LecturesintheLatticeBoltzmannMethod

PauloCesarPhilippi1Professor

MechanicalEngineeringDepartmentFederalUniversityofSantaCatarina88040900FlorianopolisSCBrazil

1philippi@lmpt.ufsc.br

PrefaceThissetoflecturesisintendedtobeafirstcourseintheLatticeBoltzmannMethod(LBM)forundergraduateandgraduatestudents,whowanttounderstandthetheoreticalfundamentalsbehindthemethod.Historically,thelatticeBoltzmannequation(LBE)hasitsoriginfromlatticegasBooleanmodelsandwasintroducedbyMcNamaraandZanetti[1],in1988,byreplacingtheBooleanvariablesinthediscretecollisionpropagationequationsbytheirensembleaverages.Higuera and Jimenez [2] proposed a linearization of the term derived from the Booleanmodels,recognizingthatthisfullformwasunnecessarilycomplexwhenthemainpurposewastoretrievethehydrodynamicequations.Following this lineof reasoning,Chen et al. [3] suggested replacing the collision termby asinglerelaxationtimeterm,followedbyQianetal.[4]andChenetal.[5],who introducedamodelbasedonthecelebratedkinetictheory ideaofBhatnagar,Gross,andKrook(BGK),[6],but adding restparticles and retrieving the correct incompressibleNavierStokesequations,withthirdordernonphysicaltermsinthelocalspeed,u.TheBGKcollisiontermdescribestherelaxationofthedistributionfunctiontoanequilibriumdistribution.Thisdiscreteequilibriumdistributionwassettledbywriting itasasecondorderpolynomialexpansionintheparticlevelocityi,withparametersadjustedtoretrievethemassdensity,thelocalvelocity,andthemomentumfluxequilibriummoments,whicharenecessaryconditionsforsatisfyingtheNavierStokesequations.Until some years ago LBMwasmostly restricted to isothermal flows. Thermal lattice BGKschemes included higherorder terms in the equilibrium distribution function, requiring toincrease the lattice dimensionality, i.e., the number of vectors in the finite and discretevelocity set i , i= 0, . . . ,b 1. Simulation of the thermal lattice Boltzmann equationwashamperedbynumericalinstabilitieswhenthelocalvelocityincreases.ThefirstthermallatticeBoltzmannmodelswere introduced inabout1990andthereareseveralreasonsthatmaybeconjectured for their failure in handling nonisothermal flows. Thermal lattice Boltzmannmodelswere firstly treatedbyAlexanderetal. [7],whoextended theQianetal. [4]secondorderequilibriumdistribution toa thirdordermodel for solving some thermohydrodynamicproblems,resultinginagoodagreementwhencomparedwithanalyticalsolutions.McNamaraandAlder[8],foundasetof13and26restrictionsthatthisexpansionmustsatisfytoretrievethe correct advectiondiffusion macroscopic equations, respectively, in two and threedimensions.Nonlineardeviations in themomentum and energy equations, in themodelofAlexander and coworkers,were found by Chen et al. [9],who introduced a fourth orderpolynomial expansion into the equilibriumdistribution, fitting adjustableparameters. Theseauthors used combinations of square lattices for satisfying the restrictions imposed by theChapmanEnskoganalysisandfounda16velocitylatticeintwodimensionsandalatticewith41velocitiesinthreedimensions.With theexceptionofMcNamaraandZanettisunconditionallystableLBE, [1],all theabovemodelshavestabilityissues.In these studies the equilibrium distribution was written as finite expansions in the localvelocitywithfreeparametersthatwereadjustedtosatisfysomemainrestrictionstoretrievethefulladvectiondiffusionequations.Consequently,thereisnoformallinkconnectingtheLBEtotheBoltzmannequation.This connectionhasbeen firstestablishedbyHeand Luo [10]whodirectlyderived the LBEfromtheBoltzmannequation forsomewidelyknownlattices D2Q9,D2Q6,D2Q7,D3Q27bythediscretizationofthevelocityspace,usingtheGaussHermiteandGaussRadauquadrature.Excludingtheabovementionedlattices,thediscretevelocitysetsobtainedbythiskindofquadraturedonotgenerateregularspacefillinglattices.

Shan et al.[11] and Philippi et al.[12], in 2006, reopened the prospect of using the latticeBoltzmannmethodtosimulatenonisothermaland/orhighKnudsennumberflowsthroughthedirectresolutionoftheBoltzmannequation,whentheyestablishedasystematiclinkbetweenthe Kinetic Theory of Gases and the lattice Boltzmann methods (LBM), determining thenecessary conditions for the discretization of the velocity space in different orders of theKnudsen number. The lattices obtained through themethod proposed by these authors, aprescribedabscissasquadrature,provedtobestableinflowsoverawiderangeofparametersbytheuseofthehighorder latticeBoltzmannschemes, leadingtovelocitysetswhich,whenused in a discrete velocity kinetic scheme, ensures accurate recovery of the highorderhydrodynamic moments and assuring increasingly higher order of isotropy of the latticetensors.It is importanttostressthatthepurposeofacontinuouskineticequation isnottosolvethefullBoltzmannequation itself,which inmost cases is, in fact,unknown,but to consistentlyretrieve themacroscopicequationsdescribing thebehaviorofaphysical system.Therefore,numerically solvingakineticequation, the frameworkofLBM,must, firstly,be thoughtasamethodforsolvingaphysicalproblem,withsomeadvantageswithrespecttoclassicalCFDduetoitsintrinsicLagrangiannature.Inthereversesense,duetoitsmesoscopicnature,akineticequation enables to reveal, or to put in evidence, the influence of a number ofmolecularprocessesonthemacroscopicbehaviorofaphysicalsystem.With the advent of ever faster computers, mesoscopic particle models, together withnumerical simulations, provide scientists with a very powerful tool to investigate newchallenges in complex flow problems, both at the fundamental and the applied levels. Therecent years have witnessed many promising advances with the development of newtheoreticalframeworksandapplicationsofthemesoscopicparticlemodels.

Nowadays LBM subjects range from the fundamentals of the kinetic theory and quantumtransport toappliedsubjects, includingsomeonessuchassuchas:magnetohydrodynamics;quantum lattice Boltzmannmodels; unstructured lattice Boltzmann equation (ULBE); nonNewtonian flows;moleculardynamics and latticeBoltzmann approaches for the analysisofprecursor films;droplet spreadingon solid surfaces;biopolymer translocation;microfluidics;ballistic aggregation and fragmentation; granular flows; combustion; hybrid methods thatcombine the latticeBoltzmannmodelwith traditional finitedifference techniques;boundaryconditions in theLBM framework;highperformancecomputingwithgraphicprocessingunit(GPU).

References

[1]G.R.McNamaraandG.Zanetti,Phys.Rev.Lett.61,2332,1988.[2]F.J.HigueraandJ.Jimenez,Europhys.Lett.9,663,1989.[3]S.Chen,H.Chen,D.Martinez,andW.H.Matthaeus,Phys.Rev.Lett.67,3776,1991.[4]Y.H.Qian,D.dHumires,andP.Lallemand,Europhys.Lett.17,479,1992.[5]H.Chen,S.Chen,andW.H.Matthaeus,Phys.Rev.A45,R5339,1992.[6]P.Bhatnagar,E.Gross,andM.Krook,Phys.Rev.94,511,1954.[7]F.J.Alexander,S.Chen,andJ.D.Sterling,Phys.Rev.E47,R2249,1993.[8]G.McNamaraandB.Alder,PhysicaA194,218,1993.[9]Y.Chen,H.Ohashi,andM.Akiyama,Phys.Rev.E50,2776,1994.[10]X.HeandL.S.Luo,Phys.Rev.E56,6811,1997.[11]P.C.Philippi,L.A.HegeleJr.,L.O.E.dosSantos,andR.Surmas,Phys.Rev.E73,56702,2006.[12]X.Shan,X.F.Yuan,andH.Chen,J.FluidMech.550,413,2006.

TableofcontentsLecture01TheBoltzmannequationLecture02TheequilibriumsolutionLecture03LBMdiscretizationLecture04BoltzmannequationfornonidealfluidsLecture05PropertiesoftherepulsiontermLecture06MacroscopicequationsLecture07PhasetransitionsLecture08VelocitydiscretizationLecture09AttributingvolumetothemoleculesLecture10LBMvariables

Lecture 1: The Boltzmann equation

Ludwig Boltzmann (1844 1906)

Matter is composed by molecules. The kinetic theory tries to understand themacroscopic properties of uids from the properties of their molecules: molec-ular mass, electrical properties, shape parameters, the mean free path and soon. Although the atomic theory of matter begun in Greece with Leucippus1 ,Democritus, and Epicurus2 , the modern kinetic theory was born with the worksof Daniel Bernouilli3 who theoretically demonstated the Boyles law, PV = cte,under a constant temperature and showed that the pressure is proportional to

1Leucippus was one of the earliest Greeks to develop the theory of atomism the ideathat everything is composed entirely of various imperishable, indivisible elements called atoms which was elaborated in greater detail by his pupil and successor, Democritus. Aristotelesand Theophrastus explicitly credit Leucippus with the invention of Atomism. Around 440 BCLeucippus founded a school at Abdera, which his pupil, Democritus, was closely associatedwith. His fame was completely overshadowed by that of Democritus, who systematized hisviews on atoms. Extracted from Wikipedia.

2Epicurus (341 BC 270 BC) was an ancient Greek philosopher and the founder of theschool of philosophy called Epicureanism. Only a few fragments and letters remain of Epi-curuss 300 written works. Much of what is known about Epicurean philosophy derives fromlater followers and commentators.For Epicurus, the purpose of philosophy was to attain the happy, tranquil life, characterized

by ataraxia, peace and freedom from fear, and aponia, the absence of pain, and by living aself-su cient life surrounded by friends. He taught that pleasure and pain are the measuresof what is good and evil, that death is the end of the body and the soul and should thereforenot be feared, that the gods do not reward or punish humans, that the universe is innite andeternal, and that events in the world are ultimately based on the motions and interactions ofatoms moving in empty space. Extracted from Wikipedia.

3Daniel Bernoulli (1700 1782) was a Dutch-Swiss mathematician and was one of themany prominent mathematicians in the Bernoulli family. He is particularly remembered forhis applications of mathematics to mechanics, especially uid mechanics, and for his pioneeringwork in probability and statistics. Bernoullis work is still studied at length by many schoolsof science throughout the world. Extracted from Wikipedia.

1

the square of the mean speed of the molecules. John James Waterston4 showedthat the temperature of a gas is directly related to the kinetic energy of the mole-cules. Rudolf Clausius5 introduced the concept of the mean free path. JamesClerk Maxwell6 introduced the velocity distribution function and established atransport theory based on the molecular properties of a gas. At the end of 19th

century, the kinetic theory received a strong impulsion with the work of LudwigBoltzmann7 who establised an evolution equation for the distribution functionand demonstrated the second law of thermodynamics for a system of particles.In the following the Boltzmann equation is derived.Let f(!x ;! ; t)!x! be the number of particles with velocities between!

and! +

! that are found at time t in the elementary volume !x between

!x e !x +!x . Consider now the development of f(!x +!x ;! +! ; t+t)in a Taylor series around (!x ;! ; t),

4John James Waterston (1811 1883) was a Scottish physicist and a pioneer of the kinetictheory of gases. A Waterstons paper submitted to the Royal Society was rejected. Some yearsafter Waterstons death, Lord Rayleigh (Secretary of Royal Society at that time) managed todig it out from the archives of the Royal Society. Finally, Watersons paper was published inthe Philosophical Transactions of the Royal Society in 1892. Extracted from Wikipedia.

5Rudolf Julius Emanuel Clausius (1822 1888), was a German physicist and mathematicianand is one of the central founders of the science of thermodynamics. By his restatement ofSadi Carnots principle known as the Carnot cycle, he put the theory of heat on a truer andsounder basis. His most important paper, On the mechanical theory of heat, published in1850, rst stated the basic ideas of the second law of thermodynamics. In 1865 he introducedthe concept of entropy. Extracted from Wikipedia.

6James Clerk Maxwell (1831 1879) was a Scottish physicist and mathematician. His mostprominent achievement was formulating classical electromagnetic theory. Maxwell also helpedto develop the MaxwellBoltzmann distribution. Maxwell is considered by many physiciststo be the 19th-century scientist who had the greatest inuence on 20th-century physics. Hiscontributions to the science are considered by many to be of the same magnitude as thoseof Isaac Newton and Albert Einstein. In the millennium poll a survey of the 100 mostprominent physicists Maxwell was voted the third greatest physicist of all time, behind onlyNewton and Einstein. On the centennial of Maxwells birthday, Einstein himself describedMaxwells work as the "most profound and the most fruitful that physics has experiencedsince the time of Newton." Extracted from Wikipedia.

7Ludwig Eduard Boltzmann (1844 1906) was an Austrian physicist famous for his foundingcontributions in the elds of statistical mechanics and statistical thermodynamics. He wasone of the most important advocates for atomic theory at a time when that scientic modelwas still highly controversial. Boltzmanns most important scientic contributions were inkinetic theory, including the Maxwell-Boltzmann distribution for molecular speeds in a gas. Inaddition, Maxwell-Boltzmann statistics and the Boltzmann distribution over energies remainthe foundations of classical statistical mechanics. They are applicable to the many phenomenathat do not require quantum statistics and provide a remarkable insight into the meaning oftemperature. Much of the physics establishment did not share his belief in the reality ofatoms and molecules a belief shared, however, by Maxwell in Scotland and Gibbs in theUnited States; and by most chemists since the discoveries of John Dalton in 1808. He hada long-running dispute with the editor of the preeminent German physics journal of his day,who refused to let Boltzmann refer to atoms and molecules as anything other than convenienttheoretical constructs. Only a couple of years after Boltzmanns death, Perrins studies ofcolloidal suspensions (19081909), based on Einsteins theoretical studies of 1905, conrmedthe values of Avogadros number and Boltzmanns constant, and convinced the world that thetiny particles really exist. Extracted from Wikipedia.

2

Figure 1: Attraction-repulsion eld around molecules.

f(!x +!x ;! +! ; t+t) = f(!x ;! ; t) + (@tf)t+(rxf) !x +

rxf ! + :::; (1)or, dividing by t with t! 0,

(@tf) +! (rxf) +!g

rxf= lim

t!0f(!x +!x ;! +! ; t+t) f(!x ;! ; t)

t;(2)

where !g = !

t is an acceleration.

Therefore, if at time t + t the particles that were in !x , with velocity !are, presently, in !x +!x transported with ! and with the velocity ! +!due to the acceleration !g , the above expresion is null,

@tf +! (rxf) +!g (rf) = 0: (3)

Nevertheless, there are (@tf)+col

!x! t particles in the volume !x , whichacquired the velocity

! in the course of the time interval t because they

have collided with other particles and some of the particles in !x , lost, in thistime interval, the velocity

! due also to the collisions they suered with other

particles.So

3

@tf +! (rxf) +!g (rf) = (@tf)+col (@tf)col : (4)

In Boltzmanns model, the particles behave like billard balls, each collisionproducing a sharp change in their velocity. A molecule may be thought as amaterial point with a force eld around it. This force eld has a strong repulsionkernel. When two molecules have a frontal collision, they will experience thisstrong repulsion when their electronic orbits begin to intercept with themselves.Around this repulsion kernel, molecules have a soft attraction eld produced byelectrostatic forces (Figure 1) .

Considering to be a length related to a parameter possible to be identiedwith the molecular diameter, the two particles will experience a repulsion forcewhen their centers are at a distance equal to (Figure 2). At this distancethe attraction and repulsion elds canceal themselves but the kinetic energy ofthe bullet with respect to the target particle is not null and the center of thebullet particle will penetrate into the -sphere, where it will be frained alongthe radial coordinate.

Figure 2: Two particles will experience a repulsion force when their centers areat a distance less or equal to .

This radial frainage will produce a deviation in the bullet trajectory depictedin Figure (3). Using the label 1 for the bullet particle, the relative velocity of thebullet with respect to the target before the collision is !g = ! 1 ! when thetarget and bullet have the velocities

! and

! 1, respectively, or,

!g 0 = ! 01! 0

4

when, as it is the case shown in Figure (3), the target and bullet have thevelocities

! 0 and

! 01.

Figure 3: Trajectory deviation of the bullet particle produced by the radialfrainage due to the repulsion forces.

The collision term takes account of these losses and gains in the particlepopulations due to radical changes of the velocity

! happening in a very small

time interval.Considering that our reference is on a single target particle with velocity

! ,

this particle will change its velocity to! 0 if it is hit by a bullet particle with

velocity! 1 during the time interval t.

Consider a single bullet particle with velocity! 1 and

g = j!g j =! 1 ! : (5)

Let the z-ordinate of our reference frame be oriented along the direction givenby the vector !g . The bullet particle will hit the target in the time interval twhen: a) the impact factor b is smaller than (Figure 4); b) it is found inthe elementary volume bd"dbgt in this time interval (Figure 5), where " is theazimuthal angle.Inside the volume bd"dbgt there are

5

Figure 4: A bullet particle will hit the target when the impact factor b is smallerthan

f!x ;! 1; t d! 1| {z } bullet part. per unit vol..

bd"dbgt| {z } ,volume in cyl. coord..

(6)

bullet particles.

Therefore, since there are f!x ;! 1; t! !x target particles inside !x ,

during the time interval t, the number of target particles that loss the velocity! , during t will be,

(@tf)coltd

! !x

=

Z! 1

Z 2"=0

Z b=0

f!x ;! ; t f !x ;! 1; t bd"dbgd! 1t! !x :(7)

considering all the possible! 1 in the velocity space and all the impact para-

meters b = b (!g ) in all the azimuthal planes ",or

6

Figure 5: Elementary volume in cylindrical coordinates.

(@tf)col =

Z! 1

Z 2"=0

Z b=0

f!x ;! ; t f !x ;! 1; t bd"dbgd! 1 (8)

In the same way,

(@tf)+col =

Z! 1

Z 2"=0

Z b=0

f!x ;! 0; t f !x ;! 01; t bd"dbg0d! 1 (9)

where! 0;! 01 must be such that

! 0;! 01 !

! ;! 1 after the collision.It is shown

in Appendix A that g0 = g. The Appendix also show how to calculate! 0;! 01

from known vaules of! ;! 1.

In its nal form the Boltzmann equation is written as,

@tf +! (rxf) +!g (rf)

=

0@Z! 1

Z 2"=0

Z b=0

24 f !x ;! 0; t f !x ;! 01; tf!x ;! ; t f !x ;! 1; t

35 bd"dbgd! 11A (10)

The acceleration !g is given by!g = !g (e) +!g (`d) (11)

7

where !g (e) is related to the external body forces and !g (`d) is the attractionforce per unit mass due to the long-range intermolecular interaction. In theBoltzmann original model, developed for rareed gases, there is no long-rangeaction among the molecules, !g (`d) = 0, resulting in a model which equation ofstate is the one for ideal gases, Pv = RT .In fact, in the classical approach, developed for rareed gases, the long-

distance attraction is included into the collision term, the molecules being freeof any interaction potential in the time interval between any two collisions,when their centers are distanced by j!x 1 !x j > . In Appendix B it is shownthat, with some simplications, in the framework of the mean eld theory, theintermolecular long range forces can be considered as in the above treatment.

8

Appendix A: Ballistics of binary collisions 8 ; 9

The idea is to nd the velocities! 0;! 01 of, respectively, the target and bullet

particles before the collision that will retrieve! ;! 1, after the collision. We

then start from! ;! 1 and will nd

! 0;! 01 in terms of the rst.

During the interaction, the target particle velocity !v evolves from ! to ! 0and the bullet particle velocity !v 1 evolves from ! 1 to ! 01 . Newtons secondlaw gives

d!xdt

= !v ; d!x 1dt

= !v 1 (12)and

md!vdt

= @@r

!rr;md!v 1dt

=@

@r

!rr

(13)

where !r = !x !x 1.From Eq. (13),

d!v +!v 1

dt= 0 (14)

which expresses the preservation of momentum during the collision,

! +

! 1 =

! 0 +

! 01 (15)

Now, by multiplying these equations by !v ;!v 1;

d

dt

m2v2 +

m

2v21

= @

@r

!rr !v !v 1

= @@r

!rrd (!x !x 1)

dt

= @

@r

!rrd!rdt

= @

@!r d!rdt

=d

dt(16)

or

d

dt

m2v2 +

m

2v21 +

= 0 (17)

8For a complete treatment of the ballistic of binary collisions see C. CercignaniMathematical Methods in Kinetic Theory, Springer (1995)

9Although important for understanding the collision details, this appendix is not amain requisite for the remaining lectures and may be avoided by the students who aremainly interested in learning LBM. The same comments for the Appendix B and C.

9

which expresses the conservation of the total energy inside a sphere of radius :Outside this sphere = 0 and

2 + 21 = 02 + 021 (18)

Let ! be the unit vector in the direction of ! 0 ! , Figure ();! 0 ! = C! (19)

or

! 0 =

! + C! (20)

or

! 01 =

! 1 C! (21)

Using these relations into Eq.(18)

2 + 21 = 2 + 2C

! ! + C2 + 21 2C

! 1 ! + C2 (22)

or ! 1 !

| {z }

!g

! = C (23)

Therefore

! 0 =

! + (!g ! )! (24)

! 01 =

! 1 (!g ! )! (25)

Subtracting the two above equations

!g 0 = !g 2 (!g ! )! (26)or

(!g 0)2 = (!g )2 4 (!g ! )2 + 4 (!g ! )2 = (!g )2 (27)meaning that the collision does not aect the norm of the relative velocity !g .From Eq. (26)

!g 0 !g| {z }g2 cos

= !g !g| {z }g2

2(!g ! )| {z }g cos

!g ! (28)

or

cos = cos 2 (29)

10

or

= 2 (30)The collision plane is given by ". So, considering the vector !g pointing

along the direction x, the direction given by the unitary vector ! will be givenby its coordinates along the axis x, y and z, Figure (3)

(cos "; sin cos "; sin sin ") (31)

Under a central eld the angular momentum is also preserved (Figure )

gt = d

dt = 2

d

dt= cte = gb (32)

or

dt =2d

gb(33)

Radial and tangential components of the relative velocity.

And from the energy conservation, Eq. (17), in the center of mass system(see Remark 1),

m

4

!v 1 !v 2 +=

m

4

:2+ 2

:

2+

=m

4g2 (34)

11

The two above equations may be rewritten as

m

4

"d

dt

2+ 2

d

dt

2#+ =

m

4g2 (35)

or using dt = 2dgb ,

m

4

"gb

2d

d

2+ 2

gb

2

2#+ =

m

4g2 (36)

m

4

"g2b2

4

d

d

2+g2b2

2

#+ m

4g2 = 0 (37)

dividing by g2b2

2

m

4

"1

2

d

d

2+ 1

#+

2

g2b2

m

4g2= 0 (38)

orm

4

"1

2

d

d

2#+m

4+

2

g2b2

m

4g2= 0 (39)

or 2

g2b2

m4g2

m4

2d

d

2=m

4(40)

or

d

d=gbm1=2

2

vuuut 14

m4 g

2

1

b

2 ()

(41)Integrating this equation between the point where the bullet particle reaches

the sphere , when = arcsinb

and the point of closest approach, where

= 0;

=gbm1=2

2

Z 0

1

2

vuuut 1m4 g

2

1

b

2 ()

d+ arcsin b

(42)

where 0 is the solution ofdd = 0, i.e., the solution of

m

4g2

1

b

2! () = 0 (43)

12

Therefore, for each! , the right hand side of the Boltzmann equation is

evaluated in accordance with the following steps:i) b is an integration variable and so

! 1. Then

!g = ! 1 ! and Eq. (42)can be used for nding the polar angle ;ii) The azimuthal angle " is also an integration variable and Eq. (31) can be

used for nding

! = (cos "; sin cos "; sin sin ") (44)iii) the particle velocities before collision that will give

! and

! 1 after

collision, are Eqs. (24) and (25)

! 0b; ";

! ;! 1

=! + cos ! (45)

! 01b; ";

! ;! 1

=! 1 cos ! (46)

iv) The distribution f can be used for evaluating f!x ;! 0; t f !x ;! 01; t

related to! and to the corresponding values of the integration variables b; ";

! 1:

Remark

(m1 +m2) v0 = m1v1 +m2v2

g = v2 v1 (47)

Therefore when the masses are identical,

v1 =2v0 + g

2

v2 =2v0 g2

(48)

so

1

2mv21 +

1

2mv22 =

m

8

4v20 + g

2 + 4v0g+m

8

4v20 + g

2 4v0g

=m

4

4v20 + g

2

(49)

13

Appendix B: The Boltzmann equation derivedfrom the Liouvillle equation

In a liquid, the mean free path has the same order of magnitude than themolecular diameter, multiple collisions are frequent and long-range interactionsare important. We begin with the Liouville equation.Consider a mechanical system of N particles. Let

fN!x 1;! 1; :::!x N ;! N ; t ; (50)

to be the joint probability of nding, at time t, dt the particle 1 at the position!x 1, d!x 1 with velocity ! 1, d! 1, the particle 2 at the position !x 2, d!x 2 withvelocity

! 2, d

! 2 and so on, until particle N at the position

!x N , d!x N withvelocity

! N , d

! N . The Liouville equation describing the dynamical evolution

of this system is given by

@tfN +

Xi

! i:@!x if

N +Xi

!g i :@! ifN = 0; (51)

where !giis the acceleration due to the force acting on particle i,

!gi= !g e

i+

NXj=1j 6=i

!gij: (52)

Force !g eiis related to the force on particle i due to an external eld and!g

ijis the force on particle i due to its interaction with particle j,

!gij= 1

m

@ (xij)

@!x i ; (53)

where xij = j!x i !x j j and is the potential energy depending, only, on thedistance between particles i and j.Joint probability fN can be integrated in the phase space!x 2;! 2; :::!x N ;! N

to give the marginal probability f1 of nding, at time t, dt the particle 1 atthe position !x 1, d!x 1 with velocity ! 1, d! 1

f1!x 1;! 1; t = Z :::Z fNd!x 2:::d!x Nd! 2:::d! N ; (54)

considering that the probability fN gives a too detailed description of the sys-tem, which is unnecessarily complex, since the dynamical evolution of an arbi-trary, but, single particle, can be a reliable description of the whole mechanicalsystem of particles, when these particles cannot be individually labelled.Integration of Eq. (51) gives for the temporal derivativeZ

:::

Z@tf

Nd!x 2:::d!x Nd! 2:::d! N = @tf1: (55)

14

Consider now the integral

Z:::

Z Xi

! i:@!x if

Nd!x 2:::d!x Nd! 2:::d! N

=

Z:::

Z ! 1:@!x 1f

Nd!x 2:::d!x Nd! 2:::d! N

+

Z:::

Z Xi=2

! i:@!x if

Nd!x 2:::d!x Nd! 2:::d! N

=! 1:@!x 1f

1 +

Z:::

Z Xi=2

! i:@!x if

Nd!x 2:::d!x Nd! 2:::d! N

=! 1:@!x 1f

1 ; (56)

because, each term

! i:@!x if

N = @!x i! ifN

; (57)

for i = 2; :::; n, since! i and

!x i are independent variables. In this manner,when this term is integrated in !x i it gives i) the ux of fN outside the boxwhere the mechanical system is contained, which must be null or ii) something

proportional to fN!x 1;! 1; :: j!x ij ! 1::::!x N ;! N ; t when there is no box

enclosing the system, which must be also null.Proceeding in a similar term with the force term,

Z:::

Z Xi

!gi:@! ifNd!x 2:::d!x Nd! 2:::d! N

=

Z:::

Z Xi

!g ei :@! ifNd!x 2:::d!x Nd! 2:::d! N

+

Z:::

Z Xi;j

!g i;j :@! ifNd!x 2:::d!x Nd! 2:::d! N

So

15

Z:::

Z Xi

!g i :@! ifNd!x 2:::d!x Nd! 2:::d! N

=

Z:::

Z!g e1:@! 1fNd

!x 2:::d!x Nd! 2:::d! N| {z }=!g e1:@! 1f1

+Xi=2

Z:::

Z!g ei :@! ifNd

!x 2:::d!x Nd! 2:::d! N| {z }=0

+X;j=2

Z:::

Z!g 1;j :@! 1fNd

!x 2:::d!x Nd! 2:::d! N

+Xi=2

Xj

Z:::

Z!g

i;j:@! ifNd!x 2:::d!x Nd! 2:::d! N| {z }=0

: (58)

The term

X;j=2

Z:::

Z!g 1;j :@! 1fNd

!x 2:::d!x Nd! 2:::d! N

=NXj=2

@! 1

Z:::

Z!g 1;jf 2

!x 1;! 1;!x 2;! 2; t d!x 2d! 2=|{z}

indistinguishable particles

(N 1)

@! 1

Z:::

Z!g

12f 2!x 1;! 1;!x 2;! 2; t d!x 2d! 2 (59)

After integration, considering f = Nf1 and changing the notation for thetarget and the incident particles, the Liouville equation becomes, for large N,

@tf +! :@!x f +

!g e:@!f

= @!

Z:::

Z!g 12;jf 2

!x ;! ;!x 1;! 1; t d!x 1d! 1= @!

Z:::

Z1

m

@ (j!x 1 !x j)@!x f

2

!x ;! ;!x 1;! 1; t d!x 1d! 1; (60)which is a Boltzmann equation for the distribution function f , with a collisionterm .This collision term will be split in two collision terms10 = sd+ld, where

sd is referred to short distance interactions, j!x 1 !x j < and ld to long10X. He and G. D. Doolen, J. Stat. Phys., 107, 1-2 (2002).

16

range interactions j!x 1 !x j > , being the distance related to the moleculardiameter below which the attraction forces change to a strong repulsion amongthe molecules.Therefore

@tf +! :@!x f +

!g e:@!f = sd +ld; (61)

where

sd = @!

Z Zj!x 1!x j

1

m

@ (j!x 1 !x j)@!x f

2

!x ;! ;!x 1;! 1; t d!x 1d! 1:(63)

Long-range term. Mean eld theory.

Consider, rst, the long-range collision term

ld =@

@!:

Z Zj!x 1!x j>

1

m

@ (j!x 1 !x j)@ (!x ) f

2!x ;! ;!x 1;! 1; t d!x 1d! 1:

(64)By making the assumption that, for j!x 1 !x j > , the molecular chaos

prevails

f2!x ;! ;!x 1;! 1 = f !x ;! ; t f !x 1;! 1; t ff1; (65)

one obtains

ld =@f!x ;! ; t@!

Z Z

j!x 1!x j>1

m

@ (j!x 1 !x j)@!x f

!x 1;! 1; t d!x 1d! 1=

@f!x ;! ; t@!

:

Zj!x 1!x j>

1

m

@ (j!x 1 !x j)@!x n (

!x 1; t) d!x 1

=1

m

@f!x ;! ; t@!

:@

@!xZj!x 1!x j>

(j!x 1 !x j)n (!x 1; t) d!x 1: (66)

17

The integrand in the above equation is the mean eld, i.e., the potentialenergy related to the integrated action of each of the n molecules placed in theneighborhood, on a single molecule at the position !x ,

m (!x ) =

Zj!x 1!x j>

(j!x 1 !x j)n (!x 1; t) d!x 1: (67)

Since the acceleration !g (`d) is the gradient of the potential energy dividedby its mass,

!g (`d) = 1m

@m@!x ; (68)

we get

ld = !g (`d) @f!x ;! ; t@!

; (69)

giving the same long-range term as previously obtained.

18

Appendix C: Short-range collision term derivedfrom Eq. (62) 11

Using the results of Appendix B, the integration of the Liouville equationgives, after dropping out the subscript index for the target particles and usingthe subscript 1 for the integration variables,

@tf(1) +

! :@!x f

(1) +!g (e) +!g (`d) :@!

f (1)

= (N 1)@!

Z Z

j!x 1!x j

Assumption 2: The external and attractive forces are very small when com-pared with the intermolecular forces during collision

@tf(2) +

! :@!x f

(2) +! 1:@!x f

(2) +!g !x !x 1 :@! f (2)

+!g !x 1!x :@! 1f (2) = 0 (74)

or, equivalently, considering the time t to be the instant immediately after thecollision

f (2)!x ;! ;!x 1;! 1; t

= f (2)!x ! t;! !g !xt;!x 1 ! 1t;! 1 !g !x 1t; tt(75)

wheret is a time interval that is small with respect to the time interval betweenany two collisions but large with respect to the collision time.Assumption 3: Molecular chaos

f (2)!x ;! ;!x 1;! 1; t

= f (2)!x ! t;! !g !xt;!x 1 ! 1t;! 1 !g !x 1t; tt

= f (1)!x ! t;! !g !xt; tt

f (1)!x 1 ! 1t;! 1 !g !x 1t; t+t (76)

where 0 < s < t.Assumption 4:It is supposed that the distribution function does not vary be-

tween points !x and !x 1 at the begining of the collision process.

Consider now the time average of Eq. (70) given

f (1) =1

t

Z t0

f (1)ds (77)

@tf (1) +! :@!x f (1) +

!g (e) +!g (`d) :@!f (1)

= (N 1)t

Z t0

Z Zj!x 1!x j

or

@tf (1) +! :@!x f (1) +

!g (e) +!g (`d) :@!f (1)

= (N 1)t

Z t0

Z Zj!x 1!x j>>>>>>:Z t0

@s

266664f (1)

!x +! (st) ;! +!g (st) ; t+ (st)

!

f (1) !x 1 +! 1 (st) ;!

1 +!g 1 (st) ; t+ (st)

!3777759>>>>=>>>>;

d!x 1d! 1dsor

@tf (1) +! :@!x f (1) +

!g (e) +!g (`d) @!f (1)

= (N 1)t

Z Z

8>>>>>>>:Z t0

@s

266664f (1)

!x +! (st) ;! +!g (st) ; t+ (st)

!

f (1) !x +! 1 (st) ;!

1 +!g 1 (st) ; t+ (st)

!3777759>>>>=>>>>;

d!x 1d! 1ds=

(N 1)t

Z Z 8>>>>>:

24 f (1) !x ! t;! !g t; ttf (1)

!x ! 1t;! 1 !g 1t; tt35

hf (1)

!x ;! ; t f (1) !x ;! 1; ti9>>>=>>>; d

!x 1d! 1

21

Since t is very small

!x ! t !x ! 1t !x (78)

tt t (79)the collision process being considered to only modify the velocities of the targetand bullet particles, respectively,

! !g t = ! 0;before the collision to ! ,

after the collision and! 1 !g 1t = ! 01 before the collision to

! 1, after the

collision.Therefore, we can rewrite Eq. (??) as

@tf (1) +! :@!x f (1) +

!g (e) +!g (`d) :@!f (1)

=(N 1)t

Z Z 24 f (1) !x ;! 0; t f (1) !x ;! 01; tf (1)

!x ;! ; t f (1) !x ;! 1; t35 d!x 1d! 1 (80)

where! 0;! 01 are the velocities of the target and bullet particles before the

colision (at time tt) that will give the velocities ! , ! 1 after the collision.When evaluating the right hand side of the above equation, the position of

the target particle !x is constant. Therefore when evaluating the integral wecan replace !x 1 by !x 1 !x , by placing the system of coordinates on the centerof the target particle.Writing the element of volume d! = d (!x 1 !x ) in cylindrical coordinates,

1

td! = d

tbdbd" = gbdbd" (81)

where !g is the velocity of the bullet particle with respect to the target particle.When N is very large N 1 N and, considering

Nf (1)!x ;! ; t = f !x ;! ; t (82)

as the number of particles that during the times between t and t+dt are expectedto be in the position between !x and !x+ d!x with velocity between ! and ! +d! ;

@tf +! :@!x f +

!g (e) +!g (`d) :@!f

=

Z Z Z 24 f !x ;! 0; t f !x ;! 01; tf!x ;! ; t f !x ;! 1; t

35 gbdbd"d! 1 (83)

22

References

[1] N. N. Bogoliubov (1946). "Kinetic Equations" (in English). Journal ofPhysics USSR 10 (3): 265 274.

[2] N. N. Bogoliubov (1946). "Kinetic Equations" (in Russian). Journal of Ex-perimental and Theoretical Physics 16 (8): 691 702.

[3] N. N. Bogoliubov, K. P. Gurov (1947). "Kinetic Equations in QuantumMechanics" (in Russian). Journal of Experimental and Theoretical Physics17 (7): 614 628.

[4] J. Yvon (1935): Theorie Statistique des Fluides et lEquation et lEquationdEtat (in French), Actes sientique et industrie. 203. Paris: Hermann.

[5] John G. Kirkwood (March 1946). "The Statistical Mechanical Theory ofTransport Processes I. General Theory". The Journal of Chemical Physics14 (3): 180. http://dx.doi.org/10.1063/1.1724117.

[6] John G. Kirkwood (January 1947). "The Statistical Mechanical Theoryof Transport Processes II. Transport in Gases". The Journal of ChemicalPhysics 15 (1): 72. http://dx.doi.org/10.1063/1.1746292.

[7] M. Born and H. S. Green (31 December 1946). "A General Kinetic Theoryof Liquids I. The Molecular Distribution Functions". Proc. Roy. Soc. A 188:10 18.

23

Lecture 2: The equilibrium solution

The Boltzmann equation

@tf +! (rxf) +

!g (e) +!g (`d) (rf) = ; (1)is an integro-diferential equation and the term,

==

0@Z! 1

Z 2"=0

Z b=0

24 f !x ;! 0; t f !x ;! 01; tf

!x ;! ; t f !x ;! 1; t35 bd"dbgd! 1

1A ; (2)gives the net balance between the molecules that acquire the velocity

! and

the ones that loss this velocity. In equilibrum conditions this balance must benull, otherwise the distribution f would vary in the course of the time and inthe space. So,

f!x ;! 0; t f !x ;! 01; t = f !x ;! ; t f !x ;! 1; t ; (3)

or

ln f!x ;! 0; t+ ln f !x ;! 01; t = ln f !x ;! ; t+ ln f !x ;! 1; t ; (4)

meaning that

ln f = ln f (eq) = collisional invariant. (5)

The collisional invariants are

m = mass, (6)

m! = momentum, (7)

1

2m2 = kinetic energy. (8)

Therefore, any linear combination of these invariants is also a collisionalinvariant,

ln f (eq) = A+!B ! + C2

= a+ b! !c

2; (9)

or

1

Figure 1: Maxwell-Boltzmann distributions for dierent values of a =q

kTm

f (eq) = eaeb! !c

2= de

b! !c

2: (10)

Parameters d, b e !c are determined from

n =

Zf (eq)d

! = number density of molecules, (11)

!u = 1n

Zf (eq)

! d! = mean molecular velocity, (12)

ec;f =1

n

Zf (eq)

1

2m! !u

2d! = mean peculiar kinetic energy.(13)

In fact, for a uid in equilibrium the quantities

n =

Zfd! ; (14)

!u =D!E=1

n

Zf! d! ; (15)

must be constants and given by replacing the distribution f by the distributionthe system has at equilibrium, i.e., f (eq).The third integral deserves a more lengthy discussion. The mean kinetic

energy per molecule is given by

ec =1

2m

2=1

n

Zf1

2m2d

! : (16)

Writting,

2 =! !u

2+ 2

! !u

!u + u2; (17)

2

it can be easily seen that this kinetic energy has two components

ec

J

molecule

=1

n

Zf1

2m! !u

2d!| {z }

=ec;f

+1

2mu2| {z }

advection energy

: (18)

The thermodynamic internal energy, e, per molecule1 , is the sum of the thepeculiar kinetic energy ec;f , i.e., the energy due to the molecular random motionof the molecules and the potential energy due to the intermolecular interactionamong these molecules,

e = ec;f +

n; (20)

where is the intermolecular potential energy per unit volume.For thermodynamic systems in equilibrium, the thermodynamic internal en-

ergy, eM , per mol, satises,

deM = cvdT| {z }energy due to molecular motion

+

T

@P

@T

v

Pdv| {z }

intermolecular potential energy

: (21)

For ideal gases, with R=universal gas constant,

P =RTv; (22)

the second term is null and,

deM = cvdT; (23)

meaning that solely the molecular motion contributes to the thermodynamicenergy. Nevertheless, for real uids, the thermodynamic energy must take theintermolecular potential energy into account. For a uid with, e.g., a van derWaals equation of state

P =RTv b

a

v2; (24)

the parameter "a" is related to the intermolecular forces and we get

1We use a bar over the symbol e, e.g., e; ec; ec;f for denoting energy per unit molecule,

e = em; e = ec

m; ec;f =

ec;fm

for denoting energy per unit mass and eM ; ec;f;M for denotingenergy per unit mol. Therefore e = ne is the thermodynamic internal energy per unit volumeand

ec;f = nec;f =

Zf1

2m! !u

2d! (19)

is the peculiar kinetic energy per unit volume.

3

deM = cvdT +a

v2dv: (25)

In both cases, the molal heat cv is a molecular parameter related to themotion degrees of freedom of the molecules2 ,

cv = cv;transl + cv;rot + cv;vib: (26)

For molecules with solely translational degrees of freedom3 (our case)

cv = cv;transl =D

2R; (27)

where D = space dimension.

2With monatomic gases, thermal energy comprises only translational motions. Transla-tional motions are ordinary, whole-body movements in 3D space whereby particles move aboutand exchange energy in collisions like rubber balls in a vigorously shaken container (see an-imation here). These simple movements in the three X, Y, and Zaxis dimensions of spacemeans individual atoms have three translational degrees of freedom. A degree of freedom isany form of energy in which heat transferred into an object can be stored. This can be intranslational kinetic energy, rotational kinetic energy, or other forms such as potential energyin vibrational modes. Only three translational degrees of freedom (corresponding to the threeindependent directions in space) are available for any individual atom, whether it is free, as amonatomic molecule, or bound into a polyatomic molecule.As to rotation about an atoms axis (again, whether the atom is bound or free), its energy

of rotation is proportional to the moment of inertia for the atom, which is extremely smallcompared to moments of inertia of collections of atoms. This is because almost all of the massof a single atom is concentrated in its nucleus, which has a radius too small to give a signicantmoment of inertia. In contrast, the spacing of quantum energy levels for a rotating objectis inversely proportional to its moment of inertia, and so this spacing becomes very large forobjects with very small moments of inertia. For these reasons, the contribution from rotationof atoms on their axes is essentially zero in monatomic gases, because the energy-spacing of theassociated quantum levels is too large for signicant thermal energy to be stored in rotationof systems such small moments of inertia. For similar reasons, axial rotation around bondsjoining atoms in diatomic gases (or along the linear axis in a linear molecule of any length)can also be neglected as a possible "degree of freedom" as well, since such rotation is similarto rotation of monatomic atoms, and so occurs about an axis with a moment of inertia toosmall to be able to store signicant heat energy.In polyatomic molecules, other rotational modes may become active, due to the much higher

moments of inertia about certain axes which do not coincide with the linear axis of a linearmolecule. These modes take the place of some translational degrees of freedom for individualatoms, since the atoms are moving in 3-D space, as the molecule rotates. The narrowingof quantum mechanically-determined energy spacing between rotational states results fromsituations where atoms are rotating around an axis that does not connect them, and thusform an assembly that has a large moment of inertia. This small dierence between energystates allows the kinetic energy of this type of rotational motion to store heat energy atambient temperatures. Furthermore (although usually at higher temperatures than are ableto store heat in rotational motion) internal vibrational degrees of freedom also may becomeactive (these are also a type of translation, as seen from the view of each atom). In summary,molecules are complex objects with a population of atoms that may move about within themolecule in a number of dierent ways (see animation at right), and each of these waysof moving is capable of storing energy if the temperature is su cient. (from Wikipedia:http://en.wikipedia.org/wiki/Heat_capacity)

3See http://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry)

4

Therefore, by integrating Eq. (25), we get for a van der Waals equation ofstate,

eM (T; v) =D

2RT +

a1v

(in J=mol), (28)

plus a constant C (T0; v0) that depends only on the initial state.This means that, in a molecular basis,

ec;f =1

n

Zf (eq)

1

2m! !u

2d! =

D

2kT (in J=molecule), (29)

where,

k =RN ; (30)

N being the Avogadro4 number and k = Boltzmann constant, Also, for a vander Waals equation of state,

e (T; n) =D

2kT + (an) (in J=molecule). (31)

Eq. (29) is, in fact, the denition of temperature for systems composed ofmolecules with only translational degrees of freedom.The intermolecular potential energy per unit volume becomes for a van der

Waal equation of state, = an2 and is always negative. When the numberdensity of molecules increases, meaning a larger number of molecules in the samevolume, this energy increases in absolute value, meaning that the molecules are

4The Avogadro constant is named after the early nineteenth century Italian scientistAmedeo Avogadro, who, in 1811, rst proposed that the volume of a gas (at a given pressureand temperature) is proportional to the number of atoms or molecules regardless of the natureof the gas. The French physicist Jean Perrin in 1909 proposed naming the constant in honorof Avogadro. Perrin won the 1926 Nobel Prize in Physics, in a large part for his work indetermining the Avogadro constant by several dierent methods.The value of the Avogadro constant was rst indicated by Johann Josef Loschmidt who, in

1865, estimated the average diameter of the molecules in air by a method that is equivalentto calculating the number of particles in a given volume of gas. This latter value, the numberdensity of particles in an ideal gas, is now called the Loschmidt constant in his honour, andis approximately proportional to the Avogadro constant. The connection with Loschmidt isthe root of the symbol L sometimes used for the Avogadro constant, and German languageliterature may refer to both constants by the same name, distinguished only by the units ofmeasurement.Accurate determinations of Avogadros number require the measurement of a single quantity

on both the atomic and macroscopic scales using the same unit of measurement. This becamepossible for the rst time when American physicist Robert Millikan measured the charge on anelectron in 1910. The charge of a mole of electrons is the constant called the Faraday and hadbeen known since 1834 when Michael Faraday published his works on electrolysis. By dividingthe charge on a mole of electrons by the charge on a single electron the value of Avogadrosnumber is obtained. Since 1910, newer calculations have more accurately determined thevalues for Faradays constant and the elementary charge. (from Wikipedia )

5

strongly linked, When the number density decreases, ! 0. For an ideal gas,the molecules are free from the intermolecular forces and = 0.

With these restrictions Eqs (11-13), give for the equilibrium distribution,the Maxwell-Boltzmann distribution

f (eq) = n m2kT

D=2e (

! !u)

2

2kTm : (32)

6

Lecture 03: Discretization using nitedierences

Consider the Boltzmann equation in the form

@tf +! (rxf) = ; (1)

where the term is given by

= (sd) !g (e) +!g (`d) (rf) ; (2)

with the short distance repulsion term written as,

(sd) =

Z! 1

Z 2"=0

Z b=0

(f 0f 01 ff1) bd"dbgd! 1: (3)

It is possible to make f = f (eq) in the evaluation of rf on the right handside of Eq. (2) without any eects on the macroscopic equations,

rf = rf (eq) = ! !u

kTm

f (eq): (4)

So we get a kinetic model for

= (sd) +!g (e) +!g (`d)

! !u

kTm

f (eq): (5)

A kinetic model means that the Boltzmann equation is replaced by a modelthat is able to retrieve the main, or some of the mains properties of the Boltz-mann equation. In LBM, the more widely used model for (sd) is the BGK [1]model1

(sd) =f (eq) f

; (6)

1"In 1951, Bhatnagar went to Harvard University, Cambridge as a Fullbright scholar fortwo years. This handsome tall scholar from India was often mistaken in the Universitycorridors for a student. Once he took his place at the lecture rostrum, the students realizedthat he was indeed a senior faculty. He lectured on mathematical theory of gases based on themathematically formidable book by S. Chapman and T.G. Cowling. At Harvard, he producedtwo very important publications (1) a book Stellar Interiors jointly with D.H. Menzel andH.K. Sen, published in the International Astrophysical Series and (2) a research paper inPhysical Review in 1954 which contained the famous BGK (Bhatnagar, Gross, Krook) model.The Bolzmann equation, governing the evolution of a state of molecules in gases, contains anextremely complicated integral term . Faced with the reality that the Boltzmann equation wastoo di cult to handle due to this collision integral term, Bhatnagar, Gross and Krook usedtheir deep understanding of relaxation process of a swarm of molecules towards an equilibriumstate to replace this term by a much simpler term free, which has since been used as alternativeto the Boltzmann equation in solving problems in rareed gas dynamics, plasma physics andthe kinetic theory itself. " [2]

1

meaning that the collision term can be thought as a relaxation term towardsthe equilibrium when a uid is in a non-equilibrium state.

Professor P.L. Bhatnagar

We will talk more about kinetic models in the following lectures. In thislecture we focus our attention on LBM discretization and consider as a knownfunction

f; f (eq)

.

Discretization means to replace the entire continuous physical space, repre-sented by the continuous variable !x by some points !x i and the entire velocityspace represented by the continuous variable

! by some velocity vectors

! i;

i = 0; :::; nb1, for which the distribution f will be calculated in dierent timesseparated by a time interval t. Figure (1) represents a 37-velocity lattice foundby Philippi et al.[3], suitable for solving non-isothermal problems.

Figure 1: The D2V37 lattice, [3].

Therefore, considering a single velocity! i from this set of nb velocities the

Boltzman equation can be written as

@tfi +! i (rxfi) = i; (7)

2

where fi = f!x ;! i; t indicates the value of f for ! = ! i, i.e., the packet of

particles with velocity! i that are found in the point

!x at time t.At the time t+ , this packet wil be,

fi (!x ; t+t) = fi (!x ; t) + t@fi

@t+1

2(t)

2 @2fi@t2

+ :::; (8)

and will be known from the value this packet had at time t, when all the timederivatives @fi@t ;

@2fi@t2 ; :::are known at time t.

This is not the case. Nevertheles, when t! 0, it is possible to neglect allthe derivatives of order 2 and higher.

fi (!x ; t+t) = fi (!x ; t) + t@fi

@t+O

(t)

2

(9)

giving

@fi@t

=fi (!x ; t+t) fi (!x ; t)

t+O (t) (10)

In the same way, for calculating the value fi in a point!x +!x , at the time

t+t

fi (!x +!x ; t+t) = fi (!x ; t) + t@fi

@t+!x @fi

@!x +O(!x )2 ; (t)2

:

(11)When

!x = ! it; (12)we get

fi

!x +! it; t+t = fi (!x ; t)+t@fi@t

+! i @fi

@!x+O

(!x )2 ; (t)2

;

(13)or

@fi@t

+! i @fi

@!x=fi

!x +! it; t+t fi (!x ; t)t

+O (j!x j ;t) :(14)

And the discrete Boltzmann equation can be written as

fi

!x +! it; t+t fi (!x ; t) = it+O (!x )2 ; (t)2 : (15)The term

3

Figure 2: Illustration of the collision-propagation scheme in LBM. The amountof particles in each direction is represented by the length of the vector pointingin that direction.

! itj!x j ; (16)

is the Courant-Fridrich-Lewy number2 , CFL, and in the present numericalscheme CFL = 1. This condition implis that at the time t + t the parti-cles with velocity

! i will be found at the site

!x + !x , at the time t + t.This scheme is at the origin of the LB method, which i an explicite numericalmethod when the interacton term i is evaluated at time t. Therefore, when theexternal and intermolecular forces are neglected and the repulsion term (sd) ismodelled using a BGK relaxation term

i =f(eq)i fi

; (17)

the LB method proceeds in accordance with two steps (Figure 2).

2 In mathematics, the CourantFriedrichsLewy condition (CFL condition) is a necessarycondition for convergence while solving certain partial dierential equations (usually hyper-bolic PDEs) numerically. (It is not in general a su cient condition.) It arises when explicittime-marching schemes are used for the numerical solution. As a consequence, the timestepmust be less than a certain time in many explicit time-marching computer simulations, other-wise the simulation will produce wildly incorrect results. The condition is named after RichardCourant, Kurt Friedrichs, and Hans Lewy who described it in their 1928 paper, [4], [5].For example, if a wave is crossing a discrete grid, then the timestep must be less than the

time for the wave to travel adjacent grid points. As a corollary, when the grid point separationis reduced, the upper limit for the time step also decreases. In essence, the numerical domainof dependence must include the analytical domain of dependence in order to assure that thescheme can access the information required to form the solution.

4

a) Collision step: the distributions fi are recalculated in each site!x i using

the local information in this site

f 0i (!x i; t+t) = fi (!x i; t) + f

(eq)i (

!x i; t) fi (!x i; t)=t

: (18)

b) Propagation step: the new values of the distributions f 0i are propagatedalong the i-directions

fi

!x +! it; t+t = f 0i (!x ; t+t) : (19)

References

[1] Bhatnagar, P.L., Gross, E.P and Krook, M., A model for collisional processesin gases I: small amplitude processes in charged and in nutral one-componentsystems, Phys. Rev. 94, 1954, 511-525.

[2] P. Prasad, 19th P.L. Bhatnagar Memorial Award Lecture -2006, delivered at 72nd Annual Conference of the IndianMathematical Society at Jabalpur, 27 - 30 December, 2007(http://math.iisc.ernet.in/~prasad/prasad/plbhatnagar.pdf)

[3] P. C. Philippi, L. A. Hegele, Jr., L. O. E. dos Santos, and R. Surmas, Phys.Rev. E 73, 056702 (2006).

[4] R. Courant, K. Friedrichs and H. Lewy, ber die partiellenDixoerenzengleichungen der mathematischen Physik, MathematischeAnnalen, vol. 100, no. 1, pages 3274, 1928.

[5] R. Courant, K. Friedrichs and H. Lewy, "On the partial dixoerence equationsof mathematical physics", IBM Journal, March 1967, pp. 215-234, Englishtranslation of the 1928 German original.

5

Lecture 04: The Boltzmann equation for non-idealuids

Consider the Boltzmann equation in the form

@tf +! @xf = (sd) +!g (`d)

! !u

kTm

f (eq) +!g (e) ! !u

kTm

f (eq) (1)

where !g (e) corresponds to the force per unit mass due to external body forces,!g (`d) to the intermolecular attraction among the molecules, which we will sup-pose to be only dependent of the distante between the centers of the moleculesand, (sd), to the collision term related to the short range repulsion forces (Fig-ure 1 ).

Figure 1: Force eld around an isolated molecule.

The term (sd) should, then, be understood as related to the repulsion eldinside the -spheres of radius and which centers are the material points usedto represent the molecules. In the surface of this sphere, this repulsion eldis exactly equilibrated by the electrostatic attraction eld and the resultingforce is null (Figure 2). Therefore, in its trajectory inside the -sphere of thetarget particle, the bullet particle will be frained and deviated from the centerof the target, transforming part of its kinetic into potential energy, which is onlyrecovered when the bullet is again in the surface of the -sphere in its exitingtrajectory

1

Figure 2: Repulsion -sphere of the target particle, indicating the surface wherethe two repulsion elds intercept with themselves.

The term

!g (`d) ! !u

kTm

f (eq) (2)

related to the long-range interaction, deserves some discussion.Consider a molecule labelled as 1 located in point !x 1 and all the molecules

labelled as 2 around it. The potential energy related to the interaction of allmolecules 2 with the single molecule 1 is gven by

1 =

Zj!x 2!x 1j>

(12)n (!x 2) d!x 2 (3)

where (12) is the potential energy (in Joules) related to the electrostatic inter-action between the particle 1 and a single particle 2.Considering that the interaction length is not too large, and that the number

of particles per unit volume n does not have strong variations, it is possible toconsider n (!x 2) = n (!x 1) inside the integration domain and we get,

2

1 = 2n (!x 1) 1

2

Zj! j>

(12)d!| {z }a

= 2an (!x 1) (4)

Where a is a positive constant, because (12) < 0. This constant dependssolely on the electrostatic eld generated by the molecules being a molecularproperty of the substance that is being analyzed.The force !g (`d) on molecule 1 will thus be

!g (`d) = r1m

(5)

Therefore the Boltzmann equation takes the form

@tf +! @xf = (sd)| {z }

repulsion

+ (`d)| {z }intermolecular attraction

+!g (e) ! !u

kTm

f (eq)| {z }external body forces

(6)

where

(`d) =2a

mrn

! !u

kTm

f (eq) (7)

represents the net gain of particles that acquire the velocity! due to the

intermolecular forces from the other particles around it.Exercise: Discretize Eq. (6) considering a single discrete velocity

! i and

a Courant-Friedrich-Lewy number, CFL = 1. Propose a method to discretizern in 2D discrete spaces (hint: use Taylor series).

Appendix: Intermolecular attraction forces1

Intermolecular forces are, essentially, electrostatic forces, since every mole-cule is a site of positive and negative electrical charges. Coulombs forceshappen among ions, in accordance with

F12 =1

40

q1q2r212

!r12

j!r 12 j; (8)

1References [2] and [1] are indicated for further reading.

3

when two charged particles q1 and q2 separated by a distance r12 , where 0 is theelectrical permittivity of vacuum equal to 8:854191012 (coulomb)2 = Nm2.The potential energy is given by

!F = r; (9)

giving

12=

1

40

q1q2r12

(10)

Coulomb forces are inversely proportional to the square of the distance be-tween any two ions and have a much longer interaction length when comparedto the other intermolecular forces. Asymmetric molecules such as H20 and CO2have a permanent dipole, the geometric center of the electronic clouds does notcoinciding with the geometric center of the positive charges (Figure 3). Theirdipole moment vector is,

! = q`!n `, (11)where ` is the distance that separates the charges and !n ` is a unitary vectorgiving the dipole direction and conventionally oriented from the negative to thepositive charge.

Figure 3: H2O and CH4 molecules. Intermolecular forces are dependent on themolecular shape.

Consider two dipoles which centers are separated by a distance r12, let !n `;1

to be given in terms of the polar 1 and azimuthal, 1 angles and!n `;2, in terms

4

of 2 and 2, Figure 4. When r12 is much larger than the dipole lenghts, thesetwo dipoles will share a potential energy given by

12 = 140

12r312

[2 cos 1 cos 2 sin 1 sin 2 cos (1 2)] (12)

Figure 4: Calculation of the potential energy between two dipoles.

When 1 = 2 = 0, the dipoles are in a single vertical plane and the potentialenergy becomes

12 = 140

12r312

[2 cos 1 cos 2 sin 1 sin 2] (13)

which attains a minimum when the dipoles are aligned, 1 = 2 = 0, the positiveend of dipole 1 being at the closest position with respect to the negative end ofdipole 2. Potential energy becomes a maximum, when 1 = 0 and 2 = .Molecules are subjected to random uctuations in their translational motion

as they collide with other molecules. Kinetic energy, Ec, is exhanged among thecolliding molecules and the rotational and vibrational modes are excited fromthe available translational energy. The potential energy, related to the elec-trostatic attraction between any two molecules is converted into kinetic energyduring the approximation. Two molecules that are permanent dipoles have aminimum in their mutual potential energy, < 0, when they are aligned. Whenthese molecules do have enough kinetic energy, i.e., when +Ec 0, they willrest aligned at their closest position. They are only uncoupled when they receiveenough energy to liberate them from the attractive electrostatic eld by, e.g.,collisions with another molecules.Collision frequency and, also, molecular kinetic energy are increased when

the temperature increases.In this manner, it may be supposed that the prob-ability that a pair of dipoles have a potential energy

12follows a Boltzmann

distribution

5

P () = ekT (14)

In this case, considering all the possible directional congurations of thispair, given by all the possible polar 1, 2 and azimuthal angles, = 1 2,the ensemble average potential energy of this couple is given by

h12i =R12e

12kT dRe

12kT d

(15)

where

d = sin 1 sin 2d1d2d (16)

It results, neglecting higher order terms,

h12i = 23

1

40

221

22

r612kT(17)

In this way, when the temperature T increases, the molecules have moreavailable kinetic energy and the congurations with a small potential energy,related to a strong electrostatic attraction, are less frequent in the ensemble ofpossible congurational states of the molecular system.The ensemble average intermolecular force between any two polar molecules

separated by !r is given byD!F 12

E= 4

1

40

221

22

r712kT!n r (18)

The dipole length and the dipolar moment of a polar molecule are subjectedto uctuations produced by electrostatic elds originated from other molecules.In the same manner, a non-polar molecule will acquire a dipolar moment, ind,under the induction of an electrical eld,

!E . We dene the molecular polariz-

ibility, , by

! ind = !E (19)giving a measure of the easiness a molecule becomes polarized, or changes itsdipolar moment, when subjected to an electrical eldWhen a polar molecule, 1, is at a distance r12 from a non-polar molecule, 2,

the electrical eld at 2 will be

!E 2 = 1

20

1r312

!n r (20)

and the dipolar moment

! ind = 220

1r312

!n r (21)

6

Considering that the induced dipole 2 has always the same orientation ofthe electrical dipole that is at its origin, the potential energy that is sharedby a polar 1 and a non-polar 2 molecule can be calculated using Eq. (13) for1 = 2 = 0, giving

12 = 12

1

(20)2

221

r612(22)

London forces

Symmetrical molecules do not have any permanent electrical dipole and theattractive forces among these molecules was rst described by London in 1930as due to very rapid uctuations of the geometric center of their molecularelectronic clouds. These uctuations originate temporary dipoles that are con-stantly changing in magnitude and direction with a characteristic electronicfrequency 0 that can be related to the refractive index n of the uid, when abeam of polarized light with frequency traverses it, following,

n = 1 +c

20 2(23)

where c is a constant.Each time a molecule has a instantaneous dipolar moment it induces a

dipolar moment in a neighbor molecule. The strenght of the interaction de-pends on the polarizibility of the rst molecule, because the strenght of theinstantaneous dipole moment depends on the loseness of the control that thenuclear charges exert on the outer electrons and also on the polarizibility of thesecond molecule, because that polarizibility depends on how readly a dipole canbe induced by another molecule. The resulting potential energy was found tobe, [2], [1],

12 = 32

1

(40)2

12r612

I1I2I1 + I2

(24)

where I is the rst ionization potential (Figure 5), i.e., the work that must beperformed to remove one electron from the uncharged molecule, I h0. Theattractive forces among non-polar molecules are also called dispersion forcesconsidering the dispersion of light in the medium due to the oscilating dipoles.Polar molecules also atract themselves with dispersion forces, but, for polarmolecules, the time average of their dipolar moment is dierent from zero (beingequal to its permanent dipolar moment).When an apolar molecule is close to a polar molecule, the second molecule

induces a dipolar moment on the rst one and the rst molecule induces auctutation on the geometrical center of the electrons of the second moleculeand, consequently a uctuation on the permanent dipolar moment of this mole-cule. The two molecules will attract themselves with a London dispersion forcesuperposed to a Debye induction force.

7

Figure 5: Ionization potential of some molecules

Summary

All these forces can be writen as

12 = B(12)

j!r 2 !r 1j6(25)

The Table below summarizes the several contributions to the potential en-ergy.

References

[1] Prausnitz, J. M. , Lichtenthaler, R. N. & de Azevedo, E. G. 1999 Molec-ular Thermodynamics of Fluid-Phase Equilibria, 3rd Edition, New Jersey:Prentice Hall.

[2] A. W. Adamson, Physical Chemistry of Surfaces, John Wiley (1997).

8

Lecture 05: Properties of the repulsion term

Consider the integral

Z

(sd)

!'!d!

=

Z Z f0f01 ff1

'!gbdbd"d

! 1d

! ; (1)

for any function '!of the molecular velocity

! .

Although (sd)! 1

is dierent from (sd)

!; the value of the integralZ

(sd)!'!d! =

Z

(sd)

! 1

'! 1

d! 1; (2)

and so Z Z f0f01 ff1

'!gbdbd"d

! 1d

!

=

Z Z f0f01 ff1

'! 1

gbdbd"d

! 1d

! : (3)

On the other hand if we change! ! 0 and, consequently, ! 1 ! 01,

Z! 0

Z! 01

Z 2"=0

Z b=0

ff1 f 0f 01

'!0g0bdbd"d

! 01d

! 0

= Z! 0

Z! 01

Z 2"=0

Z b=0

f0f01 ff1

'!0gbdbd"d

! 01d

! 0; (4)

because g = g0. Since the intermolecular eld is a central eld (Figure 1) wecan also show, (see Lecture 01, Appendix A) that d

! 01d

! 0 = d

! 1d

! meaning

that the transformation! 1;

! ! ! 01;

! 0has a Jacobian, J = 1:Changing

! 1;!

by! 01;

! 0 is equivalent to invert the collision term and under a central eld

the gain of particles with velocities! 1;

! when they have the velocities

! 01;!

0 before the collision is the same gain of particles with velocities! 01;

! 0 when

they have the velocities! 1;

! before the collision.

Therefore

Z

(sd)'

!d! =

Z

(sd)'

! 1

d!

= Z

(sd)'

! 0d! =

Z

(sd)'

! 01d! ; (5)

1

gg

a

f

q

rz

b

s

b

gg

gg

aa

f

q

rz

b

s

b

Figure 1: Trajectory of the bullet under a central eld.

or Z

(sd)'

!d! =

1

4

Z Z

(sd)

'+ '1 '

0 '01d! : (6)

When '!is a collisional invariant, '

!= m;m

! ; 12m

2, we will get,Z

(sd)'

!d! = 0: (7)

This property is important and is directly related with the preservation ofmass, momentum and kinetic energy in collisions.It is easy to show that this property is satised by the BGK relaxation term

(sd) =f (eq) f

; (8)

meaning that, when '!is a collisional invariant,Zf (eq) f

'!d! = 0: (9)

2

Lecture 06: Macroscopic equationsMoments of the distribution f

For a given point !x , the distribution f gives the amount of particles withvelocities between

! and

! + d

! . When this distribution is integrated in

the velocity space, we loss this detailed information and get statistical averagevalues, which are the macroscopic variables of interest. The rst moments ofthe distribution f are:

density , =Zmfd

! ; (1)

momentum per unit volume, !u =Zm! fd

! ; (2)

mean kinetic energy per unit volume, ec =Z1

2m2fd

! ; (3)

Writing

2 =! !u

2+ 2

! !u

!u + u2; (4)

we see that the kinetic energy has two parts,

ec =

Zf1

2m! !u

2d!| {z }

=ec;f

+1

2u2| {z }

advection energy

(5)

The rst part, ec;f , is due to the random motion of the molecules around thelocal velocity !u . This kinetic energy is dierent from zero even when the localvelocity !u = 0. This kinetic energy is called the mean peculiar kinetic energyof the molecules and can be used to dene the thermodynamic temperature, T ,by requiring

1

n

Zf1

2m! !u

2d! =

1

n

Zf (eq)

1

2m! !u

2d! =

D

2kT (6)

since the thermodynamic internal energy per molecule for an ideal gas, withsolely translational degrees of freedom in equilibrium is known, from statisticalmechanics1 , to be D2 kT .

When '!is a collisional invariant, '

!= m;m

! ; 12m

2, the mo-ments Z

f'!d! (7)

1See the link: http://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry)

1

are equilibrium momentsZf'!d! =

Zf (eq)'

!d! (8)

in the sense that, in equilibrium, these are the only moments that show a macro-scopic manifestation: density, local velocity (or momentum) and temperature.We will, now, consider the moments of interest in non-equilibrium conditions.

The tensor

=

Z(m) fd

! (9)

represents the rate with which the momentum m is transferred along the di-rection due to the molecules that move with a non-null velocity component

along this direction. In the SI units,

=

kg(ms )2

m3 =kg(m

s2)m

m3 =Nm2 , rep-

resenting a force along the direction per unit area, orthogonal to (Figure1)

Figure 1: Tensor

Writting

= ( u) u

+ u + u uu (10)

it can be seen that

=

Zm ( u)

u

fd! + uu (11)

Now, let the distribution f be decomposed in two parts,

f = f (eq) + f (neq); (12)

considering the second part, f (neq), as a perturbation of f (eq).

2

By dening,!C =

! !u 2kTm

1=2 (13)we see that,

Zm ( u)

u

f (eq)d

!

= nm m2kT

D=2 Ze (u)22kT

m ( u) u

d!

= nm m2kT

D=22kTm

D2 +1

ZeC

2

CCd!C

=nm

D=2

2kT

m

ZeC

2

CCd!C

=2nkT

D=2

ZeC

2 C2

3d!C

| {z }

=D=2

2

= P0 (14)

whereP0 = nkT (15)

is the thermodynamic pressure of an ideal gas.Each molecule has its own velocity

! . When the uid is at rest !u =D!

E= 0, and m indicates the momentum m that is transported in the

-direction by a single molecule. In equilibrium, since the MB distribution isisotropic (independent of the direction), the rate with which the momentum istransferred into a direction dierent from by the whole set of molecules is zero.We have solely transfer of momentum in the same direction of this momentumand this transfer is independent of the direction (a consequence of isotropy).When these molecules transport momentum in the x-direction and nd a bar-rier in their trajectory they return with the same x-velocity (elastic collisions)indicating a force on the barrier given by the net balance of momentum alongthe x-direction,

3

Z 11

dz

Z 11

dy

Z 10

mxxf(eq)dx| {z }

molcules moving to the right

Z 11

dz

Z 11

dy

Z 10

mxxf(eq)dx| {z }

molcules moving to the left

=

Z 11

dz

Z 11

dy

Z 10

mxxf(eq)dx

+

Z 11

dz

Z 11

dy

Z 01

mxxf(eq)dx

=

Z 11

dz

Z 11

dy

Z 11

mxxf(eq)dx = P = nkT (16)

Consider now the non-equilibrium moment,Zm ( u)

u

f (neq)d

! = (17)

representing a diusive ux of momentum which appears in non-equilibriumconditions. For a uid at rest, !u = 0, we get

=

Zmf

(neq)d! = 0 (18)

Because, in this case, is an equilibrium momentZmfd

! =

Zmf

(eq)d! = nkT (19)

In 2D channel ows, for xy be something dierent from zero, the velocityux must vary along the direction y, in which case

xy =

Zm(x ux)| {z }

Cx

y uy

| {z }Cy

f (neq)d! (20)

is related to the diusive transfer of the x momentum along the y-direction.Thisgives rise to the concept of viscosity. Viscosity is caused by the transfer ofmomentum between two planes sliding parallel to one another but at dierentrates, and this momentum is transferred by molecules moving between the planes(Figure 2). Molecules from the faster plane move to the slower plane and tendto speed it up.We get for the tensor

, Eq. (11)

= nkT + + uu (21)

4

Figure 2: Faster molecules in plane y transfer their x-momentum when movingto plane y + dy, while slower molecules in the plane y + dy transfer their x-momentum when moving to plane y. This gives rise to a frainage tangentialforce in the plane y and to an acceleration force in the plane y + dy. When thetwo planes move with the same velocity, xy = 0.

Another moment of interest is the one giving the ow of kinetic energytransported by the molecules,

! =Zf1

2m2

! d! (22)

Decomposing

! = !u +

! !u

(23)

we get,

! = !uZf1

2m2d

! +

Zf1

2m2

! !u

d!

=

ec;f +

1

2u2

!u

+

Zf1

2m

! !u

2+ 2

! !u

!u + u2

! !u

d! (24)

Decomposing f = f (eq) + f (neq)

5

Zf1

2m

! !u

2+ 2

! !u

!u + u2

! !u

d!

=

Zf (eq)

1

2m

2664! !u 2| {z }!0

+ 2! !u

!u| {z }

!P0!u

+ u2|{z}!0

3775! !u d!

+

Zf (neq)

1

2m

2664! !u 2 + 2! !u !u| {z }!b !u

+ u2|{z}!0

3775! !u d!= P0

!u + b !u +!q (25)where !q is the heat ow vector,

!q =Zf (neq)

1

2m! !u

2 ! !u

d! (26)

This vector indicates how much of kinetic energy related to the relativemouvement of the molecules is transferred into a direction due to the relativevelocity of the molecules along that direction. This ux is diusive and for auid at rest it is

!q =Zf (neq)

1

2m2

! d! (27)

The term P0!u , related to the compression work and the term b !u will be

discussed later.

Macroscopic equations

Consider the Boltzmann equation

@tf +! @xf = (sd)| {z }

repulsion

+ (`d)| {z }intermolecular attraction

+!g (e) ! !u

kTm

f (eq)| {z }external body forces

(28)

where

(`d) =2a

mrn| {z }

!g (`d)

! !u

kTm

f (eq) (29)

Mass conservation

6

Multiplying the Boltzmann equation by m and integrating in the velocityspace, remarking that Z

m@tfd! = @t

Zmfd

! = @t (30)

because the time t and the molecular velocities! are independent variables.

On the other hand,Zm! @xfd! = rx

Zm! fd

! = rx (!u ) (31)

since the molecular velocities are also independent of !x .For the repulsion term Z

m(sd)d! = 0 (32)

because m is a collisional invariant.For the force terms

Zm!g

! !u

kTm

f (eq)d! = 0

because !g depends only on the position, independently if !g is the externalbody or the intermolecular force and the MB distribution is an even function of! !u .Therefore we get,

@t+r (!u ) = 0 (33)which is the mass conservation equation.

Momentum balance equationMultiplying the Boltzmann equation by m

! and integrating in the velocity

space we will get

@t (u) + @

= !g (e) + !g (`d) (34)

which is the momentum balance equation (do that as an exercise).Replacing

from Eq. (21)

@t (u) + @ (uu + P0) = !g (e) + !g (`d) @ (35)

The term P0=nkT corresponds to the thermodynamic pressure for a systemof material points without intermolecular attraction. The term

!g (`d) = 2amrn = 2anrn = r an2 (36)

Therefore, Eq. (35) can be written as

7

@t (u) + @

0B@uu + nkT an2| {z }=P

1CA = g(e) @ (37)The term

P = nkT|{z}repulsive part

an2|{z}attractive part

(38)

is the thermodynamic pressure for a system of material points with intermolec-ular attraction forces.The momentum balance equation may also be written in the form,

@t (u) + @ (uu) = g(e) @P @ (39)

or, using the mass conservation equation,

@t (u) + u@ (u) = g(e) @P @ (40)

It should be observed that Eq. (38) is not still the van der Waals equationof state,

P =RTv b

a

v2=

nkT

1 bn an2 (41)

because our molecules are being represented by material points without volume,which coresponds to the parameter b in the van der Waals equation of state.

8

Lecture 07: Phase transitions

The surface tension counteracts the weight of a small liquid drop.

When a liquid has a phase transition, the interface liquid-vapor is too muchthin to enable the density n to be considered constant through it, as we havedone in the Lecture 04, when calculating the potential energy on a point !x 1 toto the interaction of a molecule in this point with all the molecules around it

1 =

Zj!x 2!x 1j>

(12)n (!x 2) d!x 2: (1)

Nevertheless, we can consider a Taylor series for n,

n (!x 2) = n (!x 1) + @n@x

(x2 x1)| {z }=

+1

2

@n

@x

@n

@x(x2 x1)| {z }

=

(x2 x1)| {z }=

+ :::;

(2)giving,

1

=

Zj!x 2!x 1j>

(12)n (!x 2) d!x 2

= 2n (!x 1) 12

Zj! j>

(12)d!| {z }=a

+@n

@x

Zj! j>

(12)d!| {z }

=0

+1

2

@n

@x

@n

@x

Zj! j>

(12)d! : (3)

1

The potential (12) depends solely on the norm of ! being an even functionof x; y and z. ThereforeZ

j! j>(12)d

! = Zj! j>

(12)2

3d! : (4)

On the other hand, in the same way as for the force parameter a, the termRj! j> (12)

2

3 d! depends only on the electrical properties of the molecules,

being a constantZj! j>

(12)2

3d! = 2

Zj! j>

(12)2

6d! = 2: (5)

So

1 = 2an (!x 1) r2n (!x 1) + :::; (6)and the intermolecular attraction force will be given by

!g (`d) = r1m

; (7)

or

!g (`d) = nr1: (8)So

!g (`d) = 2anrn+ nr r2n : (9)In this case, the kinetic equation will be written as

@tf +! @xf = (rep)| {z }

repulsion

+ (`d)| {z }intermolecular attraction

+!g (e) ! !u

kTm

f (eq)| {z }external body forces

; (10)

with

(`d) = !g (`d) ! !u

kTm

f (eq)

=

2a

mrn+

mr r2n

! !u

kTm

f (eq): (11)

The momentum balance equation will be also modied. Considering that,

2

nr r2n= n@ (@ (@n))

= @ [n (@ (@n))] @ (@n) @n= @ [n (@ (@n))] @ [@n@n] + (@n) @@n; (12)

and

(@n) @@n =1

2@ [@n@n] ; (13)

we get

!g (`d) = 2anrn+ r nr2n+ 12r (rn rn) r (rnrn)

= r an2 + nr2n+ 1

2 (rn rn)

b (rnrn) : (14)The momentum balance equation will thus have the form

@t (!u ) +r: (!u!u )

= !g (e)| {z }external forces

rPs|{z}scalar pressure

r b| {z }viscous forces

r bS| {z }surface forces

;(15)

where the scalar pressure is given by

Ps = nkT|{z}repulsion

an2|{z}attraction| {z }

P=thermodynamic pressure

nr2n 12rn rn| {z }

surface pressure

; (16)

and the surface tensor is given by

bS = rnrn: (17)The surface forces have a role solely in the interface liquid vapor where the

gradients are strong.

3

Lecture 09: Velocity discretization

The velocity discretization is a critical step in deriving the lattice Boltzmannequation from the continuous Boltzmann equation since it is intended, in thisstep, to replace the entire continuous velocity space

fMB

! 0

=

1

2

m

kT0

D=2e

202

1Xn=0

1

n!a(n)rn (n;

!u 0;)H(n)rn! 0

; (4)

where! 0 =

kT0m

1=2! is a dimensionless velocity, !u 0 =

kT0m

1=2!u is a localdimensionless velocity

=T

T0 1: (5)

The rst Hermite polynomial tensors are

H(0) = 1;H(1) = 0;;H(2) = 0;0; ;H(3) = 0;0;0;

1

3

0; + 0; + 0;

;

H(4) = 0;0;0;0; 16

0;0; + 0;0; + 0;0;+0;0; + 0;0; + 0;0;

(6)

+1

3( + + ) ; (7)

Using the orthogonality of the Hermite polynomial, the tensors parametersa(n)rn (n;

!u 0;) can be found as

a(n)rn =

kT0m

D=2 ZH(n)rn

! 0

fMBd! 0: (8)

The rst parameters in the Hermite development are

a(0) = n;

a(1) = nu0;;

a(2) = n (u0;u0; +) ;

a(3) = n

u0;u0;u0; +

1

3 (u0; + u0; + u0;)

;

a(4) = n

(p)

0BB@u0;u0;u0;u0;

+ 16

u0;u0; + u0;u0; + u0;u0;+u0;u0; + u0;u0; + u0;u0;

+ 13

2 ( + + )

1CCA : (9)

2

Now, the Hermite expansion of the MB distribution is truncated by removingall Hermite polynomial tensors H(n)rn with order n > N . Indeed, it can be shown,[1], [2], [3], that the number b of discrete velocities used in LBE is directly relatedto the order of approximation of the nite sum f (p)

MB;Nto the full MB distribution

f (p)MB.Therefore, solving a discrete LBE gives an approximation to the full kinetic

equation, whose accuracy can be as high as it can be achieved by increasing theorder of approximation of the Hermite expansion and, consequently, the numberb of discrete velocities used in the representation of the continuous velocity space.Nevertheless, since the main interest in numerically solving a LBE is to

describe the behaviour of a physical system whose macroscopic equations areknown, or supposed to be known, the expansion of the MB distribution in anite set of orthogonal polynomial tensors of order N of approximation has theimportant feature of preserving the moments

'(p)

of order smaller or equal

than N , ZfMB;N

'd! =

ZfMB'd! : (10)

Using this property by replacing fMB

by fMB;N

in Eq. (3), the followingequation is obtained,

1

(2)D=2

Ze

202 H(n)rn

! 0

H(m)rm

! 0

d! 0

=b1Xi=0

WiH(n)rn! 0;i

H(m)rm

! 0;i

; (11)

for all n;m smaller or equal than N . By dening the inner product in thediscrete space (N) that maps the velocity space onto the real numbers as

(f; g) =

nb1Xi=0

Wif! 0;i

g! 0;i

; (12)

it can be seen that Eq. (11) requires that the norm and orthogonality of Hermitepolynomial tensors are preserved in the discrete space (N) for all n;m smalleror equal than N . This means that the discrete velocities

! 0;i and weights Wi

must be chosen in such a manner as to satisfy these conditions.In Philippi et al., [1], it is shown that when the discrete velocity space is

invariant under =2 rotations and reections about the x, y and z axis, thenorm preservation of the Hermite polynomial tensors in (N) is a necessary andsu cient condition for the orthogonality among these functions.This is impor-tant, since this property reduces the discretization problem to nd the weightsWi and poles

! 0;i satisfying solely the norm restrictions, Eq.(11), when n = m.

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If a set !e i, i = 0; ::::; nb 1, of lattice vectors is chosen, the discretizationproblem reduces to nd the weightsWi and a scaling factor A, such that

! 0;i =

A!e i, by solving the norm preservation condition, Eq. (11), with n = m.Considering that the poles !e i are previously known this method was called

quadrature with prescribed abcissas. When the set !e i, i = 0; ::::; nb 1, isregular, the lattice is space lling and enables the use of collision-propagationschemes in the numerical solution of the LBE. When this set is not regular,another numerical method, such as nite dierence, must be used. Surmas etal., [5] and Shan, [3], show several discrete regular and non-regular velocity setsin one, two and three dimensions, obtained with this method.

Finaly, from Eqs. (1) and (3), using a truncated expansion of fMB

! i

,

Eq (4) it is seen that the projections of the MB distribution along the discretepoles

! 0;i are,

fi;eq = Wiem2i2kT0

2kT0m

D=2fMB

! i

= Wi

NXn=0

1

n!a(n)rn (n;

!u 0;)H(n)rn! 0;i

(13)

= Wi

a(0)H(0) (A!e i) + a(1) H(1) (A!e i) +

1

2a(2)H(2) (A!e i) + :::

:(14)

Example: For 1D problems let, e0 = 0; e1 = 1 and e1 = 1. Consider asecond order equilibrium distribution, with N = 2

H(0) = 1; (15)H(1)x = 0x = Aei;x; (16)H(2)xx = 20x 1 = A2e2i;x 1: (17)

The constants a, W0;W1 are determined by solving

b1Xi=0

Wi

hH(n)rn

! 0;i

i2=

1

(2)D=2

Ze

202

hH(n)rn

! 0

i2d! 0; (18)

giving

4

W0 +W1 +W1

=1

(2)D=2

Z 11

e202 d0 (19)

W0 (Ae0)2+W1 (Ae1)

2+W1 (Ae1)

2

=1

(2)D=2

Z 11

e202 20d0 (20)

W0A2e20 1

2+W1

A2e21 1

2+W1

A2e21 1

2=

1

(2)D=2

Z 11

e202

20 1

2d0; (21)

and the discrete equilibrium distributions are

f0;eq

= W0

a(0)H(0) (A!e 0) + a(1) H(1)x (A!e 0) +

1

2a(2)xxH(2)xx (A!e 0)

= W0n

1 + u0A e0 + 1

2(u0u0 +)

A2e0e0 1

= W0n

1 1

2

u20 +

; (22)

f1;eq

= W1

a(0)H(0) (A!e 1) + a(1) H(1) (A!e 1) +

1

2a(2)H(2) (A!e 1)

= W1n

1 + u0A e1 + 1

2(u0u0 +)

A2e1e1 1

= W1n

1 + u0A+

1

2

u20 +

A2 1 ; (23)

f1;eq = W1

a(0)H(0) (A!e 1) + a(1) H(1) (A!e 1) +

1

2a(2)H(2) (A!e 1)

= W1n

1 + u0A e1 + 1

2(u0u0 +)

A2e1e1 1

= W1n

1 u0A+ 1

2

u20 +

A2 1 : (24)

Example:The D2Q9 lattice (Figure 1)The Hermite polynomials are

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Figure 1: The D2Q9 lattice

H(0) = 1; (25)H(1)x = 0x = Aei;x; (26)H(1)y = 0y = Aei;y; (27)H(2)xx = 20x 1 = A2e2i;x 1; (28)H(2)yy = 20y 1 = A2e2i;y 1; (29)H(2)xy = 0x0y = A2ei;xei;y: (30)

The constants a, W0;W1;W2 are determined by solving

b1Xi=0

Wi

hH(n)rn

! 0;i