Con en s lis s a ailable a ScienceDi ec
In e na ional Jou nal o Mul iphase Flow
jou nal homepage: www.else ie .com/loca e/ijmul low
T ansien bubble ising in he p esence o a su ac an a e y low
concen a ions
D. Fe nández-Ma ínez a, M.G. Cabezas a,∗, J.M. López-He e a b, M.A. He ada a,
J.M. Mon ane o b
aDepa amen o de Ingenie ía Mecánica, Ene gé ica y de los Ma e iales and Ins i u o de Compu ación Cien í ica A anzada (ICCAEx), Uni e sidad de
Ex emadu a, E-06006 Badajoz, Spain
bE.T.S.I., Dep o. de Ingenie ía Ae oespacial y Mecánica de Fluidos, Uni e sidad de Se illa, Camino de los Descub imien os s/n 41092, Spain
A R T I C L E I N F O
Keywo ds:
Bubble ising
Su ac an
Dynamic adso p ion laye
A B S T R A C T
We s udy he o ma ion o he dynamic adso p ion laye when a bubble is eleased in a ank con aining
wa e wi h a iny amoun o su ac an . The in luence o he so p ion kine ic cons an s is examined by
compa ing he expe imen s wi h Sodium Dodecyl Sul a e (SDS) and T i on X-100. The expe imen s allowed
us o de e mine he pa ame e condi ions ha lead o a s able bubble ising and o alida e he simula ion.
A simple scaling analysis and he simula ion show ha he o ma ion o he dynamic adso p ion laye can
be spli in o h ee phases cha ac e ized by dispa a e ime scales. The mechanisms con olling hose phases
a e su ac an con ec ion, adso p ion–deso p ion, and di usion. The amoun o su ac an adso bed on o he
in e ace inc eases mono onously h oughou he h ee phases. The expe imen s and he simula ion show ha
he ising eloci y eaches a maximum a imes o he o de o 𝑘−1
𝑑 (𝑘𝑑 is he deso p ion cons an ) when he
dynamic adso p ion laye is p ac ically o med. This occu s e en when only aces o su ac an a e p esen
in he liquid. The non-mono onous beha io o he maximum su ac an su ace concen a ion is explained in
e ms o he e e se low in he ea o he bubble igh a e he bubble elease. This wo k con ibu es o
he unde s anding o he complex in e play be ween hyd odynamics and su ac an anspo and kine ics o e
bubble ising.
1. In oduc ion
Many echnological and indus ial p ocesses in ol e he in e ac ion
be ween gases and liquids. Among hem, we can men ion was e-wa e
ea men , chemical (Jakobsen, 2014) and biochemical (Do an, 2013)
eac o s, nuclea enginee ing (Lahey, 1990), and me allu gical bub-
ble column eac o s. The hea and mass ans e ac oss he in e ace
o bubbles domina e he phenomena occu ing in hese p ocesses.
This in e acial ans e is conside ably in luenced by he size, shape,
ajec o y, and eloci y o hese bubbles.
The p ocesses men ioned abo e in ol e la ge popula ions o bub-
bles in e ac ing among hem. Single-bubble dynamics se e as a basis
o analyzing complex mul i-bubble sys ems. The ising o a bubble
in s ill wa e is a pa adigma ic p oblem ha has been analyzed o
se e al cen u ies and con inues o cap u e he a en ion o many e-
sea che s oday (Sa man, 1956; Sanada e al., 2008; T ipa hi e al.,
2015; Cano-Lozano e al., 2016a,b; Ku e e al., 2021; Bonne is e al.,
2023).
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (M.G. Cabezas).
When a bubble is eleased in a liquid ba h con aining su ac an ,
he su ac an molecules adso b on o he ee su ace du ing he bubble
ising. The molecules a e ad ec ed by he ou e cu en owa d he ea
o he bubble. Su ac an accumula es in ha egion, whe e deso p ion
conside ably inc eases. This p ocess esul s in an e en dis ibu ion o
su ac an o e he in e ace, p oducing a Ma angoni s ess ha sub-
s an ially al e s he o ces ac ing on he bubble (Takagi and Ma sumo o,
2011).
The bubble ising in he p esence o a su ac an is a complex
phenomenon in which luid-dynamic and physicochemical p ocesses
a e coupled, especially o la ge non-sphe ical bubbles. The bubble
shape change implies a ia ions o he in e acial a ea, which induces
so p ion p ocesses coun e ac ing he in e ace expansion o comp es-
sion (Le ich, 1962). In e ace expansion/comp ession, so p ion kine -
ics, and con ec ion o e he bubble su ace compe e o es ablish he so-
called dynamic adso p ion laye (Dukhin e al., 1998, 2015; Ulagana han
e al., 2016; Zawala e al., 2023).
h ps://doi.o g/10.1016/j.ijmul iphase low.2025.105205
Recei ed 30 Sep embe 2024; Recei ed in e ised o m 28 Feb ua y 2025; Accep ed 28 Feb ua y 2025
In e na ional Jou nal o Mul iphase Flow 188 (2025) 105205
A ailable online 8 Ma ch 2025
0301-9322/© 2025 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license (
h p://c ea i ecommons.o g/licenses/by/4.0/ ).
D. Fe nández-Ma ínez e al.
The e olu ion o he bubble eloci y in he p esence o a su ac an
signi ican ly di e s om ha in clean wa e (Sa man, 1956; Sanada
e al., 2008; T ipa hi e al., 2015; Cano-Lozano e al., 2016a,b; He ada
and Egge s, 2023; Bonne is e al., 2023). Fo su icien ly la ge su ac-
an concen a ions, he bubble eloci y eaches i s maximum alue
a e ini ial accele a ion. Then, he bubble decele a es un il a pla eau
alue ( he e minal eloci y) is a ained. This non-s eady ising p ocess
co esponds o he majo pa o he bubble ajec o y in many p ac ical
si ua ions. The non-mono onous dependence o he bubble eloci y on
he e ical posi ion ( ime) is usually e e ed o as he local eloci y
p o ile (K zan and Malysa, 2002; K zan e al., 2007; K ach and Finch,
2010; Dukhin e al., 2015).
The local eloci y p o ile o a ising bubble is a inge p in o he
dynamic adso p ion laye ’s ansien beha io . I is s ongly a ec ed
by he adso p ion o su ace-ac i e species a he bubble su ace. Fo
his eason, he bubble pa h allows one o ob ain in o ma ion abou
he su ac an so p ion kine ics ha no o he echnique can p o ide,
pa icula ly when comp ising p o eins a a ious sol en condi ions,
such as pH and ionic s eng h. Adso p ion and deso p ion occu a he
same localiza ion in mos expe imen al con igu a ions. Con e sely, he
su ac an is essen ially adso bed on o he on mobile pa o a ising
bubble, while deso p ion occu s mainly wi hin he ea s agnan cap.
This ea u e may a o he s udy o he adso p ion kine ics o la ge
molecules (Dukhin e al., 2015).
The e is abundan expe imen al in o ma ion abou su ac an s’ e -
ec on he ising single bubble. Fo ins ance, i is well known ha
he heigh and wid h o he eloci y p o ile maximum dec ease as
he su ac an concen a ion inc eases (K zan e al., 2007). K ach and
Finch (2010) ound ha he ins an aneous alues o he bubble eloci y
and aspec a io we e co ela ed in hei expe imen s. This sugges s
ha su ac an s can essen ially a ec bubble ise eloci y h ough he
bubble shape. Fdhila and Duine eld (1996) showed ha he bubble
slowdown occu s when he su ac an co e age eaches app oxima ely
hal he bubble su ace. The s eady-s a e eloci y was independen o
he su ac an concen a ion o su icien ly la ge concen a ions (Sam
e al., 1996; Zhang and Finch, 2001).
Se e al s udies ha e examined he e ec s o speci ic su ac an s. Fo
ins ance, adding a su ac an o 𝛽-lac oglobulin (Ulagana han e al.,
2014) solu ions o a ying he pH o bo ine se um albumin solu-
ions (Zawala e al., 2010) d ama ically al e he eloci y p o ile o
a ising bubble. The ime o es ablishing an immobile igid su ace
laye a he ising bubble su ace becomes sho e wi h inc easing
pH (Ulagana han e al., 2016). The bubble mobili y is mo e in luenced
han he bubble shape when adding sodium dodecyl sul a e (SDS) o
a pseudo-plas ic liquid (Tzounakos e al., 2004). Tagawa e al. (2014)
s udied he bubble pa h ins abili y in 1-pen anol and T i on X-100 so-
lu ions, obse ing ha he d ag o ce mono onically inc eases wi h he
su ac an concen a ion, while he li o ce showed a non-mono onic
beha io . Ce yl ime hyl ammonium b omide (CTAB) dissol ed in ul-
apu e wa e exhibi s wo dis inc eloci y s ages, whe eas Tween 80
solu ions display a h ee-s age eloci y p o ile (Luo e al., 2022).
The in o ma ion in all he expe imen al s udies is limi ed o he
ime-dependen bubble shape and eloci y. In e p e ing hese expe i-
men al da a equi es heo e ical models o desc ibe he uns eady dy-
namic adso p ion laye .
App oxima e analy ical me hods ha e been p oposed o desc ibe
he uns eady dynamic adso p ion laye and i s in luence on bubble is-
ing (Zholko skij e al., 2000). The esul s only apply o sphe ical shapes
and low Reynolds numbe s (Zholko skij e al., 2000). In he quasi-
s eady app oxima ion, he ins an aneous eloci y has been assumed
o be he s eady eloci y co esponding o he amoun o he solu e
accumula ed a ha ins an (Dukhin e al., 2015). P e ious s udies ha e
used his heo y o ob ain he pa ame e s o adso p ion–deso p ion
kine ics om he local eloci y p o ile measu ed in he expe imen s. Fo
non-sphe ical bubbles and la ge Reynolds numbe s, accu a e ansien
nume ical simula ions a e equi ed o p oduce eliable esul s.
Mos nume ical s udies ocus mainly on he inal s a iona y (s a-
ble) (Mes el, 1994; Lakshmanan and Eh ha d, 2010; Li and Mao,
2001; Takemu a, 2005; Fuku a e al., 2008; Tasoglu e al., 2008;
Ken heswa an e al., 2023; Dani e al., 2022) o oscilla ing (uns a-
ble) (Pesci e al., 2018; Fa soiy e al., 2024) egime eached by
he ising bubble. The ini ial s age ollowing he bubble elease has
ecei ed less a en ion. T ansien axisymme ic simula ions show ha
he bubble a ains a maximum eloci y be o e slowing down o i s
s eady-s a e eloci y (Liao and McLaughlin, 2000). A speed compa able
o he s eady-s a e speed in pu e wa e is eached e en i he bubble
is con amina ed be o e eleasing i (Liao and McLaughlin, 2000). The
elas ici y numbe and he bulk Pecle numbe signi ican ly a ec he
ini ial ansien mo ion as well (Tasoglu e al., 2008).
Cueno e al. (1997) conside ed a simpli ied dynamic p oblem in
which he bubble emained sphe ical and i s eloci y was cons an
h oughou he con amina ion p ocess, ocusing on he physicochemical
p ocesses neglec ed in many p e ious app oaches. They desc ibed he
empo al e olu ion o he ele an in e acial quan i ies o show he
o ma ion o he dynamic adso p ion laye . The ising o a bubble
co e ed by T i on X-100 was simula ed by decoupling he solu ion o
he luid low and mass ans e unde di e en app oxima ions (Zhang
e al., 2001). The bes esul s we e ob ained assuming he s agnan
cap model and su ac an ans e o he in e ace con olled by di u-
sion. Ma sumo o e al. (2006) showed he in luence o he adso p ion
and deso p ion cons an s on he local eloci y p o ile and he empo al
e olu ion o he d ag coe icien . The heigh and wid h o he eloci y
p o ile maximum dec ease as he adso p ion (deso p ion) cons an
inc eases (dec eases).
Pesci e al. (2018) conduc ed h ee-dimensional di ec nume ical
simula ions o a bubble ising in a liquid con aining a soluble su ac-
an . The nume ical esul s ag eed wi h hei expe imen s (Pesci e al.,
2017). They concluded ha he ini ial ansien s age is e y sensi i e
o he ini ial su ace concen a ion. The quasi-s eady s a e o he ise
eloci y is eached wi hou adso p ion and deso p ion being necessa ily
in equilib ium.
Rubio e al. (2024) ha e ecen ly explained how aces o su ac an
can signi ican ly change he s eady ( e minal) egime o bubble ising.
The iny su ace ension a ia ion occu s wi hin an ex emely hin,
di usi e su ace bounda y laye . This p oduces a Ma angoni s ess
con ined wi hin ha laye ha immobilizes he ee su ace and d as i-
cally inc eases he ou e iscous s ess only in he di usi e laye . The
dynamic adso p ion laye desc ibed by Rubio e al. (2024) quali a i ely
di e s om hose desc ibed in o he wo ks unde di e en condi ions,
in which he Ma angoni and ou e iscous s esses a e sp ead o e a
conside able po ion o he bubble su ace.
In his wo k, we will s udy nume ically how he singula dynamic
adso p ion laye appea ing a e y low su ac an concen a ions (Ru-
bio e al., 2024) g ows o e ime. We will analyze he empo al e o-
lu ion o he ele an in e acial quan i ies and explain he e ec o
su ac an con ec ion, kine ics, and di usion on he g ow h o he
dynamic adso p ion laye .
The nume ical esul s o Ma sumo o e al. (2006) o de o mable
bubbles a e es ic ed o small Reynolds and Pecle numbe s. Ou
bounda y- i ed nume ical me hod is simila o ha employed in ha
wo k. We use a spec al colloca ion echnique (Kho ami, 1989) o
accumula e he g id poin s nex o he in e ace, which allows us o
esol e he ex emely hin su ac an bounda y laye on he ou e side
o he bubble su ace. In his way, we sol e he comple e model (Ma -
sumo o e al., 2006) e en o ealis ic alues o he Reynolds and Pecle
numbe s.
I is well known ha he su ac an monolaye enhances he pa h
ins abili y, educing he c i ical adius abo e which helical and zig-
zagging ajec o ies a e obse ed (Tagawa e al., 2014; Rubio e al.,
2024). Non-axisymme ic ins abili ies a e no allowed in ou axisym-
me ic simula ions. Fo his eason, we will conduc expe imen s o
de e mine he pa ame e condi ions leading o a s aigh pa h (ax-
isymme ic low) and o alida e ou nume ical solu ions o hose
condi ions.
In e na ional Jou nal o Mul iphase Flow 188 (2025) 105205
2
D. Fe nández-Ma ínez e al.
Fig. 1. Ske ch o he nume ical domain. The blue and ed ou e bounda ies co espond
o he inle and non- e lec ing bounda y condi ions, espec i ely. (Fo in e p e a ion o
he e e ences o colo in his igu e legend, he eade is e e ed o he web e sion
o his a icle.)
2. Me hods
2.1. Go e ning equa ions
Conside a bubble o adius 𝑅= [3𝑉∕(4𝜋)]1∕3 (𝑉 is he bubble
olume), densi y 𝜌(𝑖), and iscosi y 𝜇(𝑖) ising in a liquid o densi y 𝜌(𝑜)
and iscosi y 𝜇(𝑜). The su ace ension o he clean in e ace is 𝜎𝑐, while
he g a i y accele a ion is 𝑔.
We dissol e a su ac an in he liquid a he concen a ion 𝑐∞. The
su ac an can be conside ed soluble because he so p ion cha ac e is ic
imes a e smalle han he hyd odynamic cha ac e is ic ime. In he
amewo k o he model conside ed he e, he su ac an p ope ies a e
he olume ic di usion coe icien 𝑜, he su ace di usion coe i-
cien 𝑠, he adso p ion and deso p ion cons an s, 𝑘𝑎 and 𝑘𝑑, and he
maximum packing densi y 𝛤∞.
The hyd odynamic equa ions a e sol ed in a cylind ical sys em o
coo dina es (𝑟, 𝑧) whose o igin solidly mo es wi h he bubble’s uppe
poin (Fig. 1). The con inui y and momen um equa ions a e
𝛁⋅𝐯(𝑗)= 0,(1)
𝜌(𝑗)𝐷𝐯(𝑗)
𝐷𝑡 = −𝜌(𝑗)(𝑔+𝑑2ℎ
𝑑𝑡2)𝐞𝐳+𝛁⋅𝝈(𝑗),(2)
whe e 𝐯(𝑗)=𝑢(𝑗)𝐞𝐫+𝑤(𝑗)𝐞𝐳 is he axisymme ic eloci y ield, he
supe sc ip s 𝑗=𝑖 and 𝑜 e e o he inne and ou e phases, espec i ely,
𝐞𝐫 and 𝐞𝐳 a e he uni ec o s along he axis 𝑟 and 𝑧, espec i ely, 𝐷∕𝐷𝑡
is he ma e ial de i a i e, ℎ=𝑍−𝑧 is he e ical posi ion o he
bubble’s uppe poin (Fig. 1),
𝝈(𝑗)= −𝑝(𝑗)𝐈+𝝉(𝑗)(3)
is he s ess enso , 𝑝(𝑗) is he hyd os a ic p essu e, 𝐈 is he iden i y
ma ix, and
𝝉(𝑗)=𝜇(𝑗)[𝛁𝐯(𝑗)+(𝛁𝐯(𝑗))𝑇](4)
is he iscous s ess enso .
The eloci y ield is con inuous (𝐯(𝑖)=𝐯(𝑜)) a he ee su ace.
The in e ace is pa ame ized in e ms o he me idional a c leng h 𝑠
(0≤𝑠≤𝑠𝑓) as 𝑟𝑠=𝑓(𝑠) and 𝑧𝑠=𝑔(𝑠), whe e (𝑟𝑠, 𝑧𝑠) is he in e ace
loca ion. He e, 𝑠𝑓 is he a c leng h co esponding o he bubble’s ea
poin (Fig. 1). The kinema ic compa ibili y condi ion eads
𝜕𝑓
𝜕𝑡 +𝑢(𝑖)𝑔′−𝑤(𝑖)𝑓′= 0,(5)
whe e he apos ophe (′) indica es he de i a i e wi h espec o 𝑠.
The equilib ium o no mal and angen ial s esses on he in e ace
leads o he equa ions
𝐞𝐧⋅(𝝈(𝑜)−𝝈(𝑖))⋅𝐞𝐧=𝜎𝜅, 𝐞𝐭⋅(𝝈(𝑜)−𝝈(𝑖))⋅𝐞𝐧=𝜏Ma,(6)
whe e 𝜎 is he local alue o he in e acial ension, 𝜅= ∇⋅𝐞𝐧 is ( wice)
he mean cu a u e, and
𝐞𝐧=𝑔′𝐞𝑟−𝑓′𝐞𝑧
(𝑓′2 +𝑔′2)1∕2 and 𝐞𝐭=𝑓′𝐞𝑟+𝑔′𝐞𝑧
(𝑓′2 +𝑔′2)1∕2 (7)
a e he uni ec o s no mal and angen ial o he in e ace, espec i ely.
In addi ion, 𝜏Ma =𝜎′ is he Ma angoni s ess. We neglec he iscous
su ace s esses in Eqs. (6) because he shea and dila a ional iscosi ies
o he su ac an monolaye a e e y small (Ponce-To es e al., 2020).
We es ic ou sel es o low su ac an concen a ions, which im-
plies ha he su ac an is p esen as monome s. The monome olu-
me ic concen a ion 𝑐(𝑜)(𝐫, 𝑡) in he ou e phase is calcula ed om he
conse a ion equa ion (C as e e al., 2009; Kalogi ou and Bly h, 2019)
𝜕𝑐(𝑜)
𝜕𝑡 +𝐯(𝑗)⋅𝛁𝑐(𝑜)=𝑜𝛁2𝑐(𝑜).(8)
Now, we model he ans e o monome s be ween he bulk and
he bubble su ace. The ne so p ion lux =𝑎−𝑑 is calcula ed as
he di e ence be ween he adso p ion 𝑎 and deso p ion 𝑑 lux. The
kine ic model
𝑎=𝑘𝑎𝑐(𝑜)
𝑠(1 − 𝛤
𝛤∞),𝑑=𝑘𝑑𝛤(9)
is adop ed o calcula e hese luxes. As men ioned abo e, 𝑘𝑎 and 𝑘𝑑 a e
he adso p ion and deso p ion cons an s, espec i ely, 𝑐(𝑜)
𝑠 is he bulk
su ac an concen a ion e alua ed a he in e ace, 𝛤 is he su ac an
su ace concen a ion ( he su ace co e age, measu ed in mols pe uni
a ea), and 𝛤∞ is he maximum packing densi y.
The su ac an su ace concen a ion 𝛤 e i ies he ad ec ion-
di usion equa ion (C as e e al., 2009)
𝜕𝛤
𝜕𝑡 +𝛁𝑠⋅(𝛤𝐯𝑠) + 𝛤(𝛁𝑠⋅𝐧)(𝐯⋅𝐞𝐧) = 𝑠∇2
𝑠𝛤+,(10)
whe e 𝛁𝑠 is he angen ial in insic g adien along he ee su ace,
𝐯𝑠=𝗜𝑠𝐯 is he ( wo-dimensional) su ace eloci y, 𝗜𝑠=𝗜−𝐞𝐧𝐞𝐧 is he
enso ha p ojec s any ec o on ha su ace, 𝗜 is he iden i y enso ,
and 𝑠 is he su ace di usion coe icien .
The dependence o he su ace ension 𝜎 on he su ace concen-
a ion 𝛤 is gi en by he Langmui equa ion o s a e (T ico , 1997)
𝜎=𝜎𝑐+𝛤∞𝑅𝑔𝑇ln (1 − 𝛤
𝛤∞),(11)
whe e 𝜎𝑐 is he su ace ension o he clean in e ace, 𝑅𝑔 is he gas
cons an , and 𝑇 is he empe a u e.
The kine ic model (9) yields he Langmui iso he m a equilib ium
(𝑎=𝑑). Combining he Langmui and Gibbs iso he ms leads o he
Langmui equa ion o s a e (11) (T ico , 1997). This means he model
(9) and (11) a e consis en . Fo his eason, we selec ed Eq. (11) o
model he SDS e ec , e en hough i may be less accu a e han i ing
he expe imen al da a (Ponce-To es e al., 2020).
We assume ha he liquid ba h is no pe u bed by he bubble in
he ups eam egion 𝑟=𝑅𝑜 and 𝑧 > 0 (Fig. 1). The e o e,
𝑤(𝑜)= − 𝑑ℎ
𝑑𝑡 , 𝑢(𝑜)= 0, 𝑝(𝑜)+𝜌(𝑜)𝑔𝑧 =cons . (12)
in ha bounda y. The non- e lec ing bounda y condi ions
𝜕𝑢(𝑜)
𝜕𝑧 =𝜕𝑤(𝑜)
𝜕𝑧 = 0 (13)
a e applied in he downs eam egion a om he bubble (𝑟=𝑅𝑜 and
𝑧 < 0) o cap u e he wake (Fig. 1). The su ac an concen a ion a
𝑟=𝑅𝑜 is 𝑐∞.
We conside he egula i y condi ions
𝑢(𝑗)=𝜕𝑤(𝑗)
𝜕𝑟 =𝜕𝑝(𝑗)
𝜕𝑟 =𝜕𝑐(𝑜)
𝜕𝑟 = 0 (14)
In e na ional Jou nal o Mul iphase Flow 188 (2025) 105205
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Table 1
P ope ies o SDS and T i on X-100. The alues o SDS we e hose conside ed by
Rubio e al. (2024). The alues o T i on X-100 we e aken om Re s. Tagawa e al.
(2014) and Lin e al. (1990).
CMC (mol/m3)𝑘𝑎 (m/s) 𝑘𝑑 (s−1)𝛤∞ (mol/m2)𝑜 (m2/s)
SDS 8.0 2.34 × 10−4 6.4 3.9 × 10−6 8.0 × 10−10
T i on X-100 0.23 1.45 × 10−4 3.3 × 10−2 2.9 × 10−6 2.6 × 10−10
a he symme y axis 𝑟= 0. The condi ion 𝑤(𝑜)= 0 a he in e ace uppe
poin allows us o calcula e he bubble’s e ical posi ion ℎ(𝑡). Finally,
we speci y he bubble’s olume h ough he equa ion
𝑉=𝜋∫𝑠𝑓
0
𝑓2𝑔′𝑑𝑠. (15)
We s a he simula ion om he hyd os a ic solu ion co esponding
o a sphe ical bubble. Conside a bubble ‘‘ins an aneously’’ in la ed in a
ank o s ill wa e wi h su ac an . The Damkohle numbe Da=𝜏𝐷∕𝜏𝑘
ela es he ime scales 𝜏𝐷 and 𝜏𝑘 co esponding o he su ac an ans-
e limi ed by di usion and so p ion, espec i ely (see he Appendix
A) (Manikan an and Squi es, 2020). Conside ing he alues shown in
he nex sec ion, he Damkohle numbe akes alues o he o de o
10 and 102 o SDS and T i on X-100, espec i ely, indica ing ha he
su ac an ans e o he bubble is limi ed by di usion in bo h cases.
In he expe imen , he bubble emained a ached o he capilla y o
app oxima ely 5 s. The su ace concen a ions o SDS and T i on X-100
a ha ins an we e simila o he equilib ium concen a ion o SDS
(see Fig. A.13 in he Appendix A). Fo his eason, we conside his
alue as he ini ial su ace concen a ion in all he simula ions.
The abo e equa ions a e nume ically in eg a ed wi h a a ian o he
bounda y- i ed spec al me hod p oposed by He ada and Mon ane o
(2016). The majo di icul y associa ed wi h a soluble su ac an is he
exis ence o a e y hin di usi e bounda y laye nex o he in e ace
o he small di usion coe icien s o mos su ac an s (la ge Pecle
numbe s). We use Chebyshe spec al colloca ion poin s o accumula e
he g id poin s nex o he in e ace (He ada and Mon ane o, 2016),
acili a ing he esolu ion o his laye (Rubio e al., 2024). A g id
sensi i i y analysis is shown in he Supplemen al Ma e ial.
2.2. Expe imen al me hod
In an expe imen , ni ogen was injec ed h ough a needle o o m
he bubble in he cen e o he ank bo om. The bubble de ached
om he needle and ose ac oss he ank un il i eached he ee
su ace. We used a i ual binocula s e eo ision sys em (Luo e al.,
2022) o image wo pe pendicula iews o he ising bubble. The
images we e p ocessed a he pixel le el (Canny, 1986) o de e mine
he bubble shape and eloci y. De ails o he expe imen al se up and he
image p ocessing analysis can be ound in he Supplemen al Ma e ial.
This expe imen al me hod was alida ed by Rubio e al. (2024) om
compa ison wi h he esul s o clean wa e ob ained by Duine eld
(1995).
We conside SDS and T i on X-100 in ou expe imen s. SDS is
an anionic su ac an wi h a molecula weigh o 288.4 g/mol, while
T i on X-100 is a nonionic su ac an wi h a molecula weigh o 647
g/mol. Table 1 shows he ele an p ope ies o hese su ac an s. The
di usion coe icien s o SDS and T i on X-100 a e commensu a e wi h
each o he . SDS is sligh ly mo e ac i e han T i on X-100, as indica ed
by he alue o 𝛤∞. The SDS adso p ion cons an is also simila o ha
o T i on X-100. In e es ingly, he deso p ion a e 𝑘𝑑 o T i on X-100 is
wo o de s o magni ude smalle han ha o SDS. Fo small su ac an
concen a ions, 𝛤=𝑘𝑎𝑐∕𝑘𝑑 a equilib ium [Eqs. (9)]. This means ha
one needs o dissol e much ewe moles o T i on X-100 in wa e
o achie e he same su ac an su ace concen a ion a equilib ium.
This is e lec ed in he c i ical micelle concen a ion, a ound 35 imes
smalle in he T i on X-100 case.
3. Expe imen al esul s
As men ioned in he In oduc ion, he goal o ou expe imen al
s udy is wo old: (i) o de e mine he pa ame e condi ions ha lead o
an axisymme ic, s aigh (s able) bubble ising and (ii) o alida e he
nume ical me hod. Fig. 2 shows he h ee ypes o ajec o ies obse ed
in ou expe imen s wi h T i on X-100. Simila esul s we e ob ained o
SDS (Rubio e al., 2024).
Fig. 2a co esponds o a bubble ollowing a quasi-s aigh ajec-
o y. The small ampli ude oscilla ions can be a ibu ed o he came a
ib a ion while mo ing (we did no obse e hose oscilla ions when he
came a was a es ). The pa h is sligh ly il ed ( he il angle is smalle
han 0.1◦), p obably due o a small asymme y in he bubble o ma ion
p ocess. Fo 𝑐∞∕𝑐cmc = 5 × 10−4 [case (b)], he bubble mo ion emains
s able un il 𝑧≃ 350 mm. Helical ins abili y de elops a la ge dis ances
om he ejec o . This expe imen highligh s he impo ance o conduc -
ing expe imen s wi h la ge anks and may explain he disc epancies
among he c i ical condi ions ound in p e ious wo ks.
The dis ance om he ejec o a which he helical ins abili y de el-
ops dec eases as he su ac an concen a ion inc eases (Fig. 2c). Fo
𝑐∞∕𝑐cmc = 3.3 × 10−3 [case (d)], he helical ins abili y e ol es owa d a
zig-zag mo ion, as shown by he p ojec ion o he bubble pa h on o he
𝑥𝑦 plane ( he ed line co esponds o he zig-zag mo ion). The zig-zag
ins abili y a ises close o he bubble ejec o o he highes su ac an
concen a ion conside ed in ou analysis [case (e)].
Figs. 3–5 compa e he e ec o SDS and T i on X-100 on he bubble
eloci y and aspec a io. We moni o he bubble eloci y and he
aspec a io o sa ely de e mine whe he he bubble has eached a
s eady mo ion.
The esul s in Figs. 3and 4 we e ob ained o he same alue
o he ela i e concen a ion 𝑐∞∕𝑐cmc o SDS and T i on X-100. Fo
small concen a ions, Eqs. (9) yield 𝛤eq∕𝛤∞= [𝑘𝑎𝑐cmc∕(𝑘𝑑𝛤∞)] 𝑐∕𝑐cmc
a equilib ium, whe e 𝑘𝑎𝑐cmc∕(𝑘𝑑𝛤∞) = 75 and 348 o SDS and T i on
X-100, espec i ely. This means ha he same alue o 𝑐∕𝑐cmc leads o
a much highe equilib ium su ace co e age 𝛤eq∕𝛤∞ and densi y 𝛤eq in
he T i on X-100 case. Howe e , he T i on X-100 equilib ium su ace
co e age is eached a much longe imes due o he much smalle
alue o he deso p ion cons an . This sugges s ha T i on X-100 mus
p oduce a smalle e ec on a ela i ely sho ime scale, bu his e ec
mus e en ually exceed ha p oduced by SDS a su icien ly la ge imes.
The esul s in Figs. 3and 4 a e consis en wi h he abo e p edic ion.
The bubble eloci y and aspec a io o SDS a e smalle han hose o
T i on X-100. Ne e heless, a c osso e is expec ed o occu beyond he
maximum heigh analyzed in he expe imen .
The esul s in Figs. 5and 6 we e ob ained o p ac ically he same
alue o he absolu e concen a ion 𝑐∞ o SDS and T i on X-100. In
his case, he adso p ion lux 𝑎≃𝑘𝑎𝑐∞ is expec ed o ake simila
alues o he wo su ac an s. This explains why he bubbles co e ed
wi h SDS and T i on X-100 exhibi simila eloci ies and aspec a ios
in he i s s age o he bubble ising. As explained in Sec ion 2.2, he
same olume ic concen a ion leads o a much highe T i on X-100
su ace concen a ion a equilib ium. This explains why he T i on X-
100 e ec s a e la ge han hose caused by SDS a he same absolu e
concen a ion (Figs. 5and 6).
The expe imen al local eloci y p o ile exhibi s he so-called ‘‘o e -
shoo ing’’ phenomenon: he eloci y eaches a maximum and subse-
quen ly dec eases o i s e minal alue. Conside he ollowing balance
o o ces ac ing on he bubble:
=𝐷0(𝑣𝑧(𝑡)) + (𝑡),(16)
whe e is he app oxima ely cons an buoyancy o ce, 𝐷0 ep esen s
he d ag o ce exe ed on he same bubble bu in he absence o
su ac an , and is he ex a d ag o ce associa ed wi h he su ac an
monolaye . In e es ingly, he ajec o ies wi h and wi hou SDS a e
p ac ically he same o 𝑧 ≲ 50 mm (Fig. 5). This implies ha ≃ 0
du ing his pa o he bubble ising p obably because he inc ease
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D. Fe nández-Ma ínez e al.
Fig. 2. Bubble ajec o y o 𝑅= 0.76 mm and 𝑐∞∕𝑐cmc = 10−4 (a), 5 × 10−4 (b), 10−3 (c), 3.6 × 10−3 (d), and 10−2 (e). The g aphs also show he p ojec ions o he ajec o ies on o
he planes (𝑥, 𝑦), (𝑥, 𝑧), and (𝑦, 𝑧). (Fo in e p e a ion o he e e ences o colo in his igu e legend, he eade is e e ed o he web e sion o his a icle.)
Fig. 3. Bubble eloci y 𝑣𝑧 and aspec a io 𝜒 as a unc ion o he e ical posi ion 𝑧 o
he cen e o g a i y o 𝑅= 0.76 mm and 𝑐∞∕𝑐cmc = 5×10−4 . The solid and dashed lines
co espond o he s able and oscilla o y pa s o he bubble ajec o y, espec i ely.
in he iscous ic ion is balanced by he dec ease in he bubble
de o ma ion ( he bubble adop s a mo e ae odynamic shape) due o he
su ac an . Fo 𝑧 ≳ 50 mm, he bubble shape ha dly changes while (i)
he iscous ic ion inc eases due o he g ow h o he Ma angoni s ess
Fig. 4. Bubble eloci y 𝑣𝑧 and aspec a io 𝜒 as a unc ion o he e ical posi ion 𝑧 o
he cen e o g a i y o 𝑅= 0.76 mm and 𝑐∞∕𝑐cmc = 10−3 . The solid and dashed lines
co espond o he s able and oscilla o y pa s o he bubble ajec o y, espec i ely.
wi hin he su ac an su ace bounda y laye and (ii) he sepa a ion
poin sligh ly displaces ups eam due o he displacemen o ha laye ,
as explained in Sec ion 4.2. This esul s in an inc ease in he d ag o ce
and a dec ease in he bubble eloci y ( he o e shoo ing phenomenon).
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D. Fe nández-Ma ínez e al.
Fig. 5. Bubble eloci y 𝑣𝑧 and aspec a io 𝜒 as a unc ion o he e ical posi ion
𝑧 o he cen e o g a i y o 𝑅= 0.66 mm and he concen a ions 𝑐∞= 8.3 × 10−4
mol/m3 o T i on X-100 and 𝑐∞= 8 × 10−4 mol/m3 o SDS. The solid and dashed lines
co espond o he s able and oscilla o y pa s o he bubble ajec o y, espec i ely.
The ed ci cles co espond o he expe imen al da a o Zhang and Finch (2001) o
𝑅= 0.70 mm and T i on X-100 a he concen a ion 𝑐∞= 7.5 × 10−4 mol/m3. The a ow
indica es he maximum in he eloci y o SDS due o he o e shoo ing phenomenon.
(Fo in e p e a ion o he e e ences o colo in his igu e legend, he eade is e e ed
o he web e sion o his a icle.)
Fig. 6. Bubble eloci y 𝑣𝑧 and aspec a io 𝜒 as a unc ion o he e ical posi ion 𝑧 o
he cen e o g a i y o 𝑅= 0.76 mm and he concen a ions 𝑐∞= 8.3×10−4 mol/m3 o
T i on X-100 and 𝑐∞= 8 × 10−4 mol/m3 o SDS. The solid and dashed lines co espond
o he s able and oscilla o y pa s o he bubble ajec o y, espec i ely.
Figs. 5and 6 allow us o analyze he in luence o he bubble
adius. The su ac an e ec inc eases as he bubble adius dec eases.
Speci ically, he e minal eloci y educ ion caused by SDS is la ge o
he 𝑅= 0.66 mm case, and T i on X-100 des abilizes he bubble pa h a
smalle alues o 𝑧 in ha case. This esul can be expec ed because he
su ac an p oduces in e acial e ec s, and he su ace- o- olume a io
inc eases as 𝑅 dec eases.
All he ajec o ies analyzed in Figs. 3–5 become uns able excep
o he case SDS a 𝑐∞= 8 × 10−4 mol/m3, whe e he bubble eaches a
s eady s a e. In he nex sec ion, we nume ically analyze he empo al
e olu ion o he dynamic adso p ion laye o ha case. The analysis
o his axisymme ic low is ealis ic because he low does no su e
om non-axisymme ic ins abili ies o e he en i e bubble ajec o y.
4. Nume ical esul s
4.1. Valida ion o he nume ical me hod
We calcula ed he nume ical solu ion o 𝜌(𝑖)= 1.2 kg∕m3, 𝜌(𝑜)=
997 kg∕m3, 𝜇(𝑖)= 1.83 × 10−5 kg/(m⋅s), 𝜇(𝑜)= 9.3 × 10−4 kg/(m s), and
𝜎𝑐= 72 mN/m. The alues o 𝛤∞ we e aken om Table 1. The di usion
coe icien 𝑜 was se equal o ha o SDS o a o he con e gence o
he nume ical me hod, which may sligh ly a ec he e minal eloc-
i y (Rubio e al., 2024). We conside ed 𝑠= 4.2 × 10−7 m2/s. The alue
o 𝑠 was inc eased wi h espec o he expec ed expe imen al alue o
a o he con e gence o he nume ical me hod. We e i ied ha his
does no signi ican ly a ec he esul . Speci ically, an inc ease in 𝑠
o he o de o 10−1 p oduces an inc ease in he e minal eloci y o
he o de o 10−5. This occu s because he Pecle numbe is su icien ly
la ge o he su ac an su ace dis ibu ion o be domina ed by he
ad ec ion and adso p ion–deso p ion kine ics (Tagawa e al., 2014).
The epo ed alues o he adso p ion and deso p ion cons an s
o mos su ac an s a e ela i ely inconsis en . Fo his eason, hese
alues a e adjus ed in he simula ion o ep oduce he bubble pa h
obse ed expe imen ally. In ac , bubble ising has been p oposed o de-
e mine he alues o hose cons an s. We ollowed di e en s a egies
o de e mine he adso p ion and deso p ion cons an s o T i on X-100
and SDS.
Fo he small su ac an concen a ions conside ed in ou analysis,
𝛤∕𝛤∞≪1, which implies ha
𝜎𝑐−𝜎≃𝐿𝑑𝑐∞𝑅𝑔𝑇(17)
a equilib ium. This equa ion shows he c i ical ole played by he de-
ple ion leng h 𝐿𝑑=𝑘𝑎∕𝑘𝑑 (o , equi alen ly, he Langmui equilib ium
adso p ion cons an 𝐾𝐿=𝐿𝑑∕𝛤∞) in he bubble dynamics. A eliable
alue o he deple ion leng h, 𝐿𝑑= 4.4 mm, has been de e mined o
T i on X-100 in p e ious s udies (P osse and F anses, 2001; Tagawa
e al., 2014). Fo his eason, i is easonable o i he nume ical bubble
pa h o he expe imen al one by a ying 𝑘𝑎 and 𝑘𝑑 while keeping 𝐿𝑑
equal o ha alue (Lin e al., 1990; Fdhila and Duine eld, 1996). We
ha e done his wi h T i on X-100.
The su ace ension measu emen o SDS does no lead o a eliable
alue o 𝐿𝑑 (Fdhila and Duine eld, 1996; P osse and F anses, 2001).
Al e na i ely, Rubio e al. (2024) ook he deso p ion cons an 𝑘𝑑 om
accu a e expe imen al measu emen s and used 𝑘𝑎 as he i ing pa am-
e e , whose expe imen al alue exhibi s mo e unce ain y. The alue
shown in Table 1 led o he bes ag eemen be ween he nume ical
e minal eloci y and he expe imen al one o he case analyzed in his
wo k. We ake ha alue o he ansien simula ions conduc ed he e.
The deple ion leng h 𝐿𝑑 ob ained om his p ocedu e is consis en wi h
he alue gi en by Fdhila and Duine eld (1996).
Fig. 7 compa es he nume ical and expe imen al local eloci y
p o iles o T i on X-100 and SDS. The nume ical esul s o T i on X-
100 we e calcula ed o 𝑘𝑎= 7.25 × 10−4 m∕s and 14.5 × 10−4 m∕s while
keeping 𝐿𝑑 cons an (𝐿𝑑= 4.4 mm), as explained abo e. The simula ion
wi h 𝑘𝑎= 14.5 × 10−4 m∕s app oxima ely cap u es he dependency o
he bubble eloci y on he e ical coo dina e. Signi ican de ia ions
a ise in he oscilla o y pa o he expe imen al bubble ajec o y, which
sugges s ha he oscilla o y ins abili y educes he a e age e ical
eloci y.
As obse ed, he bubble ajec o y conside ably depends on 𝑘𝑎 o
a ixed alue o he deple ion leng h 𝐿𝑑. This con as s wi h he ime
e olu ion o he su ace co e age when an ini ially clean sphe ical
bubble emains a es in a liquid ba h, which depends on 𝐿𝑑=𝑘𝑎∕𝑘𝑑
(no on 𝑘𝑎 and 𝑘𝑑 sepa a ely) in he di usion-con olled limi applicable
o ou case (see he Appendix A). The e o e, he so p ion cons an s
en e ou p oblem sepa a ely due o su ac an con ec ion caused by
he bubble mo ion.
The esul s o SDS show a good ag eemen be ween he simula ion
and he expe imen . Fo 𝑧 ≳ 80 mm, bo h he bubble eloci y and he
aspec a io ake app oxima ely cons an alues (Fig. 5), indica ing ha
a quasi-s eady egime has been eached. In his egime, he bubble e-
loci y is signi ican ly smalle han ha in pu e wa e . This di e ence is
accu a ely cap u ed by he simula ion. The o e shoo ing phenomenon
is mo e no iceable in he expe imen . This sugges s ha he o ma ion
o he su ac an su ace bounda y laye is sligh ly delayed wi h e-
spec o i s nume ical coun e pa . This delay may occu because he
su ace di usion coe icien is smalle han he alue conside ed in he
simula ion.
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D. Fe nández-Ma ínez e al.
Fig. 7. Bubble eloci y 𝑣𝑧 as a unc ion o he e ical posi ion 𝑧 o he cen e o g a i y
o T i on X-100 (uppe g aph) and SDS (lowe g aph). The expe imen s wi h T i on
X-100 we e conduc ed o 𝑅= 0.76 mm and 𝑐∞= 8.3 × 10−4 mol/m3. The expe imen s
wi h SDS we e ob ained o 𝑅= 0.66 mm and 𝑐∞= 8 × 10−4 mol/m3. The solid and
dashed lines co espond o he s able and oscilla o y pa s o he expe imen al bubble
ajec o y, espec i ely. The symbols a e he simula ion esul s. The simula ion esul s
o T i on X-100 we e calcula ed wi h 𝑘𝑎= 7.25 × 10−4 m∕s and 14.5 × 10−4 m∕s. In he
wo cases, 𝐿𝑑= 4.39 mm. The esul s o SDS we e ob ained wi h he alues o 𝑘𝑎 and
𝑘𝑑 shown in Table 1.
Table 2
Dimensionless numbe s de ined based on he cha ac e is ic quan i ies wi hou
su ac an .
𝜌=𝜌(𝑖)∕𝜌(𝑜)𝜇=𝜇(𝑖)∕𝜇(𝑜)𝐵=𝜌(𝑜)𝑔𝑅2∕𝜎𝑐A =𝜌(𝑜)2𝑔𝑅3∕𝜇(𝑜)2
1.20 × 10−3 1.96 × 10−2 5.92 × 10−2 3.24 × 103
4.2. G ow h o he dynamic adso p ion laye
This sec ion analyzes nume ically he g ow h o he dynamic ad-
so p ion laye o med when a bubble is eleased in wa e con aining
a su ac an a a e y low concen a ion. Speci ically, we conside he
same con igu a ion as ha s udied by Rubio e al. (2024) in he s eady
egime, a bubble wi h 𝑅= 0.66 mm in wa e con aining SDS a he
concen a ion 𝑐∞= 8 × 10−4 mol/m3 (see Figs. 5and 7). Tables 2
and 3 show he alues o he dimensionless numbe s cha ac e izing he
p oblem. In his sec ion, leng h, ime, eloci y, lux, and s esses a e
measu ed in e ms o 𝑅, 𝑡𝑐=𝑅∕𝑣𝑡, 𝑣𝑡, 𝑣𝑡𝑐∞, and 𝜌(𝑜)𝑣2
𝑡, espec i ely.
Fig. 8 shows he su ac an monolaye e olu ion du ing he bubble
ising. Since he bubble shape changes o e ime, he pola angle
measu ed om he bubble’s cen e o g a i y also changes, and i s
use as he independen a iable is con using. Fo his eason, we ha e
selec ed ins ead he su ace a ea 𝑎 measu ed om he no h pole. The
su ac an su ace concen a ion 𝛤 sha ply inc eases in he ea o he
bubble. The maximum o 𝛤(𝑎) displaces owa d he bubble equa o
un il i eaches he app oxima ely cons an loca ion 𝑎∕(4𝜋𝑅2)≃0.9,
whe e 𝛤 becomes almos en imes he ini ial concen a ion 𝛤0. This
la ge inc ease in 𝛤 occu s wi hin a e y hin di usi e laye , p oducing
a signi ican su ace ension g adien e en hough his quan i y changes
in less han 1%. The su ace ension g adien gi es ise o a Ma angoni
s ess 𝜏Ma h ee o de s o magni ude la ge han he angen ial iscous
s ess in a su ac an - ee bubble (Rubio e al., 2024). Al hough he
Ma angoni s ess is con ined wi hin he su ace bounda y laye , i
immobilizes pa o he bubble’s sou h hemisphe e and signi ican ly
educes he e minal eloci y (Fig. 5).
The esul s o he ne so p ion lux =𝑎−𝑑 show ha su ac an
adso bs on o (deso bs om) he in e ace in on o (behind) he
di usi e laye o e he en i e bubble mo ion. The hyd os a ic p essu e
dis ibu ions o e he bubble su ace e lec he bubble accele a ion.
P essu e is buil up in on o he bubble as he bubble eloci y
inc eases, while a nega i e gauge p essu e a ises a ound he equa o ,
la ening he bubble (Fig. 9).
Fig. 8. Su ac an su ace concen a ion 𝛤, su ace ension 𝜎, su ace eloci y 𝑣, ne
so p ion lux =𝑎−𝑑, Ma angoni s ess 𝜏Ma, and hyd os a ic p essu e 𝑝 as a unc ion
o he a ea 𝑎.
Fig. 9. E olu ion o he bubble shape.
Fig. 10 shows he e olu ion o he main quan i ies desc ibing he
bubble dynamics. The symbols indica e he ins an s co esponding o
he p o iles in Fig. 8. In clean wa e , he bubble eloci y mono onously
inc eases un il i asymp o ically eaches i s e minal alue. Howe e ,
he eloci y in ou simula ion eaches a maximum a 𝑡∕𝑡𝑐∼ 50 (𝑡∼ 0.1
s) and hen sligh ly dec eases ( he o e shoo ing phenomenon) (Fig.
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D. Fe nández-Ma ínez e al.
Table 3
Dimensionless numbe s de ined in ol ing he physical p ope ies o he su ac an monolaye .
Pe =𝑅𝑣𝑡∕𝑜Pe𝑠=𝑅𝑣𝑡∕𝑠𝐾𝑎=𝑘𝑎𝑐cmc𝑅∕(𝛤∞𝑣𝑡)𝐾𝑑=𝑘𝑑𝑅∕𝑣𝑡Ma=𝛤∞𝑅𝑔𝑇∕𝜎𝑐𝑐∞∕𝑐cmc
2.57 × 1054.89 × 1021.02 1.36 × 10−2 0.132 10−4
Fig. 10. Bubble e ical posi ion 𝑧, eloci y 𝑣𝑧, mean su ac an concen a ion 𝛤,
maximum su ac an concen a ion 𝛤max, no malized ne so p ion a e 𝛷, bubble su ace
a ea 𝐴, and aspec a io 𝜒 as a unc ion o ime 𝑡. The symbols indica e he ins an s
co esponding o he p o iles in Fig. 8. The dashed ho izon al lines indica e he
e minal alues. The a ow indica es he maximum eloci y due o he o e shoo ing
phenomenon.
10b). This dec ease is p oduced by he ex a d ag esul ing om he
la e g ow h o he dynamic su ac an laye . Fo 𝑡 ≳ 0.1 s, he e is s ill
ne su ac an adso p ion a he in e ace (Fig. 10e). The concen a ion
jump a he su ac an su ace bounda y laye g ows. The Ma angoni
s ess peak becomes highe and wide (Fig. 8e), inc easing he no mal
ou e iscous s ess ha con ibu es o he d ag (Rubio e al., 2024).
Addi ionally, he su ac an su ace bounda y laye sligh ly displaces
ups eam (Fig. 8a), shi ing he sepa a ion poin in he same di ec ion
and enla ging he low-p essu e a ea in he wake ha also con ibu es
o he d ag (Fig. 8 ). These wo e ec s a e esponsible o he bubble
decele a ion o 𝑡∕𝑡𝑐≳50 (𝑡 ≳ 0.1 s).
The mean su ac an concen a ion 𝛤 mono onously inc eases
(Fig. 10c) due o he con inuous ne adso p ion o su ac an . As
explained below, he sha p inc ease in 𝛤max o 𝑡∕𝑡𝑐≲5 (𝑡 ≲ 0.01 s)
(Fig. 10d) e lec s he in ense su ac an su ace con ec ion igh a e
he bubble elease. To analyze his aspec o he p oblem, we calcula e
he no malized ne so p ion a e
𝛷=1
4𝜋𝑅2𝛤0
𝑑
𝑑𝑡 (𝐴𝛤 ),(18)
whe e 𝐴 is he bubble su ace a ea. 𝛷 is he ime de i a i e o he su -
ac an mass apped in he bubble, 𝐴𝛤 , ela i e o i s ini ial alue. As
explained below, i eaches i s maximum alue a 𝑡∕𝑡𝑐≃ 7.64 (𝑡≃ 0.016
s) (Fig. 10e), when he bubble on (clean) su ace has g own. Fig.
10 shows ha he bubble su ace sligh ly expands (𝐴 inc eases) du ing
he bubble accele a ion and comp esses du ing he decele a ion. This
expansion/comp ession is accompanied by a signi ican change in he
aspec a io 𝜒 (Fig. 10g), as also obse ed in Fig. 9. The ins an aneous
alues o he bubble eloci y and aspec a io a e co ela ed, as K ach
and Finch (2010) obse ed in hei expe imen s.
Fou ime scales desc ibe he o ma ion o he dynamic adso p ion
laye . The as es p ocess is con ec ion o he ea pa o he bubble
o he su ac an adso bed be o e he bubble is eleased. The ime
scale o his mechanism is 𝑅∕𝑣𝑧𝑐 ∼ 10−2 s, whe e 𝑣𝑧𝑐 ≃ 0.1 m∕s is
he cha ac e is ic ising eloci y du ing he bubble accele a ion. This
i s p ocess is ollowed by ha in luenced by adso p ion–deso p ion
kine ics. The adso p ion ime scale is 𝑡𝑎= (𝑘𝑎𝑐𝑠∕𝛤)−1. The ini ial
su ac an concen a ion is app oxima ely ha a equilib ium. Then,
𝑐𝑠∕𝛤≃𝑘𝑑∕𝑘𝑎, and, he e o e, 𝑡𝑎=𝑘−1
𝑑. We conclude ha he adso p ion
ime scale is commensu a e wi h he deso p ion cha ac e is ic ime 𝑡𝑑=
𝑘−1
𝑑∼ 0.1 s, which sugges s ha bo h mechanisms compe e wi h each
o he du ing he same phase o he dynamic adso p ion laye o ma ion.
The su ac an di usion go e ns he las s age o his p ocess. The
su ace and olume ic di usion imes a e 𝑡𝐷𝑠 =𝑅2∕𝑠∼ 1 s and
𝑡𝐷𝑜 =𝑅2∕𝑜∼ 500 s, espec i ely.
The ime scales men ioned abo e can be obse ed when analyzing
he maximum su ac an concen a ion 𝛤max and mean su ac an con-
cen a ion 𝛤. The su ac an adso bed on o he in e ace a 𝑡= 0 s is
swep owa d he ea o he bubble e y as . A 𝑡∕𝑡𝑐= 3.64 (𝑡= 0.008
s), when he bubble has a eled a dis ance smalle han i s adius (Fig.
10a), he maximum su ac an concen a ion 𝛤max has inc eased o e
i e imes (Fig. 10d). Howe e , he mean su ac an concen a ion 𝛤
has emained almos cons an (Fig. 10c). This con i ms ha su ac an
con ec ion domina es he p ocess on he ime scale 𝑅∕𝑣𝑧𝑐 ∼ 10−2 s.
Now, we analyze he e olu ion o he su ac an su ace dis ibu ion
du ing he con ec ion-domina ed s age o he p ocess. Fo his pu pose,
we ha e e-plo ed in Fig. 11 he p o iles shown in Fig. 8, sepa a ing
hem in se e al g aphs and zooming in o he ea o he bubble. The
maximum su ac an concen a ion is ini ially loca ed a he bubble
bo om (see Fig. 11-a1 o 𝑡∕𝑡𝑐<6.44 (𝑡 < 14 ms)). The su ace
ension g adien p oduces a Ma angoni s ess 𝜏Ma ha signi ican ly
slows he su ac an -loaded pa o he in e ace. This s ess becomes
la ge enough o e e se he in e ace mo ion in he bubble ea a
𝑡∕𝑡𝑐≃ 5.24 (𝑡≃ 11 ms) (Fig. 11-a3). The e e se low anspo s
he su ac an away om he bubble bo om, making 𝛤max dec ease
e en hough he e is ne adso p ion o su ac an , as indica ed by he
inc ease in 𝛤 (Fig. 10c). A 𝑡∕𝑡𝑐≃ 8.83 (𝑡≃ 19 ms), he su ac an
concen a ion p o ile exhibi s he shape cha ac e is ic o he s eady
solu ion. Speci ically, mo e han 80% o he bubble su ace is almos
clean, and a hin di usi e bounda y laye sepa a es his egion om
he su ac an -loaded ea (Fig. 8a). The Ma angoni s ess peak in he
In e na ional Jou nal o Mul iphase Flow 188 (2025) 105205
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D. Fe nández-Ma ínez e al.
Fig. 11. Su ac an su ace concen a ion 𝛤, su ace ension 𝜎, su ace eloci y 𝑣 and ne so p ion lux =𝑎−𝑑, Ma angoni s ess 𝜏Ma, and hyd os a ic p essu e 𝑝 as a unc ion
o 𝑎.
su ace bounda y laye p ac ically immobilizes he su ac an -loaded
pa o he in e ace (Fig. 8c).
The adso p ion–deso p ion o su ac an con ols he second phase
o he dynamic adso p ion laye g ow h. The low d ags he su ac an
owa d he ea o he bubble. The une en su ac an concen a ion
enhances adso p ion in he almos -clean egion and a o s deso p ion
in he loaded ea (Fig. 11-b4). This esul s in a posi i e ne so p ion
lux un il he bubble eaches i s s eady eloci y, as indica ed by he
con inuous inc ease in 𝛤 (Fig. 10c).
The ne so p ion a e 𝛷 [Eq. (18)] ini ially g ows (Fig. 10e) as
he clean pa o he bubble su ace enla ges. This occu s un il he
Ma angoni s ess peak almos immobilizes he su ac an -loaded egion
a 𝑡∕𝑡𝑐≃ 6.44 (𝑡≃ 14 ms) (Fig. 11-a3). Then, he bubble de o ma ion
inc eases he su ace a ea (Fig. 10 ). This allows o he g ow h o he
su ac an -loaded egion. This e ec and he inc ease in he su ac an
concen a ion a o deso p ion, educing he ne so p ion a e. A e
he maximum eloci y and de o ma ion a e eached (𝑡∕𝑡𝑐>50, 𝑡 >
0.1 ms), he sligh eco e y o he sphe ical shape (Fig. 9) con ibu es
o enla ging he loaded a ea a he expense o he clean egion (Fig. 11-
c1), which also educes he ne adso p ion a e. The p ocess con inues
un il eaching he delica e balance be ween he su ac an con ec ion,
di usion, and so p ion co esponding o he s eady egime (Rubio e al.,
2024).
Fig. 12 shows he olume ic su ac an concen a ion in he s a-
iona y s a e as a unc ion o he dis ance 𝑛 no mal o he in e ace a
he in e ace poin s indica ed in he igu e. In Fig. 12a, hese poin s a e
loca ed ups eam o he momen um bounda y laye sepa a ion. In his
case, a hin su ac an bounda y laye o ms wi h a hickness 𝛿 e i ying
he scaling law 𝛿∕𝑅∼Pe−1∕2 (Rubio e al., 2024). As can be obse ed,
he nume ical esul s a e consis en wi h he p edic ion 𝛿∕𝑅∼ 0.002 o
ha law. The in e ace poin s conside ed in Fig. 12b a e wi hin he
su ac an wake. Di usion anspo s he su ac an molecules away
om he in e ace.
5. Conclusions
We analyzed he ansien bubble ising in he p esence o a su -
ac an a e y low concen a ions. The expe imen s o 𝑅= 0.76 mm
and 𝑐∞∕𝑐cmc ≥5×10−4 o T i on X-100 showed ha a helical ins abili y
e en ually de elops. The helical ins abili y e ol ed owa d a zig-zag
mo ion as he T i on X-100 concen a ion inc eased. Fo he same
ela i e concen a ions 𝑐∞∕𝑐cmc o T i on X-100 and SDS, T i on X-100
p oduces a smalle e ec on a ela i ely sho ime scale, bu his e ec
e en ually exceeds ha p oduced by SDS a su icien ly la ge imes. The
T i on X-100 e ec s a e la ge han hose caused by SDS a he same
absolu e concen a ion. The magni ude o he su ac an e ec inc eases
as he bubble adius dec eases o bo h T i on X-100 and SDS. The
expe imen s allowed us o de e mine he pa ame e condi ions ha lead
o an axisymme ic, s aigh (s able) bubble ising. The bubble mo ion
emained s able o 𝑅= 0.66 mm and he concen a ion 𝑐∞= 8 × 10−4
mol/m3 o SDS. The expe imen al esul s sa is ac o ily ag eed wi h he
ansien nume ical solu ion o his case.
We nume ically analyzed he g ow h o he dynamic adso p ion
laye o SDS o 𝑅= 0.66 mm and 𝑐∞= 8 × 10−4 mol/m3. The e ical
eloci y o a bubble in clean wa e inc eases mono onously. Con e sely,
he bubble eloci y in ou simula ion a ains a maximum a a ound
𝑡∕𝑡𝑐≃ 65 (𝑡≃ 0.1 s). This occu s despi e he iny su ac an concen-
a ion conside ed in ou analysis, which sugges s ha he so-called
‘‘o e shoo ing’’ phenomenon occu s o any su ac an concen a ion.
Bo h he Ma angoni s ess con ined in he bounda y laye and he
inc ease in he low-p essu e egion a ea a e esponsible o he bubble
decele a ion. The bubble su ace sligh ly expands du ing he bubble ac-
cele a ion and comp esses du ing he decele a ion. A non-mono onous
a ia ion o he aspec a io accompanies his expansion/comp ession.
The amoun o su ac an in he monolaye con inuously inc eases o e
he bubble mo ion.
One can dis inguish ou consecu i e phases in he o ma ion o
he dynamic adso p ion laye : he almos ins an aneous con ec ion o
In e na ional Jou nal o Mul iphase Flow 188 (2025) 105205
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