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published wo k see 10.1017/j m.2016.496.
Unde conside a ion o publica ion in J. Fluid Mech. 1
Maximum d op adius and c i ical Webe
numbe o splashing in he dynamical
Leiden os egime
Guillaume Riboux†& Jos´
e Manuel Go dillo
´
A ea de Mec´anica de Fluidos, Depa amen o de Ingenie ´ıa Ae oespacial y Mec´anica de
Fluidos, Uni e sidad de Se illa, A enida de los Descub imien os s/n 41092, Se illa, Spain.
(Recei ed ?? and in e ised o m ??)
A oom empe a u e, when a d op impac s agains a smoo h solid su ace a a eloci y
abo e he so called c i ical eloci y o splashing, he d op loses i s in eg i y and agmen s
in o iny d ople s iolen ly ejec ed adially ou wa ds. Below his c i ical eloci y, he d op
simply sp eads o e he subs a e. Splashing is also epo ed o occu o solid subs a e
empe a u es abo e he Leiden os empe a u e, TL, o which a apo laye p e en s he
d op om ouching he solid. In his case, he splashing mo phology di e s om he one
epo ed a oom empe a u e because, hanks o he p esence o he gas laye , he shea
s esses ac ing on he liquid can be neglec ed. Ou pu pose he e is o p edic , o wall
empe a u es abo e TL, he c i ical Webe numbe o splashing as well as he maximum
sp eading adius. Fi s , making use o Bounda y In eg al Simula ions, we calcula e bo h
he ime e olu ion o he liquid eloci y as well as he heigh o he shee which is ejec ed
angen ially o he subs a e. These esul s a e hen used as bounda y condi ions o
he one dimensional mass and momen um equa ions desc ibing he dynamics o he im
limi ing he expanding liquid shee . Ou p edic ions o bo h he maximum sp eading
adius and o he c i ical Webe numbe o splashing, a e in good ag eemen wi h
expe imen al obse a ions.
1. In oduc ion
The unde s anding o he sp eading o he b eak up p ocesses o a d op impac ing
on o a solid subs a e is an a ea o cu en ac i e esea ch because o i s ele ance in a
numbe o echnological applica ions such as coa ing, cleaning, cooling, and combus ion
(Josse and & Tho oddsen 2016). I is known ha , a oom empe a u e, d op splashing
does no only depend on he adius R, on he impac ing eloci y Vand on he densi y
ρ, he iscosi y µand he in e acial ension σo he liquid, bu also on he ma e ial
p ope ies o he gas, on he gas p essu e (Xu e al. 2005) and on he physicochemical
p ope ies o he subs a e (Duez e al. 2007). Fo he case o d ops impac ing a smoo h
and d y solid subs a e a oom empe a u e illus a ed in igu e 1, i can be obse ed
ha once he d op ouches he solid a T= 0, a small bubble -which has no in luence on
he subsequen dynamics- is en apped nea he axis o symme y. Subsequen ly, a he
ejec ion ime Te, a hin shee o liquid is expelled om he adial posi ion A(Te), wi h
π A2(Te) he a ea o he we ed egion. Then, i he impac eloci y is below he c i ical
eloci y o splashing, V < V ∗, he ejec ed lamella sp eads angen ially along he solid;
howe e , i V > V ∗, he edge o he liquid shee dewe s he solid as a consequence o he
li o ces exe ed by he su ounding gas (Riboux & Go dillo (2014), om now on RG14).
Then, capilla y and Rayleigh–Taylo dis u bances g ow in he azimu hal di ec ion o he
†Email add ess o co espondence: g ib[email p o ec ed]
2Guillaume Riboux & Jos´e Manuel Go dillo
im, causing i s disin eg a ion in o d ops. A oom empe a u e, splashing will occu
only i he liquid on is able o dewe he solid. Howe e , when he empe a u e o he
subs a e is abo e TL, wi h TL he Leiden os empe a u e, he liquid ne e ouches he
subs a e, wi h independence o he alue o he impac eloci y because, unde hese
condi ions, he d op le i a es on i s own apo (Shi o a e al. 2016). Consequen ly, he
c i ical eloci y o splashing will s ongly depend on whe he he solid empe a u e is
below o abo e TL, as i has been ecen ly epo ed by S aa e al. (2015). In addi ion, as
a consequence o he ic ionless mo ion o he liquid wi h he subs a e, he maximum
sp eading diame e o impac ing d ops inc eases o empe a u es abo e TLwi h espec
o he case o d op impac a oom empe a u e (Las akowski e al. 2014; Wilde man
e al. 2016).
In his con ibu ion we aim o p edic , o empe a u es o he subs a e abo e TL,
he maximum sp eading adius o he d op as well as he c i ical impac eloci y abo e
which he d op disin eg a es in o smalle d ople s. Mo eo e , o impac eloci ies la ge
han he c i ical one, we will also p o ide esul s o he eloci ies and o he diame e s
o he iny d ople s ejec ed. Fo his pu pose, he low ield wi hin he ejec ed lamella
is modeled using a ballis ic app oxima ion and nex , making use o in eg al balances o
mass and momen um, he dynamics o he im is desc ibed. A simila s udy was ca ied
ou by Riboux & Go dillo (2015) ( om now on RG15), whe e he analy ical exp essions
o he liquid eloci y and he heigh o he liquid shee deduced in RG14 we e used
as bounda y condi ions o he equa ions desc ibing he low in he expanding lamella.
One o ou main con ibu ions he e is ha hese analy ical exp essions a e subs i u ed
by uni e sal ime-dependen unc ions calcula ed nume ically using a Bounda y In eg al
Me hod (BIM). We ind ha , o sho imes a e impac , he simula ions a e in excellen
ag eemen wi h he heo e ical p edic ions in RG14 bu , o imes T∼R/V , hese
nume ical esul s depa om hose in RG14 and also om he equa ions p o ided in
Roisman (2009); Egge s e al. (2010). The di e ences ound a e essen ial o co ec ly
p edic he expe imen s.
Po en ial low nume ical simula ions will be ca ied ou using he Bounda y In eg al
me hod desc ibed in Rod ´ıguez–Rod ´ıguez e al. (2006); Go dillo & Gekle (2010). This
ype o code is adequa e o simula e he splashing o d ops o subs a e empe a u es
abo e TLsince: i) he o ici y in he alling d op is ini ially ze o and ii) he shea
s esses ac ing in he ci cula egion o adius A(T) below he d op -see igu e 1- a e
negligible because, in he dynamic Leiden os egime, a hin apo laye p e en s he
liquid om ouching he wall. The e o e, he p oduc ion o o ici y is es ic ed o
egions nea he ee su ace whe e he in e acial cu a u e is highes , e.g., he egion
whe e he impac ing d op mee s he ejec ed liquid shee . Consequen ly, he low ield
a e he impac is mos ly i o a ional excep in e y na ow egions localized nea he
ee su ace. Mo i a ed by his ac , in his con ibu ion, we app oxima e he eloci y ield
wi hin he d op as =∇Φ, wi h Φ he eloci y po en ial sa is ying he impene abili y
condi ion a Z= 0, ∂Φ/∂Z = 0. In addi ion, hanks o he symme y o he low wi h
espec o he plane Z= 0, is calcula ed he e as he esul o he head on collision o
wo d ops o iden ical adii Rmo ing wi h espec i e eloci ies Vezand −Vez. He e, ez
indica es he uni ec o poin ing in he opposi e di ec ion o ha o he alling d op.
The compa ison o he expe imen al images co esponding o an e hanol d op impac ing
a solid subs a e a oom empe a u e wi h he nume ical p o iles is p o ided in igu e 1.
Excep in he egion nea he edge o he lamella, he ag eemen be ween expe imen s and
nume ical esul s, is ema kable. The disc epancies in he posi ion o he im obse ed in
igu e 1 a e due o he ac ha , a oom empe a u e, he iscous shea s esses a he
wall con ibu e o u he decele a e he ad ancing on . We used ou own expe imen s,
Maximum d op adius and splashing c i e ium in he dynamical Leiden os egime 3
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
(
a
)
(
b
)
(
c
)
(
d
)
(
e
)
A
(
T
)
(
)
(
g
)
(
h
)
(
i
)
(
a
)
(
b
)
(
c
)
(
d
)
(
e
)
(
)
(
g
)
Figu e 1. Top: The sequence o images illus a e an e hanol d op o ini ial adius R=1.03 mm
impac ing agains a d y smoo h solid su ace a oom empe a u e wi h a eloci y V=1.69 m·s−1
-We = ρV 2R/σ = 100 and Re = ρV R/µ = 1360- a imes (a) ≃0, (b) = 0.10, (c) = 0.19, (d)
= 0.29, (e) = 0.38, ( ) = 0.48, (g) = 0.57, (h) = 0.76, (i) = 0.95 wi h =TV/R. The
b igh line ep esen s he esul o he Bounda y In eg al Simula ion, which is ca ied ou using
a numbe o nodes N ha inc eases dynamically in ime (N(T= 0) = 501). Nodes a e clus e ed
in he egions wi h he highes cu a u e. The adius o he we ed a ea A(T)/R =a( ) = √3 ,
(Riboux & Go dillo 2014) is indica ed in (e). Bo om: The igh column is a zoom o he egion
whe e he lamella is ejec ed o di e en ins an s o ime, namely, (a) = 0.016, (b) 0.049, (c)
0.082, (d) 0.164, (e) 0.246, ( ) 0.328 and (g) 0.491. Fo e e ence, he le column illus a es he
nume ical esul s o he same alues o . The absence o iscous shea s esses causes he im
o low as e in he nume ical simula ions han in he expe imen s.
pe o med a oom empe a u e, o compa e wi h he nume ical esul s since we could
no ind in he li e a u e expe imen al images o d ops impac ing a ho subs a e in he
Leiden os egime wi h enough spa io- empo al esolu ion.
4Guillaume Riboux & Jos´e Manuel Go dillo
Figu e 2. The po en ial low nume ical esul s s a imposing he ini ial we ed adius,
a= sin θ0≈π/36 wi h θ0= 5◦. The inse (a) shows he e-g idding p ocedu e used o s a-
bilize he nume ical simula ion, consis ing in placing nodes in be ween he nodes o he p e ious
ime s ep. The inse (b) is a zoom o he egion om which he lamella is ejec ed a h ee di e -
en ins an s o ime: e, = 0.15 and = 0.35. The igu e also illus a es he we ed adius a( )
as well as he adial posi ion o he im, ( ), and i s eloci y, ( ). The inse also shows he
de ini ion o heigh o he lamella, h( , ), no o be con used wi h he wid h o he im, h ( ).
0 0.03 0.06 0.09 0.12 0.15 0.18
0
0.2
0.4
0.6
0.8
1
e
(
a
)
(
)
a
(
)

3
˙
(
)=
(
e
)
101102103
We
10-2
10-1
100
e
∝
We
−
2
/
3
(
b
)
Figu e 3. (a) The igu e shows he nume ical esul s o bo h a( ) and he adial posi ion o he
edge lamella ( ) in he case o We = 100. The e ical line indica es he ejec ion ime p edic ed
by equa ion (2.2) and he dashed line ep esen s he p edic ed posi ion o he im, ejec ed om
=√3 ewi h an ini ial eloci y ( e) = 1/2p3/ e, wi h egi en in equa ion (2.2). (b) The
ejec ion imes p edic ed by he Bounda y In eg al Simula ions a e such ha e∝We−2/3, in
ag eemen wi h equa ion (2.2).
2. Po en ial low simula ions
Following he no a ion in RG14, lowe case a iables will be used in wha ollows o e e
o dimensionless a iables, which a e cons uc ed he e using as scales o eloci y, leng h
and p essu e V,Rand ρV 2, espec i ely. Since he F oude numbe F = V2/gR 1
and he shea s esses ac ing on he liquid a e negligible, he only ele an dimensionless
pa ame e cha ac e izing he splashing o d ople s in he Leiden os egime is he Webe
numbe , de ined he e as We = ρV 2R/σ.
The mos ele an e en aking place a e he impac o a d op in he dynamic Leiden-

Maximum d op adius and splashing c i e ium in he dynamical Leiden os egime 5
os egime is ha a e, wi h e he ejec ion ime -see igu e 2-, an ex emely hin liquid
shee o ini ial hickness h ( e) is expelled angen ially o he subs a e wi h an ini ial
eloci y ( e). In RG14 an algeb aic equa ion o ewas deduced based on he ollowing
ac s: i) p io o he ejec ion o he lamella, he ime e olu ion o he adius o he we ed
a ea is a( ) = √3 †, ii) a e, he eloci y o he ip o he lamella is equal o he eloci y
o he we ed adius, i.e., ( e) = ˙a( e) wi h do s deno ing ime de i a i es and iii) he
lamella can only be ejec ed i i s ip ad ances as e han he adius o he we ed a ea.
We concluded in RG14 ha ecan be calcula ed sol ing he algeb aic equa ion
√3
2Re−1 −1/2
e+ We−1= ¨a h2
=c2 3/2
ewi h c= 1.1,(2.1)
which exp esses he ac ha he ejec ion ime is he ins an a which he decele a ion
o he edge o he lamella coincides wi h he decele a ion o he we ed a ea, ¨a. Fo wall
empe a u es abo e TL, he edge o he lamella is no decele a ed by he ac ion o iscous
shea s esses and, he e o e, se ing o in ini y he Reynolds numbe Re in equa ion (2.1)
yields,
e= (c2We)−2/3.(2.2)
Using he analy ical exp essions o bo h ( e) and h ( e) de i ed in RG14 as well as
he esul in equa ion (2.2), i can be concluded ha ( e) = 1/2p3/ e∝We1/3and
h ( e) = √12 3/2
e/π ∝We−1. Figu e 3 con i ms he esul in equa ion (2.2), e∝We−2/3.
Once he lamella is ejec ed, in RG14 we also deduced ha , o > eand 1, bo h
he liquid eloci y and he heigh o he liquid laye a =a( ) = √3 , which is he adial
posi ion whe e he d op mee s he lamella, a e espec i ely gi en by a= 2˙a=p3/ and
ha=√12 3/2/(3π). Figu e 4 con i m hese heo e ical p edic ions o We ⩾100. Mo e
p ecisely, igu e 4 shows ha , while he analy ical exp ession o ais alid o a bi a y
imes, he co esponding analy ical exp ession o hadepa s om he nume ical esul s
o ⩾0.1 (see igu e 4c). The e o e, he equa ions desc ibing he liquid eloci y in he
lamella u( , ) and i s heigh h( , ) a =√3 i.e., a he adial posi ion om which he
liquid shee is ejec ed o imes > e, a e espec i ely gi en by
u( =a, ) = a=p3/ , and
(h( =a, ) = ha=√12 3/2/(3π) o < 0.1
h( =a, ) = P( ) o ⩾0.1,
(2.3)
wi h
P( ) =
5
X
i=0
(0.1pi) iand p0=−2.453 ×10−3, p1= 1.321, p2=−1.176,
p3= 0.4943, p4=−0.1047, p5= 8.89 ×10−3
(2.4)
a polynomial which is i ed o he nume ical esul s.
The ad hoc adial eloci y ield wi hin he d op p oposed by bo h Roisman (2009) and
Egge s e al. (2010) is u( , ) = /( +τ), wi h τan adjus able o de uni y cons an . While
he s agna ion poin ype o low wi hin he impac ing d op hypo hesized by Roisman
(2009) and Egge s e al. (2010) is a good app oxima ion o he eal low ield o ∼O(1),
o imes such ha > e, 1, he eloci y ield a ≃√3 does no co espond o a
†This esul was de i ed o he e y i s ime by Riboux and Go dillo in RG14 using he
linea iza ion o he bounda y condi ions o he po en ial low (Wagne 1932).
6Guillaume Riboux & Jos´e Manuel Go dillo
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
0
1
2
3
4
5
6
7
8
9
a
(
a
)
We=100
∗
We=100
We=300

3
/
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
0
1
2
3
4
5
6
7
8
9
ha
(
b
)
×
10
−
2
(

12
/
3
π
)
3
/
2
P
(
)
[1]
[2]
0 0.05 0.1 0.15 0.2
0
4
8
12
ha
×
10
−
3
(
c
)
We=100
∗
(

12
/
3
π
)
3
/
2
P
(
)
Figu e 4. (a) Compa ison be ween he heo e ical exp ession a= 2˙a=p3/ deduced in
RG14 and he nume ical esul . The good ag eemen be ween heo y and expe imen s depic ed
in his igu e is independen o he Webe numbe whene e We ⩾100. Two o he simula ions
shown a e ca ied ou using N(T= 0) = 501 whe eas in he case ma ked wi h an as e isk
[∗], N(T= 0) = 1001. (b) Compa ison o he heigh o he lamella a =√3 p edic ed by
he models in Egge s e al. (2010) ([1]) and Roisman (2009) ([2]) wi h he nume ical esul .
(c) Compa ison be ween he heo e ical exp ession ha=h(a( ), ) = (√12/3π) 3/2deduced in
RG14, alid o 1 and he nume ical esul . The e ical line indica es he ins an a which
we ha e se he ansi ion be ween he esul p edic ed by he po en ial low heo y and he
i ing polynomial P( ) in equa ion (2.4).
s agna ion poin low (see RG14 o de ails). In e es ingly, ou heo y p edic s a adial
eloci y a =a( ) = √3 ,u( =a( ), ) = a( )/ =p3/ o 1 which is, by
coincidence, he adial eloci y co esponding o a s agna ion poin o low o he ype
u= / pa icula ized a =√3 . This is why igu e 4ashows a ai ly good be ween
he analy ical exp ession a=p3/ and he nume ical esul s o a bi a y alues o
. Figu e 4balso shows he alues o ha( ) = h( =√3 , ) p edic ed by he model in
Roisman (2009),
h( , ) = 8 η
( +τ)2exp −6η 2
( +τ)2,(2.5)
wi h η= 0.39, τ= 2 ×0.25 and wi h a cons an equal o 8 because he e we de ine
dimensionless leng hs using he d op adius ins ead o i s diame e , as well as he esul s
p edic ed by he model in Egge s e al. (2010),
h( , ) = 1
( +τ)2H(x)
H(x) = 3.19
(1 + C x2)6wi h x( , ) =
+τ







(2.6)
wi h C= 0.604 and τ= 1 (Las akowski e al. 2014). Clea ly, nei he he model by
Roisman (2009) no ha by Egge s e al. (2010) is in ag eemen wi h he nume ical
esul s.
Maximum d op adius and splashing c i e ium in he dynamical Leiden os egime 7
0.1
0.2
z
(
a
)
0.1
0.2
z
(
b
)
0.1
0.2
z
(
c
)
0.1
0.2
z
(
d
)
a
(
)
0.1
0.2
z
(
e
)
0.1
0.2
z
(
)
ha
(
)
0.1
0.2
z
(
g
)
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
0.1
0.2
z
(
h
)
Figu e 5. Compa ison, o We = 100, be ween he heigh o he lamella p edic ed by he model
-ballis ic equa ions (3.1) subjec ed o he bounda y condi ions gi en in equa ion (2.3)- and he
nume ical esul . The e ical line illus a es he adial posi ion =a( ) = √3 o se e al
ins an s o ime as well as he heigh o he lamella a =√3 ,ha( ) = h( =√3 , ); ais he
liquid eloci y a =a( ) = √3a, namely a=u( =√3 , ). (a) = 0.19, (b) 0.25, (c) 0.32,
(d) 0.45, (e) 0.55, ( ) 0.70, (g) 0.90 and (h) = 1.10.
3. P edic ion o he c i ical Webe numbe o splashing and o he
maximum sp eading adius o d ople s impac ing in he
Leiden os egime
Fo a gi en ins an o ime > e, he lamella ex ends om he posi ion whe e he
d op mee s he liquid shee i.e., =√3 , down o = ( ), wi h ( ) indica ing he
adial posi ion o he im. In RG15, we showed ha bo h he liquid eloci y u( , ) and
he heigh o he liquid laye wi hin he lamella, h( , ), can be calcula ed using he pai
o equa ions (Go dillo & Gekle 2010; Ville maux & Bossa 2011),
Du
D = 0 and D ln ( h)
D =−∂ u
∂ ,(3.1)
wi h D/D ≡∂/∂ +u ∂/∂ he ma e ial de i a i e. The o me o he wo equa ions
in (3.1) exp esses ha luid pa icles conse e hei eloci ies wi hin he lamella om
=√3 down o he adial posi ion ( ) whe e he im is loca ed; he la e , is he
con inui y equa ion. The pai o equa ions in (3.1) ep esen ing he ballis ic mo ion o
luid pa icles along he lamella a e sol ed subjec ed o he ini ial condi ions gi en in
equa ion (2.3) by means o he Lag angian Nume ical Me hod al eady used in RG15.
The esul s o he model o h( , ), shown in igu e 5, a e in excellen ag eemen wi h
he nume ical simula ions.
The adial posi ion o he edge o he lamella, ( ), as well as i s hickness, h ( ), a e
deduced applying he in eg al balances o mass and momen um a he im (Taylo 1959;
8Guillaume Riboux & Jos´e Manuel Go dillo
Culick 1960),
d
d = ,
π
4
dh2
d = [u( , )− ]h( , ),
π h2
4
d
d = [u( , )− ]2h( , )−2 We−1.















(3.2)
The sys em o equa ions (3.2) is sol ed subjec ed o he ollowing ini ial condi ions,
=a( e) = √3 e,
= ( e) = 1/2p3/ e,
h =h ( e) = √12 3/2
e/π ,







(3.3)
wi h egi en by equa ion (2.2).
Figu e 6ashow he solu ion o he sys em o equa ions (3.1)–(3.3) o a pa icula
alue o he Webe numbe , whe eas igu e 6billus a es he compa ison be ween he
p edic ed maximum sp eading adius and he expe imen al da a in T an e al. (2012).
As i was poin ed ou in Wilde man e al. (2016), he e a e wo clea ly di e en ia ed
egions in he plo max s We o igu e 6b: he impac is elas ic o We .10 since he
ini ial kine ic ene gy is ans o med in o su ace ene gy, as i appa en om he capilla y
wa es de eloping a he in e ace o he impac ing d op whe eas, o We &10, he d op
sp eading p ocess is domina ed by ine ia. I is discussed in Wilde man e al. (2016) ha
app oxima ely one-hal o he ini ial kine ic ene gy is dissipa ed in he sudden expansion
connec ing he liquid shee wi h he im o We ⩾10, a ac which is implici ly aken
in o accoun in he mass and momen um conse a ion equa ions (3.2). In spi e o he
ac ha he sys em o equa ions (3.2) is analogous o ha used in Roisman (2009);
Egge s e al. (2010); Ville maux & Bossa (2011); Las akowski e al. (2014), he c ucial
di e ence be ween he p esen s udy and p e ious ones which is ha , in ou case, we do
no impose a speci ic o m o he eloci y ield wi hin he lamella. Ins ead, since luid
pa icles conse e hei eloci ies wi hin he liquid shee , ou sys em o equa ions (3.2)
subjec ed o he eal ini ial condi ions gi en in (2.3), wi h ecalcula ed sel -consis en ly
h ough equa ion (2.2), p o ides wi h he co ec exp essions o bo h he liquid eloci y
u( ) and he heigh o he lamella h( ) ups eam o he o oidal im, as i can be in e ed
om he good ag eemen exis ing be ween expe imen s and p edic ions depic ed in igu e
6b.
Ano he o he ad an ages o no imposing an ad-hoc eloci y ield wi hin he lamella
es s on he ac ha bo h he c i ical Webe numbe and he diame e s and he eloci ies
o he d ople s ejec ed o impac eloci ies V > V ∗, which s ongly depend on h( ), can
be calcula ed sel -consis en ly. Indeed, in RG15, ollowing he ideas in Ville maux & Bossa
(2011); Agbaglah e al. (2013), we deduced a c i e ium o he disin eg a ion o he edge
o he liquid shee based on he ac ha he d ople s composing he sp ay esul om
he ampli ica ion in he azimu hal di ec ion o Rayleigh–Taylo and capilla y ins abili ies.
The g ow h a es o he dis u bances de eloping in he azimu hal di ec ion o he o oidal
im a e highly a enua ed as a consequence o he simul aneous g ow h o i s hickness. In
consequence, we concluded ha d ops will only be ejec ed when he ime cha ac e izing
he adial g ow h o he im, Th= (R/V ) h= (1/H dH /dT)−1, is subs an ially la ge
han he capilla y ime Tc= (R/V ) c=ρ H3
/8σ1/2. The b eakup ime bis ixed
a he ins an a which c/ h≃0.085 o he easons explained in RG15, which a e
ep oduced he e o he sake o cla i y: i) he cha ac e is ic ime o g ow h o a capilla y