- Handbook of Mathematical Fluid Dynamics.pdf

Preface
The motion of fluids has intrigued scientists since antiquity and we may say that the
field of mathematical fluid dynamics originated more than two centuries ago. In 1755
Euler [2] gave a mathematical formulation of the principle of conservation of mass in
terms of a partial differential equation. In 1823 Cauchy [1] described conservation of linear
and angular momentum by PDEs. Material symmetry and frame invariance were used by
Cauchy [1] and Poisson [9] to reduce the constitutive equations. The dissipative effects of
internal frictional forces were modeled mathematically by Navier [8], Poisson [9], Saint-
Venant [11] and Stokes [12].
In the 19th century no sharp distinction was drawn between mathematicians and
physicists as we sometime see in more recent times. The formulation of the equations
of fluid motion could be considered as either mathematics or physics. The first work
in fluid dynamics that has a "modern" mathematical flavor may have been done by
Riemann in 1860 on isothermal gas dynamics [10]. He raised and solved the eponymous
problem. Riemann recognized the mathematical nature of the entropy. This notion led
him to his duality method for solving the non-characteristic Cauchy problem for linear
hyperbolic equations. Surprisingly, his paper did not generate the immediate interest of his
contemporaries. What we now call the Cauchy problem for a PDE and the search for its
solution did not have the significance that it is accorded nowadays. Even Poincar6 did not
raise that kind of question in his Th~orie des tourbillons.
For this reason, the birth of Mathematical Fluid Dynamics, in the sense that is commonly
accepted nowadays, must be dated circa 1930. Local-in-time existence of solutions for the
Euler equation of incompressible perfect fluids is proved by Lichtenstein [5] in 1925/28.
Then in 1933 Wolibner [13] proves their persistence. Last, Leray's fundamental analysis of
the Navier-Stokes equations for an incompressible fluid is published in 1934 [3]. As much
as Riemann, Leray developed new mathematical tools which proved to have independent
interest: e.g., weak solutions (that we now call Leray's solutions in this context) and
topological degree (a joint work with Schauder [4]).
Since the 1930s, the interest that mathematicians devote to fluid dynamics has
unceasingly increased. Leading people, such as J. Hadamard, A.N. Kolmogorov, J. von
Neumann and J. Nash made decisive contributions. In 1994, P.-L. Lions was awarded a
Fields medal after his breakthrough on the Boltzmann equation (with R. DiPerna) and on
the Navier-Stokes system of an isentropic fluid (see, for instance, [6]). Today, the topic
displays such a variety of models and questions that thousands of scientists, among them
many mathematicians, focus their research on fluid dynamics.

vi Preface
Because of the intense activity and the rapid increase of our knowledge, it appeared
desirable to set up a landmark. Named "The Handbook of Mathematical Fluid Dynamics",
it is a collection of refereed review articles written by some of the very best specialists
in their discipline. The authors were also chosen for the high quality of their expository
style. We, the editors, are much indebted to our colleagues who enthusiastically accepted
this challenge, and who made great efforts to write for a wide audience. We also thank the
referees who worked hard to ensure the excellent quality of the articles.
Of course, the length of these articles varies considerably since each topic can be narrow
or wide. A few of them have the appearance of a small book. Their authors deserve special
thanks, for the immense work that they achieved and for their generosity in choosing to
publish their work in this Handbook.
At the begining of our editorial work, we decided to restrict the contents to mathematical
aspects of fluid dynamics, avoiding to a large extent the physical and the numerical aspects.
We highly respect these facets of fluid dynamics and we encouraged the authors to describe
the physical meaning of their mathematical results and questions. But we considered that
the physics and the numerics were extremely well developed in other collections of a
similar breadth (see, for instance, several articles in the Handbook of Numerical Analysis,
Elsevier, edited by P. Ciarlet and J.-L. Lions). Furthermore, if we had made a wider choice,
our editing work would have been an endless task!
This has been our only restriction. We have tried to cover many kinds of fluid
models, including ones that are rarefied, compressible, incompressible, viscous or
inviscid, heat conducting, capillary, perfect or real, coupled with solid mechanics or with
electromagnetism. We have also included many kinds of questions: the Cauchy problem,
steady flows, boundary value problems, stability issues, turbulence, etc. These lists are
by no mean exhaustive. We were only limited in some places by the lack, at present, of
mathematical theories.
Our first volume is more or less specialized to compressible issues. There might be valid
mathematical, historical or physical reasons to explain such a choice, arguing, for instance,
for the priority of Riemann's work, or that kinetic models are at the very source of almost
all other fluid models under various limiting regimes. The truth is more fortuitous, namely
that the authors writing on compressible issues were the most prompt in delivering their
articles in final form. The second and third volumes will be primarily devoted to problems
arising in incompressible flows.
Last, but not least, we thank the Editors at Elsevier, who gave us the opportunity
of making available a collection of articles that we hope will be useful to many
mathematicians and those beyond the mathematical community. We are also happy to thank
Sylvie Benzoni-Gavage for her invaluable assistance.
Chicago, Lyon
September 2001
Susan Friedlander and Denis Serre
susan@math.uic.edu
denis.serre @umpa.ens-lyon.fr

Preface vii
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
A.-L. Cauchy, Bull. Soc. Philomathique (1823), 9-13; Exercices de Math6matiques 2 (1827), 42-56, 108-
111; 4 (1829), 293-319.
L. Euler, M6m. Acad. Sci. Berlin 11 (1755), 274-315; 15 (1759), 210-240.
J. Leray, J. Math. Pures Appl. 12 (1933), 1-82; 13 (1934), 331-418; Acta Math. 63 (1934), 193-248.
J. Leray and J. Schauder, Ann. Sci. Ecole Norm. Sup. (3) 51 (1934), 45-78.
L. Lichtenstein, Math. Z. 23 (1925), 89-154; 26 (1926), 196-323; 28 (1928), 387-415; 32 (1930), 608-725.
P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vols. 1, 2, Oxford Univ. Press (1998).
J. Nash, Bull. Soc. Math. France 90 (1962), 487-497.
C.L.M.H. Navier, M6m. Acad. Sci. Inst. France 6 (1822), 375-394.
S.D. Poisson, J. Ecole Polytechnique 13 (1831), 1-174.
B. Riemann, G6tt. Abh. Math. C1.8 (1860), 43-65.
B. de Saint-Venant, C. R. Acad. Sci. Paris 17 (1843).
G.G. Stokes, Trans. Cambridge Philos. Soc. 8 (1849), 207-319.
W. Wolibner, Math. Z. 37 (1933), 698-726.

List of Contributors
Blokhin, A., Sobolev Institute of Mathematics, Novosibirsk, Russia (Ch. 6)
Cercignani, C., Politecnico di Milano, Milano, Italy (Ch. 1)
Chen, G.-Q., Northwestern University, Evanston, IL (Ch. 5)
Fan, H., Georgetown University, Washington DC (Ch. 4)
Feireisl, E., Institute of Mathematics AV (?R, Praha, Czech Republic (Ch. 3)
Galdi, G.E, University of Pittsburgh, Pittsburgh, PA (Ch. 7)
Slemrod, M., University of Wisconsin-Madison, Madison, WI (Ch. 4)
Trakhinin, Yu.,Sobolev Institute of Mathematics, Novosibirsk, Russia (Ch. 6)
Villani, C., UMPA, ENS Lyon, Lyon, France (Ch. 2)
Wang, D., University of Pittsburgh, Pittsburgh, PA (Ch. 5)

CHAPTER 1
The Boltzmann Equation and Fluid Dynamics
C. Cercignani
Dipartimento di Matematica, Politecnico di Milano, Milano, Italy
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. The basic molecular model ......................................... 4
3. The Boltzmann equation ........................................... 5
4. Molecules different from hard spheres ................................... 9
5. Collision invariants .............................................. 10
6. The Boltzmann inequality and the Maxwell distributions ......................... 12
7. The macroscopic balance equations ..................................... 13
8. The H-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
9. Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10. The linearized collision operator ...................................... 21
11. Boundary conditions ............................................. 22
12. The continuum limit ............................................. 25
13. Free-molecule and nearly free-molecule flows ............................... 33
14. Perturbations of equilibria .......................................... 36
15. Approximate methods for linearized problems ............................... 38
16. Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
17. Polyatomic gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
18. Chemistry and radiation ........................................... 52
19. The DSMC method ............................................. 57
20. Some applications of the DSMC method .................................. 61
21. Concluding remarks ............................................. 63
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
HANDBOOK OF MATHEMATICAL FLUID DYNAMICS, VOLUME I
Edited by S.J. Friedlander and D. Serre
9 2002 Elsevier Science B.V. All rights reserved

The Boltzmann equation andfluid dynamics 3
1. Introduction
We say that a gas flow is rarefied when the so-called mean free-path of the gas molecules,
i.e., the average distance covered by a molecule between to subsequent collisions, is not
completely negligible with respect to a typical geometric length (the radius of curvature
of the nose of a flying vehicle, the radius of a pipe, etc.). The most remarkable feature of
rarefied flows is that the Navier-Stokes equations do not apply. One must then resort to the
concepts of kinetic theory of gases and the Navier-Stokes equations must be replaced by
the Boltzmann equation [43].
Thus the Boltzmann equation became a practical tool for the aerospace engineers, when
they started to remark that flight in the upper atmosphere must face the problem of a
decrease in the ambient density with increasing height. This density reduction would
alleviate the aerodynamic forces and heat fluxes that a flying vehicle would have to
withstand. However, for virtually all missions, the increase of altitude is accompanied
by an increase in speed; thus it is not uncommon for spacecraft to experience its peak
heating at considerable altitudes, such as, e.g., 70 km. When the density of a gas decreases,
there is, of course, a reduction of the number of molecules in a given volume and, what is
more important, an increase in the distance between two subsequent collisions of a given
molecule, till one may well question the validity of the Euler and Navier-Stokes equations,
which are usually introduced on the basis of a continuum model which does not take into
account the molecular nature of a gas. It is to be remarked that, as we shall see, the use of
those equations can also be based on the kinetic theory of gases, which justifies them as
asymptotically useful models when the mean free path is negligible.
In the area of environmental problems, the Boltzmann equation is also required.
Understanding and controlling the formation, motion, reactions and evolution of particles
of varying composition and shapes, ranging from a diameter of the order of 0.001 gm to
50 gm, as well as their space-time distribution under gradients of concentration, pressure,
temperature and the action of radiation, has grown in importance, because of the increasing
awareness of the local and global problems related to the emission of particles from electric
power plants, chemical plants, vehicles as well as of the role played by small particles in the
formation of fog and clouds, in the release of radioactivity from nuclear reactor accidents,
and in the problems arising from the exhaust streams of aerosol reactors, such as those
used to produce optical fibers, catalysts, ceramics, silicon chips and carbon whiskers.
One cubic centimeter of atmospheric air at ground level contains approximately 2.5 x
1019 molecules. About a thousand of them may be charged (ions). A typical molecular
diameter is 3 x 10-10 m (3 x 10-4 gm) and the average distance between the molecules
is about ten times as much. The mean free path is of the order of 10-8 m, or 10-2 l,tm.
In addition to molecules and ions one cubic centimeter of air also contains a significant
number of particles varying in size, as indicated above. In relatively clean air, the number
of these particles can be 105 or more, including pollen, bacteria, dust, and industrial
emissions. They can be both beneficial and detrimental, and arise from a number of natural
sources as well as from the activities of all living organisms, especially humans. The
particles can have complex chemical compositions and shapes, and may even be toxic
or radioactive.

4 C. Cercignani
A suspension of particles in a gas is known as an aerosol. Atmospheric aerosols are of
global interest and have important impact on our lives. Aerosols are also of great interest
in numerous scientific and engineering applications [175].
A third area of application of rarefied gas dynamics has emerged in the last quarter of
the twentieth century. Small size machines, called micromachines, are being designed and
built. Their typical sizes range from a few microns to a few millimiters. Rarefied flow
phenomena that are more or less laboratory curiosities in machines of more usual size can
form the basis of important systems in the micromechanical domain.
A further area of interest occurs in the vacuum industry. Although this area existed
for a long time, the expense of the early computations with kinetic theory precluded
applications of numerical methods. The latter could develop only in the context of the
aerospace industry, because the big budgets required till recently were available only there.
The basic parameter measuring the degree of rarefaction of a gas is the Knudsen number
(Kn), the ratio between the mean free path )~ and another typical length. Of course,
one can consider several Knudsen numbers, based on different characteristic lengths,
exactly as one does for the Reynolds number. Thus, in the flow past a body, there are
two important macroscopic lengths: the local radius of curvature and the thickness of the
viscous boundary layer 8, and one can consider Knudsen numbers based on either length.
Usually the second one (Kn~ = )~/8), gives the most severe restriction to the use of Navier-
Stokes equations in aerospace applications.
When Kn is larger than (say) 0.01, the presence of a thin layer near the wall, of thickness
of the order )~ (Knudsen layer), influences the viscous profile in a significant way.
This and other effects are of interest in both high altitude flight and aerosol science; in
particular they are all met by a shuttle when returning to Earth. Another phenomenon of
importance is the formation of shock waves, which are not discontinuity surfaces, but thin
layers (the thickness is zero only if the Euler model is adopted).
When the mean free path increases, one witnesses a thickening of the shock waves,
whose thickness is of the order of 6)~. The bow shock in front of a body merges with
the viscous boundary layer; that is why this regime is sometimes called the merged layer
regime by aerodynamicists. We shall use the other frequently used name of transition
regime.
When Kn is large (few collisions), phenomena related to gas-surface interaction play an
important role. They enter the theory in the form of boundary conditions for the Boltzmann
equation. One distinguishes between free-molecule and nearly free-molecule regimes. In
the first case the molecular collisions are completely negligible, while in the second they
can be treated as a perturbation.
2. The basic molecular model
According to kinetic theory, a gas in normal conditions (no chemical reactions, no
ionization phenomena, etc.) is formed of elastic molecules rushing hither and thither at
high speed, colliding and rebounding according to the laws of elementary mechanics.
Monatomic molecules of a gas are frequently assumed to be hard, elastic, and perfectly
smooth spheres. One can also consider these molecules to be centers of forces that move

according to the laws of classical mechanics. More complex models are needed to describe
polyatomic molecules.
The rules generating the dynamics of many spheres are easy to describe: thus, e.g., if
no body forces, such as gravity, are assumed to act on the molecules, each of them will
move in a straight line unless it happens to strike another molecule or a solid wall. The
phenomena associated with this dynamics are not so simple, especially when the number
of spheres is large. It turns out that this complication is always present when dealing with
a gas, because the number of molecules usually considered is extremely large: there are
about 2.7.1019 in a cubic centimeter of a gas at atmospheric pressure and a temperature
of 0~
Given the vast number of particles to be considered, it would of course be a hopeless task
to attempt to describe the state of the gas by specifying the so-called microscopic state, i.e.,
the position and velocity of every individual sphere; we must have recourse to statistics. A
description of this kind is made possible because in practice all that our typical observations
can detect are changes in the macroscopic state of the gas, described by quantities such as
density, bulk velocity, temperature, stresses, heat-flow, which are related to some suitable
averages of quantities depending on the microscopic state.
3. The Boltzmann equation
The exact dynamics of N particles is a useful conceptual tool, but cannot in any way
be used in practical calculations because it requires a huge number of real variables (of
the order of 102~ The basic tool is the one-particle probability density, or distribution
function P(1)(x, ~, t). The latter is a function of seven variables, i.e., the components of
the two vectors x and ~ and time t.
Let us consider the meaning of p(1) (x, ~, t); it gives the probability density of finding
one fixed particle (say, the one labelled by 1) at a certain point (x, ~) of the six-dimensional
reduced phase space associated with the position and velocity of that molecule. In order
to simplify the treatment, we shall for the moment assume that the molecules are hard
spheres, whose center has position x. When the molecules collide, momentum and kinetic
energy must be conserved; thus the velocities after the impact, ~'l and ~'2, are related to
those before the impact, ~l and ~2, by
~' In (~1 '
1 =~1--n 9 --~2)]
'= In (~1 '
~2 ~2 "+- n 9 -- ~2)]
(3.1)
where n is the unit vector along ~1 -- ~t Note that the relative velocity
1"
V --" ~1 -- ~2 (3.2)
satisfies
V'= V - 2n(n. V), (3.3)

6 C. Cercignani
i.e., undergoes a specular reflection at the impact. This means that if we split V at the point
of impact into a normal component Vn, directed along n and a tangential component Vt
(in the plane normal to n), then Vn changes sign and Vt remains unchanged in a collision.
We can also say that n bisects the directions of V and -W = -(~t 1 - ~i)"
Let us remark that, in the absence of collisions, p(1) would remain unchanged along
the trajectory of a particle. Accordingly we must evaluate the effects of collisions on the
time evolution of p(1). Note that the probability of occurrence of a collision is related to
the probability of finding another molecule with a center at exactly one diameter from the
center of the first one, whose distribution function is p(1). Thus, generally speaking, in
order to write the evolution equation for p(1) we shall need another function, p(2), which
gives the probability density of finding, at time t, the first molecule at Xl with velocity ~1
and the second at X2 with velocity ~2; obviously p(2) = p(2) (Xl, x2, ~ 1, ~2, t). Hence p(1)
satisfies an equation of the following form:
Op(1) Op(l)
-Jr-~1" -- G - L. (3.4)
Ot OX1
Here L dXl d~l dt gives the expected number of particles with position between Xl and
x 1-+-dx 1and velocity between ~1 and ~1-+d~ 1which disappear from these ranges of values
because of a collision in the time interval between t and t + dt and G dxl d~ 1dt gives the
analogous number of particles entering the same range in the same time interval. The count
of these numbers is easy, provided we use the trick of imagining particle 1 as a sphere at
rest and endowed with twice the actual diameter 0- and the other particles to be point
masses with velocity (~i - ~l) = Vi. In fact, each collision will send particle 1 out of the
above range and the number of the collisions of particle 1 will be the number of expected
collisions of any other particle with that sphere. Since there are exactly (N - 1) identical
point masses and multiple collisions are disregarded, G - (N - 1)g and L -- (N - 1)/,
where the lower case letters indicate the contribution of a fixed particle, say particle 2. We
shall then compute the effect of the collisions of particle 2 with particle 1. Let x2 be a point
of the sphere such that the vector joining the center of the sphere with x2 is an, where n
is a unit vector. A cylinder with height [V. n[ dt (where we write just V for V2) and base
area dS = 0-2 dn (where dn is the area of a surface element of the unit sphere about n) will
contain the particles with velocity ~2 hitting the base dS in the time interval (t, t + dt); its
volume is 0-2 dn[V. n[ dt. Thus the number of collisions of particle 2 with particle 1 in the
ranges (Xl, Xl + dxl), (~1, ~1 + d~l), (X2, X2 + dx2), (~2, ~2 + d~2), (t,t +dt) occuring at
points of dS is p(2)(Xl, x2,/~ 1, ~2, t) dxl d/~1d~2~ dn[V2 9n[ dt. If we want the number of
collisions of particle 1 with 2, when the range of the former is fixed but the latter may have
any velocity/~ 2 and any position x2 on the sphere (i.e., any n), we integrate over the sphere
and all the possible velocities of particle 2 to obtain:
1dxl d/~1dt
= dxl d~l dt f R3fu- P(2)(Xl' Xl + 0-n'/~l' ~2' t)lV" nl0-2dnd~2, (3.5)

TheBoltzmannequationandfluiddynamics 7
where B- is the hemisphere corresponding to V. n < 0 (the particles are moving one
toward the other before the collision). Thus we have the following result:
L--(N-1)O-ZJR3 ft3- P(Z)(xl'xl+o-n'~jl'~2't)](~2-~l)'nld~zdn"
(3.6)
The calculation of the gain term G is exactly the same as the one for L, except for the fact
that we have to integrate over the hemisphere B +, defined by V. n > 0 (the particles are
moving away one from the other after the collision). Thus we have:
G--(N-1)O-2 fR3 f13+P(2)(Xl'Xl+o-n'~l'~2't)l(~2-~l)'nld~2dn"
(3.7)
We can now insert in Equation (3.4) the information that the probability density p(2) is
continuous at a collision; in other words, although the velocities of the particles undergo
the discontinuous change described by Equations (3.1), we can write"
p(2) (x1, ~ 1, x2, ~2, t) -- p(2)(x1, ~1 -- n(n. V), x2, ~2 -+-n(n. V), t)
if Ix1 - x21 -- o-. (3.8)
For brevity, we write (in agreement with Equations (3.1))"
~fl -- ~1 -- n(n. V), ~2 -- ~2 -+- n(n. V). (3.9)
Inserting Equation (3.8) in Equation (3.5) we thus obtain:
G--(N- 1)o 2 JR3 ft~+ p(2)(Xl, x, + o-n, ~'1,~2, t)[(~2 - ~l)" n[ d~2 dn
(3.10)
which is a frequently used form. Sometimes n is changed into -n in order to have the same
integration range as in L; the only change (in addition to the change in the range) is in the
second argument of p(2), which becomes Xl - o-n.
At this point we are ready to understand Boltzmann's argument. N is a very large number
and o- (expressed in common units, such as, e.g., centimeters) is very small; to fix the ideas,
let us consider a box whose volume is 1 cm 3 at room temperature and atmospheric pressure.
Then N ~ 1020 and o- ~ 10-8 cm. Then (N - 1)o-2 ~ No 2 ~ 104 cm2 -- 1 m2 is a sizable
quantity, while we can neglect the difference between Xl and Xl -4-o-n. This means that the
equation to be written can be rigorously valid only in the so called Boltzmann-Grad limit,
when N --+ cxz,o- --+ 0 with No 2 finite.

8 C. Cercignani
In addition, the collisions between two preselected particles are rather rare events. Thus
two spheres that happen to collide can be thought to be two randomly chosen particles and
it makes sense to assume that the probability density of finding the first molecule at x l with
velocity ~l and the second at x2 with velocity ~2 is the product of the probability density
of finding the first molecule at Xl with velocity ~1 times the probability density of finding
the second molecule at x2 with velocity ~2. If we accept this we can write (assumption of
molecular chaos):
P(2)(Xl, ~1, x2, ~2, t ) -- p(1)(Xl,~l,t)p(1)(x2,~2, t) (3.11)
for two particles that are about to collide, or, letting a = 0
P(2)(Xl, ~1, Xl + o'n, ~2, t ) = P(1)(Xl,l~l,t)p(1)(Xl,~2, t)
for (~2 - ~1)" n < 0. (3.12)
Thus we can apply this recipe to the loss term (3.4) but not to the gain term in the form (3.5).
It is possible, however, to apply Equation (3.12) (with ~'1,~2' in place of ~1, ~2) to the
form (3.8) of the gain term, because the transformation (3.9) maps the hemisphere 13+
onto the hemisphere B-.
If we accept all the simplifying assumptions made by Boltzmann, we obtain the
following form for the gain and loss terms:
L L ' t)](~ -~).nld~2dn ,
G = Na 2 p(1) (Xl, ~:1,t) p(1) (Xl, ~2' 2 1
3 -
(3.13)
L--Nff2 fR3 fl~- P(1)(Xl'l~l't)p(1)(Xl'~2't)[(~2-~l)'ll]d~2dll" (3.14)
By inserting these expressions in Equation (3.6) we can write the Boltzmann equation in
the following form:
Op(1) Op(1)
+ ~.
at OXl
s 't)
= Na 2 [P(1)(Xl,l~'l,t)p(1)(Xl,l~2 '
3 -
-- P(1)(Xl,l;1,t)p(1)(Xl,l~2, t)]](l;2 - ~1)" n] d~2 dn. (3.15)
We remark that the expressions for ~'1 and ~2t given in Equations (3.1) are by no means the
only possible ones. In fact we might use a different unit vector to, directed as V', instead
of n. Then Equations (3.1) is replaced by:
, - 1
, 1
~J2- ~- ~l~Jl- ~J21to,
(3.16)

where ~ = 89
(~j1+ ~2) is the velocity of the center of mass. The relative velocity V satisfies
v' : ~lVI. (3.17)
The recipes (3.13) and (3.14) can be justified at various levels of rigor [36,113,39,47].
We finally mention that we have for simplicity neglected any body force acting on the
molecules, such as gravity. It is not hard to take them into account; if the force per unit
mass acting on the molecules is denoted by X, then a term X. 0p(1)/0~1 must be added to
the left-hand side of Equation (3.8).
4. Molecules different from hard spheres
In the previous section we have discussed the Boltzmann equation when the molecules
are assumed to be identical hard spheres. There are several possible generalizations of this
molecular model, the most obvious being the case of molecules which are identical point
masses interacting with a central force, a good general model for monatomic gases. If
the range of the force extends to infinity, there is a complication due to the fact that two
molecules are always interacting and the analysis in terms of "collisions" is no longer
possible. If, however, the gas is sufficiently dilute, we can take into account that the
molecular interaction is negligible for distances larger than a certain a (the "molecular
diameter") and assume that when two molecules are at a distance smaller than a, then no
other molecule is interacting with them and the binary collision analysis considered in the
previous section can be applied. The only difference arises in the factor o-21(~2 - ~1)" nl
which turns out to be replaced by a function of V = I~2 -- ~ I I and the angle 0 between n
and V ([39,35,42]). Thus the Boltzmann equation for monatomic molecules takes on the
following form:
Op(l) Op(1)
+ ~.
Ot Oxl
=NfR3 it3_[V(1)(x1 , ~tl,/) e(1)(x1 , ~j~, t)
- e(1)(Xl,~,t)p(1)(Xl,~2, t)]n(o, 1~2 -- ~1) d~2d0 d~, (4.1)
where e is the other angle which, together with 0, identifies the unit vector n. The function
B(O,V) depends, of course, on the specific law of interaction between the molecules. In
the case of hard spheres, of course
B(O,1~2 - ~ll) = cos0 sin01~2 - ~ll- (4.2)
In spite of the fact that the force is cut at a finite range cr when writing the Boltzmann
equation, infinite range forces are frequently used. This has the disadvantage of making
the integral in Equation (4.1) rather hard to handle; in fact, one cannot split it into the
difference of two terms (the loss and the gain), because each of them would be a divergent

10 C. Cercignani
integral. This disadvantage is compensated in the case of power law forces, because one
can separate the dependence on 0 from the dependence upon V. In fact, one can show [39,
35] that, if the intermolecular force varies as the n-th inverse power of the distance, then
B(O, 1/~2-/~11) = ffi(O)1/~2- I~11
(n-5)/(n-1), (4.3)
where fl(O) is a non-elementary function of 0 (in the simplest cases it can be expressed
by means of elliptic functions). In particular, for n = 5 one has the so-called Maxwell
molecules, for which the dependence on V disappears.
Sometimes the artifice of cutting the grazing collisions corresponding to small values
of l0 - zr/2l is used (angle cutoff). In this case one has both the advantage of being able
to split the collision term and of preserving a relation of the form (4.3) for power-law
potentials.
Since solving of the Boltzmann equation with actual cross sections is complicated, in
many numerical simulations use is made of the so-called variable hard sphere model in
which the diameter of the spheres is an inverse power law function of the relative speed V
(see [43]).
Another important case is when we deal with a mixture rather than with a single gas.
In this case we have n unknowns, if n is the number of the species, and n Boltzmann
equations; in each of them there are n collision terms to describe the collision of a molecule
with other molecules of all the possible species [43,39].
If the gas is polyatomic, then the gas molecules have other degrees of freedom in addition
to the translation ones. This in principle requires using quantum mechanics, but one can
devise useful and accurate models in the classical scheme as well. Frequently the internal
energy Ei is the only additional variable that is needed; in which case one can think of
the gas as of a mixture of species [43,39], each differing from the other because of the
value of Ei. If the latter variable is discrete we obtain a strict analogy with a mixture;
otherwise we have a continuum of species. We remark that in both cases, kinetic energy is
not preserved by collisions, because internal energy also enters into the balance; this means
that a molecule changes its "species" when colliding. This is the simplest example of a
"reacting collision", which may be generalized to actual chemical species when chemical
reactions occur. The subject of mixture and polyatomic gases will be taken up again in
Section 16.
5. Collision invariants
Before embarking in a discussion of the properties of the solutions of the Boltzmann
equation we remark that the unknown of the latter is not always chosen to be a probability
density as we have done so far; it may be multiplied by a suitable factor and transformed
into an (expected) number density or an (expected) mass density (in phase space, of
course). The only thing that changes is the factor in front of Equations (3.1) which is no
longer N. In order to avoid any commitment to a special choice of that factor we replace
NB(O, V) by B(O, V) and the unknown P by another letter, f (which is also the most
commonly used letter to denote the one-particle distribution function, no matter what its

The Boltzmann equation and fluid dynamics 11
normalization is). In addition, we replace the current velocity variable ~1 simply by ~ and
2 by ~,. Thus we rewrite Equation (4.1) in the following form:
Of Of ~ fB (f'ft*-ff*)B(O V)d~ dOde,
o-; +~~x= ~ _ ' *
(5.1)
where V - [~ - ~, 1.The velocity arguments ~i and ~, in f' and f,~ are of course given by
Equations (3.1) (or (3.15)) with the suitable modification.
The right-hand side of Equation (5.1) contains a quadratic expression Q(f, f), given
by:
Q(f' f)= fR3fs2 (f' f',- ff,)B(O, V)dl~,dOde. (5.2)
This expression is called the collision integral or, simply, the collision term and the
quadratic operator Q goes under the name of collision operator. In this section we study
some elementary properties of Q. Actually it turns out that it is more convenient to study
the slightly more general bilinear expression associated with Q(f, f), i.e.:
1~ fs (f'g1*+g' f*- fg*-gf*)13(O' V)dl;*dOde"
Q(f' g) = -2 3 2
(5.3)
It is clear that when g = f, Equation (5.3) reduces to Equation (5.2) and
Q(f, g) -- Q(g, f). (5.4)
Our first aim is to indicate a basic property of the eightfold integral:
fR Q(f, g)r
3
where f, g and ~b are functions such that the indicated integrals exist and the order of
integration does not matter. Simple manipulations (see [43,39,35]) give the following
result:
n O(f, g)~b(~) d/~
,, , ,
= _ (f g, + g f, - fg, - gf,)
8 3 3 -
x (4)+ 4), - dp'- r V) dl~,dl~dOde. (5.6)

12 C. Cercignani
This relation expresses a basic property of the collision term, which is frequently used. In
particular, when g = f, Equation (5.6) reads
R3 Q(f' f)r
= (f f, - ff,)(dp + ~b, - ~b' - ~bl,)B(0, V) d~j d~jd0 de. (5.7)
3 3 - *
We now observe that the integral in Equation (5.6) is zero independent of the particular
functions f and g, if
r + r = r + r (5.8)
is valid almost everywhere in velocity space. Since the integral appearing in the left-hand
side of Equation (5.7) is the rate of change of the average value of the function 4~ due
to collisions, the functions satisfying Equation (5.8) are called "collision invariants". It
can be shown (see, e.g., [39]) that a continuous function 4~has the property expressed by
Equation (5.8) if and only if
~b(~) =a +b.~ +cl~l 2, (5.9)
where a and c are constant scalars and b a constant vector. The assumption of continuity
can be considerably relaxed [5,40,6]. The functions 7t0 = 1, (Tel, 7t2, ~P3)= ~, 7t4 = I~12
are usually called the elementary collision invariants; they span the five-dimensional
subspace of the collision invariants.
6. The Boltzmann inequality and the Maxwell distributions
In this section we investigate the existence of positive functions f which give a vanishing
collision integral:
= (f f, - f f,) 13(0, V) d~,, dOde - O.
3 -
(6.1)
In order to solve this equation, we prove a preliminary result which plays an important
role in the theory of the Boltzmann equation: if f is a nonnegative function such that
log f Q(f, f) is integrable and the manipulations of the previous section hold when
q~= log f, then the Boltzmann inequality:
fR log f Q(f, f) d~j ~<0 (6.2)
holds; further, the equality sign applies if, and only if, log f is a collision invariant, or,
equivalently:
f = exp(a + b. ~j+ cl~j12). (6.3)

To prove Equation (6.2) it is enough to use Equation (4.11) with r = log f:
~3 f Q(f' f) d~
log
-l fRfB l~ V)dtjdtj de
3 - ~ *
and Equation (6.2) follows thanks to the elementary inequality
(6.4)
(z - y)log(y/z) <~0 (y, z e R+). (6.5)
Equation (6.5) becomes an equality if and only if y = z; thus the equality sign holds in
Equation (6.2) if and only if:
f' f~, = f f, (6.6)
applies almost everywhere. But, taking the logarithms of both sides of Equation (6.6), we
find that r = log f satisfies Equation (5.8) and is thus given by Equation (5.9). f = exp(r
is then given by Equation (6.3).
We remark that in the latter equation c must be negative, since f must be integrable.
If we let c = -/3, b = 2fly (where v is another constant vector) Equation (6.3) can be
rewritten as follows:
f -- A exp(-fi I~ - vl2), (6.7)
where A is a positive constant related to a, c, Ibl2 (/3,v, A constitute a new set of con-
stants). The function appearing in Equation (3.7) is the so called Maxwell distribution or
Maxwellian. Frequently one considers Maxwellians with v = 0 (nondrifting Maxwellians),
which can be obtained from drifting Maxwellians by a change of the origin in velocity
space.
Let us return now to the problem of solving Equation (6.1). Multiplying both sides by
log f gives Equation (6.2) with the equality sign. This implies that f is a Maxwellian,
by the result which has just been proved. Suppose now that f is a Maxwellian; then
f = exp(r where r is a collision invariant and Equation (6.6) holds; Equation (6.1)
then also holds. Thus there are functions which satisfy Equation (6.1) and they are all
Maxwellians, Equation (6.7).
7. The macroscopic balance equations
In this section we compare the microscopic description supplied by kinetic theory with
the macroscopic description supplied by continuum gas dynamics. For definiteness, in this
section f will be assumed to be an expected mass density in phase space. In order to obtain
a density, p = p (x, t), in ordinary space, we must integrate f with respect to ~:
P = s f d/~. (7.1)

14 C. Cercignani
The bulk velocity v of the gas (e.g., the velocity of a wind), is the average of the molecular
velocities ~ at a certain point x and time instant t; since f is proportional to the probability
for a molecule to have a given velocity, v is given by
fR31~f dl~
v = fR 3f d/~ (7.2)
(the denominator is required even if f is taken to be a probability density in phase
space, because we are considering a conditional probability, referring to the position x).
Equation (7.2) can also be written as follows:
pv -- ~3/~ f d/~ (7.3)
or, using components:
pvi -- fR3 ~if d/~ (i = 1, 2, 3). (7.4)
The bulk velocity v is what we can directly perceive of the molecular motion by means of
macroscopic observations; it is zero for a gas in equilibrium in a box at rest. Each molecule
has its own velocity ~ which can be decomposed into the sum of v and another velocity
c =/~ - v (7.5)
called the random or peculiar velocity; c is clearly due to the deviations of ~ from v. It is
also clear that the average of c is zero.
The quantity PVi which appears in Equation (7.4) is the i-th component of the mass flow
or, alternatively, of the momentum density of the gas. Other quantities of similar nature are:
the momentum flow
mij -- fR3 ~i~j f d~ (i, j -- 1, 2, 3); (7.6)
the energy density per unit volume:
1URI~J12f d/~; (7.7)
W~ 3
the energy flow:
1 fR ~il/~12fd~
ri -- -2 3
(i, j = 1, 2, 3). (7.8)

Equation (7.6) shows that the momentum flow is described by the components of a
symmetric tensor of second order. The defining integral can be re-expressed in terms of
c and v. We have [43,39,35]:
mij =- pvi l)j -+-Pij, (7.9)
where:
pij -- fR3 CiCj f d~j (i, j -- 1, 2, 3) (7.10)
plays the role of the stress tensor (because the microscopic momentum flow associated
with it is equivalent to forces distributed on the boundary of any region of gas, according
to the macroscopic description).
Similarly one has [43,39,35]:
1
w -- plvl 2 + pe, (7.11)
where e is the internal energy per unit mass (associated with random motions) defined by:
1 s le12fd~j. (7.12)
pe-- -~ 3
and:
(') k
,,
ri -- pvi _v_2 -+-e + vj Pij -+-qi
j=l
(i = 1, 2, 3), (7.13)
where qi are the components of the heat-flow vector:
if
qi -- -~ ci ICl2f d~j. (7.14)
The decomposition in Equation (7.13) shows that the microscopic energy flow is a sum of
a macroscopic flow of energy (both kinetic and internal), of the work (per unit area und
unit time) done by stresses, and of the heat-flow.
In order to complete the connection, as a simple mathematical consequence of the
Boltzmann equation, one can derive five differential relations satisfied by the macroscopic
quantities introduced above; these relations describe the balance of mass, momentum and
energy and have the same form as in continuum mechanics. To this end let us consider the
Boltzmann equation
Of Of
+/~. - Q (f, f) (7 15)
at "

16 C. Cercignani
If we multiply both sides by one of the elementary collision invariants ~Pc~ (a =
0, 1, 2, 3, 4), defined in Section 4, and integrate with respect to ~, we have, thanks to
Equation (1.15) with g = f and 4~= ~Pc~"
fR TtO~(l~)Q(f,f) dl~= O,
3
(7.16)
and hence, if it is permitted to change the order by which we differentiate with respect to t
and integrate with respect to ~"
3
Ot ~/af d~ + E ~xi ~i~a f d~ -- 0
i=1
(a -- 1, 2, 3, 4). (7.17)
If we take successively ot -- 0, 1, 2, 3, 4 and use the definitions introduced above, we obtain
3
Op i~1 O
a---t -~" . ~X i ( p v i ) = O,
(7.18)
3
0 0
Ot (pvj) -+-Z -~xi (pvivj -+-Pij) - 0
i=1
(j = 1, 2, 3), (7.19)
)
Ot -2plvle + pe
-[- ~ pl)i Ivl 2 + e + UjPij -+"qi = O.
i=1 j=l
(7.20)
The considerations of this section apply to all the solutions of the Boltzmann equation.
The definitions, however, can be applied to any positive function for which they make
sense. In particular if we take f to be a Maxwellian in the form (5.7), we find that the
constant vector v appearing there is actually the bulk velocity as defined in Equation (7.2)
while fl and A are related to the internal energy e and the density p in the following way:
fl = 3/(4e), A- p(4yre/3) -3/2. (7.21)
Furthermore the stress tensor turns out to be diagonal (Pij -- (2pe)6ij, where ~ij is the
so-called "Kronecker delta" (= 1 if i -- j;= 0 if i 5~ j)), while the heat-flow vector is
zero.
We end this section with the definition of pressure p in terms of f; p is nothing else
than 1/3 of the spur or trace (i.e., the sum of the three diagonal terms) of pij and is thus
given by:
1 f. [elZfd,~" (7.22)
P=3 3

If we compare this with the definition of the specific internal energy e, given in
Equation (3.11), we obtain the relation:
2
p = -=pe. (7.23)
.5
This relation also suggests the definition of temperature, according to kinetic theory,
T = (2e)/R, where R is the gas constant equal to the universal Boltzmann constant k
divided by the molecular mass m. Thus:
1 s lel2fd/~" (7.24)
T = 3,oR 3
8. The H-theorem
Let us consider a further application of the properties of the collision term Q(f, f) of the
Boltzmann equation:
Of of
+ 1~.-2- = Q(f, f). (8.1)
Ot ox
If we multiply both sides of this equation by log f and integrate with respect to ~j, we
obtain:
07-[ 0
Ot + -~x "J = S, (8.2)
where
7-[= fR3 f log f ds
e, (8.3)
J = fR3 f~f log f d/~, (8.4)
S = fR3 log f Q(f, f) d~. (8.5)
Equation (8.2) differs from the balance equations considered in the previous section
because the right-hand side, generally speaking, does not vanish. We know, however, that
the Boltzmann inequality, Equation (5.2), implies:
S~<0 and S=0 iff fisaMaxwellian. (8.6)
Because of this inequality, Equation (8.2) plays an important role in the theory of
the Boltzmann equation. We illustrate the role of Equation (8.6) in the case of space-

18 C. Cercignani
homogeneous solutions. In this case the various quantities do not depend on x and
Equation (8.2) reduces to
07-t
= S ~<0. (8.7)
Ot
This implies the so-called H-theorem (for the space homogeneous case): 7-/is a decreasing
quantity, unless f is a Maxwellian (in which case the time derivative of 7-/ is zero).
Remember now that in this case the densities p, pv and pe are constant in time; we can
thus build a Maxwellian M which has, at any time, the same p, v and e as any solution
f corresponding to given initial data. Since 7-/decreases unless f is a Maxwellian (i.e.,
f = M), it is tempting to conclude that f tends to M when t ~ cx~. This conclusion
is, however, unwarranted from a purely mathematical viewpoint, without a more detailed
consideration of the source term S in Equation (8.7), for which [47] should be consulted.
If the state of the gas is not space-homogeneous, the situation becomes more complicated.
In this case it is convenient to introduce the quantity
H- fs2 7-/dx, (8.8)
where s is the space domain occupied by the gas (assumed here to be time-independent).
Then Equation (8.2) implies
dt ~< J. n &r, (8.9)
~2
where n is the inward normal and dcr the surface element on OS2. Clearly, several situations
may arise (see [43] and [47] for a detailed discussion).
It should be clear that H has the properties of entropy (except for the sign); this
identification is strengthened when we evaluate H in an equilibrium state (see [43,39,
35]) because it turns out to coincide with the expression of a perfect gas according to
equilibrium thermodynamics, apart from a factor -R. A further check of this identification
is given by an inequality satisfied by the right-hand side of Equation (8.9) when the gas is
able to exchange energy with a solid wall bounding S2 (see Section 11 and [43,39,47]).
9. Model equations
When trying to solve the Boltzmann equation for practical problems, one of the major
shortcomings is the complicated structure of the collision term, Equation (4.2). When one
is not interested in fine details, it is possible to obtain reasonable results by replacing the
collision integral by a so-called collision model, a simpler expression J (f) which retains
only the qualitative and average properties of the collision term Q(f, f). The equation for
the distribution function is then called a kinetic model or a model equation.

The most widely known collision model is usually called the Bhatnagar, Gross and
Krook (BGK) model, although Welander proposed it independently at about the same time
as the above mentioned authors [14,173]. It reads as follows:
,l(f) -- v[~(l~) - f (l~)], (9.1)
where the collision frequency v is independent of ~ (but depends on the density p and the
temperature T) and q~ denotes the local Maxwellian, i.e., the (unique) Maxwellian having
the same density, bulk velocity and temperature as f:
-- p(2yr RT) -3/2exp[- I/~- vI2/(2RT)]. (9.2)
Here p, v, T are chosen is such a way that for any collision invariant ~pwe have
~3 r d~ -- JR3 r d~. (9.3)
It is easily checked that, thanks to Equation (9.3):
(a) f and q~ have the same density, bulk velocity and temperature;
(b) J (f) satisfies conservation of mass, momentum and energy; i.e., for any collision
invariant:
s ~(/~)J(f) d~ -- O; (9.4)
(c) J (f) satisfies the Boltzmann inequality
fRlog f J(f) d/~ ~<0
3
(9.5)
the equality sign holding if and only if, f is a Maxwellian.
It should be remarked that the nonlinearity of the BGK collision model, Equation (9.1),
is much worse than the nonlinearity in Q(f, f); in fact the latter is simply quadratic in
f, while the former contains f in both the numerator and denominator of an exponential,
because v and T are functionals of f, defined by Equations (6.2) and (6.27).
The main advantage in the use of the BGK model is that for any given problem one
can deduce integral equations for p, v, T, which can be solved with moderate effort on a
computer. Another advantage of the BGK model is offered by its linearized form (see [43,
113,35]).
The BGK model has the same basic properties as the Boltzmann collision integral, but
has some shortcomings. Some of them can be avoided by suitable modifications, at the
expense, however, of the simplicity of the model. A first modification can be introduced in
order to allow the collision frequency v to depend on the molecular velocity, more precisely
on the magnitude of the random velocity c (defined by Equation (6.5)), while requiting that
Equation (9.4) still holds. All the basic properties, including Equation (9.5), are retained,
but the density, velocity and temperature appearing in q~ are not the local ones of the gas,

20 C. Cercignani
but some fictitious local parameters related to five functionals of f different from p, v, T;
this follows from the fact that Equation (9.3) must now be replaced by
fR3 v(Icl)~r(~J)cP(~J)d~J- fR3 v(Icl)!#(~j)f(~j) d~j. (9.6)
A different kind of correction to the BGK model is obtained when a complete agreement
with the compressible Navier-Stokes equations is required for large values of the collision
frequency. In fact the BGK model has only one parameter (at a fixed space point and time
instant), i.e., the collision frequency v; the latter can be adjusted to give a correct value for
either the viscosity/z or the heat conductivity x, but not for both. This is shown by the fact
that the Prandtl number Pr = lZ/CpX (where Cp is the specific heat at constant pressure)
turns out [39,35] to be unity for the BGK model, while it is about to 2/3 for a monatomic
gas (according to both experimental data and the Boltzmann equation). In order to have a
correct value for the Prandtl number, one is led [87,62] to replacing the local Maxwellian
in Equation (9.1) by
rP(I~) = p(rc)-3/2(detA)I/e exp(-(~j - v). [A(/~- v)]), (9.7)
where A is the inverse of the matrix
A-1 = (2RT/Pr)I- 2(1 - Pr)p/(p Pr), (9.8)
where Iis the identity and p the stress matrix. If we let Pr -- 1, we recover the BGK model.
Only recently [3] this model (called ellipsoidal statistical (ES) model) has been shown
to possess the property expressed by Equation (9.5). Hence the H-theorem holds for the
ES model.
Other models with different choices of q5 have been proposed [151,35] but they are not
so interesting, except for linearized problems (see [43,39,35]).
Another model is the integro-differential model proposed by Lebowitz, Frisch and
Helfand [114], which is similar to the Fokker-Planck equation used in the theory of
Brownian motion. This model reads as follows:
3 [02f
J (f)= D ~ -UfT~2-~-~ - o,)y]],
RT O~k
(9.9)
where D is a function of the local density p and the local temperature T. If we take D
proportional to the pressure p = pRT, Equation (9.9) has the same kind of nonlinearity
(i.e., quadratic) as the true Boltzmann equation.
The idea of kinetic models can be naturally extended to mixtures and polyatomic gases
[151,127,81,43].

10. The linearized collision operator
On several occasions we shall meet the so-called linearized collision operator, related to
the bilinear operator defined in Equation (5.3) by
Lh = 2M -1Q(Mh, M), (10.1)
where M is a Maxwellian distribution, usually with zero bulk velocity. When we want
to emphasize the fact that we linearize with respect to a given Maxwellian, we write LM
instead of just L.
A more explicit expression of Lh reads as follows
Lh--f~ fB M*(h'+h~*-h*-h)B(O'V)dl~*dn'
3 +
(10.2)
where we have taken into account that M'M', = MM,. Because of Equation (4.10) (with
Mh in place of f, M in place of g and g in place of q~),we have the identity:
f~t3 MgLh dl~
= lf, f, (h'+h', . . . .
h h,)(f+f, g g,)
4 3 3 +
x 13(0, V) d/~, d/~dn. (10.3)
This relation expresses a basic property of the linearized collision term. In order to make
it clear, let us introduce a bilinear expression, the scalar product in the Hilbert space of
square summable functions of ~ endowed with a scalar product weighted with M:
(g, h) -- f~3 ~hM d~j, (10.4)
where the bar denotes complex conjugation. Then Equation (1.7) (with ~, in place of g)
gives (thanks to the symmetry of the expression in the right-hand side of Equation (10.3)
with respect to the interchange g r h):
(g, Lh) -- (Lg, h). (10.5)
Further:
(h, Lh) <~0 (10.6)
and the equality sign holds if and only if
hf + hi, - h - h, = 0, (10.7)
i.e., if and only if h is a collision invariant.

22 C. Cercignani
Equations (10.5) and (10.6) indicate that the operator L is symmetric and non-positive
in the aforementioned Hilbert space.
11. Boundary conditions
The Boltzmann equation must be accompanied by boundary conditions, which describe
the interaction of the gas molecules with the solid walls. It is to this interaction that one
can trace the origin of the drag and lift exerted by the gas on the body and the heat transfer
between the gas and the solid boundary.
The study of gas-surface interaction may be regarded as a bridge between the kinetic
theory of gases and solid state physics and is an area of research by itself. The difficulties
of a theoretical investigation are due, mainly, to our lack of knowledge of the structure
of surface layers of solid bodies and hence of the effective interaction potential of the
gas molecules with the wall. When a molecule impinges upon a surface, it is adsorbed
and may form chemical bonds, dissociate, become ionized or displace surface molecules.
Its interaction with the solid surface depends on the surface finish, the cleanliness of the
surface, its temperature, etc. It may also vary with time because of outgassing from the
surface. Preliminary heating of a surface also promotes purification of the surface through
emission of adsorbed molecules. In general, adsorbed layers may be present; in this case,
the interaction of a given molecule with the surface may also depend on the distribution
of molecules impinging on a surface element. For a more detailed discussion the reader
should consult [39,110] and [41].
In general, a molecule striking a surface with a velocity ~ reemerges from it with a
velocity ~ which is strictly determined only if the path of the molecule within the wall
can be computed exactly. This computation is very hard, because it depends upon a great
number of details, such as the locations and velocities of all the molecules of the wall and
an accurate knowledge of the interaction potential. Hence it is more convenient to think in
terms of a probability density R(~ ~--+ ~; x, t; r) that a molecule striking the surface with
velocity between ~ and ~j~+ d~~at the point x and time t will re-emerge at practically
the same point with velocity between ~ and ~ + d~ after a time interval r (adsorption or
sitting time). If R is known, then we can easily write down the boundary condition for the
distribution function f (x, ~, t). To simplify the discussion, the surface will be assumed to
be at rest. A simple argument ([43,39,35]) then gives:
f(x,/~, t)I/~ .n[
= dr R(~j' --+ ~; x, t; r)f(x, ~', t - r)l~'' n[ d~'
'.n<O
(x ~ 0S'2,~. n > 0). (11.1)
The kernel R can be assumed to be independent of f under suitable conditions which
we shall not detail here [39,110,41]. If, in addition, the effective adsorption time is small
compared to any characteristic time of interest in the evolution of f, we can let r = 0 in

the argument of f appearing in the right-hand side of Equation (3.4); in this case the latter
becomes:
f (x, ~j, t)I~j" nl
= f~'.n<0e(~t ~/~; x, t)f(x,/~', t)l~'" nl d~j' (x 6 Y2, ~ 9n > 0), (11.2)
where
fO G
R(~j' --+ ~j; x,t)= drR(~j' --+ ~j; x,t; r). (11.3)
Equation (11.2) is, in particular, valid for steady problems.
Although the idea of a scattering kernel had appeared before, it is only at the end of
1960's that a systematic study of the properties of this kernel appears in the scientific
literature [35,110,41]. In particular, the following properties were pointed out [36,35,110,
41,34,108,48,109,37]:
(1) Non-negativeness, i.e., R cannot take negative values:
R(~j' --+ ~j; x, t; r) ~>0 (11.4)
and, as a consequence:
R(se' -+/~; x, t) ~>0. (11.5)
(2) Normalization, if permanent adsorption is excluded; i.e., R, as a probability density
for the totality of events, must integrate to unity:
R(/~' --+ ~; x,t" r)dse -- 1
dr '.n~>0 (11.6)
f~ R (~' ~ se; x, t) d~ -- 1. (11.7)
'-n>/O
(3) Reciprocity; this is a subtler property that follows from the circumstance that the
microscopic dynamics is time reversible and the wall is assumed to be in a local equilibrium
state, not significantly disturbed by the impinging molecule. It reads as follows:
les'.nIMw(~')R(es' ~ se;x, t; r)= I~ .nlMw(~)R(-~ --~ -se'; x, t; r) (11.8)
les'. nlMw(fj')R(es' ~/~; x,t) = I/~"nlMw(es)R(-es -+ -/~'; x,t). (11.9)

24 C.Cercignani
Here Mw is a (non-drifting) Maxwellian distribution having the temperature of the wall,
which is uniquely identified apart from a factor.
We remark that the reciprocity and the normalization relations imply another property:
(3') Preservation of equilibrium, i.e., the Maxwellian Mw must satisfy the boundary
condition (11.1):
Mw(~)l~. nl = dr
'.n<O
R(~j' --+/~;x, t; r)Mw(/~')l~j'' nl d~j' (11.10)
equivalent to:
Mw(~)l~" nl = f~'.n<O R(8' ~ ~; x, t)Mw (/~')l/~'. nl dS'. (11.11)
In order to obtain Equation (11.10) it is sufficient to integrate Equation (11.8) with respect
to ~' and r, taking into account Equation (11.6) (with -~ and -~' in place of ~' and
~, respectively). We remark that one frequently assumes Equation (11.10) (or (11.11)),
without mentioning Equation (11.8) (or (11.9)); although this is enough for many purposes,
reciprocity is very important when constructing mathematical models, because it places a
strong restriction on the possible choices. A detailed discussion of the physical conditions
under which reciprocity holds has been given by B~irwinkel and Schippers [10].
The basic information on gas-surface interaction, which should be in principle obtained
from a detailed calculation based on a physical model, is summarized in a scattering kernel.
The further reduction to a small set of accommodation coefficients can be advocated for
practical purposes, provided this concept is firmly related to the scattering kernel (see [43,
39,47] for further details).
In view of the difficulty of computing the kernel R(~j' --~ ~) from a physical model
of the wall, frequently one constructs a mathematical model in the form of a kernel
R (~' ~ ~j) which satisfies the basic physical requirements expressed by Equations (11.5),
(11.7), (11.9) and is not otherwise restricted except by the condition of not being too
complicated.
One of the simplest kernels is
R(~j' ~ ~j) = ctMw(~)1~. nl + (1 - ct)3(~ - ~' + 2n(~' 9~)). (11.12)
This is the kernel corresponding to Maxwell's model [121], according to which a fraction
(1 -ct) of molecules undergoes a specular reflection, while the remaining fraction c~ is
diffused with the Maxwellian distribution of the wall Mw. This is the only model for the
scattering kernel that appeared in the literature before the late 1960's. We refer to original
papers and standard treatises for details on more recent models [110,41,34,108,48,109,37,
10,121,135,91,100,112,99,68,174,38,54,53,22,43,39], among which the most popular in
recent years has been the so-called Cercignani-Lampis (CL) model.
It is remarkable that, for any scattering kernel satisfying the three properties of
normalization, positivity and preservation of equilibrium, a simple inequality holds. The

latter was stated by Darrozbs and Guiraud [70] who also sketched a proof. More details
were given later [37,39]. It reads as follows
fin--- fR~'3 n flog f d~<~-(2RTw)-lfR ~
.3 nl~ 12fd~j (x ~ 0S'2). (11.13)
Equality holds if and only if f coincides with Mw (the wall Maxwellian) on 0S-2 (unless
the kernel in Equation (11.2) is a delta function). We remark that if the gas does not slip
upon the wall, the right-hand side of Equation (11.13) equals -qn/(RTw) where qn is the
heat-flow along the normal, according to its definition given in Section 5. If the gas slips on
the wall, then one must add the power of the stresses Pn "v to qn. In any case, however, the
right-hand side equals qn
(w), where q(nw)is the heat-flow in the solid at the interface, because
the normal energy flow must be continuous through the wall and stresses have vanishing
power in the solid, because the latter is at rest. If we identify the function H introduced in
Section 8 with -rl/R (where r/is the entropy of the gas), the inequality in Equation (11.13)
is exactly what one would expect from the Second Principle of thermodynamics.
12. The continuum limit
In this section we investigate the connection between the Navier-Stokes equations and
the Boltzmann equation by using a method, which originated with Hilbert [85] and
Enskog [74].
The discussion is made complicated by the various possible scalings. For example, if we
denote by (~,,7) the microscopic space and time variables (those entering in the Boltzmann
equation) and by (x, t) the macroscopic variables (those entering in the fluid dynamical
description), we can study scalings of the following kind
= s-ix, (12.1)
7 - s -~ t, (12.2)
where ot is an exponent between 1 and 2. For ot -- 1, this is called the compressible scaling.
If c~ > 1, we are looking at larger "microscopic" times. We now investigate the limiting
behavior of solutions of the Boltzmann equation in this limit.
Notice first that the compressible Euler equations,
8tp + div(pv) = O,
Ot(pvi) + div pvvi + -~peei - 0,
1 1 5
Ot[p(e + -~lvl2)] + div[pv(-~lvl2 + -~e)] - 0,
(12.3)

26 C. Cercignani
are invariant with respect to the scaling t -+ e-it, x -+ e-1 x. Here, ei denotes the unit
vector in the i-th direction and p is related to p and e -- 3RT/2 by the state equation for a
perfect gas.
To investigate how these equations change under the scalings (12.1)-(12.2), let
v (x, t)-
p (x, t) = p( -lx,
Te(x,t)--T(e-lx,
y=c~- 1,
(12.4)
where (p, v, T) solve the compressible Euler equations (12.3). We easily obtain
Otpe + div(peve) = O, (12.5)
1 62(1_c~ )
Otve + (ve" Ox)ve = pe OxPe" ' (12.6)
2Te(0x. ve)=0
OtTe + (ve . Ox)Te + -~ (12.7)
The scaling of the bulk velocity field ve in (12.4) is done in a dimensionally consistent
way.
We expect that the continuum limit of the Boltzmann equation under the scaling (12.1)-
(12.2) will be given by the asymptotic behavior of (pe, ve, Te), satisfying (12.3), in the
limit e --+ 0. We will now investigate this limit.
To this end, let r/= e2(~ and expand
Vr/ ~ ve __ VO -+- ?/Vl -+-//2V2 @ "'',
pO = pe _ PO + ~lPl + r/2p2 -+- 99",
T o =--T e= To + oT1 + 02T1 +'".
If we collect the terms of order 0-1 in (12.4), we have
Oxpo -0 (12.8)
and the terms of order 77
~ give
OtPo + div(povo) = O,
Otvo + (vo-0x)Vo = -Oxp______j_l, (12.9)
Po
2
OtTo + (vo. 0x)To + ~ Todiv vo -- 0.

From (12.8) and the perfect gas law p0 - p0T0, which we assume to hold at zeroth order,
it follows that poTo is constant as a function of the space variables. The first and third
equations in (12.9) now imply
2
Ot(poTo)+ div(p0T0v0) -- - ~-p0T0divv0. (12.10)
.5
As poTois only a function of t, say A(t), Equation (12.10) implies that
3AI
divv0--5 (t)/A(t). (12.11)
Under suitable assumptions, Equation (12.11) implies that A1(t) = 0 and then divv0 = 0.
This is the case, for example, if we are in a box with nonporous walls, because then the
normal component of v0 is 0 and we can use the divergence theorem. A similar argument
applies to the case of a box with periodicity boundary conditions; or if we are in the entire
space 9~3 and the difference between v0 and a constant vector decays fast enough at infinity.
Assuming that we have conditions which imply that div v0 = 0, we easily get from the
continuity equation that P0 will be independent of x if the initial value is, and the same
for To. But with this knowledge Equations (12.9) then actually entail that P0 and To are
constant.
Therefore, if the initial conditions are "well prepared" in the sense that v~(x, 0) is a
divergence-free vector field and p~ (x, 0), Te(x, 0) are constant, we expect as a first-order
approximation for p~, v~, T ~ the solution of the equations
div v0 - 0,
Otvo + (vo. Ox)VO- - ~
OxPl (12.12)
p0
which are the incompressible Euler equations. This limit, known as the "low velocity
limit", is well known at the macroscopic level. We refer to Majda's book [120] for
references and a detailed discussion. The variable r/-1 enters into the theory as the square
of the speed of sound. If this parameter is large compared to typical speeds of the fluid, then
the incompressible model is well suited to describe the time evolution, provided that the
initial velocity field is divergence-free and the initial density and temperature are constant.
The incompressible fluid limit was met in the last section in connection with the Stokes
paradox. In fact, it seems that this limit and the derivation of the steady incompressible
Navier-Stokes equations from the Boltzmann equation were first considered in connection
with the flow past a body at small values of the Mach number [32]. We remark that in the
steady case div v0 - 0 and P0 and To turn out to be constant without using the boundary
conditions.
Let us examine the kinetic picture as described by the Boltzmann equation. The above
discussion suggests that if ot e (1,2), then in the scaling (12.1)-(12.2) the solutions
of the Boltzmann equation will converge to a local Maxwellian distribution whose
parameters satisfy the incompressible Euler equations. This assertion can actually be
proved rigorously [71,9].

28 C. Cercignani
For ct = 2 something special happens. Of course, the incompressible Euler equations are
invariant under the scaling (12.1)-(12.3); however, for c~= 2 the incompressible Navier-
Stokes equations,
Otv + (v. 0x)V= -Oxr + v A v,
0x. v = 0 (12.13)
(where r = p/p and v = Iz/p is the kinematic viscosity) are also invariant under the same
scaling. It is therefore of great interest to understand whether the Boltzmann dynamics
"chooses" in this limit the Euler or the Navier-Stokes evolution.
We can expect that the answer is Navier-Stokes. In other words, considering larger times
than those typical for Euler dynamics (e-zt instead of e-c~t, c~ < 2), dissipation becomes
nonnegligible. We can see an illustration of this behavior in the following example:
consider two parallel layers of fluid moving with velocities v and v + 6v. Suppose that
we want to decide whether there is any momentum transfer between these layers (which is
expected for the Navier-Stokes equations, but not for the Euler equations). The momentum
transfer can in principle be affected by the trend to thermalization typical of the Boltzmann
collision term, but a scaling argument shows that it is proportional to e~-Z6v, with the
consequence that it remains only relevant for c~= 2.
These considerations can be put on a rigorous basis, and, of course, the viscosity
coefficient can be computed in terms of kinetic expressions (see [43,39,35]).
The incompressible Navier-Stokes equations can be derived from the Boltzmann
equation if the time interval is such that smooth solutions of the continuum equations
exist. The tool which yields this result is a truncated Hilbert expansion and one gets local
convergence for the general situation [71] or global and uniform convergence if the data
are small in a suitable sense [9].
Let us consider now the cases in which compressibility is not negligible. If we scale
space and time in the same way, the scaled Boltzmann equation becomes:
Otf e + I~. Oxfe = _1Q(fe, fe). (12.14)
E
We will use the abbreviation Dt f := Otf + ~ 9Oxf. Of course, we expect that
eDtf e -+ 0 as e --+ O, (12.15)
and if
fe __+ fo, (12.16)
the limit f0 must satisfy
Q(fO, f0) = o. (12.17)

This implies, as we know, that f0 is a local Maxwellian distribution:
f~
p(x,t) ( I~-v(x,t)l 2)
(2zrRT(x, t))3/2 exp - 2RT(x, t) "
(12.18)
The fields (p, x, T), which characterize the behavior of the local Maxwellian distribution
M in space and time, are expected to evolve according to continuum equations which we
are going to derive. First, let us emphasize again that these fields are varying slowly on
the space-time scales which are typical for the gas described in terms of the Boltzmann
equation.
From the conservation laws (I. 12.36)
f g/uQ(f, f)d~j-O (c~=O ..... 4), (12.19)
we readily obtain, as we know,
f ~Po~(Ot
f + I~. Oxf) d/~ --0. (12.20)
This is a system of equations for the moments of f which is in general not closed. However,
if we assume f = M and use the identities (for M they are identities; for a general f they
are definitions given in (I.12.3-7), where e = 3RT)
-- f M ds
e, (12.21)1
, v--fMseds
e, (12.21)2
3 1 if 2M ,
w = -~pRT + ~plv] 2- ~ s
e ds
e (12.21)3
we readily obtain from (12.20) that
Otp + div(pv) = O, (12.22)
Ot(pvi) + div(fM~i) --O, (12.23)
(3 1)l(f )
Ot -~peZ+ ~p[vl 2 + ~div M~jI~I
2 --0. (12.24)
These equations are nothing but Equations (12.19)-(12.21), specialized to the case
of a Maxwellian distribution. This is of crucial importance if we want to write

30 C. Cercignani
Equations (12.23) and (12.24) in closed form. To do so we have to express f M~i and
f M~ I~12in terms of the fields (p, v, T). To this end, we use the elementary identities:
f M(~j - vj)(~i - vi)d~ 16ijpRT,
f M(f~ - v) I(/~- v)l 2d/~ = O,
(12.25)
which transform Equation (12.23) into
Ot(pvi ) + div(pvvi ) = --Oxi p, (12.26)
with
p = p R T. (12.27)
Equation (12.27) is the perfect gas law. Obviously, the quantity p defined by Equa-
tion (12.27) has the meaning of a pressure.
Recalling that the internal energy e is related to temperature T by
3
e = -~RT, (12.28)
using this and Equation (12.25) we transform Equation (12.24) into:
( 1 )) ( ( 1 ))
Ot p(e + ~lvl 2 + div pv e + ~lvl 2 - -div(pv). (12.29)
The set of Equations (12.22), (12.26) and (12.29) express conservation equations for mass,
momentum and energy respectively and can be rewritten as the Euler equations (12.3).
For smooth functions, an equivalent way of writing the Euler equations in terms of the
field (p, v, T) is
Otp + div(pv) = 0,
1
Otv + (v. Ox)V+ -Oxp = O, (12.30)
P
2
OtT + (v. ax)T + =Tax. v =0.
.5
However, in this form we lose the conservation form as given in (12.3), in which the
time derivative of a field equals the negative divergence of a current which is a nonlinear
function of this field.
Before going on, some comments on our limits are in order, because one might suspect
an inconsistency in the passage from a rarefied to a dense gas. Recall that the Boltzmann
equation holds in the Boltzmann-Grad limit (Nor 2 = O(1)). In the continuum limit, we

have to take Nff 2 -- 1/e ~ oo. This, at first glance, seems contradictory, but there is really
no problem. The Boltzmann equation holds for a perfect gas, i.e., for a gas such that the
density parameter 6 - N~r3/V, where V is the volume containing N molecules, tends to
zero. The parameter
1 Nff 2
-- -- N1/3S 2/3 (12.31)
Kn V 2/3
may tend to zero, to cxz or remain finite in this limit. These are the three cases which occur
if we scale N as 3-m (m ~>0), for m < 2, m > 2 and m = 2 respectively. In the first case
the gas is in free-molecular flow and we can simply neglect the collision term (Knudsen
gas), in the second we are in the continuum regime which we are treating here, and we
cannot simply "omit" the "small" term, i.e., the left-hand side of the Boltzmann equation,
because the limit is singular. In the third case the two sides of the Boltzmann equation are
equally important (Boltzmann gas) and this is the case dealt with before for solutions close
to an absolute Maxwellian distribution.
In spite of the fact that we face a singular perturbation problem, Hilbert [85] proposed
an expansion in powers of e. In this way, however, we obtain a Maxwellian distribution
at the lowest order, with parameters satisfying the Euler equations and corrections to this
solution which are obtained by solving inhomogeneous linearized Euler equations [85,43,
39,35]. In order to avoid this and to investigate the relationship between the Boltzmann
equation and the compressible Navier-Stokes equations, Enskog introduced an expansion,
usually called the Chapman-Enskog expansion [74,39,35]. The idea behind this expansion
is that the functional dependence of f upon the local density, bulk velocity and internal
energy can be expanded into a power series. Although there are many formal similarities
with the Hilbert expansion, the procedure is rather different.
As remarked by the author [35,39], the Chapman-Enskog expansion seems to introduce
spurious solutions, especially if one looks for steady states. This is essentially due to the
fact that one really considers infinitely many time scales (of orders e, e2..... e n .... ).
The author [35,39] introduced only two time scales (of orders e and e2) and was able
to recover the compressible Navier-Stokes equations. In order to explain the idea, we
remark that the Navier-Stokes equations describe two kinds of processes, convection and
diffusion, which act on two different time scales. If we consider only the first scale we
obtain the compressible Euler equations; if we insist on the second one we can obtain
the Navier-Stokes equations only at the price of losing compressibility. If we want both
compressibility and diffusion, we have to keep both scales at the same time and think of f
as
f(x, g;, t) -- f (ex, ~,, et, e2t). (12.32)
This enables us to introduce two different time variables tl - et, t2 - e2t and a new space
variable Xl - ex such that f - f(xl, ~, tl, t2). The fluid dynamical variables are functions

32 c. Cercignani
of xl, tl, t2, and for both f and the fluid dynamical variables the time derivative is given
by
O Of 2 Of
m = e~ + e ~. (12.33)
Ot Otl Ot2
In particular, the Boltzmann equation can be rewritten as
Of 62 Of
+ + e!~"Oxf = Q(f, f)
Otl -~2
If we expand f formally in a power series in e, we find that at the lowest order f is
a Maxwellian distribution. The compatibility conditions at the first order give that the
time derivatives of the fluid dynamic variables with respect to tl is determined by the
Euler equations, but the derivatives with respect to t2 are determined only at the next
level and are given by the terms of the compressible Navier-Stokes equations describing
the effects of viscosity and heat conductivity. The two contributions are, of course, to be
added as specified by (12.33) in order to obtain the full time derivative and thus write the
compressible Navier-Stokes equations.
It is not among the aims of this article to describe the techniques applied to and the
results obtained from the computations of the transport coefficients, such as the viscosity
and heat conduction coefficients, for given molecular interaction. For this we refer to
standard treatises [64,86,75].
The results discussed in this section show that there is a qualitative agreement between
the Boltzmann equation and the Navier-Stokes equations for sufficiently low values of the
Knudsen number.
There are however flows where this agreement does not occur. They have been especially
studied by Sone [154]. New effects arise because the no-slip and no temperature jump
boundary condition do not hold. In addition to the thermal creep induced along a boundary
with a nonuniform temperature, discovered by Maxwell, two new kinds of flow are induced
over boundaries kept at uniform temperatures. They are related to the presences of thermal
stresses in the gas.
The first effect [154,153,138] is present even for small Mach numbers and small
temperature differences and follows from the fact that there are stresses related to the
second derivatives of the temperature (see Section 15). Although these stresses do not
change the Navier-Stokes equations, they change the boundary conditions; the gas slips on
the wall, and thus a movement occurs even if the wall is at rest. This effect is particularly
important in small systems, such as micromachines, since the temperature differences are
small but may have relatively large second derivatives; it is usually called the thermal stress
slip flow [154,153,138].
The second effect is nonlinear [101,155] and occurs when two isothermal surfaces do
not have constant distance (thus in any situation with large temperature gradients, in the
absence of particular symmetries). In fact, if we assume that in the Hilbert expansion
the velocity vanishes at the lowest order, i.e., the speed is of the order of the Knudsen
number, the terms of second order in the temperature show up in the momentum equation.

These terms are associated with thermal stresses and are of the same importance as those
containing the pressure and the viscous stresses. A solution in which the gas does not move
can be obtained if and only if:
grad T A grad(I grad TIe) = 0. (12.34)
Since Igrad T I measures the distance between two nearby isothermal lines, if this quantity
has a gradient in the direction orthogonal to grad T, the distance between two neighboring
isothermal lines varies and we must expect that the gas moves.
These effects may occur even for sufficiently large values of the Knudsen number; they
cannot be described, however, in terms of the local temperature field. They rather depend
by the configuration of the system. They should not be confused with flows due to the
presence of a temperature gradient along the wall, such as the transpiration flow [139] and
the thermophoresis of aerosol particles [162].
Numerical examples of simulations of this kind of flow are discussed in [43].
13. Free-molecule and nearly free-molecule flows
After discussing the behavior of a gas in the continuum limit, in this section we consider
the opposite case in which the small parameter is the Knudsen number (or the inverse of
the mean free path).
By analogy with what we did in the previous section, we might be tempted to use a series
expansion of the form (12.2), albeit with a different meaning of the expansion parameter.
This, however, does not work in general, for a reason to be presently explained. The factor
multiplying the gradient of f in Equation (5.1) takes all possible values and hence also
values of order e; thus we should expect troubles from the molecules travelling with low
speeds, because then certain terms in the left-hand side can become smaller than the fight-
hand side, in spite of the small factor e. This is confirmed by actual calculations, especially
for steady problems.
Let us now consider the limiting case when the collisions can be completely neglected.
This, by itself, does not pose many problems.
The Boltzmann equation (in the absence of a body force) reduces to the simple form
Dtf = atf + I~ . Oxf =0. (13.1)
Since the molecular collisions are negligible, the gas-surface interaction discussed in
Section 11 plays a major role. This situation is typical for artificial satellites, since the
mean free path is 50 meters at 200 kilometers of altitude.
The general solution of Equation (13.1) is in terms of an arbitrary function of two vectors
g(., .):
f (x, l~, t) = g(x- I~t, ~,). (13.2)
In the steady case, Equation (13.1) reduces to
I~. Oxf = 0, (13.3)

34 C. Cercignani
and the general solution becomes:
f (x, ~, t) = g(x A l~, l~). (13.4)
Frequently it is easier to work with the property that f is constant along the molecular
trajectories than with the explicit solutions given by Equations (13.3)-(13.4).
The easiest problem to deal with is the flow past a convex body. In this case, in fact,
the molecules arriving at the surface of the wall have an assigned distribution function f~,
usually a Maxwellian distribution with the density p~, bulk velocity v~, and temperature
Too, prevailing far away from the body, and the distribution function of the molecules
leaving the surface is given by the boundary conditions. The distribution function at any
other point P, if needed, is simply obtained by the following rule: if the straight line
through P having the direction of ~ intersects the body at a point Q and ~j points from
Q towards P, then the distribution function at P is the same as that at Q; otherwise it
equals f~.
Interest is usually confined to the total momentum and energy exchanged between the
molecules and the body, which, in turn, easily yield the drag and lift exerted by the gas on
the body and the heat transfer between the body and the gas.
In practice, the temperature of a body is determined by a balance of all forms of heat
transfer at the body surface. For an artificial satellite, a considerable part of heat is lost by
radiation and this process must be duly taken into account in the balance.
The results take a particularly simple form in the case of a large Mach number since
we can let the latter go to infinity in the various formulas. One must, however, be careful,
because the speed is multiplied by sin 0 in many terms and thus the aforementioned limit
is not uniform in 0. Thus the limiting formulas can be used, if and only if, the area where
S sin 0 ~< 1 is small.
The standard treatment is based on the definition of accommodation coefficients, but
calculations based on other models are available [39,50,49].
The case of nonconvex boundaries is, of course, more complicated and one must solve an
integral equation to obtain the distribution function at the boundary. If one assumes diffuse
reflection according to a Maxwellian, the integral equation simplifies in a considerable
way, because just the mass flow at the boundary must be computed [39].
In particular the latter equation can be used to study free-molecular flows in pipes
of arbitrary cross section with a typical diameter much smaller than the mean free path
(capillaries). If the cross section is circular the equation becomes particularly simple and
is known as Clausing's equation [39].
The perturbation of free-molecular flows is not trivial for steady problems because of
the abovementioned non-uniformity in the inverse Knudsen number. If one tries a na'fve
iteration, the singularity arising in the first iterate may cancel when integrating to obtain
moments (cancellation is easier, the higher is the dimensionality of the problem, because a
first-order pole is milder, if the dimension is higher). The singularity is always present and,
although it may be mild, it can build up a worse singularity when computing subsequent
steps. The difficulties are enhanced in unbounded domains where the subsequent terms
diverge at space infinity. The reason for the latter fact is that the ratio between the mean
free path )~ and the distance d of any given point from the body is a local Knudsen

number which tends to zero when d tends to infinity; hence collisions certainly arise in
an unbounded domain and tend to dominate at large distances. On this basis we are led to
expect that a continuum behavior takes place at infinity, even when the typical lengths
characterizing the size of the body are much smaller than the mean free path; this is
confirmed by the discussion of the Stokes paradox for the steady linearized Boltzmann
equation (see [43,39,35]).
Both difficulties are removed by the so-called collision iteration: the loss term is partly
considered to be unknown in the iteration, thus building an exponential term which controls
the singularity. The presence of the latter is still felt through the presence of logarithmic
terms in the (inverse) Knudsen number. In higher dimensions this is multiplied by a power
of (Kn)-1 which typically equals the number of space dimensions relevant for the problem
under consideration in a bounded domain. In particular the dependence upon coordinates
will show the same singularity (we can think of local Knudsen numbers based on the
distance from the nearest wall); as a consequence first derivatives will diverge at the
boundary in one dimension and the same will occur for second, or third derivatives, in two,
or three, space dimensions, respectively. In an external domain we have, in addition to the
low speed effects, the effect of particles coming from infinity, which actually dominates. In
particular in one dimension (half-space problems) the terms coming from iterations are of
the same order as the lowest order terms; actually for a half-space problem there is hardly
a Knudsen number (the local one is an exception). In two dimensions the corrections in the
moments are of order Kn-1 log Kn. In three dimensions a correction of order Kn-2 log Kn
is preceded by a correction of order Kn -1 .
Care must be exercised when applying the aforementioned results to a concrete
numerical evaluation, as mentioned above. In fact, for large but not extremely large
Knudsen numbers (say 10 ~< Kn ~< 100)logKn is a relatively small number, although
log Kn ---> oQ for Kn ---> oo. Hence terms of order log Kn/Kn, though mathematically
dominating over terms of order 1/Kn are of the same order as the latter for practical
purposes. As consequence, the two kinds of terms must be computed together if numerical
accuracy is desired for the aforementioned range of Knudsen numbers.
Related to this remark is the fact that any factor appearing in front of Kn in the argument
of the logarithm is meaningless unless the term of order Kn -I is also computed. This
is particularly important when the factor under consideration depends upon a parameter
which can take very large (or very small) values (typically a speed ratio). Thus Hamel
and Cooper [70,85] have shown that the first iterate of the integral iteration is incapable
of describing the correct dependence upon the speed ratio and have applied the method
of matched asymptotic expansions [81] to regions near a body and far from a body. In
particular, for the hypersonic flow of a gas of hard spheres past a two-dimensional strip,
they find for the drag coefficient
elog e]
CD = CDf.m. 1 + 2zr ' (13.5)
where the inverse Knudsen number e is based on the mean free path )~ = 7r3/2cr2nooS w
(or is the molecular diameter and S~, = S~(Tw/Too), whereas n~ and S~ are the number
density at infinity).

36 C. Cercignani
If we consider infinite-range intermolecular potentials, then we have fractional powers
rather than logarithms.
All the considerations of this section have the important consequence that approximate
methods of solution which are not able to allow for a nonanalytic behavior for Kn ~ c~
produce poor results for large Knudsen numbers.
14. Perturbations of equilibria
The first steady solutions other than Maxwellian to be investigated were perturbations of
the latter. The method of perturbation of equilibria is different from the Hilbert method
because the small parameter is not contained in the Boltzmann equation but in auxiliary
conditions, such as boundary or initial conditions. The advantage of the method is that we
can investigate problems in the transition regime, provided differences in temperature and
speed are moderate.
Let us try to find a solution of our problem for the Boltzmann equation in the form
oo
f =Zenfn,
n=O
(14.1)
where at variance with previous expansions e is a parameter which does not appear in
the Boltzmann equation. In addition f0 is assumed from the start to be a Maxwellian
distribution.
By inserting this formal series into Equation (5.1) and matching the various orders in e,
we obtain equations which one can hope to solve recursively:
0tfl + ~j" 0xfl = 2Q(fl, f0), (14.2)1
j-1
Otf j + 1~. Oxfj = 2Q(f j, fo) + Z Q(J~' f j-i),
i=1
(14.2)j
where, as in Section 4, Q(f, g) denotes the symmetrized collision operator and the sum is
empty for j = 1.
Although in principle one can solve the subsequent equations by recursion, in practice
one solves only the first equation, which is called the linearized Boltzmann equation. This
equation can be rewritten as follows:
Oth + Ij 9Oxh -- LMh, (14.3)
where LM denotes the linearized collision operator about the Maxwellian M, i.e., LMh =
2Q(M, Mh)/M, h = fl/M (see Section 10). We shall assume, as is usually done with

little loss of generality, that the bulk velocity in the Maxwellian is zero and we shall denote
the unperturbed density and temperature by P0 and To.
Although the equation is now linear, and hence all the weapons of linear analysis
are available, it is far from easy to solve for a given boundary value problem, such as
Couette flow. Yet it is possible to gain an insight on the behavior of the general solution
of Equation (14.3) (see [43,39,35]). This insight gives the following picture for a slab
problem, provided the plates are sufficiently far apart (several mean free paths). There are
two Knudsen layers near the boundaries, where the behavior of the solution is strongly
dependent on the boundary conditions, and a central core (a few mean free paths away
form the plates), where the solution of the Navier-Stokes equations holds (with a slight
reminiscence of the boundary conditions). If the plates are close in terms of the mean free
path, then this picture does not apply because the core and the kinetic layers merge.
One can give evidence for the above statements just in the case of the linearized
Boltzmann equation, but there is a strong evidence that this qualitative picture applies to
nonlinear flows as well, with a major exception. In general, compressible flows develop
shock waves at large speeds and these do not appear in the linearized description. As
already remarked, these shocks are not surfaces of discontinuity as for an ideal fluid,
governed by the Euler equations, but layers of rapid change of the solution (on the scale
of the mean free path). One can obtain solutions for flows containing shocks from the
Navier-Stokes equations, but, since they change significantly on the scale of the mean free
path, they are inaccurate. Other regions where this picture is inaccurate are the zones of
high rarefaction, where nearly free-molecular conditions may prevail, even if the rest of
the flow is reasonably described in terms of Navier-Stokes equations, Knudsen layers and
shock layers.
The theory of Knudsen layers can be essentially described by the linearized Boltzmann
equation. The main result concerns the boundary conditions for the Navier-Stokes
equations. They turn out to be different from those of no-slip and no temperature jump.
In fact, the velocity slip turns out to be proportional to the normal gradient of tangential
velocity and the temperature jump to the normal gradient of temperature. When one can
use the Navier-Stokes equations but must use the slip and temperature-jump boundary
conditions, one talks of the slip regime; this typically occurs for Knudsen numbers
between 10-1 and 10-2.
Subtler phenomena may occur if the solutions depend on more than one space
coordinate. The most important change with respect to traditional continuum mechanics
is the presence of the term with the second derivatives of temperature in the expression
of the stress deviator and of the term with the second derivatives of bulk velocity in the
expression of the heat flow. These terms were already known to Maxwell [121]. In recent
times, their importance has been stressed by Kogan et al. [101] and by Sone et al. [155] (as
already mentioned in Section 12).
Even in fully three-dimensional problems the solution of the linearized Boltzmann
equation reduces to the sum of two terms, one of which, h8, is important just in the
Knudsen layers and the other, hA, is important far from the boundaries. The latter has a
stress deviator and a heat flow with constitutive equations different from those of Navier-
Stokes and Fourier. In spite of this, the bulk velocity, pressure, and temperature satisfy
the Navier-Stokes equations when steady problems are considered. In fact, when we take

38 C. Cercignani
the divergence of the heat flow vector a term proportional to the Laplacian of v vanishes,
thanks to the continuity equation, and thus just a term proportional to the temperature
gradient survives; then, taking the divergence of the stress, a term grad(AT) vanishes,
because of the energy equation. Yet, the new terms in the constitutive relations may produce
physical effects in the presence of boundary conditions different from those of no-slip and
no temperature jump. In fact, we must expect the velocity slip to be proportional to the
shear stress and the temperature jump to the heat flow.
15. Approximate methods for linearized problems
Linearization combined with the use of models lends itself to the use of analytical methods,
which turn out to be particularly useful for a preliminary analysis of certain problems.
Closed form solutions are not so frequent and are practically restricted to the case of half-
space problems [43,39,35]. The latter, in turn, are useful to investigate Knudsen layers and
compute the slip and temperature jump coefficients.
The use of BGK or similar models permits reducing the solution of Boltzmann's integro-
differential equation in phase-space to solving integral equation in ordinary space. This is
obtained because in the BGK model the distribution function f occurs only in two ways:
explicitly in a linear, simple way and implicitly through a few moments (appearing in
the local Maxwellian and the collision frequency). Then one can express f in terms of
these moments by integrating a linear, simple partial differential equation; then, using the
definitions of these moments and the expression of f one can obtain integral equations for
the same moments [43,39,35]. These equations can be solved numerically in a much easier
way than the Boltzmann equation. This is particularly true in the linearized case.
The integral equation approach lends itself to a variational solution. The main idea of
this method (for linearized problems) is the following. Suppose that we must solve the
equation:
12h=S, (15.1)
where h is the unknown,/2 a linear operator and S a source term. Assume that we can form
a bilinear expression ((g, h)) such that ((s h)) = ((g, Eh)),for any pair {g, h} in the set
where we look for a solution. Then the expression (functional):
J(/~) -- ((h, C/~)) - 2((S,/Tt)) (15.2)
has the property that if set/t - h + r/, then the terms of first degree in ~ disappear and
J(h) reduces to J(h) -+-((r/,/2r/)) if and only if h is a solution of Equation (15.1). In other
words if r/is regarded as small (an error), the functional in Equation (15.2) becomes small
of second order in the neighborhood of h, if and only if h is a solution of Equation (15.1).
Then we say that the solutions of the latter equation satisfy a variational principle, or
make the functional in Equation (15.2) stationary. Thus a way to look for solutions of
Equation (15.1) is to look for solutions which make the functional in Equation (15.2)
stationary (variational method).

The method is particularly useful if we know that ((0,/20)) is non-negative (or non-
positive) because we can then characterize the solutions of Equation (15.1) as maxima or
minima of the functional (15.2). But, even if this is not the case, the property is useful. First
of all, it gives a non-arbitrary recipe to select among approximations to the solution in a
given class. Second, if we find that the functional J is related to some physical quantity,
we can compute this quantity with high accuracy, even if we have a poor approximation to
h. If the error 0 is of order 10%, then J will be in fact computed with an error of the order
of 1%, because the deviation of J (/t) from J (h) is of order 02, as we have seen.
The integral formulation of the BGK model lends itself to the application of the
variational method [58]. Thus in the case of Couette the functional is related to the stress
component p12 which is constant and gives the drag exerted by the gas on each plate. Thus
this quantity can be computed with high accuracy [58,43].
This method can be generalized to other problems and to the more complicated mod-
els [39]. It can also be used to obtain accurate finite ordinate schemes, by approximating
the unknowns by trial functions which are piecewise constant [44].
In the case of the steady linearized Boltzmann equation, Equation (14.3), a similar
method can be used. Let us indicate by Dh the differential part appearing in the left-hand
side (Dh = ~ 90xh for steady problems) and assume that there is a source term as well
(an example of a source occurs in linearized Poiseuille flow, see [43,39,35]) and write our
equation in the form:
Dh-Lh--S. (15.3)
If we try the simplest possible bilinear expression
f0L g(x, s
e)h(x,/j) dx d,~ (15.4)
((g,h)) -- 3
and we use it with Eh = Dh - Lh we cannot reproduce the symmetry property ((Eg, h)) =
((g, Eh)). It works for Lh but not for Dh. There is however a trick [33] which leads to the
desired result.
Let us introduce the parity operator in velocity space, P, such that P[h(~j)] -- h(-~).
Then we can think of replacing Equation (15.3) by
PDh- PLh = PS (15.5)
because this is completely equivalent to the original equation. In addition, because of the
central symmetry of the molecular interaction PLh = LPh and the fact that we had no
problems with L is not destroyed by the fact that we use P. On the other hand we have by
a partial integration:
((g, PDh)) --((PDg, h)) + ((g+, Ph-)) B -((Pg-,h+)) B. (15.6)

- Handbook of Mathematical Fluid Dynamics.pdf

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