o in he de e mina ion o he pe o mance o he cell [8,9].
In he modeling o phenomena associa ed wi h he mul i-phase flow in he p esence o a solid subs a e, one o he
majo challenges is o deal wi h he mo ing bounda y o he h ee-phase (gas/liquid/solid) in e ace, he so-called con ac -
line, using an app op ia e condi ion [10,11]. Theo e ical in es iga ions o he mo emen o he con ac -line [12,13]imply
ha he classical con inuum-le el hyd odynamics along wi h he con en ional no-slip condi ion a he solid su ace lead o
an unbounded eloci y g adien and consequen ly a singula i y in he s ess a he con ac -line. The con en ional app oach
o alle ia e his singula i y is o ake in o accoun a slip condi ion in he icini y o he con ac -line [14,15], o which he e
is also some e idence om molecula dynamics simula ions [16–18].
Employing he slip condi ion in he con ex o he con inuum hyd odynamics allows o a heo e ical solu ion o he
iscous bending phenomenon and leads o he well-es ablished Cox’s ela ion [19], which gi es a co ela ion be ween he
appa en mac oscopic con ac -angle and he mic oscopic con ac -angle. Mo e ecen ly, i was shown ha alle ia ing he
*Co esponding au ho a : Uni e si a Poli ècnica de Ca alunya (UPC), 08034 Ba celona, Spain.
E-mail add esses: [email p o ec ed] (M.R. Hashemi), pa el. yzhako[email p o ec ed] (P.B. Ryzhako ), [email p o ec ed] (R. Rossi).
h ps://doi.o g/10.1016/j.jcp.2021.110480
0021-9991/©2021 The Au ho (s). Published by Else ie Inc. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
s ess singula i y can esul in a complemen o he hyd odynamic heo y; Zhang and Mohseni [20] explo ed he possibil-
i y o in eg a ing he singula s ess in he close icini y o he con ac -line in o de o ob ain a model o he dynamic
mic oscopic con ac -angle.
Besides he hyd odynamic heo y ha ocuses on he phenomena a he con inuum le el, molecula –kine ic heo y [21]
has also been acqui ed o de i e a model o he mo ing con ac -line. I was shown ha he esul ing model is consis en
wi h he esul s o he molecula dynamics simula ions [22,23]. Bo h he Cox’s ela ion and he molecula –kine ic model
ha e been examined by fi ing he expe imen ally obse ed co ela ion be ween he con ac -angle and he con ac -line
eloci y [24–26].
I had been e ealed ha depending on he ea u es o he se o expe imen s, one model o ano he p o ides a be e
ma ch [27–29]. This can be explained as a esul o he ac ha he hyd odynamic heo y accoun s o he iscous dissipa-
ion while he molecula –kine ic heo y ocuses on he ene gy dissipa ion in a e y close icini y o he con ac -line [26].
Thus, depending on he flow configu a ion and he eloci y o he con ac -line, ei he o hese mechanisms is dominan and
he beha io can be be e cha ac e ized wi h he espec i e model. Based on he expe imen al esul s, due o he ambigui y
in de e mining he unde lying physics and he lack o a sys ema ic app oach o de e mine cons i u i e pa ame e s [30,20],
i is no a s aigh o wa d ask o decide which heo y (and he esul ing) model should be employed. The e o e, in o -
de o exploi he p os o bo h he heo ies, combined models we e p oposed [31–35], in which he ic ional con ac -line
slip is aken in o accoun as well as he iscous dissipa ion. Recen ly, u ilizing a se ies o molecula dynamics simula ions,
Fe nández-Toledano e al. [36]s a ed ha he hyd odynamic heo y is a eliable means o co ela ing he appa en (ex-
pe imen ally measu able) con ac –angle and he mic oscopic con ac –angle, while he molecula –kine ic heo y go e ns he
dynamic mic oscopic con ac –angle. This confi ms he a ionale o de eloping combined models like he one p oposed by
Pe o and Pe o [31].
In he con ex o he nume ical modeling o he dynamics o he con ac -line, he u iliza ion o he gene alized Na ie -
slip condi ion [37–39]is a iable choice [40]. Being based on he combina ion o he Na ie -slip condi ion on he solid
subs a e and he ic ional mo emen o he con ac -line due o he unbalanced Young s ess, i is consis en wi h he
molecula dynamics simula ions [37,39] and he he modynamic p inciples [30,41] o modeling he we ing phenomena.
The gene alized Na ie -slip condi ion has so a been applied in he nume ical simula ion o a ious cases in ol ing mo ing
con ac -line [42–45]. A nume ically di e en , bu undamen ally simila app oach is he di ec imposi ion o a ic ion o ce
a he con ac -line along wi h he s anda d Na ie -slip condi ion [46]. In he nume ical modeling, i is also possible o im-
pose he no-slip condi ion on he solid su ace while he o ce singula i y is ci cum en ed by modi ying he con en ional
o mula ion [39]; as a no able choice, di usion can be in oduced as he mechanism unde lying he con ac -line mo e-
men [47] simila o he di use in e ace me hods [48–50]. Ne e heless, his app oach is ou o he scope o he p esen
pape and will no be u he discussed he e.
Besides he u ilized slip condi ion, one o he undamen al issues wi h he compu a ional me hods applied o he mo ing
con ac -line p oblem is he mesh-dependence o he esul s [51,52]. A physical and a nume ical ac o , a leas pa ially,
esponsible o his issue a e he un esol ed sub-g id hyd odynamics and he in e acial o ce smoo hing, espec i ely.
In he icini y o he con ac -line, hyd odynamic mechanisms ac a a small leng h-scale which, e en being a beyond
he molecula –scale, canno be adequa ely esol ed unless a p ohibi i e efinemen o he compu a ional mesh is pe -
o med [53]. The hyd odynamic heo y is a means o ci cum en he need o such efinemen [54] and helps imp o ing he
mesh-independence o he nume ical esul s [55–57]. On he o he hand, con en ional nume ical me hods ypically u ilize
a nume ically smoo h ep esen a ion o he physically localized su ace ension [58–60] ollowing he so-called “con inuum
o ce app oach” [61]. In he p esence o he mo ing con ac -line, he unbalanced Young s ess is also smoo hed ou o ac
simila o a body o ce cen alized a he con ac -line [62,45]. This app oach is associa ed wi h an a ificial hickness o he
in e ace, which is usually se equal o he leng h o a ew compu a ional cells o he bes pe o mance. The e o e, fixing
he a io o his smoo hing leng h o he cell size [45], a highly efined mesh is necessa y in he icini y o he in e ace and
he con ac -line in o de o minimize he e o . A emedy o his issue is o u ilize a compu a ional mesh ha is fi ed o
he liquid-gas in e ace, e.g. [63,64,46]. Howe e , such an app oach may esul in se e ely de o med meshes and equi es a
equen emeshing, which d ama ically inc eases he compu a ional cos s, pa icula ly in 3D. Mo eo e , in case o a se e e
opological change in he liquid phase, his class o app oaches may lead o ambigui ies in he ecogni ion o he liquid
bounda y.
In his wo k, a nume ical me hod is p esen ed ha by alle ia ing he abo e men ioned issues, p o ides easonably ac-
cu a e esul s on a he coa se meshes. The p e iously in oduced p essu e-en iched fini e elemen /le el-se model o he
wo-phase flows [65]is u he de eloped by inco po a ing he equi emen s o he mo ing con ac -line p oblems. The
simplified o m o he molecula –kine ic model is implemen ed along wi h he Na ie -slip condi ion ha ac s on he solid
subs a e. Following he me hodology p esen ed by Buscaglia and Ausas [66], he implemen a ion o he mo ing con ac line
condi ion is done by e ising he a ia ional o mula ion o he me hod. In o de o make he o e all nume ical algo i hm
consis en , he le el-se smoo hing p ocedu e [65]is also modified by in oducing a bounda y condi ion ha is compa ible
wi h he con ac line condi ion. To accoun o he sub-elemen al hyd odynamics, he simplified o m o Cox’s ela ion [19]
is used unde he condi ion o a small capilla y numbe . In addi ion, his ela ion is applied only once he con ac angle
eaches he alue wi hin a h eshold o he equilib ium con ac angle. This ensu es ha he con ac line eloci y is lim-
i ed and consequen ly, he Reynolds numbe is small. Ne e heless, in o de o emo e hese limi a ions, a mo e gene al
hyd odynamic model [67,54]should be acqui ed ha is a subjec o u u e de elopmen s. In his wo k, an elemen spli -
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M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
ing p ocedu e [65]is pe o med a each s ep, which enables ep esen ing in e ace wi h ze o- hickness. Consequen ly, he
e ms associa ed wi h he mo ing con ac -line model a e in eg a ed along he cu e ep esen ing he con ac -line while he
su ace ension ac s locally a he in e ace. I mus be no ed ha such domain spli ing is ully exploi ed by inco po a ing
an en iched fini e elemen space, which enables p essu e (g adien ) discon inui y wi hin an elemen .
In he ollowing sec ion, he go e ning equa ions including he con ac -line condi ion a e fi s discussed and hen imple-
men ed in he a ia ional o m. Then, he cus omized e sion o he le el-se me hod is b iefly desc ibed and he addi ional
bounda y condi ion equi ed o he smoo hing p ocedu e is in oduced. The pe o mance o he p esen me hod is e ified
by compa ing he esul wi h he heo e ical ela ion be ween he oo p in adius and he con ac angle o a d ople sp ead-
ing wi h a sphe ical-cap shape [68]a a small Bond numbe [69]. The esul s a e u he alida ed agains he expe imen al
da a published by Se eno e al. [26] o a d ople o liquid squalane ha is sp eading on a solid silica subs a e. The deg ee
o mesh-(in)dependence o he esul s is shown o bo h es -cases. All simula ions p esen ed in his wo k a e conduc ed
o h ee-dimensional compu a ion domains.
2. Nume ical me hod
The momen um and mass conse a ion equa ions o a fluid sys em can be w i en as
ρ∂u
∂ +u·∇u=ρb+∇·σin ,(1)
and
∂ρ
∂ +∇·(ρu)=0in,(2)
espec i ely. I should be no ed ha in his wo k, he homogeneous fluid domains (liquid and gas) a e conside ed o be
incomp essible and consequen ly, Eq. (2) educes o ∇·u =0in each phase. The fluid domain,  ⊂Rd, is bounded by
bounda y ∂ ⊂Rd−1, whe e ddefines he numbe o spa ial dimensions. This se o equa ions is subjec o he ini ial
condi ion
u(x,0)=u0in ,(3)
Di ichle
u(x, )=uDon ∂D,(4)
and Neumann
T(x, )=TNon ∂N,(5)
bounda y condi ions. The ac ion ec o is calcula ed as T =n ·σwi h he o al s ess enso , σ, being ob ained om he
New onian cons i u i e equa ion
σ=−pI+μ∇u+∇uT.(6)
He e, nis a uni ec o no mal o ∂and poin ing o he ou side o .
2.1. Mul i-phase flow
Le us conside a sys em consis ing o wo immiscible fluids and a solid subs a e (see Fig. 1). Then, he domain can
be sepa a ed in o 1and 2wi h  =(1∩2)and  =(1∪2). The sepa a ing in e ace is a cons i uen pa o
bo h ∂1and ∂2, while i coincides wi h he solid subs a e only a he con ac -line ∂ =(∂ ∩), whe e he h ee
phases (bo h fluids 1 and 2along wi h he solid subs a e) come in o con ac and h ee su ace ensions, γ, γ1s, and γ2s,
ac simul aneously on he fluid 1-fluid 2, fluid 1-solid, and fluid 2-solid in e aces, espec i ely (see Fig. 2).
Being in e nal o he fluid domain , he in e acial condi ions can be in e p e ed as a jump in he ac ion due o he
su ace ension
T(x, )=−γκnin on ,(7)
and con inui y o he eloci y field
u(x, )=0on,(8)
whe e nin is he no mal o he in e ace, , and o any a iable A he jump ope a o eads A =A1−A2wi h subsc ip s
1 and 2deno ing he alue in he co esponding phase domains.
3
M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
Fig. 1. Schema ic o he fluid domain =1∪2.
Fig. 2. Schema ic o a d ople con ac ing a solid su ace. Liquid-gas, liquid-solid, and gas-solid su ace ensions wi h espec i e coefficien s o γ, γ1s, and
γ2sa e depic ed in his figu e.
A he con ac –line o he equilib ium s a e [70](θ=θY), Young’s ela ion [71]s a es ha [10,72]
γcos(θY)+γ1s=γ2s.(9)
The e o e, one can simply w i e cos(θY) =(γ2s−γ1s)/γ. In case he configu a ion de ia es om he equilib ium, he unbal-
anced Young s ess ( o ce pe uni leng h) is defined as [32,50]
τY=γ[cos(θY)−cos(θ)].(10)
He e, τYcan be in e p e ed as he ne (e ec i e) ension ha ac s pa allel o he solid subs a e a he con ac -line and is
esponsible o i s mo emen . Based on he molecula –kine ic heo y [21], he mo emen o he con ac -line is associa ed
wi h an ene gy dissipa ion ha is usually e e ed o as a ic ion o ce ac ing on a mo ing con ac -line [33,39,50]. Deno ing
he slip– eloci y associa ed wi h he mo emen o he con ac –line wi h uslip , his unde lying mechanism can be ep esen ed
by [25,36]
uslip =2k0λsinh λ2τY
2kBTon ∂,(11)
whe e pa ame e s k0and λa e he cha ac e is ic equency and he a e age dis ance o he ( andom he mal) molecula
displacemen s in he icini y o he con ac –line, espec i ely. In Eq. (11), kBis he Bol zmann cons an and Tdeno es he
absolu e empe a u e. In i s simples o m, i he a gumen o sinh in Eq. (11)is small, he o mula o he molecula –kine ic
heo y eads
τY=ζuslip on ∂,(12)
wi h ζ=kBT/k0λ3 ep esen ing he coefficien o ic ion a he con ac –line [26]. Fu he mo e, in o de o a oid he
singula i y in he icini y o he con ac -line [73], he no-slip condi ion on he solid subs a e is subs i u ed by he Na ie -
slip bounda y condi ion ha can be o mula ed as [39,66,74]
ns·u=0on∂s,(13)
and
Is·T=−βIs·u=−βuon ∂s,(14)
whe e nsis he no mal o solid subs a e ∂s, and Is=(I−ns⊗ns)deno es he su ace uni enso wi h Ibeing he
iden i y enso . In his wo k, he slip condi ion (13)is implemen ed using he local o a ion o he unknown eloci ies a
solid su ace ∂sas discussed in [75].
4
M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
Fig. 3. Schema ic o he compu a ionally ep oduced and he physically expec ed in e ace.
I is wo h men ioning ha he combina ion o (12) and (14)is essen ially equi alen o he so-called “gene alized Na ie
bounda y condi ion” [39]. Ano he impo an poin o men ion is ha so a , no sys ema ic app oach has been in oduced
o a p io i de e mina ion o pa ame e s βand ζ o be used in a nume ical simula ion [45]. In sec ion 3.2, i is shown ha
o he p esen me hod, ζcan be se acco ding o he co esponding pa ame e ob ained by fi ing he expe imen al da a
by a compa able model (e.g. see [26]).
2.1.1. Sub-elemen hyd odynamics
Conside ing he p ac ical difficul ies in compu a ionally esol ing he hyd odynamics in he icini y o he con ac -line
wi h mic ome e leng h-scales [76,53,77], he well-es ablished hyd odynamic heo y is u ilized o inco po a e he sub-
elemen a ia ion o he con ac angle ha occu s due o he so-called “ iscous bending” phenomenon [25,1](see Fig. 3).
In his wo k, he o mula ion is based on he simplified linea o m [68]o he asymp o ic solu ion o he hyd odynamic
heo y [19]as
θ3=θnum3−9Ca ln(he
lmic o
), (15)
whe e he capilla y numbe is defined as Ca =uclμ/γand lmic o is he mic oscopic slip leng h-scale. I heis conside ed o be
equal o he leng h-scale associa ed wi h he con en ional expe imen al measu emen s o he con ac -angle, ln(he/lmic o) ∼
10 would be expec ed [25,68]. I is wo h no ing ha he simul aneous inco po a ion o Eqs. (12) and (15)leads o he
simplified o m o he combined molecula –kine ic/hyd odynamic model p oposed by Pe o and Pe o [31,26].
The o iginal Cox’s ela ion [19]is alid o Ca 1 and small Reynolds numbe while i s simplified o m in Eq. (15)
can be u ilized in cases o a small con ac angle, θ<3π/4, wi h a anishing iscosi y a io, μ2/μ11 (conside ing
μ2 o he su ounding fluid 2)[68]. Fo he es -cases sol ed in his pape , Eq. (15)is applied only o Ca <0.3 and
θnum −θY<2π/10. The la e condi ion p e en s he applica ion o Eq. (15)in si ua ions ha a la ge di e ence be ween
he dynamic con ac -angle and θYleads o a a he la ge con ac -line eloci y and consequen ly, a ai ly la ge Reynolds
numbe . In o de o alle ia e his condi ion, one can ollow he app oach p esen ed in [67]; howe e , in o de o keep he
simplici y o he o mula ion, i is no implemen ed in his wo k.
Al hough i is known ha he mic oscopic leng h-scale lmic o is in he o de o one nanome e , i is gene ally ob ained
by pe o ming a p ope da a-fi ing [26,68]. In his sense, lmic o is added o he lis o unknown model pa ame e s [35]
along wi h βand ζ. Fo he cases conside ed in his wo k, mic oscopic leng h-scale is se o lmic o =10−9m ha gi es
ln(he/lmic o) ∼10 o he employed compu a ional meshes. Nume ical simula ions also show ha sligh a ia ion o lmic o
does no lead o any significan changes in he esul s. Combining Eq. (15)wi h he gene alized Na ie condi ion, Yamamo o
e al. has also epo ed ha lmic o ∼10−9mled o he mos sa is ac o y esul s in hei capilla y ise simula ions [56].
2.2. Va ia ional o mula ion
The a ia ional o m o he momen um equa ion (1)can be w i en o he whole fluid domain as [65]


ρ∂u
∂ +u·∇u·wd=

ρb·wd+

p∇·wd
−

μ∇u+∇uT:∇wd+
∂
T·wd(∂), (16)
whe e wis a es unc ion in H1()d ha anishes a he Di ichle bounda y condi ions. Fo sepa a e incomp essible fluid
domains, 1and 2, he a ia ional o m o he con inui y equa ion (2) becomes
5

M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
Fig. 4. Schema ic o a possible cu in a e ahed al elemen . The in e ace, e, is shaded by yellow and he ma ching aces a e ma ked wi h he same colo .
(Fo in e p e a ion o he colo s in he figu e(s), he eade is e e ed o he web e sion o his a icle.)


qρ(∇·u)d=0,(17)
wi h qbeing a es - unc ion in L2(). The bounda y in eg al e m ∂T ·wd(∂)on he igh –hand–side o Eq. (16)
essen ially includes he Neumann bounda y (5), in e acial (7), and Na ie -slip (14) condi ions as well as he su ace ension
along wi h he molecula –kine ic model (11)ac ing a he con ac line. Conside ing uni ec o s in and sbeing angen ial
o he in e ace and he solid subs a e, espec i ely (as shown in Fig. 2), one has Is· in =− cos(θ) sand consequen ly, he
molecula –kine ic model (11)can be ew i en as
(γ2s−γ1s) s+γIs· in −2kBT
λ2sinh−1uslip
2k0λ s=0on∂.(18)
Subs i u ing he co esponding ela ions in o Eq. (16), one ob ains


ρ∂u
∂ +u·∇u·wd=

ρb·wd+

p∇·wd
−

μ∇u+∇uT:∇wd+
∂N
TN·wd(∂)
−
∂s
βu·wd(∂)−

γκnin ·wd
+
∂
[(γ2s−γ1s) s+γIs· in
−2kBT
λ2sinh−1uslip
2k0λ s·wd(∂). (19)
He e, he slip– eloci y a he con ac –line eads uslip = s·u. Simpli ying he molecula –kine ic model (11) o i s linea
o m (12), one ob ains

∂
2kBT
λ2sinh−1uslip
2k0λ s·wd(∂)=
∂
ζ( s·u) s·wd(∂). (20)
Fo he sake o simplici y and in o de o acili a e compa isons wi h he e e ences chosen in he p esen wo k (whe e ζis
p o ided), he linea app oxima ion (Eq. (20)) is used i no men ioned o he wise.
I mus be no ed ha a simila a ia ional o mula ion o he con ac line dynamics has been de i ed by Buscaglia
and Ausas [66]using he p inciple o i ual wo k. Con en ionally, he a ia ional o mula ion is de i ed by smoo hing he
su ace ensions based on he con inuum o ce app oach (see [42] o example).
In his wo k, he accu a e in eg a ion o he e ms appea ing in he a ia ional o mula ion (19)is done by spli ing
o he cu elemen s. In Fig. 4, his p ocedu e is schema ically shown o a sample elemen cu by he in e ace. Elemen al
in eg a ion domains e,cu
1and e,cu
2a e spli in o e ahed a o acili a e he in eg a ion. The in eg a ion o he e ms
associa ed wi h he elemen al in e ace (e), con ac -line (∂e), and solid subs a e (∂e
s) a e pe o med by u ilizing he
quad a u e poin s as schema ically illus a ed in Fig. 5. By employing a high–o de ( wo poin s o line-segmen s, h ee
poin s o iangles, and ou poin s o e ahed a) Gaussian quad a u e, one can assu e ha he in eg a ion p ocedu e does
no in oduce u he e o o he solu ion (i.e. he numbe o Gauss poin s is sufficien o he in eg a ion o unc ions up o
hi d–o de ). The con en ional al e na i e o he elemen spli ing p ocedu e is he inco po a ion o a smoo hed nume ical
6
M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
Fig. 5. Schema ic o a possible cu in a e ahed al elemen con ac ing he solid su ace. ∂eis ma ked wi h a ed solid line and quad a u e poin s a e
ep esen ed by black do s.
app oxima ion o he del a unc ion; in he con inuum o ce app oach, his is essen ially needed o o mula e he su ace
ension and he con ac –line model. In he p esen app oach, due o he employmen o he spli ing me hodology, such an
app oxima ion is no equi ed and he associa ed e o s a e alle ia ed.
The p esen ed o mula ion is implemen ed wi hing he amewo k o he s abilized p essu e en iched fini e elemen
me hod p oposed in [65]. Wi hin elemen e, he s anda d fini e elemen app oxima ion o he flow a iables eads
u(x, )=
I∈Ne
uI( )Ne
I(x), (21)
and
p(x, )=
I∈Ne
pI( )Ne
I(x), (22)
whe e Nedeno es he se o associa ed nodes and Ne
Iis he shape unc ion co esponding o node I. Howe e , using
he s anda d fini e elemen app oxima ion, i is impossible o cap u e he in a-elemen discon inui y in he p esence o
ma e ial in e aces; in he con ex o mul i-phase flows [65], his is he sou ce o he so-called “spu ious cu en s”. In o de
o esol e his issue, he p essu e app oxima ion wi hin an elemen cu by he in e ace can be en iched by accoun ing o
a “jump” as
p(x, )=
I∈Ne,cu
pI( )Ne,cu
I(x)+
I∈Ne,cu
pe,cu
I,en ( )Ne,cu
I,en (x), (23)
wi h en iched nodal p essu e pe,cu
I,en being local o he cu elemen .
In his wo k, en iched shape unc ion NI,en is cons uc ed based on s anda d con inuous shape unc ion NIas
NI,en (x)=NI(x)i (xI∈1and x∈2)o (xI∈2and x∈1)
0else (24)
Using his se o en iched shape unc ions, bo h he jump in he p essu e and discon inui y in i s g adien can be cap-
u ed wi hin a cu elemen . A e in oducing he en ichmen e ms, he a ia ional mul iscale me hodology wi h he
well–es ablished algeb aic sub-g id scale s abiliza ion [78]along wi h a special small–cu ea men app oach is u ilized
o s abilize he me hod as p oposed in [65]. The momen um equa ion is hen linea ized using he gene alized New on’s
me hod and sol ed along wi h he mass conse a ion equa ion in a ully implici monoli hic manne . One o he ema k-
able ea u es o his en ichmen p ocedu e is ha upon he c ea ion o he local elemen al sys em o equa ions, p essu e
condensa ion p ocedu e [65]is pe o med a he elemen al le el, hus, omi ing he in oduc ion o he addi ional en iched
p essu e deg ees o eedom. The e o e, he deg ees o eedom, and consequen ly, he size o he assembled global sys em
o equa ions is he same as ha o he s anda d fini e elemen me hod.
2.3. Le el-se
In he p esen me hod, he e olu ion o he in e ace is cap u ed using he le el-se me hod [79], which is based on
he in oduc ion o he con inuous unc ion φ ha ep esen s he signed dis ance o he in e ace. The le el-se unc ion is
con ec ed acco ding o he eloci y field by sol ing
∂φ
∂ +u·∇φ=0in.(25)
In he p esen wo k, his pu e con ec ion equa ion is s abilized ollowing he me hodology p oposed by Codina [80]. The
le el-se unc ion g adually loses i s egula i y due o i s de ia ion om a dis ance unc ion [81] and high equency noise
7
M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
(oscilla o y in e ace) [82]. The fi s p oblem can be esol ed by equen eini ializa ion o he le el-se unc ion in a way
ha ∇φ ≈1is sa isfied [83]. Due o he hype bolic na u e o he con en ional le el-se eini ializa ion o mula ion, i
is necessa y o ake in o accoun he so-called “blind-spo egion” in he icini y o he solid su ace [84]. None heless, in
he p esen wo k, he ma ching le el-se eini ializa ion p ocedu e p oposed by Elias e al. [85]is pe o med o he whole
domain once in e e y 50 ime-s eps.
Following he idea p esen ed in [86], he high equency oscilla ions can be e ec i ely cu ed by sol ing a di usion
equa ion o he le el-se unc ion as
˜
φ−ε∇2˜
φ=φin ,(26)
whe e ˜
φand φa e he smoo hed (non-oscilla o y) and o iginal le el-se unc ions, espec i ely. He e, ε=5 ×103 h2
e, wi h
 being he size o he ime-s ep and he he elemen size. In he absence o con ac wi h a solid, Eq. (26)can be sol ed
wi hou in oducing any specific bounda y condi ion [86,82,65]. In he p esen me hod, a Neumann bounda y condi ion is
implemen ed on he solid subs a e as
ns·∇˜
φ=ns·∇φon ∂s.(27)
Combining Eqs. (10), (12), and (15),
θnum =cos−1ζ
γuslip +cos(θY)3
+9Caln(he
lmic o
)1/3
,(28)
a he cu elemen s, bounda y condi ion (27)is subs i u ed by
ns·∇˜
φ=−∇φcos(θnum)on ∂e,cu
s.(29)
I should be no ed ha in case o he applica ion o he ull o m o he molecula –kine ic model, Eq. (28)should be
ew i en inco po a ing Eq. (11).
The main sho coming o he p esen ed le el-se smoo hing scheme is he p obabili y o a sligh d ople sh inkage. As
p oposed in [65], his issue can be esol ed by pe o ming a co ec ion s ep as
φI=˜
φI−1
NI
NI

J˜
φJ−φJ,(30)
whe e NIis he numbe o nodes J ha a e connec ed o node I. In his wo k, in o de no o pe u b he con ac angle,
a modified co ec ion p ocedu e is p oposed by sepa a ing he se o nodes in e io o he fluid domain om hose ha lie
on he solid subs a e, i.e.
J∈ ∂si I∈( ∂s)
∂si I∈∂s(31)
Abo e, all he ing edien s o he p oposed me hod a e de ailed. The summa y o he o e all s a egy is p esen ed in
Algo i hm 1.
3. Resul s
The p oposed nume ical me hod is implemen ed wi hin KRATOS Mul iphysics [87]an open-sou ce amewo k o mul i-
physics compu a ions. The second o de backwa d di e ence (BDF2) ime in eg a ion is applied o he flow equa ions and
he C ank–Nicolson scheme is used o ime-ma ching o he le el-se con ec ion equa ion. Algeb aic mul ig id lib a y
(AMGCL [88]) was used o sol e he linea sys em o equa ions using he GMRES(m) me hod (wi h es a pa ame e
m =40). The con e gence ole ance o he linea sol e is se o 10−9, while a ela i e ole ance o 10−5is conside ed
o check he con e gence o eloci y and p essu e.
In he ollowing, he pe o mance o p oposed nume ical me hod is fi s e ified by compa ing he simula ion esul s
wi h he heo e ical ela ion ob ained o he oo p in adius o a liquid d ople sp eading on a solid subs a e a small
Bond numbe s. The me hod is u he alida ed agains he expe imen al da a published in he li e a u e o a millime e -
sized squalane d ople sp eading on a subs a e o silicone wa e . In he end, he capabili y o he me hod is assessed by
simula ing a d ople apped inside conical po es. In all cases sol ed in his pape , g a i y g=9.8m/s2ac s in he nega i e
z–di ec ion, and 2is composed o ai wi h ρ=1.0kg/m3and μ =1.0 ×10−5Pa s. Fo he sake o con enience, he
con ac -angle is epo ed in deg ees in he es o his pape .
Rema k. Be o e assessing he esul s o he p oposed me hod, i is wo h o p o ide an insigh o he compu a ional cos s
associa ed wi h i s applica ion: using a mesh wi h ∼500 K elemen s, he o al un– ime pe ime–s ep is a ound 62 s, o
which almos 80% co esponds o he wo-phase flow sol e , 4% o he le el–se con ec ion, 8% o he le el–se smoo hing
p ocedu e, and abou 8% is consumed o he le el–se e-ini ializa ion p ocedu e.
8
M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
Algo i hm 1: Summa y o he p oposed me hod.
Inpu : u0, uD, TN, and φ0
Ou pu : uI, pI, and φI; node I∈
1n =1
2 =0
3while < un- ime do
4sol e Eq. (25) o φ(n+1/2)
Iwi h hal ime-s ep
5i n ={50,100,150,...
} hen
6 eini ialize φ
7do smoo hing acco ding o Eqs. (26)and(30)wi h condi ions (27)and(29)
8calcula e cu a u e
9 o all elemen s edo
10 i e ∩ =∅ hen
11 do elemen spli ing
12 calcula e con ac angle
13 c ea e elemen al sys em o equa ions
14 do assembling he Linea Sys em o Equa ions (LSE)
15 sol e LSE o
u(n+1)
I,p(n+1)
I
16 sol e Eq. (25) o φ(n+1)
Iwi h hal ime-s ep
17 upda e n =n +1
18 upda e =n
Fig. 6. Schema ic o he ini ial configu a ion o he liquid d ople inside a solid box.
3.1. Ve ifica ion wi h heo y
I a d ople e ains i s sphe ical-cap shape du ing sp eading on a solid su ace, one can w i e a co ela ion be ween
he oo p in adius and he ins an aneous con ac -angle based on he mass conse a ion o an incomp essible liquid. The
esul ing co ela ion eads as ( ) = (θ( ))wi h [33]
(θ)=3V
π
[1+cos(θ)]sin(θ)
[1−cos(θ)][
2+cos(θ)]1/3
.(32)
S a ing om θ(0) =π/2, he a io o he e minal adius Y o he ini ial adius o he d ople R0is
Y
R0=2[1+cos(θY)]sin(θY)
[1−cos(θY)][2+cos(θY)]1/3
.(33)
The basic assump ion o a sphe ical-cap d ople is alid i he Bond numbe (Bo =ρ1gR2
0/γ) is small o equi alen ly he
heigh o he d ople is smalle han he capilla y leng h-scale (lc∼√γ/ρ1g)[69,33,68]. This condi ion indica es ha g a i y
is domina ed by he capilla y o ce and he e o e, has a negligible e ec on he d ople dynamics. No e ha his assump ion
is ques ionable o fluids wi h la ge iscosi y, e.g. o polyme ic liquids [33].
He e, a liquid d ople wi h an ini ially hemisphe ical shape (ini ial con ac -angle o θ0=90◦) and an ini ial adius o
R0=1.5mmis sp eading on a solid subs a e. The sys em is confined in a box filled by ai wi h no-slip la e al and op
bounda ies. The schema ic o he whole sys em is shown in Fig. 6. The dimensions a e L =W=8 mm and H=3mm, liquid
iscosi y is μ1=1.0 ×10−3Pa s, densi y is ρ1=920 kg/m3, and he liquid-ai su ace ension is γ=4.26 ×10−2N/m. This
gi es a Bo =0.48 o equi alen ly a capilla y leng h-scale o lc=2.2mm. The equilib ium con ac -angle is se o θY=58◦
and he esul s a e ob ained using β=103Pa s/m and ζ=1.0Pas, no ing ha his example does no in en o ep oduce
any eal-wo ld expe imen .
9
M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
Fig. 22. P essu e con ou s o α=30◦.A =0.2s, he sys em has almos eached i s equilib ium configu a ion.
an accep able mesh-con e gence was obse ed. The esul s we e also compa ed o bo h he s uc u ed and uns uc u ed
meshes and a good ag eemen was e ealed. Fu he mo e, he s aigh o wa d employmen o he p oposed me hod o
simula e a d ople apped in a (closed) conical po e, sugges s he applicabili y o he de eloped nume ical ool o po e-
scale mul i-phase flows. I mus be no ed ha in his wo k no mesh- efinemen s a egy was u ilized o locally inc ease he
esolu ion close o he d ople in e ace.
One o he in e es ing ea u es o he p esen me hod was ha in o de o ob ain physically meaning ul esul s, he
con ac -line dissipa ion coefficien was se acco ding o he co esponding pa ame e ha was ob ained by fi ing he linea
Pe o ’s model in o he expe imen al da a. This alle ia es he ambigui y associa ed wi h he se ing o his pa ame e in he
app oaches ely on he gene alized Na ie -slip condi ion. Howe e , u he in es iga ion wi h a wide ange o liquid/solid
ma e ials is necessa y o u he suppo his affi ma ion, which would be he opic o a sepa a e esea ch.
Gene ally, du ing he ini ial s age o he d ople sp eading, ine ial e ec s a e a he significan and he e o e, he alidi y
o he simplified model used in he p esen wo k o esol e he sub-elemen al hyd odynamics becomes dubious. The e o e,
in o de o inc ease he accu acy while cap u ing he sp eading wi h a fini e ine ia, a mo e sophis ica ed hyd odynamic
model ha also inco po a es he e ms appea ing a fini e Reynolds numbe can be acqui ed. This is a subjec o u u e
de elopmen s.
16

M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
Fig. 23. P essu e con ou s o α=60◦.A =0.14s, he sys em has almos eached i s equilib ium configu a ion.
In o de o imp o e he coupling be ween he momen um equa ion and he e ol ing in e ace ha is ep esen ed by
he le el-se unc ion, in his wo k he le el-se con ec ion equa ion is spli in ime as shown in Algo i hm 1. Nume ical
simula ions showed ha such spli ing could posi i ely a ec he accu acy o he me hod and alle ia e he need o an
excessi e di usi e le el-se smoo hing o egula ize he in e ace. Ne e heless, u he in es iga ions a e needed o quan i y
his imp o emen .
CRediT au ho ship con ibu ion s a emen
Mohammad R. Hashemi: Concep ualiza ion, Me hodology, So wa e, W i ing – o iginal d a . Pa el B. Ryzhako : Concep-
ualiza ion, Funding acquisi ion, Me hodology, Supe ision, W i ing – e iew & edi ing. Ricca do Rossi: Concep ualiza ion,
Me hodology, Supe ision, W i ing – e iew & edi ing.
Decla a ion o compe ing in e es
The au ho s o he p esen wo k decla e ha hey ha e no conflic o in e es s.
Acknowledgemen s
This wo k was pe o med wi hin he amewo k o AMADEUS p ojec (“Ad anced Mul i-scAle moDEling o coupled
mass anspo o imp o ing wa e managemen in Uel cellS”, e e ence numbe PGC2018-101655-B-I00) suppo ed by
he Minis e io de Ciencia, Inno ación y Uni e sidades o Spain. The au ho s also acknowledge financial suppo o he men-
ioned Minis y ia he “Se e o Ochoa P og amme” o Cen es o Excellence in R&D ( e e ence: CEX2018-000797-S) gi en
o he In e na ional Cen e o Nume ical Me hods in Enginee ing (CIMNE).
17
M.R. Hashemi, P.B. Ryzhako and R. Rossi Jou nal o Compu a ional Physics 442 (2021) 110480
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