Modelling and numerics for respiratory aerosols

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Modelling and numerics for respiratory aerosols

Laurent Boudin, Céline Grandmont, Alexander Lorz, Ayman Moussa

To cite this version:

Laurent Boudin, Céline Grandmont, Alexander Lorz, Ayman Moussa. Modelling and numerics for

respiratory aerosols. Communications in Computational Physics, Global Science Press, 2015, 18 (3),

pp.723-756. �10.4208/cicp.180714.200415a�. �hal-01044590�

LAURENTBOUDIN,CÉLINEGRANDMONT,ALEXANDERLORZ,ANDAYMANMOUSSA

Abstra t. Inthiswork, we rstdealwith themodellingandthe dis retization ofan

aerosol evolvingintheair,intherespirationframework,withinadomainwhi h anbe

xed ormoving. We alsoinvestigate basi numeri al properties ofthe numeri al ode

whi hwasdevelopped,andalsofo usontheinuen eoftheaerosolontheairow.

1. Introdu tion

Theevolutionofdropletsorparti lesinasurroundinguidisaphenomenonen ountered

inseveralareas,rangingfrommedi ine(aerosoltherapy)tomotorindustry(transportand

ombustion ofpetrol). Intheparti ular aerosoltherapy ase, invivo observationsof drug

delivery in the airways indu e several di ulties. For instan e, aerosol deposition maps

require heavy experimental proto ols, whi h annot be easily repeated, and the obtained

measurementsmaynot bea urate enough.

Consequently,the hoi e ofphysi allyrelevantmodels,and thereafterthedesign of

sta-ble,e ientnumeri almethodsallowinsili o experimentswhi h anprovideawiderange

ofresultsforvariousphysi alsituationsandparameters (typeofaerosols,surrounding

u-ids,pathologi al state...).

Several kinds of modelling are available to des ribe the aerosol movement in a uid.

Two ofthem aredis ussed quite indetailin[31 ℄: one an onsiderindividual parti leson

theonehand, or a olle tionof parti leson theotherhand.

Two-phase models onsider the olle tion of droplets or parti les as a uid and study

the evolution, for instan e, of aerosol on entration in the ambient uid. Those models

aremost ertainlyadaptedinthe asewhenthevolume fra tiono upiedbythedispersed

phase is not negligible with respe t to the volume fra tion o upied by the surrounding

uid [34, 7, 36, 17℄. Unfortunately, su h models do not allow an a urate des ription of

parti ledeposition. Thisiswhyweshallonly fo uson spraymodels inwhi hthedis rete

aspe tofthedispersed phaseiskept.

Following parti les as individuals is the other lassi al strategy, see [9, 15, 16 , 39, 46 ℄

for instan e. Nevertheless, des ribing the behaviour of su h a (very) large number of

parti lesmayleadto bothte hni al andnumeri al di ultiesifone triestokeepthetra k

of ea h individual traje tory. For instan e, the Atomiser po ket aeroneb GO from DTF

1

Corporationhasthefollowing hara teristi s: airowrateof

0.3

mL/min,average(inmass)

diameter equal to

3.6 µ

m. Hen e, this nebulizer allows the inje tion of

10

10

parti les in

oneminute.

Inthis ontextofverynumerousparti les,andsin ethevolume o upiedbytheaerosol

remains negligible in the human airways, the formalism of statisti al physi s and kineti

theoryisespe iallywell-tted. Thistypeof ouplingwasrstintrodu ed byO'Rourke[40 ℄

or Williams [43℄ and is now quite often used to model aerosol transport in the lung, see

[10,27,13℄. Asfortheintera tionbetween theaerosolandthesurroundinguid,following

a nomen lature introdu ed by O'Rourke (see also [22 ℄), we assume the spray to be thin.

Thismeans that

•

theaerosolvolume fra tioninthemixture remainsnegligible;

Date:July17,2014.

ThisworkwaspartiallyfundedbytheANR-08-JCJC-013-01proje theadedbyC.Grandmontandthe

ANR-10-BLAN-1119proje theadedbyM.Filo he.

1

•

there arenointera tions between theaerosol parti les;

•

theaerosol an have anee t on theuid, asaresponseto thedragfor eexerted

bythe uidontheparti les.

Notethat, fortwo-phase modelsaswell asODE ones,the aerosolretroa tion on theuid

isseldomtaken into a ount,up to our knowledge. Inthesame ontext ofaerosol kineti

modelling we onsider hereafter, [27 ℄ presents a study of aerosol transport in thetra hea

where theretroa tion istakeninto a ount.

The aerosol is then des ribed by a distribution fun tion whi h satises a Vlasov-type

equation. The uid is assumed to be homogeneous, Newtonian, in ompressible, and an

bedes ribedusingtheNavier-Stokesequations,see[28℄ forinstan e. Thephysi aldomain

anbeeither xedor timedependent.

The aerosoland uidare oupledthroughtwo terms: theparti lea eleration,

depend-ing on the relative velo ity of the parti le in the uid, and theretroa tion for e applied

by theaerosol on the uid. Consequently,we have to deal with a strong oupling of two

types of equations: one at the ma ros opi s ale (Navier-Stokes), one at the mesos opi

s ale (Vlasov). The existen e of solutions of the obtained system in a xed domain has

been investigatedbyvariousauthors [30,6,11,45 ℄. Themainissueinthese mathemati al

studiesliesinthefa tthat thesystemis nonlinearand strongly oupled.

The numeri al strategy fa es the same di ulty together with the dierent level of

des riptions for both phases (ma ro/mesos opi ). We hoose an expli it time-advan ing

s heme, whi h allows to solve theuid and aerosolparts in a staggeredway. The system

isthus un oupledintheapproximationpro edure, reminis ent fromtheexisten eproofin

[11℄. For thespa edis retization, anite element pro edureanda parti le-in- ell method

arerespe tively usedto approximate theuidvelo ity and pressure, and thedistribution

fun tion. Themovingdomain aseishandledthanksto thearbitraryLagrangian-Eulerian

(ALE)method,see[38 ℄ inanite element ontext.

This wholeapproximations heme wasimplementedinthe C++library LifeV.

In this arti le, we aim to investigate the range of parameters for whi h our s heme is

numeri ally stable and a urate. In parti ular, we study the inuen e of the retroa tion

for e. Its expli it treatment may indu e unphysi al instabilities due to large parti le

ve-lo ities. In the human lung, if therapeuti aerosols seem not to require the retroa tion

term inmost standard situations, itis not the asefor polluting parti les, whose average

volume is larger. We also onsider the moving domain ase, whi h is usually not taken

into a ount,and isa rststep towardsthe bron hialwallmotion.

On e thefullaerosol-uidmodelhasbeen des ribed,we present the onsidered

numeri- almethodandthenfo uson three ases: axeddomainwithoutor withretroa tion and

amovingdomainwithoutretroa tion. Inea h ase,we studythenumeri al sensivitywith

respe t to various parameters (time step, mesh size, initial datum, parti le

representativ-ity...).

2. Model

In the upper airways, we an safelyassume that theair is Newtonian and

in ompress-ible,thus governedbythein ompressibleNavier-Stokesequations. During therespiration

pro ess,some airwaywallsmaybetime-dependent. Thus ourequationswill be onsidered

inamovingdomain. Whenwefo usonaerosolsinhumanairways,thenumberofparti les

an be signi ant, whereasthe volume o upied bytheaerosol remainssmall. A lassi al

strategy instatisti al me hani s then onsistsin des ribing thespray behavior thanksto

onesingle kineti equation.

To model our problem, we are led to ouple both types of equations to obtain a

uid/kineti systemwe present inthenextsubse tions.

2.1. Geometries. In our study, a typi al uid domain, denoted by

Ω

t

, is a ylinder or

into three (not ne essarily onne ted) subsets: theinlet

Γ

t

, theoutlet

Γ

t

and the wall

Γ

wall

t

. In most situations,

Γ

in

t

and

Γ

out

t

do not depend on time, be ause they are hosen

as arti ial boundaries. On the ontrary, the motion of

Γ

wall

t

is driven by physiologi al phenomena.

Γ

in

t

Γ

wall

t

Γ

out

t

Ω

t

Figure 1. Bran hgeometry witha moving wall.

Inthesequel

n

t

will denotetheunitve tor going out andnormalto

∂Ω

t

.

2.2. Fluid equations. Sin e, in our framework, the uid is Newtonian, in ompressible

and homogeneous, the uid mass density

̺

u

remains onstant. We denote

ν

the uid

kinemati vis osity and

η = ̺

u

ν

its dynami one. The ow is lassi ally des ribed by

its velo ity eld

u

(t, x) ∈ R

3

and the pressure

p(t, x) ∈ R

, where

t ≥ 0

is the time and

x

= (x

1

, x

2

, x

3

) ∈ R

3

is theposition. Itis governedbythefollowing equations:

̺

u

[∂

t

u

+ (u · ∇

x

)u] = −∇

x

p + η∆

x

u

+ F ,

t ∈ R

+

,

x

∈ Ω

t

,

(2.1)

∇

x

· u = 0,

t ∈ R

+

,

x

∈ Ω

t

,

(2.2)

where

F

isave toreldrepresentingthefor esa tingontheuid(gravity,aerosol

retroa -tion). Moreover,totakeintoa ount thefa tthatthedomainitself anmove,we onsider

given time-indexed open sets

Ω

t

of

R

3

. If we denote, for any

t ≥ 0

, the displa ement

A

t

: Ω

0

→ Ω

t

fromtheinitial(referen e) onguration,weshallassumethat

(t, x) 7→ A

t

(x)

isassmoothasneeded, see [38,19 ℄.

2.3. Aerosol equation. The distribution fun tion

f : R

+

× R

3

_{× R}

3

_{→ R}

+

depends on

time

t

and position

x

of the parti les, but also on their velo ity

v

. In fa t,

f

an also

depend on parti le radius, temperature or other relevant quantities, as seen in [40 , 43 ℄.

Letusemphasize thatwedonottakeintoa ount anyphenomenonmodifyingtheaerosol

distribution regarding radius (no ollision, no abrasion, et .) or other physi al quantities

(no signi ant temperature variation during the breathing pro ess, for instan e). This

ensures that the initial radius distribution is onserved with respe t to time. Therefore,

forthesake ofsimpli ity,theaerosolis hosenmonodispersed insize,meaning that

r

isa

parameter,see Remark2 below. Ea h parti le isassumedto remainspheri al and hen e,

its mass

m

is onstant and satises

m = 4πr

3

_̺

aer

/3

, where

̺

aer

is the onstant volume

massofea h parti le.

The distribution fun tionsolvestheVlasovequation, i.e.

∂

t

f + v · ∇

x

f + ∇

v

· (af ) = 0,

(2.3)

where

a(t, x, v)

is thea elerationeldundergone by theaerosol.

Remark 1. One an understand

f

in two ways. It an be seen as

•

a number density:

f (t, x, v) dx dv

isthe number of droplets lo ated in the

elemen-tary volume of the phase spa e, entred at

(x, v)

attime

t

;

•

a probability density: if the zero-th moment of

f

equals

1

,

f (t, ·, ·)

is the density

Remark 2. We an take into a ount the ase where the parti les have dierent radii.

Indeed, if the uid velo ity is given, all the terms involving the parti les linearly depend

on

f

. Hen e, several distribution fun tions of dierent radii may be superponed to model

asize-polydispersed distribution. Thishas impli ationsin the numeri al omputations, see

Remark 6 in Se tion 3.

2.4. Intera tionbetween theuidandtheaerosol. Theterms

F

and

a

muststillbe

dened. In what follows,thegravitational ee ts will be negle ted be ause we only fo us

on uid-aerosol intera tion. Both terms

F

and

a

model a drag for e (or a eleration)

between theuidandtheaerosol. WerefertoAppendixAfor adis ussionaboutthedrag

for eexpression. We hereusetheStokes law, whi h allows to write,for any

t

and

x

,

a(t, x, v) =

6πηr

m

(u(t, x) − v),

(2.4)

F

(t, x) = −m

Z

R

3

f (t, x, v) a(t, x, v) dv.

(2.5)

Whetherwe take

F

intoa ount ornotinthismodelisarealissue: itis alledtheaerosol

retroa tion ontheuid.

2.5. Initialand boundary onditions. Equations(2.1) (2.5)mustbesupplementedby

initial andboundary onditions. Re allthat for any

t ≥ 0

,themotionof theboundaryis

given bythedispla ement

A

t

: Ω

0

→ Ω

t

ofthe wholedomain, fromthe initial (referen e)

onguration,

(t, x) 7→ A

t

(x)

being as smooth asneeded. The velo ity of thedomain at

thepoint

x

∈ ∂Ω

t

ishen e given by

w(t, x) := ˙

A

t

(A

−1

t

(x)).

We assumethattheuidvelo ityt

w

on

Γ

wall

t

u

= w

on

Γ

wall

t

.

(2.6)

Of ourse,whenthedomainisxed,

w

≡ 0

andwehaveahomogeneousDiri hletboundary

onditionfor

u

on

Γ

wall

0

.

On both the inlet(s) and the outlet(s), we an assign Diri hlet or Neumann boundary

onditions,for instan e, if

u

in

t

: Γ

in

t

→ R

3

,we an hoose

u

= u

in

t

on

Γ

in

t

,

(∇

x

u

+ (∇

x

u)

T

) · n

t

− pn

t

= 0

on

Γ

out

t

.

(2.7)

For a more realisti modelling in the airow ontext, one an propose, as in [28 , 8℄ the

following strategy. In the proximal areas, the airow is omputed thanks to the

Navier-Stokesequations whereas, inthedistal part,itis des ribedbyawell hosen0Dboundary

onditions,takingthediaphragm motioninto a ount.

We hooseanabsorption boundary onditionfor theaerosolon thewall. Inthekineti

formalism,itwrites

(v − w) · n

t

< 0 ⇒ f = 0,

on

Γ

wall

t

× R

3

.

(2.8)

Remark3. The boundary onditionson

Γ

wall

t

areof ourse onsistent withthe respiration

framework: indeed, the wall is oated with mu us and the aerosol parti les deposit on the

wall whenthey hitit.

Consider

u

0

: Ω

0

→ R

3

and

f

0

: Ω

0

× R

3

_{→ R}

+

asinitialdata, i.e.

u(0, x) = u

0

(x),

x

∈ Ω

₀

,

f (0, x, v) = f

₀

(x, v),

x

∈ Ω

₀

,

v

∈ R

3

.

(2.9)

In a xed domain, with homogeneous Diri hlet boundary onditions for the uid and

absorptionfor thespray,on

∂Ω

0

,we have thefollowing proposition.

Proposition4. Assumethat

Ω

t

= Ω

0

forany

t ≥ 0

,

u

= 0

on

∂Ω

0

and

f = 0

on

∂Ω

0

×R

3

Proof. Multiplyrespe tively(2.3)by

mv

2

_/2

and(2.1)by

u

. Thenintegratetheout oming

equalitiesrespe tively on

Ω

0

× R

3

and

Ω

0

,toget

d

dt

m

2

Z Z

Ω

0

×R

3

f (t, x, v) v

2

dv dx +

1

2

Z

Ω

0

̺

u

|u(t, x)|

2

_dx

!

= −

Z Z

∂Ω

0

×

R

3

f (t, x, v)v · n

0

dv dx ≤ 0,

sin e

f

isnonnegative. Thepreviousinequalityensuresthatthetotalkineti energyofthe

oupledsystemde reases.



Remark 5. If one onsiders other kinds of uid boundary onditions, su h as (2.7) , the

previous result does not hold and no energy bound an be derived. This la k of energy

estimate omesfromthe Neumann boundary onditions for the Navier-Stokes system, and

mayalso leadto numeri al instabilities,see [25℄ for a review on this topi .

The nonlinearity in the Navier-Stokes equations and the strong oupling between the

Vlasov and Navier-Stokes equations are the two major di ulties about the full system

(2.1)(2.7) from both mathemati al and numeri al viewpoints. System (2.1) (2.9) was

mathemati allyinvestigatedinaxeddomain(globalintimeexisten eofweaksolutions)

in [30, 6 , 11, 45℄. In the two-dimensional ase, uniqueness was also investigated [44 ℄.

Notethat theapproximation strategy usedin theexisten e result established in[11 ℄ and

espe ially the un oupling pro ess between the kineti and uid equations inspired the

numeri al s heme presented inthenextse tion.

3. Numeri al s heme

Weherepropose atime-advan ings heme to solvethestrongly oupled problem(2.1) 

(2.7). We rst un ouple the uid and aerosol problems and solve the uid part with a

retroa tion sour e term oming from the previous time step. Then we solve the kineti

part, usingthe updated uidvelo ityto ompute the dragfor e. Note thatwe may need

timesub y lingof thekineti partto getan a urate value oftheretroa tion term.

The aerosolis omputed thanksto a parti le-in- ell (PIC) method [18, 20 ,21, 42℄. For

theuid, weusea Lagrangeniteelement methodasso iatedto anarbitrary

Lagrangian-Eulerian(ALE)approa h[32 ,23 ℄,tohandlethemovingaspe t. Thosemethodsarebriey

dis ussed inthenext subse tions. Themost intri ate part ofthis s heme probablylies in

the oupling between (2.1)and (2.3) through(2.5) .

Sin e the omputational domain an move, we need to dene the domain mapping

A

t

: Ω

0

→ R

3

and the asso iated velo ity

w(t, ·) : Ω

t

→ R

3

. For instan e, at ea h time

t

,

A

t

an be omputed fromtheboundariesmovement asasolution toa Poisson problem

seton

Ω

t

.

Besides, our work is embedded in the C++ nite element library LifeV

2

, whi h

previ-ouslyownednumeri altoolsto handleboth xedand movingmeshes, most lassi alnite

element methods, and oered solvers for biologi al ows (Navier-Stokes, Dar y, et .). If

more details are needed, the reader is invited to refer to [37℄, see also [38 , 19 ℄ about the

uidsolvers.

Inthefollowing,weshalldenote

T

thenaltimeof omputation,and onsideraregular

subdivision

(t

n

)

0≤n≤N

of

[0, T ]

withastep

∆t > 0

.

3.1. The Navier-Stokes equations. Mostfeatures of theuidsolverwe present below

arestandard. Nevertheless,we brieyexplain how thewhole omputation ishandled.

We dis retize the Navier-Stokes equations (2.1) (2.2) written in theALE onservative

form[38 ℄. Withour boundary onditions (2.6) (2.7) ,itis given, for any

t ∈ [0, T ]

,by

2

FreesoftwareunderLGPLli ense,jointlydevelopedinfourinstitutions: É olePolyte hniqueFédérale

̺

u

d

dt

Z

Ω

t

u

· β(t, x) dx + ̺

u

Z

Ω

t

[(u − w) · ∇

x

u] · β dx

− ̺

u

Z

Ω

t

(div w)u · β dx + η

Z

Ω

t

∇

x

u

: ∇

x

β

dx +

Z

Ω

t

p(div β) dx =

Z

Ω

t

F

· β dx,

Z

Ω

t

(div u) µ(t, x) dx = 0,

where

β

and

µ

are suitable test fun tions (transported by the ALE mapping

A

t

from

referen etest fun tions), satisfying

β

= 0

onthewalland on theinlets.

We hoose a ba kward Euler s heme and a semi-impli it treatment of the onve tive

term. In the simpler ase of a xed domain, it redu es to a standard semi-impli it Euler

s heme.

Forthespa edis retization, we useaLagrangenite element method. For a

hara ter-isti size

h > 0

, onsider atetrahedri mesh of

Ω

0

,denoted by

T

₀

h

=

M

h

[

i=1

K

_0,i

h

,

whereea h

K

h

0,i

isatetrahedron. For ea h

t

,thetetrahedri mesh

T

h

t

of

Ω

t

istheunionof

M

h

tetrahedra whose verti es aretransported fromthereferen e onguration

T

h

0

by the

dis reteALE mapping

A

h

t

. This mappingshould preserve thetetrahedri stru tureof the

mesh,see[38 ℄ for details.

Then we dene the basis fun tions, at ea h time

t

n

,

(ϕ

n

k

)

1≤k≤N

h

for velo ities, and

(ψ

n

_ℓ

)

1≤ℓ≤P

h

for pressures,transportedfrom

P1 − P1

referen ebasisfun tionson

T

h

0

. On e

again,

A

h

t

shouldpreserve the hosennite element spa e. Duetothe hoi e ofa

P1 − P1

setting, we useastabilized formulation, thatwe do notdetail here.

Then, at time

t

n

,we approximate theunknowns on

T

h

t

n

by

u

n

(x) =

N

h

X

k=1

u

n

_k

ϕ

n

_k

(x),

p

n

(x) =

P

h

X

ℓ=1

p

n

_ℓ

ψ

_ℓ

n

(x),

forany

x

∈ Ω

t

n

. Theunknowns thenbe omethefollowing olumn ve tors

U

n

= (u

n

₁

, . . . , u

n

_N

_h

)

T

,

Π

n

= (p

n

₁

, . . . , p

n

_P

_h

)

T

.

Consequently, from

t

n

to

t

n+1

,we have to solvethefollowing linearsystem

D

n+1

_(B

n+1

₎

T

B

n+1

₀

 U

n+1

Π

n+1



=

F

n

0



+

̺

u

∆t

M

n

_U

n

0



,

(3.1) where

B

n

=



−

Z

Ω

tn

ψ

_i

n

∇

x

· ϕ

n

_j

dx



1≤i≤P

h

,1≤j≤N

h

,

M

n

=

Z

Ω

tn

ϕ

n

_i

· ϕ

n

_j

dx



1≤i,j≤N

h

,

F

n

=

Z

Ω

tn

F

n

· ϕ

n

_i

dx



T

1≤i≤N

h

,

D

n+1

=

̺

u

∆t

M

n+1

_{− ̺}

u

C

n+1/2

_{+ ηA}

n+1

_,

with

A

n

=

Z

Ω

tn

∇

x

ϕ

n

_i

: ∇

x

ϕ

n

_j

dx



1≤i,j≤N

h

,

C

n+1/2

=

N

h

X

k=1

(u

n

_k

− w

n+1

_k

)

"

Z

Ω

_tn+1



ϕ

n+1

_k

· ∇

x

ϕ

n+1

_j



· ϕ

n+1

_i

dx

#!

1≤i,j≤N

h

.

Thedenitionof

F

n

on

Ω

t

n

is givenin3.3below.

Thesemi-impli ittreatmentofthe onve tivetermappearsintheexpressionof

C

n+1/2

,

where

u

n

k

is usedwith thebasis fun tionsat time

t

n+1

. In the same way, notethat (3.1)

involves nodal quantities at time

t

n+1

. In parti ular, ve tors in the right-hand side of

(3.1), are dened using the nite element oordinates of quantities on

Ω

t

n

, and further

transported bytheALE mapping on

Ω

t

n+1

(on the orrespondingnodes).

3.2. The Vlasov equation. Unlikethenite element method,theparti le method does

not provide an approximation of

f

on the mesh nodes. More pre isely, the distribution

fun tionis omputed asaweightedsum ofDira masses inthepositions and velo ities of

thenumeri al parti les, i.e.,ina measuresense,

f (t, x, v) =

N

num

X

p=1

ω

p

δ

x

_p

_(t)

⊗ δ

v

_p

_(t)

(x, v),

wherethenumberofnumeri alparti les

N

num

isinitially hosenbytheuser,

ω

p

is alledthe

representativityofnumeri alparti le

p

,and

t 7→ (x

p

(t), v

p

(t))

isthetraje tory,inthephase

spa e, of

p

. Hen e, we just have to ompute the hara teristi s to get an approximation

of

f

from itsinitial datum. Notethat, assoonas

x

p

(t)

doesnot belongto

Ω

t

anylonger,

thatmeans thatparti le

p

isdepositedon

Γ

wall

t

or went out of thedomain through

Γ

in

t

or

Γ

out

t

. Froma numeri al pointof view,

N

num

isnot modiedbut theoutgoing parti lesare

nottreated anymore.

Forpra ti alpurposes,

N

num

isverysmallwithrespe ttotheaveragenumberofphysi al

aerosol parti les

N

aero

, whi h makes the omputations mu h less ostly. In fa t, we an

write

N

aero

=

N

num

X

p=1

ω

p

,

whi h givesan orderofmagnitude of

ω

p

. Forinstan e, ifwe hoose

N

aero

= 10

10

(see2.3) and

N

num

= 10

3

, we ould set

ω

p

= 10

7

for all numeri al parti les. The relevan eof su h

a hoi e isdis ussed later inSe tion 5.

Aslongastheparti le

p

remainsinthedomain,its oordinatesinthespa ephasesolve

theCau hy problem

˙

x

p

(t) = v

p

(t),

v

˙

p

(t) = a(t, x

p

(t), v

p

(t)),

0 ≤ t ≤ T,

with hosen initial data

x

p

(0)

and

v

p

(0)

. Thisproblem is solved witha rst-order

semi-impli itEuler s heme

v

n+1

_p

= v

n

_p

+ ∆t ˜

a

n+1/2

,

x

n+1

_p

= x

n

_p

+ ∆t v

n+1

_p

,

(3.2)

where

a

˜

n+1/2

is thedraga eleration dependingon quantities at both times

t

n

and

t

n+1

.

Thisisexplainedinthenextsubse tion,be auseitisrelatedtothe ouplingbetweenboth

phases.

Eventually, we must emphasize that this method needs averaging, sin e it relies on

statisti alphysi s. Indeed,ifthe parti lesareinje ted witha uniform distribution at

Γ

in

t

,

we must pro eed with several initial numeri al distributions and take the omputations

average to lo atethedepositionareas, for instan e.

Remark 6. As we already stated, we only des ribe the s heme (and the model) for a

monodispersed (in radius) aerosol. It is of ourse possible to onsider numeri al parti les

withvariousradii. Hen e, it ispossibletoreprodu e theradius distributionof the parti les

intheaerosol. Itisveryusefulwhenoneaprioriknowshowan aerosol nebulizergenerates

3.3. The oupling. Let us now fo us on the oupling between the Vlasov and

Navier-Stokes equations. We here hoose to use an expli it time mar hing s heme, so that the

uid and aerosol parts are basi ally solved on e per time step. It enables to redu e the

omputational ost. Nevertheless,itmayleadtostabilityissuesweinvestigate afterwards.

Note that, asalready stated, the proof of existen e of weak solutions in [11 ℄ is based on

thesamekindofde ouplingstrategy. Thewholepro essisquitesimilartotheonesin[40 ℄

or[27 ℄.

We assumethat, at time

t

n

,

U

n

,

Π

n

,

(x

n

p

)

1≤p≤N

num ,

(v

n

p

)

1≤p≤N

num and

F

n

are known. Of ourse,when

n = 0

,

F

0

mustberst omputedfromtheinitialdatabeforestartingthe

timeloop. Then we solve(3.1) to obtain

U

n+1

and

Π

n+1

. Next,to advan etheaerosolin

time,we ompute a dis reteparti le a eleration denotedby

a

˜

n+1/2

and dened by

˜

a

n+1/2

=

6πηr

m

(˜

u

n+1

_(x

n

p

) − v

n+1

p

),

where the quantity

u

˜

n+1

_(x

n

p

)

is the uid velo ity at the position of parti le

p

, at the

previous time step. Sin e the uid velo ity is only known at mesh points, we need to

interpolate itinorder toevaluate

u

˜

n+1

_(x

n

p

)

. Notethat

x

n

p

∈ Ω

t

n

andthat

U

n+1

liesin

Ω

t

n+1

. First,parti le

p

maynotbein

Ω

t

n+1

any

more. Inthis ase, theparti le is onsidered to be deposited, hen e takingthe boundary

onditionof

f

on

Γ

wall

t

into a ount. Se ond,wehavetondoutinwhi h ellparti le

p

is,

anditsasso iatedbary entri oordinates. Thisispossible thanksto alo ating algorithm

[26℄. Note that, apart from the initial lo ating pro esses, this algorithm should onverge

in a very small number of iterations, so it is not so ostly. Third, we perform a linear

interpolationof

U

n+1

on themeshpointsat time

t

n+1

to get

u

˜

n+1

_(x

n

p

)

. On eweknow

a

˜

n+1/2

,we ompute

v

n+1

p

and

x

n+1

p

thanksto(3.2) ,andeventually

F

n+1

, as

F

n+1

(x) = −m

N

num

X

p=1

ω

p

6πηr

m

u

˜

n+1

_(x

n+1

p

) − v

n+1

p

δ

x

n+1

p

(x),

x

∈ Ω

t

n+1

.

Notethat, to ompute

F

attime

t

n+1

,wedonotuse

a

˜

n+1/2

. Consequently,thenumeri al

total momentum isnot onservedanymore,whi h mayimply stabilityissues.

Remark7. The uidtime stepmaynotbe suitabletopre iselyfollowthedis reteparti le

traje tories. As seenon Figure 2, without sub y ling, the parti le goes a ross several ells

duringasametimestep,whereasthehighuidvelo ityonthebla knodesspinstheparti le

traje tory. In this ase, one should introdu e a time sub y ling strategy.

Remark8. Inorderfortheretroa tiontermtobe more a urate, wemayusethesmallest

sub y le time stepforthe uidphase too. Indeed, ea h parti le maybe able toex hangeits

momentumwith the uidin ea h ell it goes a ross.

In what follows,we numeri ally study a ura yand stability properties ofour s heme.

Sin ewe aim to model theaerosol deposition in thehuman lung, the airis hosen asthe

ambient gas at temperature

310

K, so that

̺

u

= 1.204 × 10

−3

g. m

−3

and

η = 1.85 ×

10

−4

g m

−1

s

−1

. Theprevious values an befound, for instan e,in[1℄.

We rst investigate in Se tion 4 the no-retroa tion ase in a xed domain. We are

interested, for instan e, in sensitivity with respe t to the mesh size, time step, initial

parti lelo ations. Then, inSe tion5,weperformthesamekindoftestsintheretroa tion

ase, again in a xed domain. We moreover fo us on representativity issues appearing

when dealing with the retroa tion term. Finally, Se tion 6 is dedi ated to the moving

domain asewithoutretroa tion. Inparti ular,weexhibitanexa t solutionto(2.1) (2.2)

and ompare it with our numeri al solution. In every ase, one of the main issues is to

a uratelypredi tthedepositionmapof theaerosol. We on lude ourstudy byusing our

numeri al ode ona typi allunggeometry.

4. Numeri al tests without retroa tion

Werstfo usonthebehaviourofournumeri al odewhenthea tionoftheparti leson

theuid isnegle ted. We onsider an orthonormal system

(O, e

1

, e

2

, e

3

)

. The

omputa-tionaldomainisa ylinderofaxis

(O, e

3

)

,length

L = 5

mandradius

R = 0.2

m, entred

at

O

. Theairow follows thePoiseuille law,withvelo ity

u

goingalong the ylinder axis,

i.e.

u(t, x) = U



1 −

x

1

2

_{+ x}

2

2

R

2



e

3

,

U ≥ 0.

In parti ular,

u

doesnot depend on

t

and

x

3

, only on the transverse oordinates

x

1

and

x

2

.

Parti les evolve in theairow. They have the same radius

r

,mass density

ρ

p

, and are

inje ted intheuidat time

T

inj

= 0.2

s withthesameinitial velo ity

V

0

. Fig.3sums up thesituation.

O

e

1

e

3

e

2

u(t, x)

uid ow

L/2

V

0

V

0

b

R

Figure 3. Initial velo ity

V

0

for the omputations : solid line for 4.1,

dashed linefor 4.2 andfollowing.

4.1. Behavioroftheparti levelo ities. Inordertovalidatethe hara teristi smethod

implementation inour ode,letus onsiderherethe asewhenthereisonlyonenumeri al

parti leintheuid. TheStokeslawallowsto obtainananalyti expressionof theparti le

dedu e one for the parti le velo ity. Let us denote by

X

(t)

and

V

(t)

the lo ation and

velo ityofthe parti leat time

t

,and hoose

V

0

= V

0

e

3

,

V

0

_{∈ R}

. Consequently,at ea h

time

t

,

V

(t)

isparallel totheline

Re

3

. Its oordinate

V (t)

alongthis lineisthengivenby

V (t) =

"

V

0

+ u(X(t)) · e

3

Z

t

T

inj

e

s/τ

τ

ds

#

e

−t/τ

,

where

τ :=

2r

2

_ρ

p

9η

.

Hen e,weget

V (t) = u(X(t)) · e

3

+ [V

0

− u(X(t)) · e

3

]e

−(t−T

inj

)/τ

,

(4.1)

aslongastheparti le remainsinthe ylinder.

4.1.1. Motionlessuid. We rst assumethatthe uiddoesnot move, i.e.

U = 0

.

Conse-quently,(4.1) be omes

V (t) = V

0

e

−(t−T

inj

)/τ

,

theuidslowsdownthe parti le. For standard physi al parameters (waterdroplet inthe

air),therelaxationtime

τ

isverysmall. Sin eweareonlyinterestedinthe odevalidation,

letus useanonrealisti massdensityfortheparti le,i.e.

10

g/ m

3

,to allow, inthesame

time,a signi ant relaxationtimeand a smallparti le Reynoldsnumber.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

particle

velocity

(cm/s)

time (s)

Analytic solution

Numerical solution

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

error particle velocity (cm/s)

time (s)

Figure 4. Analyti /numeri al parti le velo ities: (a)plots,(b) error.

Choosing

V

0

_{= 5}

m/s and

T

inj

= 0.2

s, we get

V (t) = 5e

−3.33(t−0.2)

for any

t ≥ T

inj .

InFig.4a, theanalyti andnumeri al parti le velo ities areplotted withrespe tto time,

andFig.4b shows theabsoluteerror between both velo ities.

4.1.2. Nonzero Poiseuille prole. We now assume that the maximum uid velo ity

U

is

positive, here

U = 3

m/s. To ompute

V (t)

for ea hparti le thanks to (4.1) ,we need to

ndthevalueoftheuidvelo ityattheparti lelo ation. Thanksto thePoiseuillelaw, it

isenough to knowthedistan e between theparti leand the ylinder axis.

Consider, for instan e, two parti les. One is lo ated on the axis and the other one

0.1

m away (in theradial dire tion). Let us take

V

0

_{= −0.1}

m/s, opposite to the uid

ow. ThenFig.5showsagaintheverygoodagreementbetweentheanalyti andnumeri al

velo ities oftheparti le.

In theremainder of this se tion, we investigate thenumeri al sensitivityof our s heme

withrespe t to various parameters of the omputations, su hasthe mesh,thetime step,

et . Westudy anaerosolwith

3000

numeri al parti lessharing thesameinitial transverse

velo ity

V

0

= V

0

e

1

with

V

0

_{= 6.5}

m/s. Fig. 3 may again allow to understand the

-0.5

0

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

particle

velocity

(cm/s)

time (s)

Analytic

Numerical

Figure 5. Analyti /numeri al velo ities foraparti le ontheaxisand

an-other one

0.1

mawayfrom theaxis.

3000

numeri al parti lesareusedfortheparti lemethod: theyareuniformlydrawninside

adiskofradius

0.16

mat

x

3

= −L/2

. All theparti leshave thesameradius, i.e.

50 µ

m.

4.2. Mesh sensitivity. We onsider seven dierent meshes, from approximately

10, 000

tetrahedrato

640, 000

. Morepre isely,we give, inTable 1,thenumber oftetrahedra and

a orresponding average ell size value(

h

) obtained asthe uberoot of theratio between

the ylindervolume andthenumberof ells.

Numberof ells

9, 954

28, 800 67, 200 80, 136 130, 200 295, 416 645, 150

h

(in m)

0.0398

0.0279

0.0211

0.0199

0.0169

0.0128

0.0099

Table 1. Various meshes understudy.

Due to omputational osts, the range of explored ell sizes is narrowed between

0.01

and

0.04

. Weperformthesimulationsforallsevendierentmesheswith

20

dierentinitial

distributions ofparti les, witha timestep

∆t = 0.0027

s. Thenest mesh omputationis

taken asthereferen eone.

First, we onsider a given initial parti le distribution. For ea h numeri al parti le, we

re over the time evolution of the distan e between its referen e position and the ones

obtainedusing oarsermeshes. Thenwe omputetheaverageofthisdistan ewithrespe t

to all the numeri al parti les. We iterate this pro ess for ea h of the 20 initial parti le

distributions, whi h allows to dene the average distan e as a statisti al quantity. For

instan e,in Fig.6,theaverage parti ledistan e between omputationson meshes80 and

640is shown, aswell astheasso iated standard deviation.

After an initial phase, we observe alinear growthof theaveragedistan e error. Hen e,

the growth ratios are quantities of interest. They an be seen as line slopes. In Fig. 7,

we plot the slope values with respe t to the typi al ell size

h

, and on e again with the

statisti alerror.

Ournumeri al s hemebehaveswitharemarkablestatisti alstability. Onthedeposition

phenomenon,thevariationoftheaveragefra tionofdepositedparti les(and thestandard

deviation) at nal time does not seem signi ant in terms of meshes, see Fig. 8. The

per entage of

54.7

% is learly a satisfying value, and we an note that using a oarser

meshseemsto overestimate thedepositionee t a little.

Eventually,when fo using onthe oarsestand thenest meshes, letus emphasizethat

themaximal distan ebetween thepositions ofthesame parti leat depositiontimeinthe

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0

0.2

0.4

0.6

0.8

1

1.2

1.4

average distance between particles (cm)

time (s)

Figure 6. Average distan ebetween the oarsestand nest meshes.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.01

0.015

0.02

0.025

0.03

0.035

0.04

slope (cm/s)

characteristic meshsize (cm)

Figure 7. Growth ratios ontheaveragedistan e error.

0.53

0.54

0.55

0.56

0.57

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

percentage of deposited particles

characteristic meshsize (cm)

Figure 8. Fra tionof depositedparti leswithrespe tto

h

.

4.3. Time step sensitivity. In order to study the sensitivity of the omputations with

respe t to the time step, we use the nest mesh from Table 1 with

645, 150

tetrahedra.

∆t/2

and

∆t

. We then ompute the distan e between the lo ations of ea h numeri al

parti leinthereferen esituation andone ofthetwo others,at ea hmultiple of

∆t

,upto

naltime

1.4

s, andtake theaverage overallthe parti les.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0

0.2

0.4

0.6

0.8

1

1.2

1.4

average distance between particles (cm)

time (s)

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0

0.2

0.4

0.6

0.8

1

1.2

1.4

average distance between particles (cm)

time (s)

Figure9. Comparisonsonaveragedparti lelo ationson(a)

∆t

and

∆t/4

,

(b)

∆t/2

and

∆t/4

.

ThenFig.9abrespe tivelyshowtheaverage(overthe

20

randomlydrawnspa einitial

distributions)positionerrorsfor

∆t/4

and

∆t

,and

∆t/4

and

∆t/2

,respe tively. Asusual,

theaverageandthestandarddeviationaredisplayed. Bothresultsaresatisfa torybe ause

theerrorsseem to have a linearbehaviourwithrespe tto physi al time

t

.

Furthermore, we an he k the number of deposited parti les does not depend on the

timestepintheprevious tests. Morepre isely, themaximaldistan ebetween the impa t

positionsin omputationswith

∆t/4

and

∆t/2

is

0.004

mandin omputationswith

∆t/4

and

∆t

is

0.008

m.

Noti e that this stability behavior for the deposition of parti les is learly related to

the sub y ling strategy mentionned in Remark 7, that we systemati ally used in these

omputations. Without the renement of the time step for the parti les, some of them

ouldignore arti ially theboundary layer ofthe uid,going through several ells inone

timestep and go out of the mesh. This would indeed be an artefa t sin e, redu ing the

timestep, theseparti les wouldstart to per eive more a uratelythevelo ityof theuid

neartheboundary,when e redu ing thedepositionratio.

4.4. Sensitivity with respe t to the initialparti le lo ations. Thistest fo uses on

thesensitivitywithrespe tto theinitial spa edistribution ofthenumeri alparti les. We

hoose

∆t = 0.0027

s and onsider every mesh from Table1. For a given initial

ongu-ration of parti les, we perturb the parti le lo ations in the following way. Ea h parti le

is randomly (with a uniform law) relo ated in a small disk of radius

h/2

entred at the

parti le initial position. The small radius value ensures that the parti le remains in the

ylinder after the perturbation. Let us emphasize that the perturbation depends on the

onsideredmesh,through

h

.

Weperform the omputationsfor

20

dierentperturbationsofthesamekind,and

even-tually ompute the maximal distan e between the lo ations of theparti les at nal time

with respe t to the referen e onguration, averaged on the

20

draws. Fig. 10 shows a

remarkablestability withrespe tto theperturbation oftheinitial distribution.

5. Numeri al tests with retroa tion

In this se tion, the parti les exert a retroa tion for e on the uid, and the oupling

between the Navier-Stokes and Vlasov equations is strong. The retroa tion ee t was

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

average distance between particles (cm)

characteristic meshsize (cm)

Figure 10. Ee tofamesh-dependingperturbationoftheinitialparti le

distribution on theaverage distan ebetween parti lesat naltime.

Thedis ussionabout

F

allowsto lassifyrespiratory aerosols following [40 ℄. They an be

verythinsprays(intera tionbetweenparti les,sprayvolumefra tionand

F

arenegligible)

orthinsprays(intera tion between parti lesandsprayvolumefra tionarenegligible, but

F

isnot).

In this se tion, the omputations for both phases are performed at ea h time step,

whereas, in Se tion 4, we were able to rst ompute the full time evolution of the uid

ow, and afterwards the movement of ea h numeri al parti le as a post-treatment. The

omputational ostof oursesigni antly in reases,sin ewenowneedtoknowtheaerosol

movement atea htimeto omputetheretroa tionsour e termintheNavier-Stokes

equa-tions (2.1) . Let us emphasize that we should onsider relevant situations in whi h the

aerosolhasasigni ant ee t ontheuid ow.

We performed the same kind of sensitivity studies as in the previous se tion. We do

not provide results regarding sensitivity with respe t to themesh, be ause they are very

similartotheonesshownin4.2, butwe hooseto present aresultregardingthetimestep

sensitivity. We also ta kle thekeyissueon parti lesrepresentativity.

5.1. Time step sensitivity. Let us mimi the test from 4.3, with the same value of

∆t = 0.0027

s. Wehereusethe

80, 136

-tetrahedronmeshforthe omputations. Fig.11ab

aneasilybe omparedto Fig.9a. Theaveragedistan ebetweenthelo ationsofparti les

omputedwith

∆t

and

∆t/4

isalittlebitbiggerwhenretroa tion ison,forbothsmalland

highrepresentativities. Nevertheless, itreally remains ofthesame order ofmagnitude.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

0

0.2

0.4

0.6

0.8

1

1.2

1.4

average distance between particles (cm)

time (s)

0

0.001

0.002

0.003

0.004

0.005

0.006

0

0.2

0.4

0.6

0.8

1

1.2

1.4

average distance between particles (cm)

time (s)

Figure 11. Comparisons on averaged parti le lo ations on

∆t

and

∆t/4

with(a)

ω

p

= 1

,(b)

ω

p

= 10

4

5.2. Representativity of the numeri al parti les. Inthis subse tion,we dis uss the

numeri al approximation oftheretroa tion term, and mainly theinuen eof theparti le

representativity. First, we deal with a given number

N

num

of numeri al parti les. We

investigate the behaviour of the total kineti energy (of the oupled system) for various

valuesoftherepresentativity. Thisimpliesthatwedonotusethesamenumberofphysi al

parti lesin ea h numeri al experiment. Next, we aim to exhibit an optimal value of the

representativityfor a given realisti total numberparti les.

5.2.1. Representativity issues for a given number of numeri al parti les. We inje t

3000

numeri al parti les with axial velo ity

3

m/s at

z = 0.2

in a disk with radius

0.12

m,

and we onsider an initially motionless uid. The boundary onditions are hosen as in

Proposition 4. Sin e retroa tion is taken into a ount, the uid velo ity should be non

zero near the parti les shortly after the inje tion. A ording to Proposition 4, the total

kineti energy should de rease. The omputations are rst performed for a hosen time

stepequalto

0.0027

s, andvariousrepresentativities. InFig. 12,weobserve thatthetotal

kineti energy does not de rease when representativity, and onsequently the number of

physi al parti les, grow.

1e-06

0.0001

0.01

1

100

10000

1e+06

1e+08

0

0.05

0.1

0.15

0.2

0.25

0.3

kinetic energy

time (s)

1e7

7e6

6e6

5e6

3e6

Figure 12. Totalkineti energy for various representativities, with

∆t = 0.0027

.

Moreover,performingthesametestforthelargertimestep

0.004

s,weobserveinFig.13

thatthetotal kineti energy doesnot de reaseanymorefor

ω

p

= 5.10

6

,whereas itdidfor

∆t = 0.0027

.

Thisnumeri albehaviourmaybeexplainedbytheexpli ittreatment oftheretroa tion

asa sour e term in the Navier-Stokes equations. This expli it treatment may indu e an

unphysi al energy produ tion. More pre isely, when omputing velo ity and pressure at

t

n+1

,thesour e term writes

F

n

(x) = −m

N

num

X

p=1

ω

p

6πηr

m

u

˜

n

_(x

n

p

) − v

n

p

δ

x

n

p

(x),

x

∈ Ω

t

n

,

where

ω

p

learly appears asakey parameter. Furthermore,the omparison between both

aseswithtwodierent timestepssuggeststheexisten eofaCFL-like onditioninvolving

1e-08

1e-06

0.0001

0.01

1

100

10000

1e+06

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

kinetic energy

time (s)

5e6

3e6

Figure 13. Total kineti energy for two  riti al representativities, with

∆t = 0.004

.

5.2.2. Representativity issues for a given number of physi al parti les. Inthis subse tion,

thenumber of physi al parti lesis onstant:

10

10

,whi h is realisti for a standard

nebu-lizer. We aim to determine how manynumeri al parti lesshould be involved to represent

those physi al parti les. By hoosing very few numeri al parti les witha huge

represen-tativityin our omputations, the spa e distribution of the aerosol parti lesmaynot well

taken into a ount. Consequently, the aerosol retroa tion, whi h only has a lo al ee t,

ouldbe underestimated, leading to a non physi al behaviour. Hen e, the numberof

nu-meri al parti lesshould be a ompromise between both thenumeri al ost and thespa e

distributionof theparti lesintheuid.

Here we onsiderthree ases:

• 100

numeri al parti les, ea h representing

10

8

physi al parti les,

• 1000

numeri al parti les,ea h representing

10

7

physi al parti les,

• 10, 000

numeri al parti les, ea hrepresenting

10

6

physi al parti les.

The

1000

and

10, 000

-parti letest asesareobtainedthankstoperturbationofthe

100

-parti le one. Morepre isely, we build a random distribution of

100

parti les ina diskof

radius

0.06

mat

z = 0.2

. Thenwe reate anew distributionbyrepla ingea hof the

100

parti les by

10

parti les randomly hosen in a disk of radius

0.06

m around them. This

pro essis repeatedto getthe

10, 000

-parti le situationfrom the

1000

one.

We he ktheuidvelo ityattwodierentmeshnodes losetothe

z

-axis,neartheinitial

lo ationoftheparti lesat

z = 0.238095

,andawayfromthissamelo ationat

z = 0.595238

.

Morepre isely, in Fig. 1415, we show the numeri al error on the uid velo ity norm at

both nodes, omparing the

1000

-parti le asewiththeother ones. Inboth situations, the

results are quite similar when dealing with

1000

and

10, 000

parti les, whereas they are

signi antly dierent when dealing with

1000

and

100

parti les. Consequently, it is rst

lear that the hoi e of

100

numeri al parti les is not relevant to represent

10

10

physi al

parti les. Se ond,sin e theerror betweenthe

1000

and

10, 000

-parti lesituationsremains

small (of order of magnitude

10

−2

m/s), the hoi e of

1000

numeri al parti les appears

asa good ompromise in termsof numeri al ost and a ura y. Note thatthe testshave

beenperformedforseveralparti ledraws,andthestandarddeviationinany aseis

upper-bounded by

10

−6

.

6. Numeri al tests in a moving domain

Let us now investigate themoving domain situation. At initial time,thedomain is the

0

2

4

6

8

10

12

14

16

0.2

0.25

0.3

0.35

0.4

0.45

0.5

fluid

velocity

difference

(cm/s)

time (s)

1000 / 10,000

1000 / 100

Figure 14. Fluidvelo ityerror neartheparti lesw.r.t. time, omparison

with

1000

numeri al parti les.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.2

0.25

0.3

0.35

0.4

0.45

0.5

fluid

velocity

difference

(cm/s)

time (s)

1000 / 10,000

1000 / 100

Figure 15. Fluidvelo ityerror away fromtheparti lesw.r.t. time,

om-parison with

1000

numeri al parti les.

onsideredtimedependent domain isgiven,for any

t

,by

Ω

t

=

(κ(t)r cos θ, κ(t)r sin θ, ζ

z

(t)) | 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, −L ≤ 2z ≤ L ,

where we set

κ(t) = exp(λt),

with

λ = log(0.9)/1.4 ≃ −0.075.

Asolution to theNavier-Stokesequations writes, using ylindri al oordinates,

u(t, r, z) = u

r

e

r

+ u

z

e

z

=

˙κ(t)

κ(t)

re

r

− 2

˙κ(t)

κ(t)

ze

z

,

p(t, r, z) = −

r

2

2

¨

κ(t)

κ(t)

−

z

2

κ(t)

2

3 ˙κ(t)

2

_{− ¨}

_{κ(t)κ(t) .}

We he kedthattheNavier-StokesALEsolvera uratelyreprodu edtheprevioussolution.

Indeed,usingthe oarsestmeshandtimestep, theerror ontheuidvelo ityisat mostof

order

7.10

−3

m/s.

We an also obtain the traje tory, in the phase spa e, of ea h numeri al parti le in a

semi-analyti al way, thanks, for instan e, to the python fun tion expm. Let us use again

thenotations

X

(t)

and

V

(t)

introdu ed in 4.1. Note that

V

does not remain parallel to

e

z

anymore. We anwrite

˙

X(t) = V (t),

V

˙

(t) = −

1

τ

(V (t) − u(t, X(t))),

rewrittenwithmatrix notationsand solved by omputinga matrixexponential,i.e.

X(t)

V

(t)



= exp(At)

X(0)

V

(0)



,

where

A =

 0

I

3

A

1

−

I

3

/2



,

A

1

=

λ

2





1 0

0

0 1

0

0 0 −2





,

andI

3

denotes theidentity matrixof

R

3

.

In what follows, we perform various numeri al experiments without retroa tion to

in-vestigate thesolver e ien y when taking themesh motioninto a ount. Our numeri al

s heme uses a de oupling method, where the uid-aerosol system and the domain are

sequentlysolvedon epertimestep. Atea h timestep, we pro eedinthefollowing way:

1. move thedomain,

2. solve theuidequation,

3. lo ateand move theparti lesthanksto (3.2) .

Other impli it or expli it strategies may have been onsidered. In parti ular, the order

of thedierent steps is a ru ial issue withrespe t to the parti le deposition. Hen e, we

he k that the hosen strategy is e ient by omparing our numeri al solution with the

semi-expli itone,see Fig.16.

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

average particle position error (cm)

time (s)

Figure 16. Comparisons on averaged parti le lo ations with the

semi-analyti al referen esolution.

We hoosethesameinitial onditionsandnumberofnumeri al parti lesasinthelower

partofpage10des ribing thetest aseina xeddomain.

6.1. Meshsensitivity. We hoose

0.0027

sasthetimestepforthistest ase. Weobserve,

onFig.17, analmost onstant deposition ratiowith respe tto themesh size.

Next we ompare, on Fig. 18, the deposited parti le lo ations at both nal time and

deposition time (forea h parti le). Note that this deposition timeis theone obtained in

thereferen esituation. Thisfa tmayexplaintheslightdieren einthedistan esobserved

forea h mesh, omparing ases aand b.

6.2. Time step sensitivity. Fig. 19 shows a very good agreement between the three

0.55

0.555

0.56

0.565

0.57

0.575

0.58

0.585

0.59

0.01

0.015

0.02

0.025

0.03

0.035

0.04

percentage of deposited particles

characteristic meshsize (cm)

Figure 17. Fra tionof deposited parti lesfor various mesh size.

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.015

0.02

0.025

0.03

0.035

0.04

average distance between particles (cm)

characteristic meshsize (cm)

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.01

0.015

0.02

0.025

0.03

0.035

0.04

average distance between particles (cm)

characteristic meshsize (cm)

Figure 18. Comparisons on averaged parti le lo ations with the

semi-analyti al referen e solution at (a) nal time, and (b) deposition time in

thereferen e situation.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.2

0.25

0.3

0.35

0.4

0.45

0.5

fraction of deposited particles

time (s)

0.000675

0.00135

0.0027

Figure 19. Fra tion of deposited parti les for various time steps:

∆t/4

,

∆t/2

and

∆t

.

7. Con lusions and prospe ts

In this paper, we proposed a model to des ribe the intera tion of a thin spray in an

possi-linear oupledVlasov-Navier-Stokes systemwasthus onsideredand expli itlydis retized

intime. For the spa e approximation, a PIC method was used for the kineti equation,

whereasan ALE niteelement method was onsidered for theuidpart. We numeri ally

demonstrated the stability of our s heme in various situations: xed or moving domains,

with or without retroa tion. In parti ular, we highlighted the existen e of a CFL-like

onditionin theretroa tion ase. Notethat our numeri al simulations showed very good

onvergen e propertiesfor thedeposition rate, withrespe tto themesh size for instan e.

One ru ialpointhereliesinthesub- y lingstrategyusedtoadvan etheparti lesintime,

enabling an a urate predi tion of thedepositionphenomenon, whi h is a ru ial issueof

ourstudy. Wealsowant tounderlinethat onsidering akineti modellingoftheaerosolis

appropriatein this ontext: a phaseeld des ription, even ifnumeri ally less ostly,does

notallowaneasydes riptionofdeposition,whereasfollowingea hindividualparti lesmay

betooexpansive,inparti ularwhenretroa tionisnotnegligible. Consequently,ourmodel

andourexpli itnumeri al strategyrepresentagood ompromiseandareagood andidate

to perform e ient in sili o experiments of aerosol deposition in omplex situationssu h

aslungmodelling.

Appendix A. About the drag for e

In [40℄, O'Rourke gives a general expression ofthe dragfor e of theair on theaerosol.

Intermsof a eleration, itreads,for any

t

,

x

and

v

,

a(t, x, v) =

πr

2

8m

̺

u

C

D

|u(t, x) − v|(u(t, x) − v),

(A.1) where

C

D

isadimensionlessparameter alledthedrag oe ient. This oe ientis

semi-empiri ally omputed using the parti le Reynolds number, whi h allows to measure the

ratio between inertia andvis osityfor es on theparti le:

Re

p

:=

2

η

̺

u

r|u − v|.

Then

C

D

,whi halsodependson

u

and

v

,isgivenbyS hillerandNaumann'slaw[14 ℄,also

usedin [24 , 27℄, in theKIVA odes [5, 4 ,2, 3,41 ℄ and for omplex uid-parti le mixtures

[12,35 ℄:

C

D

=











24

Re

p



1 +

1

6

Re

p

2/3



,

ifRe

p

< 1000,

0.424,

ifRe

p

≥ 1000.

When Re

p

is small, whi h is the ase in the respiration framework, one an also use

C

D

= 24/

Re

p

, see [29 ℄. This formula immediately allows to re over the Stokes law from

(A.1).

Appendix B. Analyti solutions to the in ompressible Navier-Stokes

equations

In this se tion, we derive analyti solutions to the in ompressible Navier-Stokes

equa-tionswithout sour e termsina pres ribed moving domain,whi h isinitiallya ylinder.

Using the ylindri al oordinates

(r, θ, z)

and the asso iated basis

(e

r

, e

θ

, e

z

)

(where

e

3

= e

z

),the ylinder at initial timeisgiven by

C =

(r cos θ, r sin θ, z) | 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, −L ≤ 2z ≤ L ,

and we an write

u

with the form

u(t, r, θ, z) = u

r

e

r

+ u

θ

e

θ

+ u

z

e

z

. Assuming that

u

doesnot depend on

θ

anymore and

u

θ

= 0

,

u

satises

u(t, r, z) = u

r

e

r

+ u

_z

e

z

.

(B.1)

Thein ompressible Navier-Stokesequations thenbe ome

∂

t

u

r

+ u

r

∂

r

u

r

+ u

z

∂

z

u

r

= −∂

r

p +

 1

r

∂

r

(r∂

r

u

r

) + ∂

2

zz

u

r

−

u

r

r

2



,

(B.2)