Settling velocity of quasi-neutrally-buoyant inertial particles

(1)

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Settling velocity

of quasi-neutrally-buoyant inertial particles

Vitesse

de

sédimentation

de

particules

inertielles

dotées

de

ﬂottabilité

quasi

neutre

Marco Martins

Afonso

∗

,

Sílvio

M.A. Gama

CentrodeMatemáticadaUniversidadedoPorto,RuadoCampoAlegre687,4169-007Porto,Portugal

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received5August2017 Accepted20November2017 Availableonline6December2017

Keywords: Fluiddynamics Inertialparticles Settlingvelocity Quasi-neutralbuoyancy Steady/periodic/cellularﬂows Browniandiffusivity Mots-clés :

Dynamiquedesﬂuides Particulesinertielles Vitessedesédimentation Flottabilitéquasineutre

Fluxstationnaires/périodiques/cellulaires Diffusivitébrownienne

We investigatethe sedimentationpropertiesofquasi-neutrally buoyant inertialparticles carried by incompressible zero-mean fluid flows. We obtain generic formulae for the terminalvelocityingenericspace-and-timeperiodic(orsteady)flows,alongwithfurther informationforflowsendowedwithsomedegreeofspatialsymmetrysuchasoddparity in the vertical direction. These expressions consistin space-time integrals of auxiliary quantities that satisfy partial differential equations of the advection–diffusion–reaction type,whichcanbesolvedatleastnumerically,sinceourschemeimpliesahugereduction oftheproblemdimensionalityfromthefullphasespacetotheclassicalphysicalspace.

r

é

s

u

m

é

Nousétudionslespropriétésdesédimentationdeparticulesinertiellesdotéesdeﬂottabilité quasi neutreet transportées par un écoulement incompressible àmoyenne nulle.Nous obtenons des formules génériques pour la vitesse terminale dans des écoulements en généralpériodiquesenespaceetentemps(oustatiques),avecd’ultérieuresinformations disponiblespour les écoulements dotésde symétries spatiales spéciﬁques,telles qu’une parité négative dans la direction verticale. Cesexpressions consistent en des intégrales spatio-temporelles de quantités auxiliaires qui obéissent à des équations aux dérivées partielles du type advection–diffusion–réaction. Ces dernières peuvent être résolues au moins numériquement, car notre procédure implique une forte réduction de la dimensionnalitéduproblème,del’espacedesphasescompletàl’espacephysiqueclassique.

*

Correspondingauthor.

E-mailaddress:marcomartinsafonso@hotmail.it(M. MartinsAfonso). https://doi.org/10.1016/j.crme.2017.11.005

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1. Introduction

Particles advected by a fluid are called“inertial” if, when studying their motion,one cannot neglectthe particle rel-ative inertia withrespect to the surroundingfluid. This isusually dueto their (small but) not negligiblesize, and/or to a mismatchbetweenthe twomass densities.Common examples arerepresented bysmall bubblesinliquids, droplets in gases,andaerosolsinagenericfluid.Thecomprehensionofthedynamicsoftheseimpuritiesisstillanopenissuefromthe theoretical,experimental,andnumericalpointsofview

[1–9]

.Implicationsarerelevantinmanyapplieddomains:plankton dynamics in biology [10], chemical reactors, spray combustion and emulsions in industrial engineering[11], planet for-mation inastrophysics [12], transport ofpollutants orﬂoaters, raininitiation andsedimentationprocesses in geophysics

[13].

Our focus isprecisely on sedimentation, with a specialattention to those situations wherethe mass–density ratio is (differentfrombut)veryclosetounity.Thisisforinstancethecaseformostlivingbeingssuspendedinanaquaticmedium. Theintuitive pictureisthefollowing:inertiacausesadeviationoftheparticlesfromtheunderlyingfluidtrajectory,which leads to inhomogeneitiesfortheparticleconcentration inregions oftheflow withdifferentdynamicalproperties,dueto thepresenceofsymmetry-breakingforcesandpreferentialdirections—inourcase,gravityalongtheverticalone.Moreover, we will also considerthe effectof Browniandiffusivity. The latteris usually neglected in mostinvestigations on inertial particles,assumingthatBrowniannoiseisverysmallforfinite-sizeparticles.However,thisisnottruefortinyparticles,and especiallyinbiophysicalapplications,wherealimitedcapacityofautonomousmovementcouldbeconsideredinthissimple fashion. Thisworkthereforerepresentsacomplementary studywithrespectto similaronesthatfocused onthelimitsof smallinertiaoroflargeBrowniandiffusivity.

OurprincipalobjectiveistoobtainanEuleriandescriptionofthesettling(i.e.fallingorrising)insteadyorperiodicﬂows startingfromthewell-knownLagrangianviewpointforparticlemotion.Despitethis,ourtheoryprovidesthewholedetailed statisticalinformationofparticlemotion.Indeed,theprobabilitydensityfunctionofhavingaparticleinagivenpositionat acertaintimeisavailablefromourapproach,atleastinaperturbativeway.However,thisimpliestheresolutionofpartial differentialequations,whichingeneralcanbeaccomplishedonlynumerically.

Thepaperisorganizedasfollows.Insection2,we definetheproblemunderinvestigation,wespecifyourassumptions andwesketchouranalyticalprocedure.Weenouncethefinalresultforgenericflowsinsection 3,andwespecializeitfor vertically-antisymmetriconesinsection4.Conclusionsandperspectivesfollowinsection 5.The

Appendix A

isdevotedto showingthedetailsofthecalculationandtorecallingthemathematicaltoolsemployed.

2. Equations

We consider avery dilute suspension ofpoint-like inertialparticles subjectto the gravitationalacceleration g andto Brownian diffusion,carriedby a fluid flow.We suppose that ourd-dimensionalincompressiblevelocity field issteadyor periodic intime(withperiod

T

),andperiodicinspacewithunit cell

P

oflinearsize

.Itisnot arestrictiontofocuson velocityﬁeldswhoseaveragevanishesover

P

:

P

dx u

(

x

,

t

)

=

0 (1)

In thisway,anydeviation ofthesettling velocity withrespect tothe value found instill fluidswillrepresent agenuine interplaybetweengravityandtheotherpropertiesofparticleandflow,andnota merestreamingorsweepingeffect.The sametechniquecanbeextendedtohandlethecaseofarandom,homogeneous,andstationaryvelocityfield

[14]

withsome non-trivialmodiﬁcations intherigorousproofsofconvergence

[15]

.Foran interestinginvestigation oftherole played by meancurrentsontheeddydiffusivityoftracers,see,e.g.,

[16–21]

.

Neglectinganypossibleinteractionwithotherparticlesorwithphysicalboundaries,andtakingintoaccountthefeedback onthetransportingﬂuidinan effectivewaybymeansofasimpliﬁedadded-masseffect,theLagrangiandynamicsreduces tothefollowingsetofstochasticdifferentialequationsfortheparticleposition

X (

t)andcovelocity

V(

t)[22,23]:

⎧

⎪

⎨

⎪

⎩

˙X(

t

)

= V(

t

)

+ β

u

(

X (

t

),

t

)

+

√

2

D

μ(

t

)

˙V(

t

)

= −

V(

t

)

− (

1

− β)

u

(

X (

t

),

t

)

τ

+ (

1

− β)

g

+

√

2κ

τ

ν(

t

)

(2)

The independentvectorialwhitenoises

μ

(t)

and

ν

(t)

inﬂuencetheparticledynamicsby meansofthecouplingconstants

D

and

κ

,whichcanbeidentiﬁedasBrowniandiffusivities

[24]

.Thepresenceoftwodifferentparametersintheequations for theposition andthe velocity willbecome clear shortly.The pure number

β

≡

3

ρ

f

/(

ρ

f

+

2

ρ

p

)

∈ [

0

,

3

]

,built fromthe

constant ﬂuid(

ρ

f) andparticle(

ρ

p) massdensities,isdubbed“added-massfactor”,becauseit takesinto accountthefact

thatanyparticlemotionnecessarilyimpliessomeﬂuidmotionaroundit,thusincreasingtheintrinsicinertia—withthesole exception ofvery heavyparticles such asaerosols ordropletsina gas(

β

0).It alsoinduces amacroscopicdiscrepancy between the particle velocity

˙X(

t) and covelocity

V(

t), which is maximum forvery light particles such asbubbles in a liquid(

β

3). Alternatively, in terms of slipvelocity—deﬁned asthe difference betweenthe particle velocity and the

(3)

localinstantaneousﬂuidvelocitysampledbytheparticle:

Y(

t)

≡ ˙X (

t)

−

u

(

X (

t),t)—thecovelocityturnsouttobe

V(

t)

=

Y(

t)

+ (

1

− β)

u

(

X (

t),t)

−

√

2

D

μ

(t)

.Finally, theStokes time

τ

in thedrag termexpresses thetypical response delayof particles to ﬂow variations, and is deﬁnedas

τ

≡

Q2

_/(

₃

_γ

_β)

_for _spherical _inertial _particles _of _radius _{Q immersed} _in _a

ﬂuid with kinematicviscosity

γ

. Note,however, that, ascustomary in inertial-particlestudies,

β

and

τ

are assumed as independentparameters,sincethelattercanbevariedevenwhentheformeriskeptﬁxedbysuitablychanging Q and

γ

. The dynamical system (2) neglects the classical contributions dueto Basset (time integration for memory/history/wake effects),Oseen(nonlinearfinite-Reynolds-numbercorrectiontothebasicStokesflow),Faxén(spatialexpansionofthefluid flowforfiniteparticlesize)andSaffman(lateralliftincaseofrotation).

Afterstatisticalaveraging of

(2)

on

μ

(t)

and

ν

(t)

[25–28],thegeneralizedFokker–Planck(or Kramers,orforward Kol-mogorov)equationforthephase-spacedensityp(x

,

v

,

t)isobtained:

∂

t

+ ∂

x

· [

v

+ β

u

(

x

,

t

)

] + ∂

v

· (

1

− β)

u

(

x

,

t

)

−

v

τ

+ (

1

− β)

g

−

D

∂

_x2

−

κ

τ

2

∂

2 v

p

=

0 (3)

Letusdenoteby

L(

x

,

v

,

t)thelinearoperatorincurlybracesontheleft-handsideof

(3)

,sothat

L

p

=

0.Forfutureuse, letusalsointroducethecorrespondingphysical-spaceconcentration,obtainedbyintegratingonthecovelocityvariable:

q

(

x

,

t

)

≡

Rd

dv p

(

x

,

v

,

t

)

(4)

Theparticleterminal velocity

[29–35]

is deﬁnedasaweighted averageoftheparticlevelocity, fromtheﬁrstequation in(2): w

≡ V(

t

)

+ β

u

(

X (

t

),

t

)

+

√

2

D

μ(

t

)

p

=

T

0 dt

T

P dx

Rd dv

[

v

+ β

u

(

x

,

t

)

]

p

(

x

,

v

,

t

)

(5)

(hereandinwhatfollows,theaverageonthetemporalperiod

T

isskippedforsteadyflows).Noticethatingeneralsuch quantity corresponds toa mean behavior andnot toan asymptoticvalue—except forthe case ofstill fluids ifBrownian diffusionisnegligible.Indeed,insideaflow,each particlecan wanderinanydirectionandfollowmoreorlesscloselythe underlyingfluid trajectory, buttheoverall evolutionofa bunchofnon-interacting particleswill consistin afalling/rising describedby w.Onthecontrary,inourmodel,thewell-known“bare”asymptoticvalueofsedimentationinstillfluidsis:

W ≡ (

1

− β)

τ

g (6)

Asprovenin

A.1

,thedeviationoftheterminalvelocityfromitsbarevaluecanberewrittenusing

(4)

as:

Z ≡

w

− W =

T

0 dt

T

P dx

Rd dv u

(

x

,

t

)

p

(

x

,

v

,

t

)

=

T

0 dt

T

P dx u

(

x

,

t

)

q

(

x

,

t

)

(7)

Now,letusfocusonparticleswhosemassdensitydiffersonlyslightly(eitherinexcessorinshortfall)fromtheﬂuidone

[36–38].Since

β

1,then1

− β

issmallbutwithanundeﬁnedsign,soweintroduceasecondsmallparameterintheform of

α

≡ |

1

− β|

1.Wealsodeﬁne

J ≡

sgn

(

1

− β)

,thus

β

=

1

−

J

α

.Itcanbeshownthat,inthissituation,itispossible to proceed analytically only ifone makes the further assumption that the Brownian-diffusioncoeﬃcient

κ

appearing in theequationfortheparticleaccelerationissmallaswell,namelywiththesameasymptoticbehaviorasthemass–density mismatch:

κ

∼

α

1;or,inotherwords,onecandeﬁneaﬁniteconstant

K ≡

κ

/

|

1

− β|

=

α

−1

_κ

_with_dimensions_of_square

length over time. Notice that no assumptionis made on the Browniandiffusivity

D

driving the particle velocity, which canthenbethoughtofasaregularizingparameter.Asiswell known,thediffusivityofatracer particle—obeying

(2)

with

τ

=

0—wouldturnouttobesimply

D +

κ

,butforinertialparticlesthesituationismoresubtleand,indeed,ouranalytical procedureworksonlyif

κ

issmalland

D

isnon-zero.Itisalsoworthmentioningthat,hadoneproceededonaLagrangian route before turning to the (Eulerian) phase-space description, the zeroth-order situation

β

=

1 would correspond to a Markovianprocessdriven byacolorednoise (Ornstein–Uhlenbeck)intheLangevin equation,asalreadydescribed in

[39]

. TheLagrangianapproachhasalsobeenfollowedin

[40]

toﬁndexactexpressions fortheparticleeddydiffusivityinshear orGaussianﬂows.

Uponrescalingthecovelocityvariableaccordingto v

→

y

≡

v

/

√

|

1

− β| =

α

−1/2_v,_the_generalized_{Fokker–Planck}

oper-atorsplitsinto:

L

=

L

(0)

₊

_α

1/2

_L

(1)

₊

_α

_L

(2)

(4)

L

(0)

_{= ∂}

t

+

u

(

x

,

t

)

· ∂

x

−

D

∂

x2

−

τ

−1

∂

y

·

y

−

K

τ

−2

∂

2y (8a)

L

(1)

₌

_y

_{· ∂}

x

+

J

[

τ

−1u

(

x

,

t

)

+

g

] · ∂

y (8b)

L

(2)

_{= −}

_J

_u

₍

_x

_,

_t

₎

_{· ∂}

x (8c)

Forthesakeofnotationalsimplicity,wedeﬁnea“gravitationalvelocityﬁeld”z

(

x

,

t)

≡

u

(

x

,

t)

+

τ

g (with

∂

x

·

u

=

0

⇒ ∂

x

·

z

=

0)andtwolinearoperators,

M

(

x

,

t

)

≡ ∂

t

+

u

(

x

,

t

)

· ∂

x

−

D

∂

x2 (9)

(advection–diffusioninphysicalspace)and

N

(

y

)

≡ ∂

y

·

y

+

K

τ

−1

∂

2y (10)

(relatedtotheOrnstein–Uhlenbeckformalism).Intermsofthem,

L

(0)

₌

_M

₍

_x

_,

_t

₎

₋

_τ

−1

_N

₍

_y

_{) ,}

_L

(1)

₌

_y

_{· ∂}

x

+

J

τ

−1z

(

x

,

t

)

· ∂

y

Ourrescalingisdictatedbythecloseanalogywiththesituationdescribed in

[35,41,42]

,wherethesmall-inertialimitwas performed.Inthatcase,thesmallquantityinthedenominatorwasthesquarerootof

τ

,whilehereitisthatof

|

1

− β|

.As shownintheappendix,theadvantage ofsuch rescalingliesinthefactthat itallowsforafulldecouplingoftherescaled covelocityfromthephysical-spacedynamics,andfortheresolutionofequationsbasedontheoperator

(10)

intermsofa basicGaussianstate.Notethatinthepresentframeworkwehavetorequestthesmallnessof

κ

explicitly,aconditionthat, onthecontrary,wassomehowimplicitinthoseworks,asexplainedin

[41]

byintroducingthenon-dimensionalStokesand Pécletnumbers(whoseproductwasrequiredtobe O(1

)

).

Itisnownaturaltoexpandthephase-spacedensityintoapowerseriesin

√

α

andtoreplaceinto

(3)

:

p

(

x

,

y

,

t

)

=

∞

I=0

α

I/2p(I)

(

x

,

y

,

t

)

implyingthat

L

(0)_p(0)

₌

₀ _(11a)

L

(0)_p(1)

_{= −}

_L

(1)_p(0) _(11b)

L

(0)_p(I)

_{= −}

_L

(1)_p(I−1)

₋

_L

(2)_p(I−2)

₍

_I

_≥

₂

₎

_(11c) 3. Resultsforperiodicincompressibleﬂows

Theterminalvelocityisaccordinglyexpandedas: w

=

∞

I=0

α

I/2w(I)

Z =

∞

I=0

α

I/2

Z

(I) (12) Since

W =

α

J

τ

g,thenw(I)

_{= Z}

(I)

_{+ δ}

I2

J

τ

g.Itcanbeshown(seeappendixfordetails)thatactuallyallthehalf-integer

orders of these expressions (corresponding to odd

I

) identically vanish, so that in practice such expansions reduce to commonanalyticalones.Moreover, onealsoseesthat w(0)

₌

₀

_{= Z}

(0)_,_i.e._particles _with_{exactly-neutral}_{buoyancy—which}

would macroscopically stand still in ﬂuidsatrest—on average donot settle either inthe presence ofour class ofﬂows. In what follows,we are going to provide the expressions forthe terminal velocity up to thesecond order, that is, w(2)

and w(4)_._Formula

₍₅₎

_can_be _manipulated_in_order_to_succeed_in_performing_the_covelocity_integrals,_and_what_is_left_are

space–timeintegralsofasetoffieldssatisfyingtheequationsoftheadvection–diffusion–reactiontypeintheconfiguration space. At workingorder, suchfieldsofourinterest aredenotedby q(0),r_i(1),q(2),s(2)_{i j} ,r_i(3),andq(4).Apartfromimposing the constancy ofq(0)

=

−d,their other partial differentialequations aresolvableanalytically only forspecificflows such asparallelones.However, suchaclassofflowisnot relevantforourscope,sincenocontributiontotheterminalvelocity arises fromthem.Nevertheless,our procedureallows foratleast anumerical resolutioningeneric flows, becauseofthe drasticreductioninthedimensionalityoftheproblemfrom2d

+

1 tod

+

1.

Postponingalldetailsto

Appendix A

,anddeﬁning

∇

i

≡ ∂

xi,weassertﬁrstofallthat:

w(2)_i

=

J

τ

gi

+

Z

_i(2)

,

Z

_i(2)

=

T

0 dt

T

P dx ui

(

x

,

t

)

q(2)

(

x

,

t

)

(13)

(5)

withq(2)introducedin

(26)

;and w(4)_i

=

Z

(4) i

=

T

0 dt

T

P dx ui

(

x

,

t

)

q(4)

(

x

,

t

)

(14) withq(4)introducedin

(30)

.

Todeterminetheorder

α

1_,_the_set_of_relevant_equations_consists_of:

(

M

+

τ

−1

)

r_i(1)

=

−d

J K

−1zi (15a)

M

q(2)

= −

K

τ

−1

∇

iri(1) (15b)

withr(1)_i introducedin

(24)

.

Toanalyze O(

α

2

₎

_too,_the_system_also_comprises:

(

M

+

2τ−1

)

s(2)_{i j}

= −∇

ir(1)_j

+

J K

−1zir(1)_j (16a)

(

M

+

τ

−1

)

r_i(3)

= −∇

iq(2)

+

J K

−1ziq(2)

+

J

u

· ∇

r(1)_i

−

K

τ

−1

∇

j

(

s(2)_ji

+

s(2)_{i j}

)

(16b)

M

q(4)

=

J

u

· ∇

q(2)

−

K

τ

−1

∇

ir(3)i (16c)

withs(2)_{i j} andr(3)_i introducedin

(26)

and

(28)

respectively.

Theconclusionsthatcanbedrawnanalyticallyatthisstagearethefollowing.Theterminalvelocityisgivenby

w

=

α

w(2)

+

α

2w(4)

+

O

(

α

3

)

(17)

withtheleadingorderfrom

(13)

representedby:

α

w(2)

= |

1

− β|

J

⎡

⎣

τ

g

+

T

0 dt

T

P dx u

(

x

,

t

)

J

−1q(2)

(

x

,

t

)

⎤

⎦

= W + (

1

− β)

u f

(

u

,

τ

,

g

,

D

)

(18)

Herewe exploitedtherelation1

− β =

J

α

andthefactthat—dueto

(15a)

—theﬁeldr(1)

_{/(J K}

−1

₎

_is_independent_of_both

J

and

K

(i.e.ofboth

β

and

κ

),sothesameindependencealsoholdsfortheﬁeld q(2)

_/J

_because_of

_(15b)

_._Therefore,_in

thelimitofquasi-neutrally-buoyantparticles(andof

κ

withthesameorderofsmallnessas

|

1

− β|

),themaincontribution to the terminal velocity isrepresented by its bare value plus a same-order deviation only dependent onthe other ﬁnite quantitiesintoplay,andwhichcanbecomputednumericallyvia

(18)

and

(15a)

–(15b).Thisleadingorderisindependentof

κ

andoverallodd in1

− β

:withalltheotherparametersﬁxed,particlesslightlyheavierthantheﬂuidsettlewithavelocity opposite totheone ofslightlylighter particles; atthisstage, noimmediateconclusioncanbe drawnon thesignofsuch adeviation.Notethatexpression

(17)

doesnotexcludethepossiblepresenceoffurthertermslinearin

|

1

− β|

,butwitha “prefactor”proportionaltoapositivepowerof

κ

,becauseinourasymptoticsthesewouldbehigher-ordercontributions.No immediatesimpliﬁcationcanbeperformedontheterm

α

2_w(4)_from

₍₁₄₎

_for_the_time_being.

4. Simpliﬁcationsforﬂowsendowedwithverticalparity

Ifa vertical-parity symmetry isimposed on the flow, further simplificationscome along (atleast forthose situations wheregravityisalignedwithone sideoftheperiodicitycell).Namely,ifatapoint x_∗ definedastheverticalreflectionof thepoint x withrespect toa referencehorizontalplane (x_∗

·

g

= −

x

·

g and x_∗

×

g

=

x

×

g),thevertical andhorizontal componentsoftheﬂowsatisfy

u

(

x_∗

,

t

)

·

g

= −

u

(

x

,

t

)

·

g and u

(

x_∗

,

t

)

×

g

=

u

(

x

,

t

)

×

g (19)

then it is possible to split all the relevant physical-space ﬁelds intotheir even andodd parts. Forinstance, ue/o

(

x

,

t)

≡

[

u

(

x

,

t)

±

u

(

x_∗

,

t)

]/

2, with a purely odd vertical component u

·

g

=

uo

·

g and (a) purely even horizontalcomponent(s) u

×

g

=

ue

×

g. The consequentequationsderived fromthe sets

(15)

and(16)aresimpler todeal with,ﬁrst ofall from

a numerical point of view as deﬁnedon a halveddomain. Analytically, it can be shown that the function f in (18)is actuallylinearin g,sothat w(2)isoverallproportionaltogravity;sincethesamecanbestatedalsofor w(4)in

(17)

,such aconclusionholdsforthewholeterminalvelocityatworkingorder.

Noticethatthiscategoryalsocomprisescellularﬂowsoftenadoptedinanalyticalandnumericalinvestigationstomimic LangmuircirculationontheoceansurfaceorlateralconvectiverollsinRayleigh–Bénardcells

[29,31,43–45,35]

.

(6)

5. Conclusionsandperspectives

We investigated the sedimentation process of quasi-neutrally buoyant inertial particles in zero-mean incompressible ﬂows.Suchparticlesareespeciallyrelevantinbiophysicalapplications,wheremostoftheaquaticmicroorganisms

[46]

have a massdensityvery similarto theone ofwater.Generalformulaehavebeen foundfortheir terminalvelocity ingeneric space-and-timeperiodic(orsteady)ﬂows,withsomeadditionalinformationavailableforﬂowsendowedwithsomedegree of spatial symmetrysuch asnegative parityin thevertical direction.These expressionsconsist in space-timeintegralsof auxiliary quantitiesthatsatisfypartialdifferentialequationsoftheadvection–diffusion–reactiontype,whichcanbesolved atleastnumerically,sinceourprocedureallowedforadrasticreductionoftheproblemdimensionalityfromthefullphase spacetotheclassicalphysicalspace.Moreover,ourexpressionsextendtherangeofvalidityofthisapproachtoanyvalueof theStokes’time—awayfrompreviousperturbativelimits—oratleasttothosesituationswherethebasicdynamicalsystem

(2) makes sense andthe (Basset, Oseen, Faxén, Saffman) correctionscan be neglected. As a byproduct, ouranalysis also providesthephysical-spaceparticleprobabilitydensityfunctiononcethesedifferentialequationsaresolved.

Amongthepossibleperspectives,ﬁrstofallwementionthestudyoftheparticleeffective—or“eddy”—diffusivity

[47–51,

42]. This can be performed by means of the multiple-scale method [52–54], and represents the following step in the investigation of higher-order effects in particle advection, including also the possibility of anomalous transport [55,56]. Whenanalyzingthepossibilityofanetdisplacementalsointhehorizontaldirection,aclearconnectionwiththeproblem ofStokes’drift arises

[57–59]

.Finally,we wouldliketoattacktheproblemofparticledispersionfollowinga point-source emission, an issue that hasalreadybeen tackled fortracers[25,60] orslightly-inertial particles

[41]

,andthat shouldbe recastinthepresentframeworkofquasi-neutralbuoyancy.

Acknowledgements

We thankAndrea Mazzino,Paolo Muratore-Ginanneschi, SemyonYakubovich, and Luca Biferalefor usefuldiscussions andsuggestions.ThisarticleisbaseduponworkfromCOSTAction MP1305,supportedby COST(EuropeanCooperationin Science andTechnology).The authorswere partiallysupported by CMUP(UID/MAT/00144/2013),whichis fundedby FCT (Portugal)withnational(MEC)andEuropeanstructuralfunds(FEDER), underthepartnershipagreementPT2020;andalso byProjectSTRIDENORTE-01-0145-FEDER-000033,fundedbyERDFNORTE2020.

Appendix A. Calculationdetails

Theﬁrstequationtoattackis

(11a)

.Thankstothefulldecouplingintheoperator

(8a)

,wecansolveitthroughvariable separation: p(0)

(

x

,

v

,

t

)

=

σ

(

y

)

q(0)

(

x

,

t

)

(20)

=⇒

1 q(0)

₍

_x

_,

_t

₎

M

(

x

,

t

)

q (0)

₍

_x

_,

_t

₎

₌

_c

₌

τ

−1

σ

(

y

)

N

(

y

)

σ

(

y

)

(21)

Lookingattheright-hand-sideequalityin

(21)

integratedonthecovelocityspace,weget:

d y cσ

(

y

)

=

τ

−1

d y

∂

y

· [

y

σ

(

y

)

+

K

τ

−1

∂

y

σ

(

y

)

] =

0

=⇒

c

=

0

=⇒

σ

(

y

)

= (

2

π

K

/

τ

)

−d/2e−y2τ/2K(chosen with unit normalization) (22)

Fromthecorrespondingleft-hand-sideequality,wededuceanadvection–diffusionequationinphysicalspace:

∂

tq(0)

(

x

,

t

)

+

u

(

x

,

t

)

· ∂

xq(0)

(

x

,

t

)

−

D

∂

x2q(0)

(

x

,

t

)

=

0 (23) Forfutureuse,weintroducethefully-symmetricpolynomials(equivalenttomultivariated-dimensionalHermite polynomi-als,andwith

+

S

denotingthesymmetrizationprocessofanytensoronitsfreeindices):

Ci j

≡

yiyj

−

K

τ

−1

δ

i j

,

Ai jk

≡

yiyjyk

−

K

τ

−1

(

yi

δ

jk

+

S

)

Bi jkl

≡

yiyjykyl

−

K

τ

−1

(

yiyj

δ

kl

+

S

)

+

K

2

τ

−2

(δ

i j

δ

kl

+

S

)

togetherwiththeGaussianweight

σ

(

y

)

,theyenjoytherelations:

N

(

y

)

σ

(

y

)

=

0

,

N

(

y

)

[

yi

σ

(

y

)

] = −

yi

σ

(

y

) ,

N

(

y

)

[

Ci j

σ

(

y

)

] = −

2

Ci j

σ

(

y

)

N

(

y

)

[

Ai jk

σ

(

y

)

] = −

3

Ai jk

σ

(

y

) ,

N

(

y

)

[

Bi jkl

σ

(

y

)

] = −

4

Bi jkl

σ

(

y

)

d y yi

σ

(

y

)

=

d y

Ci j

σ

(

y

)

=

d y

Ai jk

σ

(

y

)

=

d y

Bi jkl

σ

(

y

)

=

0 along with

d y

σ

(

y

)

=

1

,

and

d y y

⊗

y

σ

(

y

)

=

K

τ

−1I

(7)

Bymakinguseoflower-orderresults,wecannowproceedtosolvethesystem

(11)

recursively,startingfrom

(11b)

:

[

M

(

x

,

t

)

−

τ

−1

N

(

y

)

]

p(1)

(

x

,

y

,

t

)

= −[

y

· ∂

x

+

J

τ

−1z

(

x

,

t

)

· ∂

y

]

p(0)

(

x

,

y

,

t

)

=

σ

(

y

)

yi

[−∂

xi

+

J K

−

1_z

i

(

x

,

t

)

]

q(0)

(

x

,

t

)

TheresolutionpassesthroughaprocessofHermitianization(verycloselyrelatedtothesecond-quantizationalgorithm

[35]

). ItconsistsinrewritingtheunknownastheproductbetweentheGaussianweightandanexpansioninapowerseriesin y

uptotheorderinquestion, inthiscasetheﬁrst,withspace–time-dependentprefactors(notice thatin

(20)

anexpansion uptoorder0,i.e.noexpansionatall,appeared):

p(1)

(

x

,

y

,

t

)

=

σ

(

y

)

[

q(1)

(

x

,

t

)

+

yir(1)_i

(

x

,

t

)

]

(24)

=⇒

[∂

t

+

u

· ∇ −

D

∇

2

]

q(1)

=

0

[∂

t

+

u

· ∇ −

D

∇

2

+

τ

−1

]

r(1)_i

= −∇

iq(0)

+

J K

−1ziq(0) (25) Resolutionof

(11c)

(for

I =

2):

[

M

(

x

,

t

)

−

τ

−1

N

(

y

)

]

p(2)

(

x

,

y

,

t

)

= −[

y

· ∂

x

+

J

τ

−1z

(

x

,

t

)

· ∂

y

]

p(1)

(

x

,

y

,

t

)

+

J

u

(

x

,

t

)

· ∂

xp(0)

(

x

,

y

,

t

)

=⇒

p(2)

(

x

,

y

,

t

)

=

σ

(

y

)

[

q(2)

(

x

,

t

)

+

yir_i(2)

(

x

,

t

)

+

Ci j

s(2)_{i j}

(

x

,

t

)

]

(26)

=⇒

⎧

⎪

⎨

⎪

⎩

[∂

t

+

u

· ∇ −

D

∇

2

]

q(2)

=

J

u

· ∇

q(0)

−

K

τ

−1

∇

ir(1)_i

[∂

t

+

u

· ∇ −

D

∇

2

+

τ

−1

]

ri(2)

= −∇

iq(1)

+

J K

−1ziq(1)

[∂

t

+

u

· ∇ −

D

∇

2

+

2τ−1

]

si j(2)

= −∇

ir(1)j

+

J K

−1zir(1)j (27) Resolutionof

(11c)

(for

I =

3):

[

M

(

x

,

t

)

−

τ

−1

N

(

y

)

]

p(3)

(

x

,

y

,

t

)

= −[

y

· ∂

x

+

J

τ

−1z

(

x

,

t

)

· ∂

y

]

p(2)

(

x

,

y

,

t

)

+

J

u

(

x

,

t

)

· ∂

xp(1)

(

x

,

y

,

t

)

=⇒

p(3)

(

x

,

y

,

t

)

=

σ

(

y

)

[

q(3)

(

x

,

t

)

+

yiri(3)

(

x

,

t

)

+

Ci j

s (3) i j

(

x

,

t

)

+

Ai jk

a (3) i jk

(

x

,

t

)

]

(28)

=⇒

⎧

⎪

⎨

⎪

⎩

[∂

t

+

u

· ∇ −

D

∇

2

]

q(3)

=

J

u

· ∇

q(1)

−

K

τ

−1

∇

ir(2)i

[∂

t

+

u

· ∇ −

D

∇

2

+

τ

−1

]

ri(3)

= −∇

iq(2)

+

J K

−1ziq(2)

+

J

u

· ∇

r(1)i

−

K

τ

−1

∇

j

(

s(2)ji

+

s (2) i j

)

[∂

t

+

u

· ∇ −

D

∇

2

+

2τ−1

]

s(3)_{i j}

= . . .

[∂

t

+

u

· ∇ −

D

∇

2

+

3τ−1

]

a(3)_{i jk}

= . . .

(29) Resolutionof

(11c)

(for

I =

4):

[

M

(

x

,

t

)

−

τ

−1

N

(

y

)

]

p(4)

(

x

,

y

,

t

)

= −[

y

· ∂

x

+

J

τ

−1z

(

x

,

t

)

· ∂

y

]

p(3)

(

x

,

y

,

t

)

+

J

u

(

x

,

t

)

· ∂

xp(2)

(

x

,

y

,

t

)

=⇒

p(4)

(

x

,

y

,

t

)

=

σ

(

y

)

[

q(4)

(

x

,

t

)

+

yir_i(4)

(

x

,

t

)

+

Ci j

s(4)_{i j}

(

x

,

t

)

+

Ai jl

a(4)_{i jl}

(

x

,

t

)

+

Bi jkl

b(4)_{i jkl}

(

x

,

t

)

]

(30)

=⇒

⎧

⎪

⎨

⎪

⎩

[∂

t

+

u

· ∇ −

D

∇

2

]

q(4)

=

J

u

· ∇

q(2)

−

K

τ

−1

∇

ir_i(3)

[∂

t

+

u

· ∇ −

D

∇

2

+

τ

−1

]

r(4)_i

= . . .

[∂

t

+

u

· ∇ −

D

∇

2

+

2τ−1

]

s(4)_{i j}

= . . .

[∂

t

+

u

· ∇ −

D

∇

2

+

3τ−1

]

a(4)_{i jk}

= . . .

[∂

t

+

u

· ∇ −

D

∇

2

+

4τ−1

]

b(4)_{i jkl}

= . . .

(31)

Notethat,forourpurpose,in

(29)

weonlyneedtoinvestigateq(3)andr(3),andin

(31)

onlyq(4).Itisalsoworthunderlining thatq(x

,

t)

=

∞_I=₀q(I)

₍

_x

_,

_t)_,_but_the_equations_for_the_q(I)_’s _necessarily_imply_the _parallel_resolution_of_the_ones_for_the r(•)’sands(•)’stoformaclosedsystemandthustocomputetheterminalvelocity.

The overall normalization of the phase-spacedensity p corresponds to an integration on the whole covelocity space (eitherintheoriginal coordinatev orintherescaled one y,whichisindifferentbecauseoftheappearanceofaJacobian) andonthespatialperiodicitycell,foranytime:

P dx

Rd dv p

(

x

,

v

,

t

)

=

1

=

P dx q

(

x

,

t

)

(32)

=⇒

P dx

Rd d y p(I)

(

x

,

y

,

t

)

= δ

_I0

=

P dx q(I)

(

x

,

t

)

(8)

For whatconcerns theinitial conditionsof p, they are morediﬃcultto implement, nevertheless itispossible toimpose them on q(0) _and_q(1)_._Indeed, _these_two _scalar _ﬁelds_satisfy _the_unforced _{advection–diffusion} _equations ₍₂₃₎ _and₍₂₅_a),

whoseuniqueperiodicsolutions(theoneweareinterestedin)aretheconstants.Thetwoexactvaluesoftheconstants—the inverseofthephysicalvolumeandzero,respectively—aredictatedby thespatialnormalization

(32)

andbythecovelocity one

(22)

:

q(0)

(

x

,

0

)

=

−d

=

q(0)

(

x

,

t

) ,

q(1)

(

x

,

0

)

=

0

=

q(1)

(

x

,

t

)

(33)

Notethata transportpropertysuchasw cannotdependontheinitialconditions,whichareactually forgottenduetothe diffusivetermintheoperator

M

.Inotherframeworkswherethisindependenceisapriorinotmet,theymustbetakenas uniformorotherwiseaveragedupon.Thispointisstrictlyrelatedtothefactthatweneglectanypossibletransientdecayin thephase-spacedensity,andweonlyfocusonitslong-termbehaviorwhichinﬂuencestheterminalvelocity.Thissteadyor periodicbehaviorofp(x

,

v

,

t)isduetothesteady/periodiccharacteroftheﬂuidﬂowu

(

x

,

t),whichistheonlynon-constant drivingagentintheevolutionequation

(3)

.

Keepingintoaccounttheexpansionsofthetermsmakingup p startingfrom

(20)

,deﬁnition

(5)

translatesinto:

w(0)

=

T

0 dt

T

P dx

Rd d y u

(

x

,

t

)

p(0)

(

x

,

y

,

t

)

=

T

0 dt

T

P dx u

(

x

,

t

)

q(0)

(

x

,

t

)

=

0 (34) w(1)

=

T

0 dt

T

P dx

Rd d y

[

u

(

x

,

t

)

p(1)

(

x

,

y

,

t

)

+

y p(0)

(

x

,

y

,

t

)

]

=

T

0 dt

T

P dx u

(

x

,

t

)

q(1)

(

x

,

t

)

=

0 (35) w(I)

=

T

0 dt

T

P dx

Rd d y

{

u

(

x

,

t

)

[

p(I)

−

J

p(I−2)

](

x

,

y

,

t

)

+

y p(I)

(

x

,

y

,

t

)

}

=

T

0 dt

T

P dx

{

u

(

x

,

t

)

[

q(I)

−

J

q(I−2)

](

x

,

t

)

+

K

τ

−1r(I−1)

(

x

,

t

)

}

(36)

(for

I ≥

2).Thevanishingofexpressions

(34)

and

(35)

isdueto

(33)

,intheformercasecoupledwith

(1)

.Becauseof(29a) and(27b)(i.e.q(3)

₍

_x

_,

_t)

₌

_0),_one_sees_that_also _w(3)

₌

_0,_and_similarly_for_all_odd

_I

_’s_in

₍₃₆₎

_by_induction.

Therelevantequationsfromthesystems

(25)

–(31)havealreadybeenreportedin

(15)

and

(16)

.Itisparticularlyuseful towritedownthetemporalevolutionofthefollowingspatialintegrals,arisingfrom(25b)and(29b)respectively:

(∂

t

+

τ

−1

)

P dx r(1)_i

(

x

,

t

)

=

J K

−1

τ

gi (37)

(∂

t

+

τ

−1

)

P dx r(3)_i

(

x

,

t

)

=

J K

−1

P dx ui

(

x

,

t

)

q(2)

(

x

,

t

)

(38)

Atemporalintegrationof

(37)

allowsustorecast

(36)

for

I =

2 intotheform

(13)

;asimilarmanipulationof

(38)

for

I =

4 leadsto

(14)

.

It is easy to show that parallel ﬂows, i.e. ﬂuid motions in which the velocity points always and everywhere in the same direction(say x1), donotaffectsedimentationifthey aresteady/periodic andincompressible—implying that u does

not dependon x1 itself—at leastat workingorder. Indeed,forsuch a class of ﬂows, all theadvective terms ofthe type u

(

x

,

t)

· ∇

p(x

,

v

,

t)vanish(alsowhenactingonother statisticalquantitiesbasedon p),becausenolong-termdependence onthespatialcoordinatex1 alignedwithu canarise—exceptforpossibletransientbehaviorsthatcanbeneglectedforour

scope.Asaconsequence,onecaneasilyprovethatallthefollowingquantitiesderivedfrom

(15)

and

(16)

vanish:

∇ ·

r(1)

(

x

,

t

)

=

q(2)

(

x

,

t

)

= ∇∇ :

s(2)

(

x

,

t

)

= ∇ ·

r(3)

(

x

,

t

)

=

q(4)

(

x

,

t

)

=

0

(9)

A.1. Proofoftheexpressionfortheterminal-velocitycorrection

Letusﬁrstlyprovetherewriting

(5)

ofthefullterminalvelocity,byexploitingthedeﬁnitionofthephase-spacedensity asanaverageofDiracdelta’soneveryrandomfactor:

p

(

x

,

v

,

t

)

≡ δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν (39)

Letus remindthat both

μ

(t)

and

ν

(t)

are whitenoises, meaning that the valuesassumedat a certain time instant are completelyuncorrelatedfromthe onesassumedatatime instantimmediatelyfollowing.Moreover,byinvokingcausality, one infersthat theinstantaneous valuesofthenoisesattime t caninﬂuencetheparticledynamicsonly atfuturetimes, butnotcomputedatt itself.Thismeansthat

μ(

t

)δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν

=

0

=

ν(

t

)δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν (40)

sincetheaveragessplitthankstotheuncorrelation,andbothnoiseshavezeromean. Forthesakeofnotationalsimplicity,let

t,x,v

• ≡

T

0 dt

T

P dx

Rd dv

•

Asaconsequence,using

(40)

and

(39)

,

V(

t

)

+ β

u

(

X (

t

),

t

)

+

√

2

D

μ(

t

)

p

=

t,x,v

[V(

t

)

+ β

u

(

X (

t

),

t

)

+

√

2

D

μ(

t

)

]δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν

=

t,x,v

[

v

+ β

u

(

x

,

t

)

]δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν

+

0

=

t,x,v

[

v

+ β

u

(

x

,

t

)

]

p

(

x

,

v

,

t

)

thankstotheproperty ofthedeltathatallows forthesubstitution

(

X (

t),

V(

t))

→ (

x

,

v

)

,andtothefact thattheselatter coordinatesareindependentofthenoises.

Letusnowcomputethedeviation

(7)

oftheterminalvelocity

(5)

fromitsbarevalue

(6)

:

Z =

t,x,v

[

v

+ β

u

(

x

,

t

)

]

p

(

x

,

v

,

t

)

− (

1

− β)

τ

g

=

t,x,v

[

v

+ β

u

(

x

,

t

)

− (

1

− β)

τ

g

]δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν

=

t,x,v

[V(

t

)

+ β

u

(

X (

t

),

t

)

− (

1

− β)

τ

g

]δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν

havingused

(32)

.Exploitingthesecondequationof

(2)

,thisrewritesas:

Z =

t,x,v

[

u

(

X (

t

),

t

)

+

√

2κν

(

t

)

−

τ

˙V(

t

)

]δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν

=

t,x,v u

(

x

,

t

)

p

(

x

,

v

,

t

)

+

0

−

τ

t,x,v

˙V(

t

)δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν

after making useof (40).Keeping (4)in mind, thedemonstration is complete ifwe prove that the addendinvolving

˙V

doesnot give anycontribution.This isachieved throughintegrations by parts (withvanishing ofthe integralsof deriva-tives,becauseofperiodicityandrapiddecayatinﬁnity)andchain-rulederivation,andtheexploitationofthematerialand functionalderivatives—d andD,respectively—andofthetranslationalinvarianceofthedelta’s:

(10)

t,x,v

˙V(

t

)δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν

=

t,x,v

d dt

[V(

t

)δ(

x

− X (

t

))δ(

v

− V(

t

))

] − V(

t

)

d dt

[δ(

x

− X (

t

))δ(

v

− V(

t

))

]

μ,ν

=

t,x,v

∂

t

V(

t

)δ(

x

− X (

t

))δ(

v

− V(

t

))

μ,ν

−

t,x,v

V(

t

)

δ(

v

− V(

t

)) ˙

X (

t

)

·

D D

X (

t

)

δ(

x

− X (

t

))

+ δ(

x

− X (

t

)) ˙

V(

t

)

·

D D

V(

t

)

δ(

v

− V(

t

))

μ,ν

=

0

+

t,x,v

V(

t

)

δ(

v

− V(

t

)) ˙

X (

t

)

· ∂

∂

x

δ(

x

− X (

t

))

+δ(

x

− X (

t

)) ˙

V(

t

)

· ∂

∂

v

δ(

v

− V(

t

))

μ,ν

=

t,x,v

∂

x

· ˙X (

t

)δ(

x

− X (

t

))δ(

v

− V(

t

))

V(

t

)

μ,ν

+

∂

v

· ˙V(

t

)δ(

x

− X (

t

))δ(

v

− V(

t

))

V(

t

)

μ,ν

=

0

+

0

2

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