# Beyond Navier–Stokes equations: Capillarity of ideal gas

A.N. Gorbana∗ and I.V. Karlinb‡ ∗Corresponding author. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>aDepartment of Mathematics,
University of Leicester, LE1 7RH Leicester, UK; bDepartment of Mechanical and
Process Engineering, ETH Zürich, CH-8092, Zürich, Switzerland

(2016)

###### Abstract

The system of Navier–Stokes–Fourier equations is one of the most celebrated
systems of equations in modern science. It describes dynamics of fluids in the
limit when gradients of density, velocity and temperature are sufficiently
small, and loses its applicability when the flux becomes so non-equilibrium
that the changes of velocity, density or temperature on the length compatible
with the mean free path are non-negligible. The question is: how to model such
fluxes? This problem is still open. (Despite the fact that the first ‘final
equations of motion’ modified for analysis of thermal creep in rarefied gas
were proposed by Maxwell in 1879.) There are, at least, three possible
answers: (i) use molecular dynamics with individual particles, (ii) use
kinetic equations, like Boltzmann’s equation, or (iii) find a new system of
equations for description of fluid dynamics with better accounting of non-
equilibrium effects. These three approaches work at different scales. We
explore the third possibility using the recent findings of capillarity of
internal layers in ideal gases and of saturation effect in dissipation (there
is a limiting attenuation rate for very short waves in ideal gas and it cannot
increase infinitely). One candidate equation is discussed in more detail, the
Korteweg system proposed in 1901. The main ideas and approaches are
illustrated by a kinetic system for which the problem of reduction of kinetics
to fluid dynamics is analytically solvable.

###### keywords:

Kinetics, Fluid dynamics, Boltzmann, Capillarity, Invariant manifold, Thermal
creep

††articletype: Review article

Published in Contemporary Physics, 58(1) (2017), 70–90

##  1 Introduction

###  1.1 Timeline (the history of equations)

Historical timelines are seldom comprehensive and are often subject to debates
but they do provide a background to examine the present. Older chronology
seems to be more reliable and less disputable (it may happen because some
details went forgotten). For understanding the situation ‘beyond
Navier–Stokes’ the following milestones from the distant past are crucial:

  * •

1687 Isaac Newton stated that the shear stress between layers of a fluid is
proportional to the velocity gradient in the direction perpendicular to the
layers.

  * •

1738 Daniel Bernoulli proved that the gradient of pressure is proportional to
the acceleration.

  * •

1759 Leonhard Euler applied Newton’s second law of motion to fluid dynamics
(no viscosity) and published the Euler equations of fluid motion.

  * •

1807 Jean Baptiste Joseph Fourier presented his heat conduction equation to
the Institut de France.

  * •

1822 Claude Navier introduced viscosity in the Euler equation.

  * •

1823 Augustin-Louis Cauchy published his general theory of stress.

  * •

1845 George Stokes published the Navier–Stokes equations.

  * •

1872 Ludwig Boltzmann published ‘Kinetic Theory of Gases’ and introduced the
Boltzmann equation for evolution of the particle distribution in the space of
possible positions and momenta.

  * •

1879 James Clerk Maxwell used Boltzmann’s kinetics to come up with additional
terms in the Navier–Stokes equations (the ‘final equations of motion’), in
order to explain the thermal creep in rarefied gases which cannot be captured
by the Navier–Stokes–Fourier equations.

  * •

1893 Johannes Diderik van der Waals introduced the thermodynamic theory of
capillarity under the hypothesis of a continuous variation of density.

  * •

1900 David Hilbert presented his prominent list of problems. The sixth Hilbert
problem includes ‘the problem of developing mathematically the limiting
processes,…, which lead from the atomistic view to the laws of motion of
continua’.

  * •

1901 Diederik Johannes Korteweg proposed the dynamics equations for fluids
with capillarity, based on van der Waals’ approach.

  * •

1917 David Enskog defended his thesis in which he gave a systematic derivation
of the Navier-Stokes-Fourier equations from the Boltzmann equation and
predicted the thermodiffusion effect in the mixtures of gases. The latter was
used in early uranium separation technologies.

  * •

1932 Steven Chapman and Steven Cowling refined the method of Enskog and wrote
the classical treatise ‘Mathematical theory of non-uniform gases’.

Thus, at the beginning of the 20th century, the main equations we intend to
discuss were invented. The Navier–Stokes–Fourier equations and the Boltzmann
equation were known and Hilbert considered reduction from kinetics (‘the
atomistic view’) to fluid mechanics as a crucially important problem. Van der
Waals proposed to include in the energy density a new term, $\sim
c(\nabla\rho)^{2}$, where $\rho$ is the density and $c$ is the capillarity
coefficient [1]. His doctorate student, Korteweg, found how this term affects
the motion of fluids, and created the dynamics of fluids with capillarity [2].

Since then the research tree has branched enormously. The van der Waals
capillarity term played an essential role in a number of important
achievements: in the Landau theory of phase transitions [3] and the
Ginzburg–Landau theory of superfluids and superconductors [4], in
Cahn–Hilliard models of phase separation [5] and in the Langer Bar-On and
Miller theory of spinodal decomposition [6], to name a few. Many new equations
of motion were invented and studied for various continuous media, and a
special discipline, rational thermodynamics, was developed to manage this
world of continuum mechanics models [7, 8]. In particular, the Korteweg
equations were slightly revised by means of rational thermodynamics [9].

The fluid dynamics equations were rigorously derived from Boltzmann’s kinetics
in some rescaling limits: if we consider a flow of an almost equilibrium gas
with very small space derivatives of velocities and density and then change
the time scale (go to ‘slow time’) then the Euler equations hold for this flow
with high accuracy. For some scaling procedures, the Navier–Stokes–Fourier
equations can be achieved as well. (Exact statements, more details, more
results and further references can be found in the book [10].)

Several families of methods were developed in theoretical physics for
deduction of fluid dynamic equations from kinetic equations. We introduce some
of them later. They usually work well for the derivation of the ‘expected’
Euler and Navier–Stokes–Fourier equations but the next step, a correction to
the Navier–Stokes equations for more non-equilibrium regimes, was far less
successful. Therefore, the most challenging goal of these methods, the correct
new post–Navier–Stokes–Fourier equations, remained unachieved.

###  1.2 The problem

In 1946 John von Neumann stated that computational fluid dynamics would make
experimental fluid dynamics obsolete [11] (see also [12]).

The computational approach to the problem as discussed by von Neumann has been
developing in many relevant works published meanwhile including e.g. very
recent publications [13] or [14].

But rephrasing the famous quotation of Lord Kelvin [15] we can say that the
beauty and clearness of the theory, which explains fluid dynamics equations as
special limits of kinetic equations, is at present obscured by a cloud.
Higher-order corrections to the Navier–Stokes–Fourier equations produced from
kinetics beyond the degenerated scaled flows demonstrate some non-physical
properties and cannot be used without regularization. Moreover, there are
relatively simple experiments with rarefied gases (known to Reynolds and
Maxwell in 1879 [16, 17]) which can be captured by neither the
Navier–Stokes–Fourier equations nor by the higher-order corrections obtained
from the Boltzmann equations by regular procedures. These are, for example,
effects of thermal creep (or transpiration), i.e. the steady motion of a
rarefied gas induced by a temperature gradient on the boundary of the flow
domain, which are demonstrated by the Crookes radiometer (the light mill, Fig.
1).

Figure 1: Crookes radiometer. It is made from a glass bulb with low air
pressure inside. Within the bulb there is a rotor with several vertical metal
vanes, which are polished on one side and black on the other. When exposed to
light the vanes turn, the dark sides retreating from the radiation source and
the light sides advancing. (Public domain picture.)

Another example of such a system is the Knudsen pump [18]. In Knudsen’s
experiments, it was a pipe with alternating narrow (diameter 1/3 mm) and wide
(diameter 10 mm) segments of 5 cm length (Fig. 2). Every second pipe joint was
heated by a special heating element. Metallic wire was used for heat removal
from other joints. The temperature difference between the heated and unheated
pipe joints was $500\text{\,}\mathrm{\SIUnitSymbolDegree}$C. For normal
pressure, there was no difference in pressure at the opposite ends of the
pipe, whereas for the pressure at one end, $p_{1}\sim 0.5$ mmHg (65 Pa) the
ratio between these pressures is $\sim 10$. Recently, this system was
revisited, both by the reproduction of experiments and by theoretical analysis
[19]. The effect was also demonstrated on two joined pipes of different inner
diameters, 12 and 24 mm, each 60 mm in length. The joint was heated to about
$500\text{\,}\mathrm{\SIUnitSymbolDegree}$C, the ends were cooled by thick
copper plates. The flux from the thinner to the thicker pipe was indicated for
the pressure 5-50 Pa. At pressure 100 Pa (and above) there was no motion
observed in this system [19].

Of course, these effects do not contradict the kinetic theory and can be
explained in the framework of Boltzmann’s kinetics and some of its
simplifications [20, 21]. The problem is in the appropriate continuum
mechanics model, which is produced systematically from kinetics and explains
thermal creeps. The prophecy of von Neumann is unfulfilled not only because
the computational tools are insufficient but also because the proper equations
for some situations remain unknown.

###  1.3 Solutions proposed: Capillarity of ideal gas

Van der Waals introduced capillarity energy for non-ideal multiphase fluids
[1, 22]. However, the terms which look similar to Korteweg’s stress tensor
have been recognized in the Chapman–Enskog expansion for non-equilibrium ideal
gas many times. Everyone can compare the Korteweg stress (equation (11) below)
with Burnett’s equations, the first post-Navier-Stokes correction derived from
the Boltzmann equation (see, for example, the Chapman and Cowling book [23]).

Nevertheless, it was not before the exact summation of the entire
Chapman–Enskog series was found [24, 25, 26, 27] that the following rule was
discovered [28, 29]: Chapman–Enskog $\Rightarrow$ Viscosity + Capillarity.

The problem with Burnett’s equations is, among others, in the wrong sign of
the higher-order correction terms leading to a parasite instability of short
waves (Bobylev’s instability [30]). Maxwell, in his attempt to describe
thermal creep, encountered the wrong sign of the effect (according to
Burnett’s equations the light mill rotates in the direction opposite to the
experimentally observed rotation).

The exact solutions demonstrate that the van der Waals capillarity emerges in
the non-equilibrium ideal gas. Therefore, Korteweg’s equations can be a
candidate for the post–Navier–Stokes–Fourier equations. The fact that the van
der Waals capillarity energy emerges from a rigorous analytic solution of the
reduction problem for kinetic models of ideal gas was unexpected.

On the other hand, the exact solutions demonstrate saturation of dissipation:
the attenuation rate of the acoustic waves has a finite limit when the wave
length tends to zero [24, 27]. In that respect, the original Korteweg’s
equation may be not sufficient: for them, the attenuation rate grows as
(frequency)2/3 (for high frequency, see (26) and Fig. 7b below). Two different
hypotheses seem to be possible at present:

  * •

For the genuine Boltzmann equation with realistic interaction between the
particles, the asymptotic of the attenuation rate of acoustic waves is the
same as for the Korteweg equations;

  * •

The proper hydrodynamics for non-equilibrium gas should include the mixed
derivatives, which look like the operator $(1-\alpha\Delta)\upartial_{t}$ in
the left hand side of the equations (in the simplest form), where $\Delta$ is
the Laplace operator.

There are some arguments in favor of the Korteweg asymptotic reported very
recently [31]. Nevertheless, the second hypothesis is strongly supported by
the exactly solved problem, as we demonstrate below.

Figure 2: Knudsen pump. Bold lines correspond to thick pipes. The direction of
flux (for open pipe ends) is indicated. For closed pipe ends, $p_{1}>p_{2}$.
Original Knudsen’s experimental data for stationary pressure are in the table.
The plot of stationary temperature difference $\delta T$ from non-heated pipe
joints is presented schematically.

###  1.4 Capillarity of ideal gas in words and numbers

In high school textbooks and popular science literature, capillarity is the
ability of a liquid to flow in narrow gaps without the assistance of, and in
opposition to, external forces such as external pressure or gravity. In
scientific literature, the term ‘capillarity’ is used in wider sense. For
example, the pressure difference across the curve interface is termed the
‘capillary pressure’. Gibbs theory of capillarity [33] carries the title
“Influence of Surfaces of Discontinuity upon the Equilibrium of Heterogeneous
Masses.” This title includes two important ideas: equilibrium and
discontinuity. On the contrary, van der Waals [1, 22] proposed the theory of
continuous transition layers which was developed later in the dynamic phase
field theory and numerics. In the limit of the thin equilibrium film, the van
der Waals theory gives results very similar to the Gibbs theory.

Gibbs studied energy and equilibria of interface surfaces. We can call them
infinitely ‘thin films’, films without thickness. Van der Waals and, later,
his student Korteweg studied continuous media with energy dependent on density
gradients. We can call the van der Waals transition layers ‘thick films’, they
have non-negligible thickness and volume. Now, we aim to demonstrate that even
very simplified piecewise continuous approximation of a thick film can give
useful insights.

Consider two chambers with fluid connected by a capillary tube (Fig. 3).
Assume that a system of heaters and coolers keeps the temperatures,
$\theta_{1}>\theta_{2}$, in these chambers constant in time. In a steady
state, there is no fluid flux through the tube but a heat flux exists and so
this is not a thermodynamic equilibrium. Two limit cases for gases are very
well known:

  * •

The Navier–Stokes limit, the mean free path $\lambda\ll d$. In this limit zero
flux implies zero pressure difference. In the steady state $P_{1}=P_{2}$;

  * •

The Knudsen limit $\lambda\gg d$, the flux in the capillary tube is
collisionless. The gas flux from the hot chamber to the cold chamber is
$const\times n_{1}\bar{v}_{1}$ and the inverse flux is $const\times
n_{2}\bar{v}_{2}$ with the same $const$ (we presume that the capillary tube is
symmetric). Here, $n_{i}$ is density, the ideal gas equation gives
$n_{i}k\theta_{i}=P_{i}$, $\bar{v}_{i}$ is the thermal velocity, proportional
to $\sqrt{k_{\rm B}\theta_{i}}$, and $k_{\rm B}$ is the Boltzmann constant.
The zero flux condition gives

$\frac{P_{1}}{\sqrt{\theta_{1}}}=\frac{P_{2}}{\sqrt{\theta_{2}}}\;\;\mbox{or}\;\;\frac{P_{2}}{P_{1}}=\sqrt{\frac{\theta_{2}}{\theta_{1}}}.$
(1)

In the Knudsen limit, we see how the steady state difference between
temperatures produces the difference between pressures. This is called the
thermomechanical effect, and the effect of the capillary flux induced by the
temperature gradient is called thermoosmosis. The ratio $P_{2}/P_{1}$ is
called the thermal transpiration ratio.

Figure 3: Two chambers connected by a capillary tube.

The thermomechanical effect and thermoosmosis were discovered in the 20th
century for various fluids, from rarefied gases to superfluid liquids and
water [34, 35]. Now, they are considered as a promising driving mechanism for
micro- or nanoscale engines [37]. There are many approaches to the theory of
these effects based on non-equilibrium thermodynamics [36]. All of them
include unknown parameters, which should be found experimentally. Here we will
demonstrate two simple models based on capillarity (perhaps, the simplest
models). First of all, assume that the thermomechanical effect is produced by
a thin layer at the wall of the capillary. The property of this surface
depends on the thermodynamic conditions and on the material of the wall and
the fluid.

In the Gibbs approach [33] we operate with a thin film (a surface). The force
in the thermomechanical effect should be proportional to the length of the
section of this surface, that is the circle, $F=\gamma\pi d$, and the pressure
difference should be (Fig. 4)

$P_{1}-P_{2}=\frac{F}{\frac{1}{4}\pi
d^{2}}=\frac{4\gamma}{d}\;\;\mbox{and}\;\;\frac{P_{2}}{P_{1}}=\frac{P_{2}d}{P_{2}d+4\gamma}.$

Figure 4: Thermal transpiration values $P_{2}/P_{1}$ of hydrogen. The simplest
thin film capillarity model (solid line) and the thick film model (bold dotted
line) versus experimental data (bold dashed line). $P_{1}$ is the pressure at
the ‘hot end’, $T_{1}=473\pm 1{\rm K}$, $P_{2}$ is the pressure at the ‘cold
end’, $T_{2}=295\pm 1{\rm K}$ (room temperature). Experimental data are taken
from [38]. The simplest capillarity model gives:
$P_{2}/P_{1}=P_{2}d/(P_{2}d+1/15)$ (fitted to data in the interval $0.01<{\rm
Kn}<0.1$). Kn (the ratio of the mean free path $\lambda$ to the diameter $d$)
is evaluated at the cold end. We used the collisional diameter of H2, $D_{\rm
H2}\approx 2.3\times 10^{-10}{\rm m}$ estimated from viscosity data using mean
free path theory. In the analysis of molecular seeds a larger diameter is used
usually, $D_{\rm H2}\approx 2.89\times 10^{-10}{\rm m}$ [41]. In the thick
film model $a\approx 0.25$ for the experimental data presented.

This model includes one unknown coefficient $\gamma$, which can be easily
evaluated from the empiric curves (Fig. 4). Of course, this simplest model
belongs to Type 5 of Peierls’ classification [32]: ‘Instructive model (No
quantitative justifications but gives insight)’ or even to Type 6: ‘Analogy
(Only some features in common)’. This model works well in comparison with
experiment (Fig. 4) for not very high Knudsen numbers ${\rm Kn}<0.1$, gives
the proper Navier–Stokes limit, but has one fundamental defect:
$P_{2}/P_{1}\to 0$ when $P_{2}d\to 0$ (or, which is the same, Kn$\to\infty$).

To improve the model, let us follow the van der Waals capillarity idea.
Consider a ‘thick film’ near the border with thickness $\varsigma$, if
$d>2\varsigma$. If $d<2\varsigma$ then the thick surface fills the whole
capillary. Assume that the thickness is proportional to the mean free path,
$\varsigma=a\lambda$. The force of the thermomechanical effect in this model
is proportional to the section of the surface orthogonal to the axis of the
capillary (to the surface area), and to the particle density (empty space does
not produce force): $F=\kappa An$, where $A$ is the area of the section. The
coefficient $\kappa$ has the physical sense of ‘energy per particle’ and
describes the energetic difference between the boundary layer of thickness
$\varsigma$ and the rest of the volume of the capillary tube. Recall that
$n\lambda=(\sqrt{2}\pi D^{2})^{-1}$, where $D$ is the collision diameter of
molecules, and $P=nk_{\rm B}\theta$. Evaluate $\lambda$ at the cold end. After
simple algebra we get the formula for the transpiration ratio in the thick
film approximation:

$\frac{P_{2}}{P_{1}}=\left\\{\begin{array}[]{cl}\frac{P_{2}d}{P_{2}d+4\kappa
a(\sqrt{2}\pi D^{2})^{-1}(1-a{\rm Kn})}&\mbox{if }a{\rm Kn}<\frac{1}{2},\\\
\frac{1}{1+\kappa(k_{\rm B}\theta_{2})^{-1}}&\mbox{if }a{\rm
Kn}\geq\frac{1}{2}.\end{array}\right.$

Here, $\kappa$ is still an unknown coefficient. The thick film fills the whole
capillary at Kn=1/(2a). It is convenient to gather all the coefficients into
two unknown numbers: coefficient $\alpha$ (a function of temperature, which
can be easily extracted from the Knudsen asymptotic) and the ratio of the film
thickness to the mean free path, $a=\varsigma/\lambda$. We express the thermal
transpiration ratio through the Knudsen number:

$\frac{P_{2}}{P_{1}}=\left\\{\begin{array}[]{cl}\frac{1}{1+\alpha{\rm
Kn}(1-a{\rm Kn})}&\mbox{if }a{\rm Kn}<\frac{1}{2},\\\
\frac{1}{1+{\alpha}/(4a)}&\mbox{if }a{\rm
Kn}\geq\frac{1}{2}.\end{array}\right.$ (2)

The thick surface model has the proper high Kn and low Kn asymptotics and can
be used for the understanding the data. Of course, it is not far from the
simple dimensional analysis and belongs to Type 5 or 6 of Peierls’
classification. It does not reveal the intrinsic physical mechanism of the
thermomechanic effect and the thermoosmosis. We just hypothesized that these
effects are connected to the energy (enthalpy) density in the layers near the
interface between gas and capillary and to the transport in these layers
(thermoosmosis slip). The models can be improved if we account carefully for
variations of the films along the tube. Solving the Korteweg equations instead
of postulating of the thick surface can provide the further improvements, but
this next step requires more technical effort. Some calculations of boundary
layers and transpiration were performed recently on the basis of Korteweg’s
equation [39]. The problem is not in an accurate calculation of these effects.
The kinetic equations or even direct molecular simulation do this job quite
well. The question is about an appropriate continuum model that describes the
bulk motion, the thermal transpiration and the other similar effects in a
unified way.

In any case, already very simple models demonstrate that the transport
phenomena of the rarefied gas near the interfaces may be described using the
idea of capillarity. We might imagine that the light mill is a close relative
of the soap boat [40] because both are moving by capillarity effects.

###  1.5 The structure of the paper

The goal of this paper is to highlight the origin of the capillarity of ideal
gas. For this purpose, we give a comprehensive introduction into modern theory
of dynamic model reduction from kinetics to fluid dynamics and analyze
approximate and recently found exact solutions of this problem.

In the next Section, we start the unhurried introduction of the main
equations: from the conservation of mass and Cauchy equations to the
Navier–Stokes–Fourier equations. In Section 2.6, Korteweg’s stress tensor is
described. After that, we discuss a simple ‘mean free path’ theory for
Korteweg’s stress. To that end, the idea of Ehrenfests’ coarse-graining is
utilized: we take the collisionless Knudsen gas with periodic equilibration to
the local equilibrium and analyze the post-Navier–Stokes equation derived by
this approach.

After that, a short but comprehensive explanation of the Chapman–Enskog
procedure is presented. We introduce the invariance equation in Section 4.2
and construct the Chapman–Enskog as a formal solution to this equation
(Section 4.3). The geometric language allows us to avoid the usual bulky
calculations while describing the main constructions.

Analysis of exactly solvable problem of reduction allows us to prove that the
capillarity effects in ideal gas is not a by-product of special approximations
and series expansions but appears as a direct consequence of kinetic
equations. In Section 5 the energy formula for hydrodynamics of non-
equilibrium ideal gas is proved. It includes the pseudodifferential
capillarity term, which coincides with the van der Waals energy of capillarity
in first approximation. Finally, hypotheses are discussed about the plausible
forms of non-equilibrium fluid mechanics.

##  2 Fluid dynamics equations

###  2.1 Hydrodynamic fields

Consider the classical fluid, which is defined by the hydrodynamic fields:
density $\rho$ (scalar), velocity $\bm{u}$ (vector), and specific internal
energy $e$, that is internal energy per unit of mass (scalar).

###  2.2 Conservation of mass

$\boxed{\frac{\upartial\rho}{\upartial t}+\sum_{j}\frac{\upartial(\rho
u_{j})}{\upartial x_{j}}=0.}$ (3)

###  2.3 Cauchy momentum equations

$\boxed{\frac{\upartial(\rho u_{i})}{\upartial t}+\sum_{j}\frac{\upartial(\rho
u_{i}u_{j})}{\upartial x_{j}}=\sum_{j}\frac{\upartial\sigma_{ij}}{\upartial
x_{j}}+f_{i}.}$ (4)

where $\bm{f}=(f_{i})$ is the body force density, $\sigma_{ii}$ ($i=1,2,3$)
are normal stresses, and $\sigma_{ij}$ ($i\neq j$) are shear stresses. The
pressure $p$ is

$p=-\frac{1}{3}{\rm tr}\bm{\sigma}=-\frac{1}{3}\sum_{i}\sigma_{ii}.$

The Cauchy stress tensor $\bm{\sigma}=(\sigma_{ij})$ ($i,j=1,2,3$) describes
the so-called contact force: Let for a small element of a body surface with
area $\Delta S$ and outer normal vector $\bm{n}=(n_{i})$ the stress tensor
$\bm{\sigma}$ be given. Then the force applied to the body through this
surface fragment is

$\Delta\bm{F}=\bm{\sigma}\cdot\bm{n}\Delta S,\mbox{ that is, }F_{i}=\Delta
S\sum_{j}\sigma_{ij}n_{j}.$

The stress tensor is symmetric (to provide conservation of angular momentum),
$\sigma_{ij}=\sigma_{ji}$.

The momentum equation (4) is the second Newton law written for a small brick
of the material: Imagine a small cubic brick (a ‘parcel’) with soft but
impenetrable boundaries and edges parallel to the coordinate axes, which moves
with the material. The outer normals of the opposite faces differ by sign
only, therefore the sum of the contact forces from all faces is
$\sum_{j}\frac{\upartial\sigma_{ij}}{\upartial x_{j}}\Delta L^{3}+o(\Delta
L^{3})$, where $\Delta L$ is the edge of the brick (use the Taylor formula for
$\sigma_{ij}$). The possibility to represent the motion of continuum as the
flight of many infinitesimal parcels with soft deformable but impenetrable
boundaries is in the essence of the mechanics of materials.

###  2.4 Energy equation

If the material body $\Omega$ experiences infinitisemal displacement
$\bm{x}\mapsto\bm{x}+\delta\bm{r(x)}$, which vanishes with derivatives on the
boundary, then the work of the contact forces is (use the Stokes formula for
integration by parts):

$\delta W=\int_{\Omega}\sum_{i,j}\frac{\upartial\sigma_{ij}}{\upartial
x_{j}}\delta
r_{i}\,{\mathrm{d}}^{3}x=-\int_{\Omega}\sum_{i,j}\sigma_{ij}\frac{\upartial\delta
r_{i}}{\upartial x_{j}}\,{\mathrm{d}}^{3}x.$ (5)

Take into account symmetry of $\sigma_{ij}$ and find that $\delta W$ depends
on the symmetric part of ${\upartial\delta r_{i}}/{\upartial x_{j}}$:

$\delta
W=-\frac{1}{2}\int_{\Omega}\sum_{i,j}\sigma_{ij}\left(\frac{\upartial\delta
r_{i}}{\upartial x_{j}}+\frac{\upartial\delta r_{j}}{\upartial
x_{i}}\right)\,{\mathrm{d}}^{3}x.$

The power is the work per time. For a moving of continuum media, the power
produced by the contact forces is

$P=-\int_{\Omega}\sum_{i,j}\sigma_{ij}\frac{\upartial u_{i}}{\upartial
x_{j}}d^{3}x=-\frac{1}{2}\int_{\Omega}\sum_{i,j}\sigma_{ij}\left(\frac{\upartial
u_{i}}{\upartial x_{j}}+\frac{\upartial u_{j}}{\upartial
x_{i}}\right)\,{\mathrm{d}}^{3}x.$ (6)

According to the Cauchy equation and the formula for power we can write the
conservation of energy

$\boxed{\frac{\upartial(\rho\epsilon)}{\upartial
t}+\sum_{j}\frac{\upartial(\rho\epsilon u_{j})}{\upartial
x_{j}}=\sum_{ij}\frac{\upartial(\sigma_{ij}u_{i})}{\upartial
x_{j}}+\sum_{i}f_{i}u_{i}-\sum_{i}\frac{\upartial q_{i}}{\upartial x_{i}},}$
(7)

where $\epsilon$ is the energy density per unit mass:
$\epsilon=e+\frac{1}{2}u^{2}$, $e$ is the specific internal energy,
$u^{2}=\sum_{i}u_{i}^{2}$, and $\bm{q}=(q_{i})$ is the vector of heat flux,
which appears because of thermal conductivity. For the perfect (monoatomic)
gas, $e=\frac{3R}{2M}\theta$, where $R$ is the ideal gas constant, $M$ is the
molar mass and $\theta$ is the absolute temperature.

###  2.5 Stress tensor and heat flux for ideal fluid: the
Navier–Stokes–Fourier equations

Cauchy equations are simple and nice, and can be applied to many materials
(and even beyond the mechanics of materials). The question is: how can we
express the stress tensor and the heat flux through the hydrodynamic fields:
density $\rho$, velocity $\bm{u}$, and specific internal energy $e$?

The simplest answer is given by the Navier–Stokes stress tensor and Fourier
heat flux:

$\boxed{\begin{split}\sigma_{ij}&=\sigma_{ij}^{E}+\sigma_{ij}^{V};\\\
\sigma_{ij}^{E}&=-p\delta_{ij};\\\
\sigma_{ij}^{V}&=\lambda\delta_{ij}\sum_{k}\frac{\upartial u_{k}}{\upartial
x_{k}}+\mu\left(\frac{\upartial u_{i}}{\upartial x_{j}}+\frac{\upartial
u_{j}}{\upartial x_{i}}\right);\\\
q_{i}&=-\kappa\frac{\upartial\theta}{\upartial x_{i}}.\end{split}}$ (8)

Here (and in what follows), $\delta_{ij}$ is Kronecker’s delta,
$\sigma_{ij}^{E}$, $\sigma_{ij}^{V}$ are elastic and viscous contributions to
the Cauchy stress tensor, the coefficients $\lambda$, $\mu$ and $\kappa$
depend on density and temperature. The thermodynamic pressure $p$ in the
elastic stress is defined through the equation of state. For example, for
ideal gas $p=nR\theta=\rho\theta R/M$, where $n$ is the molar density
($n=\rho/M$). The standard assumption is that the viscous tensor has zero
trace and, therefore, $\lambda=-\frac{2}{3}\mu$. Strictly speaking, this is
just an assumption and the mechanical pressure should not necessarily coincide
with the thermodynamic pressure and may have a viscous component.

With the constitutive equations (8) we can immediately write the
Navier–Stokes–Fourier equations for the density, velocity and temperature of
ideal gas ($p=\frac{R}{M}\theta$, $e=\frac{3R}{2M}\theta$):

$\begin{split}&\frac{\upartial\rho}{\upartial t}+\sum_{j}\frac{\upartial(\rho
u_{j})}{\upartial x_{j}}=0;\\\ &\frac{\upartial(\rho u_{i})}{\upartial
t}+\sum_{j}\frac{\upartial(\rho u_{i}u_{j})}{\upartial
x_{j}}+\frac{R}{M}\frac{\upartial(\rho\theta)}{\upartial x_{i}}\\\
&=\frac{\upartial}{\upartial x_{i}}\left(\lambda\sum_{k}\frac{\upartial
u_{k}}{\upartial x_{k}}\right)+\sum_{j}\frac{\upartial}{\upartial
x_{j}}\left[\mu\left(\frac{\upartial u_{i}}{\upartial x_{j}}+\frac{\upartial
u_{j}}{\upartial x_{i}}\right)\right]+f_{i};\\\
&\frac{\upartial(\rho[\frac{3R}{2M}\theta+\frac{1}{2}u^{2}])}{\upartial
t}+\sum_{j}\frac{\upartial(\rho[\frac{3R}{2M}\theta+\frac{1}{2}u^{2}]u_{j})}{\upartial
x_{j}}\\\ &+\rho\theta\frac{R}{M}\sum_{i}\frac{\upartial u_{i}}{\upartial
x_{i}}=\lambda\left(\sum_{i}\frac{\upartial u_{i}}{\upartial
x_{i}}\right)^{2}\\\ &+\frac{\mu}{2}\sum_{ij}\left(\frac{\upartial
u_{i}}{\upartial x_{j}}+\frac{\upartial u_{j}}{\upartial
x_{i}}\right)^{2}+\sum_{i}f_{i}u_{i}+\sum_{i}\frac{\upartial}{\upartial
x_{i}}\left(\kappa\frac{\upartial\theta}{\upartial x_{i}}\right).\end{split}$
(9)

The power produced by the viscous stress is

$P=-\lambda\left(\sum_{i}\frac{\upartial u_{i}}{\upartial
x_{i}}\right)^{2}-\frac{\mu}{2}\sum_{ij}\left(\frac{\upartial u_{i}}{\upartial
x_{j}}+\frac{\upartial u_{j}}{\upartial x_{i}}\right)^{2}.$

If $\mu>0$ and $3\lambda+2\mu\geq 0$ then $P\leq 0$ (i.e. viscosity is
friction, indeed).

According to the model classification proposed by R. Peierls [32], the
Navier–Stokes viscous stress tensor is a typical example of the linear
response models together with Ohm’s and Hooke’s laws, Fourier’s law and many
others. ‘This refers to a situation in which one is interested, by definition,
in the response of a system to some parameter in the limit in which this
parameter may be treated as infinitesimal.’

###  2.6 Fluids with capillarity: van der Waals energy and Korteweg stress
tensor

In 1877, Gibbs published the theory of capillarity based on the idea of
surfaces of discontinuity [33]. He introduced and studied thermodynamics of
two-dimensional objects – surfaces. On the contrary, van der Waals proposed
the theory of capillarity using the hypothesis that density of the body varies
continuously at and near the transition layer and the energy depends on the
gradient of density [1] (just square of gradient is added). This is an
energetic ‘penalty’ for high gradients, which does not allow discontinuities
to appear and defines the thickness and the energy of the layer between
phases. Modern ‘phase field’ approaches are also based on this idea [42, 43].
After van der Waals, the standard approach to the continuous theory of
capillarity is: represent the Gibbs free energy of a body $\Omega$ as a sum,
$G=G_{0}+G_{K}$, where

$G_{0}=\int_{\Omega}g_{0}(\rho,\theta)\,{\mathrm{d}}^{3}x;\;\;G_{K}=\int_{\Omega}K(\rho,\theta)\sum_{i}\left(\frac{\upartial\rho}{\upartial
x_{i}}\right)^{2}\,{\mathrm{d}}^{3}x.$ (10)

Here, $K(\rho,\theta)$ is the capillarity coefficient. Korteweg found the
corresponding $G_{K}$ addition to the stress tensor:

$\boxed{\sigma^{K}_{ij}=\rho\left[\sum_{k}\frac{\upartial}{\upartial
x_{k}}\left(K(\rho,\theta)\frac{\upartial\rho}{\upartial
x_{k}}\right)\right]\delta_{ij}-K(\rho,\theta)\frac{\upartial\rho}{\upartial
x_{i}}\frac{\upartial\rho}{\upartial x_{j}}.}$ (11)

Detailed analysis of this stress tensor and its various modifications from the
rational thermodynamics point of view is performed by Dunn and Serrin [9].
Thermodynamics requires to complement the Korteweg stress tensor by the
additional contributions to the internal energy and the heat flux:

$\boxed{e^{K}=\frac{1}{2\rho}\left(K-\theta\frac{\upartial
K}{\upartial\theta}\right)\sum_{i}\left(\frac{\upartial\rho}{\upartial
x_{i}}\right)^{2};\;\;q^{K}_{i}=K(\rho,\theta)\rho\frac{\upartial\rho}{\upartial
x_{i}}\sum_{j}\frac{\upartial u_{j}}{\upartial x_{j}}.}$ (12)

To modify the Navier–Stokes equation we should just add the Korteweg stress to
the elastic and viscous stresses, and also add the capillarity contributions
to the energy and to the heat flux

$\sigma_{ij}=\sigma_{ij}^{E}+\sigma_{ij}^{V}+\sigma^{K}_{ij};\;\;\epsilon=e+\frac{1}{2}u^{2}+e^{K};\;\;q_{i}=-\kappa\frac{\upartial\theta}{\upartial
x_{i}}+q^{K}_{i}.$ (13)

With these additions, the Navier–Stokes–Fourier equations for ideal gas (9)
turn into Korteweg equations for gas with capillarity. Now we have to
understand how it may happen that the ideal gas gains capillarity.

##  3 Kinetics

###  3.1 Stress tensor for collisionless gas

Collisionless gas consists of particles which are moving without collisions
between them. It is described by the one-particle distribution function
$f(\bm{x},\bm{v},t)$, where $\bm{x}$ is particle’s position, $\bm{v}$ is
particle’s velocity, and $t$ is time. The integral of the density over a
domain $\Omega$ in the 6-dimensional space of positions and velocities is the
number of particles in $\Omega$:
$\int_{\Omega}f(\bm{x},\bm{v},t)d^{3}x\,{\mathrm{d}}^{3}v=N(t)$.

Time evolution of $f(\bm{x},\bm{v},t)$ is given by a simple explicit
expression:

$\boxed{f(\bm{x},\bm{v},t)=f(\bm{x}-\bm{v}t,\bm{v},0).}$ (14)

It satisfies the advection equation

$\frac{\upartial f(\bm{x},\bm{v},t)}{\upartial t}+\sum_{i}v_{i}\frac{\upartial
f(\bm{x},\bm{v},t)}{\upartial x_{i}}=0.$ (15)

The hydrodynamic variables are (for particles of unit mass):

$\rho=\int
f(\bm{x},\bm{v},t)\,{\mathrm{d}}^{3}v;\;\rho\bm{u}=\int\bm{v}f(\bm{x},\bm{v},t)\,{\mathrm{d}}^{3}v;\;\rho\epsilon=\frac{1}{2}\int
v^{2}f(\bm{x},\bm{v},t)\,{\mathrm{d}}^{3}v.$ (16)

Integration of the advection equation (15) in $v$ gives the conservation of
mass equation (3). If we multiply the advection equation by $v_{i}$ then the
integration gives the momentum equation and after multiplication with
$\frac{1}{2}v^{2}$ the integration produces the energy equation:

$\frac{\upartial\rho}{\upartial t}+\sum_{j}\frac{\upartial(\rho
u_{j})}{\upartial x_{j}}=0;\;\frac{\upartial\rho u_{i}}{\upartial
t}+\sum_{j}\frac{\upartial\Pi_{ij}}{\upartial
x_{j}}=0;\;\frac{\upartial\rho\epsilon}{\upartial t}+\sum_{j}\frac{\upartial
Q_{j}}{\upartial x_{j}}=0.$ (17)

Here, tensor $\Pi_{ij}$ of momentum flux and vector $Q_{i}$ of energy flux are
defined as moments of the distribution $f$ :

$\Pi_{ij}=\int
v_{i}v_{j}f(\bm{x},\bm{v},t)\,{\mathrm{d}}^{3}v;\;\;Q_{i}=\frac{1}{2}\int
v_{i}v^{2}f(\bm{x},\bm{v},t)\,{\mathrm{d}}^{3}v.$ (18)

It is convenient to separate the flux with the mean flow velocity $\bm{u}$
from other forms of transport. For this purpose, the central moments of $f$
are convenient. Simple algebra gives:

$\displaystyle\rho
e=\frac{1}{2}\int(\bm{v-u})^{2}f(\bm{x},{v},t)\,{\mathrm{d}}^{3}v,\;\;\rho\epsilon=\rho
e+\frac{1}{2}\rho u^{2};\;$ (19)
$\displaystyle\sigma_{ij}=-\int(v_{i}-u_{i})(v_{j}-u_{j})f(\bm{x},\bm{v},t)\,{\mathrm{d}}^{3}v,\;\;\Pi_{ij}=u_{i}u_{j}-\sigma_{ij};$
(20) $\displaystyle
q_{i}=\frac{1}{2}\int(v_{i}-u_{i})(\bm{v-u})^{2}f(\bm{x},{v},t)\,{\mathrm{d}}^{3}v,\;\;Q_{i}=q_{i}-\sum_{j}\sigma_{ij}u_{j}+u_{i}\rho\epsilon$
(21)

(the minus sign in the definition of $\sigma$ appears just to provide
similarity to Cauchy stress). Recall that here for simplicity, we take unit
mass of the particles and measure $\rho$ in number of particles per unit
volume. To return to the standard absolute temperature $\theta$ we have to
notice that $e=\frac{3}{2}k_{\rm B}\theta$, where $k_{\rm B}$ is Boltzmann’s
constant.

With usage of central moments, the transport equations look very similar to
the qCauchy and energy equations:

$\begin{split}&\frac{\upartial\rho}{\upartial t}+\sum_{j}\frac{\upartial(\rho
u_{i})}{\upartial x_{i}}=0;\\\ &\frac{\upartial(\rho u_{i})}{\upartial
t}+\sum_{j}\frac{\upartial(\rho u_{i}u_{j})}{\upartial
x_{j}}=\sum_{j}\frac{\upartial\sigma_{ij}}{\upartial x_{j}};\\\
&\frac{\upartial(\rho\epsilon)}{\upartial
t}+\sum_{j}\frac{\upartial(\rho\epsilon u_{j})}{\upartial
x_{j}}+\sum_{j}\frac{\upartial q_{j}}{\upartial
x_{j}}=\sum_{ij}\frac{\upartial(\sigma_{ij}u_{i})}{\upartial
x_{j}}.\end{split}$ (22)

The standard definition of pressure gives:

$p=-\frac{1}{3}\sum_{i}\sigma_{ii}=\frac{1}{3}\int(\bm{v}-\bm{u})^{2}f(\bm{x},\bm{v},t)\,{\mathrm{d}}^{3}v=\frac{2}{3}\rho
e.$ (23)

Particles of collisionless gas do not interact and there are no forces. Motion
of such a gas cannot be represented as motion of continuum. Nevertheless, the
Cauchy stress tensor can be defined through the momentum flux and the Cauchy
transport equation holds (22).

The elastic part (pressure) of the stress tensor for collisionless gas has the
very common form (23). It is not surprising that the rest of the stress tensor
does not satisfy the Navier–Stokes constitutive relations (8). Nevertheless,
Ehrenfests’ idea of coarse-graining allows us to deduce, step by step, the
Euler equations, the Navier–Stokes–Fourier equations and the Korteweg
equations for fluid motion from the simple model of collisionless gas.

###  3.2 Ehrenfests’ coarse-graining: from collisionless gas to Korteweg fluid
dynamics through coarse-graining

In 1911, Paul and Tanya Ehrenfest in their paper for the scientific
Encyclopedia [44] discussed emergence of irreversibility and introduced a
special operation, that of coarse-graining. This operation transforms a
probability density in phase space into a ‘coarse-grained’ density. That is a
piece-wise constant function, a result of density averaging in cells. The size
of cells is assumed to be small, but finite, and does not tend to zero. The
coarse-graining (‘shaking’) models thermalisation of the system by
uncontrollable and weak interaction with surroundings.

We generalize Ehrenfests’ idea of coarse-graining by combining the genuine
motion with the periodic partial equilibration. The result is Ehrenfests’
chain (Figure 5) [45]. The general theory with the proof of the entropy
production formula and various examples can be found elsewhere [46, 47]. Here
we just define Ehrenfests’ chain for the collisionless gas and present the
results of the coarse-graining.

First of all, for each value of hydrodynamic variables, the equilibrium
distribution should be defined: $f^{\rm eq}(\bm{v}|\rho,\bm{u},e)$. It is the
Maxwellian distribution. In the units used in this section (we count neither
moles nor grams but particles) the Maxwellian is

$f^{\rm eq}(\bm{v}|\rho,\bm{u},e)=\rho\left(\frac{3}{4\pi
e}\right)^{3/2}\exp\left(-\frac{3(v-u)^{2}}{4e}\right).$ (24)

For the given hydrodynamic fields, $\rho(\bm{x}),\bm{u}(\bm{x}),e(\bm{x})$, we
use the local Maxwellian, $f^{\rm
eq}(\bm{v}|\rho(\bm{x}),\bm{u}(\bm{x}),e(\bm{x}))$, that is a distribution in
six-dimensional space of positions and velocities, which is the Maxwellian for
each $\bm{x}$.

Define Ehrenfests’ chain with time step $\tau$. Let the initial values of the
hydrodynamic fields be given, $\rho(\bm{x},0),\bm{u}(\bm{x},0),e(\bm{x},0)$.
The initial distribution is

$f(\bm{x},\bm{v},0)=f^{\rm
eq}(\bm{v}|\rho(\bm{x},0),\bm{u}(\bm{x},0),e(\bm{x},0)).$

Start the advection from this distribution. After time $\tau$ it approaches
the distribution $f(\bm{x}-\bm{v}\tau,\bm{v},0)$. The hydrodynamic fields
$\rho(\bm{x},\tau),\bm{u}(\bm{x},\tau),e(\bm{x},\tau)$ are defined using the
moments of this distribution: $\rho(\bm{x},\tau)=\int
f(\bm{x}-\bm{v}\tau,\bm{v},0)\,{\mathrm{d}}^{3}v$, etc.

If the hydrodynamic fields
$\rho(\bm{x},k\tau),\bm{u}(\bm{x},k\tau),e(\bm{x},k\tau)$ are given then

$f(\bm{x},\bm{v},k\tau)=f^{\rm
eq}(\bm{v}|\rho(\bm{x},k\tau),\bm{u}(\bm{x},0),e(\bm{x},k\tau)),$

and the hydrodynamic fields
$\rho(\bm{x},(k+1)\tau),\bm{u}(\bm{x},(k+1)\tau),e(\bm{x},(k+1)\tau)$ are
defined using the moments of the distribution
$f(\bm{x}-\bm{v}\tau,\bm{v},k\tau)$. Ehrenfests’ chain is the sequence
$\rho(\bm{x},k\tau),\bm{u}(\bm{x},k\tau),e(\bm{x},k\tau)$ ($k=0,1,2,\ldots$)
(Figure 5). The hydrodynamic fields do not change in the equilibration steps
and their trajectories between the equilibration jumps are just ‘shadows’ of
the advection (14). Equilibration returns the distribution to the local
Maxwellian form and free flight destroys this form.

Figure 5: Ehrenfests’ chain. Two spaces are schematically represented: the
space of distribution functions (the upper rectangle) and the space of
hydrodynamic fields (the bottom parallelogram). The vertical lines with
bidirectional arrows illustrate coupling between the hydrodynamic fields and
the corresponding local Maxwellians. The coarse-graining is the projection of
a distribution function onto the corresponding local Maxwellian with
preservation of the hydrodynamic fields.

As a result, a discrete-time dynamical system is defined in the space of
hydrodynamic fields. We can look for a continuous-time system which transforms
$\rho(\bm{x},k\tau),\,\bm{u}(\bm{x},k\tau),\,e(\bm{x},k\tau)$ into
$\rho(\bm{x},(k+1)\tau),\,\bm{u}(\bm{x},(k+1)\tau),e(\bm{x},(k+1)\tau)$ during
time $\tau$.

Let us describe formally this procedure. Denote five hydrodynamic fields as
$M(t)=\\{\rho(\bm{x},\,\bm{v},t),\,\bm{u}(\bm{x},\bm{v},t),\,e(\bm{x},\bm{v},t)$.
We aim to find a closed system $\frac{\upartial}{\upartial t}M=J_{\tau}(M)$,
where $J_{\tau}$ is a (nonlinear) operator, which depends on the coarse-
graining time $\tau$ as a parameter, and $J_{\tau}(M)$ is also a set of five
fields (scalar, vector, and scalar again). Let operator $m$ map the
distribution $f$ onto hydrodynamic fields: $M=m(F)$ (16). Select an initial
state: $M(0)=M_{0}$ and $f_{0}$ is the corresponding local Maxwellian,
$f_{0}=f^{\rm eq}(\bm{x},\bm{v}|M_{0})$. Represent the result of advection
during time $\tau$ as a power series in $\tau$:

$f(\tau)=f_{0}+\sum_{k=1}^{\infty}\frac{\tau^{k}}{k!}(-1)^{k}(\bm{v},\bm{\nabla})^{k}f_{0}.$

Find the power series for the result of hydrodynamic motion according to a
hypothetical equation $\dot{M}=J_{\tau}(M)$:

$M(t)=M(0)+tJ_{\tau}+\sum_{k=2}^{\infty}\frac{t^{k}}{k!}\left.\frac{{\mathrm{d}}^{k-1}J_{\tau}(M)}{{\mathrm{d}}t^{k-1}}\right|_{M=M_{0}},$

where the sequence of derivatives
$\frac{{\mathrm{d}}^{k-1}J_{\tau}(M)}{{\mathrm{d}}t^{k-1}}$ is calculated by
iterations of the chain rule from the equation $\dot{M}=J_{\tau}(M)$. We use
this series for $t=\tau$.

Recall that $J_{\tau}(M)$ depends also on $\tau$ as on a parameter. Represent
this dependence as a power series as well and substitute it into the series
for $M(t)$ $(t=\tau)$. The matching condition should hold:
$M(\tau)=m(f(\tau))$ (Fig. 6). Coefficients at the same powers of $\tau$ in
$M(\tau)$ and $m(f(\tau))$ should coincide. As a result, we recover the
macroscopic vector field $J_{\tau}$ order-by-order.

Figure 6: Matching condition for hydrodynamic equations. Two spaces are
schematically represented: the space of distribution functions (upper
rectangle) with the manifold of local Maxwellians and the space of
hydrodynamic fields (the bottom parallelogram). The vertical lines with
bidirectional arrows illustrate coupling between the hydrodynamic fields and
the corresponding local Maxwellians. The matching condition is:
$M(\tau)=f^{\rm eq}(\bm{x}-\bm{v}\tau,\bm{v}|M_{0})$.

The zeroth-order terms give the Euler equations and the first–order terms give
the Navier–Stokes–Fourier equations with viscosity and heat conductivity
proportional to $\tau\rho e$ with constant coefficients ($\rho e$ is the
density of internal energy): $\lambda=-\frac{2}{3}\mu$,
$\mu=\frac{1}{3}\tau\rho e$ [47]. These formulas recall the Maxwell mean free
path estimations of kinetic coefficients: $\mu\approx\frac{1}{3}\rho
l\bar{v}=\frac{1}{3}\rho\tau_{f}\bar{v}^{2}$ (where $\bar{v}$ is the thermal
velocity and $\tau_{f}$ is the mean free time, see, for example [48]). The
thermal velocity for Maxwellian distribution is $\bar{v}\sim\sqrt{4e/3}$ (that
is the most probable velocity, whereas the average magnitude of the velocity
of the particles is $\frac{2}{\sqrt{\pi}}\bar{v}$). Therefore, the coarse-
graining time $\tau$ can be considered as the analogue of the mean free time
$\tau_{f}$ with a constant coefficient of order 1.

It is worth mentioning that viscosity and thermal conductivity grow with the
coarse-graining time, and dissipative processes become more intensive with the
growth of the mean free path. Collisions in gas delay diffusive transport and
decrease kinetic coefficients.

Thus, the simple model of non-equilibrium gas based on Ehrenfests’ coarse
graining produces the Navier–Stokes–Fourier equations in the first order of
the coarse-graining time $\tau$. This model has only one free parameter,
$\tau$, and, therefore, the viscosity and the thermal conductivity are
proportional (and Prandtl number Pr=1). This is not a miracle, and just
demonstrates that the Navier–Stokes–Fourier equations provide an unavoidable
first dissipative correction to the Euler equations. The problem is in the
next approximation. In the second order of $\tau$, the coarse graining
produces the Korteweg terms from the collisionless gas.

Consider the linearized fluid dynamics near a state with $\bm{u}=0$,
$\rho=\rho_{0}$, $e=e_{0}$. Introduce the dimensionless deviation from this
state:

$M_{0}=\frac{\rho-\rho_{0}}{\rho_{0}},\,M_{i}=\frac{u_{i}}{\bar{v}}\;(i=1,2,3),\,M_{4}=\frac{3}{2}\left(\frac{\rho-\rho_{0}}{\rho_{0}}+\frac{e-e_{0}}{e_{0}}\right),$

where $\bar{v}=\sqrt{\frac{4}{3}e_{0}}$. Use the new space scale in which
$\bar{v}=1$: $x_{\rm new}=x/\bar{v}$. In these variables, the linearized
second approximation in $\tau$ for the coarse-grained free flight advection
is:

$\begin{split}&\frac{\upartial M_{0}}{\upartial
t}=\sum_{i=1}^{3}\frac{\upartial M_{i}}{\upartial x_{i}};\\\ &\frac{\upartial
M_{i}}{\upartial t}=-\frac{1}{3}\frac{\upartial M_{4}}{\upartial
x_{i}}+\frac{\tau}{4}\sum_{j=1}^{3}\frac{\upartial}{\upartial
x_{j}}\left(\frac{\upartial M_{i}}{\upartial x_{j}}+\frac{\upartial
M_{j}}{\upartial x_{i}}-\frac{2}{3}\delta_{ij}\sum_{k=1}^{3}\frac{\upartial
M_{k}}{\upartial x_{k}}\right)\\\ &\qquad+\tau^{2}\frac{\upartial}{\upartial
x_{i}}\left(\frac{1}{8}\Delta M_{0}+\frac{89}{108}\Delta
M_{4}\right)\;(i=1,2,3);\\\ &\frac{\upartial M_{4}}{\upartial
t}=-\frac{5}{2}\sum_{k=1}^{3}\frac{\upartial M_{k}}{\upartial
x_{k}}+\frac{5\tau}{2}\sum_{k=1}^{3}\Delta
M_{4}+\tau^{2}\frac{59}{9}\Delta\left(\sum_{k=1}^{3}\frac{\upartial
M_{k}}{\upartial x_{k}}\right).\end{split}$ (25)

Here, $\Delta=\sum_{k=1}^{3}\frac{\upartial^{2}}{\upartial x_{k}^{2}}$ is the
Laplace operator. Terms without $\tau$ correspond to the linearized Euler
equation, the first order terms give the Navier–Stokes–Fourier dissipation
(viscosity and thermal conductivity). Terms of the second order in $\tau$
correspond to Korteweg’s stress tensor (in equations for $M_{i}$, $i=1,2,3$)
and to the contribution of capillarity into heat flux (in equation for
$M_{4}$).

The parameter $\tau$ can be eliminated from equations (25) by rescaling.
Recall that in these equations space and time are measured by the same time
units (to provide $\bar{v}=1$). Let us select the new time and space unit
$\tau$. In this scale, $\tau=1$ in (25). From some point of view, this means
that the equations are the same for all values $\tau>0$. For example, it is
sufficient to analyse stability just for one value $\tau=1$.

Let us look for the solutions of equations (25) in the form
$M_{j}=A_{j}\exp(\lambda t+i\bm{kx})$. Here, $\bm{k}$ is the wave vector and
real parts of $\lambda$ describe dissipation. It is necessary for stability
that ${\rm Re}\lambda\leq 0$ for all real vectors $\bm{k}$. The characteristic
equation for $\lambda(k)$ has five roots (Fig. 7). For all of them ${\rm
Re}\lambda<0$ ($k\neq 0$). In the short wave asymptotic of $\lambda$
($k^{2}\to\infty$) the roots are:

$\frac{\lambda_{1,2}}{k^{2}}=-\frac{1}{4};\;\frac{\lambda_{3}}{k^{2}}=O(1/k^{2})\to
0;\;\frac{\lambda_{4,5}}{k^{2}}=-\frac{17}{12}\pm i|k|\sqrt{\frac{59\cdot
89}{9\cdot 108}}.$ (26)

$None$ $None$

Figure 7: Dispersion curves for the second approximation (25). a) Dependence
of attenuation rates on $|k|$: solid $\lambda_{1,2}$, dashed $\lambda_{3}$,
dotted $\lambda_{4,5}$; b) Curves $\lambda(k)$ on complex plane for
$\lambda_{4,5}$

The second approximation in $\tau$ for the coarse-grained collisionless gas
provides a sort of ‘mean free path model’ of capillarity effects and Korteweg
stress in ideal gas. It seems to be surprising that such a simple approach
generates the Korteweg equations (with some restrictions on the relationships
between parameters, like Pr=1).

Nevertheless, two question remain: (i) are there capillarity effects in more
realistic models of non-equilibrium gas, and (ii) what will happen in the
coarse-grained collisionless gas beyond the second order in $\tau$? Of course,
if we summarize the whole power series then the solutions of the resulting
continuum mechanics equations will go exactly through the points of Erenfests’
chain but what will happen on the way?

The more realistic model of non-equilibrium gases is well-known, that is the
Boltzmann equation.

##  4 From Boltzmann kinetics to fluid dynamics: model reduction

###  4.1 The model reduction problem

Let us consider kinetic equations which describe the evolution of a one-
particle gas distribution function $f(\bm{x},\bm{v},t)$

$\boxed{\frac{\upartial f}{\upartial t}+\sum_{i}v_{i}\frac{\upartial
f}{\upartial x_{i}}=\frac{1}{\rm Kn}Q(f).}$ (27)

The only difference from the free flight advection (15) is the collision
operator $Q(f)$ in the right hand part of (27). For the Boltzmann equation,
$Q$ is a quadratic operator and, therefore, the notation $Q(f,f)$ is often
used. Kn is the Knudsen number, which is a dimensionless parameter defined as
$\mathrm{Kn}=\frac{l}{L}$, where $l$ is the mean free path and $L$ is
‘representative physical length’ scale. It aims to measure how important the
microscopic effects are (associated with $l$) at the macroscopic scale ($L$).

The collision term $Q(f)$ in (27) describes the change in distribution $f$ due
to collisions. The term $Q(f)$ does not contribute directly into the time
derivatives of the hydrodynamic variables, $\rho=\int f\,{\mathrm{d}}^{3}v$,
$\bm{u}=\int\bm{v}f\,{\mathrm{d}}^{3}v$ and
$e=\frac{1}{2}\int(\bm{v-u})^{2}f\,{\mathrm{d}}^{3}v$ because, due to the
mass, momentum and energy conservation in collisions:

$\int\\{1;\bm{v};\bm{v}^{2}\\}Q(f)\,{\mathrm{d}}^{3}v=0.$

For the space-uniform distributions the collision operator in (27) provides
relaxation to Maxwellian distribution (24) [49] (about the general problem of
existence and stability for the Boltzmann equations we refer readers to [50]).

Therefore, the following qualitative ‘nonrigorous picture of the Boltzmann
dynamics’ [51] is expected for the solutions: (i) the collision term goes
quickly almost to its equilibrium (the system almost approaches a local
equilibrium) and during this fast initial motion the changes of hydrodynamic
variables are small, (ii) after that the distribution function is defined with
high accuracy by the hydrodynamic variables (if they have bounded space
derivatives). The relaxation of the collision term almost to its equilibrium
is supported by monotonic entropy growth (Boltzmann’s $H$-theorem). This
qualitative picture is illustrated in Fig. 8.

Figure 8:  Fast–slow decomposition. Bold dashed lines outline the vicinity of
the slow manifold where the solutions stay after initial layer. The projection
of the distributions onto the hydrodynamic fields and the parametrization of
this manifold by the hydrodynamic fields are represented. (A derivative work
based on Fig. 1 from [59].)

Perhaps, McKean gave the first clear explanation of the problem as a
construction of a ‘nice submanifold’ where ‘the hydrodynamical equations
define the same flow as the (more complicated) Boltzmann equation does’ [52].
He presented the problem by a diagram and we reproduce his idea in slightly
revised form in Fig. 9. How to find this ‘nice submanifold’? Perhaps, the
first task is to write an equation for them (for more detail we refer to the
chapter ‘Invariance equation’ of the book [53]).

Figure 9: McKean diagram. All the model reduction approaches aim to create a
lifting operation, from the hydrodynamic variables to the relevant
distributions on the invariant manifold. IM stands for Invariant Manifold. The
part of the diagram in the dashed polygon is commutative, that is any
superposition of operations following the arrows does not depend on the path
but only on the start and end points. (A derivative work based on Fig. 2 from
[59].)

###  4.2 Invariance equation

The invariance equation expresses the fact that the vector field is tangent to
the manifold. In 1892, A.M. Lyapunov introduced and studied this equation in
his doctoral thesis ‘The general problem of the stability of motion’ (the
Lyapunov ‘auxiliary theorem’ [55], which is recently used in various
applications, from model reduction in chemical kinetics to optimal control
problems [56, 57]).

The essence of the invariance equation and the Chapman–Enskog method become
more transparent in the abstract form when the bulky details of the genuine
Boltzmann equation do not obscure the simple geometric sense. Let us consider
an equation in a domain $U$ of a normed space $E$ with analytical right hand
side

$\upartial_{t}f=J(f).$ (28)

We call $f$ a microscopic variable. A space of macroscopic variables (moment
fields) is defined with a linear map $m:f\mapsto M$ ($M$ are macroscopic
variables). Assume that the image of $m$ is the whole space of $M$ (i.e. $m$
is surjective). We are looking for an invariant manifold (McKean’s nice
manifold’) parameterized with the macroscopic fields $M$. For such manifolds
we use the notation $\bm{f}_{M}$. The self-consistency condition
$m(\bm{f}_{M})=M$ is necessary: the manifold is parametrized by its own value
of macroscopic variables.

For the reduction of Boltzmann’s kinetics hydrodynamics, the microscopic
variable is the one-particle distribution function $f$ and the macroscopic
variables are hydrodynamic fields. In extended irreversible thermodynamics
(EIT) the larger sets of macroscopic variables are considered [58].

The invariance equation for $\bm{f}_{M}$ is

$\boxed{J(\mbox{\boldmath$f$}_{M})=(D_{M}\mbox{\boldmath$f$}_{M})m(J(\mbox{\boldmath$f$}_{M})).}$
(29)

Here, the differential $D_{M}$ of $\mbox{\boldmath$f$}_{M}$ is calculated at
the point $M=m(\mbox{\boldmath$f$}_{M})$.

Equation (29) means that the time derivative of $f$ on the manifold
$\mbox{\boldmath$f$}_{M}$ can be calculated by a simple chain rule: calculate
the derivative of $M$ using the map $m$,
$\dot{M}=m(J(\mbox{\boldmath$f$}_{M}))$, then write that on the invariant
manifold the time dependence $\mbox{\boldmath$f$}(t)$ can be expressed through
the time dependence of $M(t)$:
$\mbox{\boldmath$f$}(t)=\mbox{\boldmath$f$}_{M(t)}$ and the time derivatives
also coincide. If we find the approximate solution to eq. (29) then the
approximate reduced model (hydrodynamics) is

$\upartial_{t}M=m(J(\mbox{\boldmath$f$}_{M})).$ (30)

The invariance equation can be represented in the form

$\upartial^{\rm micro}_{t}\mbox{\boldmath$f$}_{M}=\upartial^{\rm
macro}_{t}\mbox{\boldmath$f$}_{M},$ (31)

where the microscopic time derivative, $\upartial^{\rm
micro}_{t}\mbox{\boldmath$f$}_{M}$ is just a value of the vector field
$J(\mbox{\boldmath$f$}_{M})$ and the macroscopic time derivative is calculated
by the chain rule,

$\upartial^{\rm
macro}_{t}\mbox{\boldmath$f$}_{M}=(D_{M}\mbox{\boldmath$f$}_{M})\upartial_{t}M$

under the assumption that dynamics of $M$ follows the projected equation (30).
For more details we refer to the review paper [59].

###  4.3 Chapman–Enskog expansion in brief

The Chapman–Enskog approach assumes the special singularly perturbed structure
of the equations and looks for the invariant manifold in a form of the series
in the powers of a small parameter $\epsilon$. A one-parametric system of
equations is considered:

$\upartial_{t}f+A(f)=\frac{1}{\epsilon}Q(f).$ (32)

The following assumptions connect the macroscopic variables to the singular
perturbation:

  * •

$m(Q(f))=0$;

  * •

for each $M\in m(U)$ the system of equations $Q(f)=0$, $m(f)=M$ has a unique
solution $\mbox{\boldmath$f$}^{\rm eq}_{M}$ (in Boltzmann kinetics it is the
local Maxwellian);

  * •

$\mbox{\boldmath$f$}^{\rm eq}_{M}$ is asymptotically stable and globally
attracting for the fast system $\upartial_{t}f=\frac{1}{\epsilon}Q(f)$ in
$(\mbox{\boldmath$f$}^{\rm eq}_{M}+\ker m)\cap U$.

Let the $\mathcal{Q}_{M}$ be the differential of the fast vector field $Q(f)$
at equilibrium $\mbox{\boldmath$f$}^{\rm eq}_{M}$:
$\mathcal{Q}_{M}=(D_{M}Q(\mbox{\boldmath$f$}))_{\mbox{\boldmath$f$}=\mbox{\boldmath$f$}^{\rm
eq}_{M}}$. For the Chapman–Enskog method it is important that
$\mathcal{Q}_{M}$ is invertible in $\ker m$.

The invariance equation for the singularly perturbed system (32) with the
moment parametrization $m$ is:

$\boxed{\frac{1}{\epsilon}Q(\mbox{\boldmath$f$}_{M})=A(\mbox{\boldmath$f$}_{M})-(D_{M}\mbox{\boldmath$f$}_{M})(m(A(\mbox{\boldmath$f$}_{M}))).}$
(33)

The self-consistency condition $m(\mbox{\boldmath$f$}_{M})=M$ gives
$m(D_{M}\mbox{\boldmath$f$}_{M})m(J)=m(J)$ for all $J$, hence,

$m[A(\mbox{\boldmath$f$}_{M})-(D_{M}\mbox{\boldmath$f$}_{M})m(A(\mbox{\boldmath$f$}_{M}))]=0.$
(34)

If we find an approximate solution $\mbox{\boldmath$f$}_{M}$ of (33) then the
corresponding macroscopic (hydrodynamic) equation (30) is

$\upartial_{t}M+m(A(\mbox{\boldmath$f$}_{M}))=0.$ (35)

Let us represent all the operators in (33) by the Taylor series (for the
Boltzmann equation $A$ is the linear free flight operator, $A=v\cdot\nabla$,
and $Q$ is the quadratic collision operator). We look for the invariant
manifold in the form of the power series:

$\mbox{\boldmath$f$}_{M}=\mbox{\boldmath$f$}^{\rm
eq}_{M}+\sum_{i=1}^{\infty}\epsilon^{i}\mbox{\boldmath$f$}^{(i)}_{M}$ (36)

with the self-consistency condition $m(\mbox{\boldmath$f$}_{M})=M$, which
implies $m(\mbox{\boldmath$f$}^{\rm eq}_{M})=M$,
$m(\mbox{\boldmath$f$}^{(i)}_{M})=0$ for $i\geq 1$. After matching the
coefficients of the series in (33), we obtain for every
$\mbox{\boldmath$f$}^{(i)}_{M}$ a linear equation

$\mathcal{Q}_{M}\mbox{\boldmath$f$}^{(i)}_{M}=P^{(i)}(\mbox{\boldmath$f$}^{\rm
eq}_{M},\mbox{\boldmath$f$}^{(1)}_{M},\ldots,\mbox{\boldmath$f$}^{(i-1)}_{M}),$
(37)

where the polynomial operator $P^{(i)}$ at each order $i$ can be obtained by
straightforward calculations from (33). Due to the self-consistency,
$m(P^{(i)})=0$ for all $i$ and the equation (37) is solvable. The first term
of the Chapman–Enskog expansion has a simple form

$\boxed{\mbox{\boldmath$f$}^{(1)}_{M}=\mathcal{Q}_{M}^{-1}(1-(D_{M}\mbox{\boldmath$f$}^{\rm
eq}_{M})m)(A(\mbox{\boldmath$f$}^{\rm eq}_{M})).}$ (38)

Many books and papers are devoted to the detailed analysis of this formula for
the Boltzmann equation and other kinetic equations after the classical book
[23]. Most of the physical applications of kinetic theory, from transport
processes in gases to modern numerical methods (lattice Boltzmann models [54])
give examples of the practical applications and deciphering of this formula.
For the Boltzmann kinetics, the zero-order approximation,
$\mbox{\boldmath$f$}^{(0)}_{M}\approx\mbox{\boldmath$f$}^{\rm eq}_{M}$
produces in projection on the hydrodynamic fields (35) the compressible Euler
equation. The first-order approximate invariant manifold,
$\mbox{\boldmath$f$}^{(1)}_{M}\approx\mbox{\boldmath$f$}^{\rm
eq}_{M}+\epsilon\mbox{\boldmath$f$}^{(1)}_{M}$, gives the compressible Navier-
Stokes equation and provides the explicit dependence of the transport
coefficients on the collision model.

The calculation of higher order terms needs nothing but differentiation and
calculation of the inverse operator $\mathcal{Q}_{M}^{-1}$, although it may be
rather bulky. The second order in $\epsilon$ hydrodynamic equations (30) are
called Burnett equations (with $\epsilon^{2}$ terms) and super-Burnett
equations for higher orders.

Alas, the Burnett equations produce non-physical effects, instability of short
waves (Bobylev’s instability) and negative viscosity at the space scale near
mean free path, i.e. close to the scales, where they are needed. What will
happen if we sum up the whole Chapman–Enskog series (at least,
hypothetically)? For Ehrenfests’ chain the sum is expected to coincide with
the chain in discrete time moments $n\tau$. For the Boltzmann equation, it
remains unclear. Let us choose a simplified system, for which the model
reduction can be performed explicitly for all time scales.

###  4.4 Exact analytic solution of reduction problem for a simple kinetic
equation

The simplest model and the starting point in our analysis is

$\boxed{\begin{split}\upartial_{t}p&=-\frac{5}{3}\upartial_{x}u,\\\
\upartial_{t}u&=-\upartial_{x}p-\upartial_{x}\sigma,\\\
\upartial_{t}\sigma&=-\frac{4}{3}\upartial_{x}u-\frac{1}{\epsilon}\sigma,\end{split}}$
(39)

To obtain this system from Boltzmann’s equation we have to select the set of
macroscopic variables: “Hydrodynamic fields plus stress tensor’. For these 10
variables (1 – density, plus 3 – momentum density, plus 1 – energy, and plus 5
– traceless symmetric stress tensor) find the representative
(quasiequilibrium) density function $f$ by conditional maximization of
entropy, $S=-\int f\ln f\,{\mathrm{d}}^{3}v$, for given values of these 10
macroscopic moments. The conditional maximization of entropy subject to given
first and second moments is a standard exercise in the MaxEnt approach. The
result is the Gaussian distribution (more precisely, it is the locally
Gaussian distribution with space-dependent parameters defined by hydrodynamic
fields). Substitute this function into Botzmann’s equation, calculate the
corresponding time derivatives of macroscopic variables and the closed
10-moment MaxEnt Grad system is ready. Linearize and study the solutions that
depend on one space coordinate $x$ with the velocity oriented along the $x$
axis. Use dimensionless variables. (Here, $\sigma$ is the dimensionless
$xx$-component of the stress tensor.)

Let us illustrate the invariance equation and the Chapman–Enskog series on the
simplest example (39).

$\displaystyle\mbox{\boldmath$f$}=\left(\begin{array}[]{c}p(x)\\\ u(x)\\\
\sigma(x)\end{array}\right),\;m=\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\end{array}\right),\;M=\left(\begin{array}[]{c}p(x)\\\
u(x)\end{array}\right),\;\ker m=\left\\{\left(\begin{array}[]{c}0\\\ 0\\\
y\end{array}\right)\right\\},$ $\displaystyle
A(\mbox{\boldmath$f$})=\left(\begin{array}[]{c}\frac{5}{3}\upartial_{x}u\\\
\upartial_{x}p+\upartial_{x}\sigma\\\
\frac{4}{3}\upartial_{x}u\end{array}\right),\;Q(\mbox{\boldmath$f$})=\left(\begin{array}[]{c}0\\\
0\\\ -\sigma\end{array}\right),\;\mathcal{Q}_{M}^{-1}=\mathcal{Q}_{M}=-1\mbox{
on }\ker m,$ $\displaystyle\mbox{\boldmath$f$}^{\rm
eq}_{M}=\left(\begin{array}[]{c}p(x)\\\ u(x)\\\
0\end{array}\right),\;D_{M}\mbox{\boldmath$f$}^{\rm
eq}_{M}=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
0&0\end{array}\right),\;\mbox{\boldmath$f$}^{(1)}_{M}=\left(\begin{array}[]{c}0\\\
0\\\ -\frac{4}{3}\upartial_{x}u\end{array}\right).\;$

We hasten to remark that (39) is a simple linear system and can be integrated
immediately in explicit form. However, that solution contains both the fast
and slow components and it does not readily reveal the slow hydrodynamic
manifold of the system. Instead, we are interested in extracting this slow
manifold by a direct method. The Chapman-Enskog expansion is thus the for this
which we shall address first.

The projected equations in the zeroth (Euler) and the first (Navier–Stokes)
order of $\epsilon$ are

$\mbox{ (Euler)
}\begin{array}[]{ll}\upartial_{t}p=-\frac{5}{3}\upartial_{x}u,\\\
\upartial_{t}u=-\upartial_{x}p;\end{array}\;\;\;\mbox{ (Navier-Stokes)
}\begin{array}[]{ll}\upartial_{t}p=-\frac{5}{3}\upartial_{x}u,\\\
\upartial_{t}u=-\upartial_{x}p+\epsilon\frac{4}{3}\upartial_{x}^{2}u.\end{array}$

It is straightforward to calculate the two next terms (Burnett and super-
Burnett ones) but let us introduce convenient notations to represent the whole
Chapman-Enskog series for (39). Only the third component of the invariance
equation (33) for (39) is non-trivial because of the self-consistency
condition (34), and we can write

$-\frac{1}{\epsilon}\sigma_{(p,u)}=\frac{4}{3}\upartial_{x}u-\frac{5}{3}(D_{p}\sigma_{(p,u)})(\upartial_{x}u)-(D_{u}\sigma_{(p,u)})(\upartial_{x}p+\upartial_{x}\sigma_{(p,u)}).$
(43)

Here, $M={(p,u)}$ and the differentials are calculated by the elementary rule:
if a function $\Phi$ depends on values of $p(x)$ and its derivatives,
$\Phi=\Phi(p,\upartial_{x}p,\upartial^{2}_{x}p,\ldots)$ then $D_{p}\Phi$ is a
differential operator,

$D_{p}\Phi=\frac{\upartial\Phi}{\upartial
p}+\frac{\upartial\Phi}{\upartial(\upartial_{x}p)}\upartial_{x}+\frac{\upartial\Phi}{\upartial(\upartial_{x}^{2}p)}\upartial^{2}_{x}+\ldots$

The equilibrium of the fast system (the Euler approximation) is known,
$\sigma_{(p,u)}^{(0)}=0$. We have already found
$\sigma_{(p,u)}^{(1)}=-\frac{4}{3}\upartial_{x}u$ (the Navier–Stokes
approximation). In each order of the Chapman–Enskog expansion $i\geq 1$ we get
from (43):

$\sigma_{(p,u)}^{(i+1)}=\frac{5}{3}(D_{p}\sigma_{(p,u)}^{(i)})(\upartial_{x}u)+(D_{u}\sigma_{(p,u)}^{(i)})(\upartial_{x}p)+\sum_{j+l=i}(D_{u}\sigma_{(p,u)}^{(j)})(\upartial_{x}\sigma_{(p,u)}^{(l)}).$
(44)

This chain of equations is nonlinear but every $\sigma_{(p,u)}^{(i+1)}$ is a
linear function of derivatives of $u$ and $p$ with constant coefficients
because this sequence starts from $-\frac{4}{3}\upartial_{x}u$ and the
induction step in $i$ is obvious.

Simple algebra gives for the Burnett term ($i+1=2$)
$\sigma_{(p,u)}^{(2)}=-\frac{4}{3}\upartial^{2}_{x}p$ and for the super
Burnett term ($i+1=3$) the
$\sigma_{(p,u)}^{(3)}=-\frac{4}{9}\upartial^{3}_{x}u$. The projected equations
have the form

$\displaystyle\begin{array}[]{ll}\upartial_{t}p=-\frac{5}{3}\upartial_{x}u,\\\
\upartial_{t}u=-\upartial_{x}p+\epsilon\frac{4}{3}\upartial_{x}^{2}u+\epsilon^{2}\frac{4}{3}\upartial_{x}^{3}p;\end{array}\mbox{
Burnett; }$ (47)
$\displaystyle\begin{array}[]{ll}\upartial_{t}p=-\frac{5}{3}\upartial_{x}u,\\\
\upartial_{t}u=-\upartial_{x}p+\epsilon\frac{4}{3}\upartial_{x}^{2}u+\epsilon^{2}\frac{4}{3}\upartial_{x}^{3}p+\epsilon^{3}\frac{4}{9}\upartial_{x}^{4}u;\end{array}\mbox{
super Burnett. }$ (50)

Compute the dispersion relation for these hydrodynamic modes. Exclude
$\epsilon$ using a new space-time scale, $x^{\prime}=\epsilon^{-1}x$, and
$t^{\prime}=\epsilon^{-1}t$. Look for wave solutions
$u=u_{k}\varphi(x^{\prime},t^{\prime})$, and
$p=p_{k}\varphi(x^{\prime},t^{\prime})$, where
$\varphi(x^{\prime},t^{\prime})=\exp(\lambda t^{\prime}+ikx^{\prime})$, and
$k$ is a real-valued wave vector. The following dispersion relations
$\lambda(k)$ are the conditions of a non-trivial solvability of the
corresponding linear system with respect to $u_{k}$ and $p_{k}$:

$\begin{split}\lambda_{\pm}&=-\frac{2}{3}k^{2}\pm\frac{1}{3}i|k|\sqrt{15-4k^{2}},\;\;\mbox{Navier
--Stokes};\\\
\lambda_{\pm}&=-\frac{2}{3}k^{2}\pm\frac{1}{3}i|k|\sqrt{15+16k^{2}},\;\;\mbox{Burnett};\\\
\lambda_{\pm}&=-\frac{2}{9}k^{2}(3-k^{2})\pm\frac{1}{9}i|k|\sqrt{135+144k^{2}+24k^{4}-4k^{6}},\;\;\mbox{super
Burnett};\end{split}$ (51)

For the super Burnet equations ${\rm Re}\lambda>0$ if $k^{2}>3$. This is an
example of Bobylev’s instability. The dispersion curves are presented in Fig.
10.

$None$
$None$

Figure 10: Dispersion curves (51) for various hydrodynamic approximations
obtained from the simple kinetic equation (39): solid - exact solution, dotted
– Navier–Stokes approximation, dashed - Burnett equation, dash and dotted –
super Burnett approximation. a) Dependence of attenuation rates on $|k|$ (for
the Navier–Stokes and Burnett curves ${\rm Re}\lambda$ coincide if
$4k^{2}<15$; they differ in ${\rm Im}\lambda$); b) Curves $\lambda(k)$ on
complex plane.

Let us analyze the structure of the Chapman–Enskog series given by the
recurrence formula (44). The terms in the series alternate: For odd
$i=1,3,\ldots$ they are proportional to $\upartial_{x}^{i}u$ and for even
$i=2,4,\ldots$ they are proportional to $\upartial_{x}^{i}p$. It follows from
the parity properties: $u$ and $\upartial_{x}$ change sign after spatial
reflection (vectors), whereas $p$ (a scalar) and $\sigma$ (a second-order
tensor) are invariant with respect to inversion. This global structure of the
Chapman–Enskog series gives the following representation of the stress
$\sigma$ on the hydrodynamic invariant manifold

$\sigma(x)=A(-\upartial_{x}^{2})\upartial_{x}u(x)+B(-\upartial_{x}^{2})\upartial_{x}^{2}p(x),$
(52)

where $A(y)$, $B(y)$ are yet unknown functions and the sign ‘$-$’ in the
arguments is adopted for simplicity of formulas in the Fourier transform. For
the stress tensor (52) the reduced equations have the form

$\boxed{\begin{split}\upartial_{t}p&=-\frac{5}{3}\upartial_{x}u,\\\
\upartial_{t}u&=-\upartial_{x}p-\upartial_{x}[A(-\upartial_{x}^{2})\upartial_{x}u+B(-\upartial_{x}^{2})\upartial_{x}^{2}p].\end{split}}$
(53)

###  4.5 Exact invariant manifold in Fourier representation

It is convenient to work with the pseudodifferential operators like (52) in
Fourier space. Let us denote $p_{k}$, $u_{k}$ and $\sigma_{k}$, where $k$ is
the ‘wave vector’ (space frequency).

The Fourier-transformed kinetic equation (39) takes the form ($\epsilon=1$):

$\begin{split}\upartial_{t}p_{k}&=-\frac{5}{3}iku_{k},\\\
\upartial_{t}u_{k}&=-ikp_{k}-ik\sigma_{k},\\\
\upartial_{t}\sigma_{k}&=-\frac{4}{3}iku_{k}-\sigma_{k}.\end{split}$ (54)

We know already that the result of the reduction should be a function
$\sigma_{k}(u_{k},p_{k},k)$ of the following form:

$\sigma_{k}(u_{k},p_{k},k)=ikA(k^{2})u_{k}-k^{2}B(k^{2})p_{k},$ (55)

where $A$ and $B$ are unknown real-valued functions of $k^{2}$.

The solution of the invariance equation amounts to finding the two functions,
$A(k^{2})$ and $B(k^{2})$. Let us rewrite the invariance equation for these
unknown functions. Compute the time derivative of $\sigma_{k}(u_{k},p_{k},k)$
in two different ways. First, use the right hand side of the third equation in
(54). We find the microscopic time derivative:

$\upartial_{t}^{\rm
micro}\sigma_{k}=-ik\left(\frac{4}{3}+A\right)u_{k}+k^{2}Bp_{k}.$ (56)

Secondly, let us use the chain rule and the first two equations in (54). We
find the macroscopic time derivative:

$\begin{split}\upartial_{t}^{\rm
macro}\sigma_{k}&=\frac{\upartial\sigma_{k}}{\upartial
u_{k}}\upartial_{t}u_{k}+\frac{\upartial\sigma_{k}}{\upartial
p_{k}}\upartial_{t}p_{k}\\\
&=ikA\left(-ikp_{k}-ik\sigma_{k}\right)-k^{2}B\left(-\frac{5}{3}iku_{k}\right)\\\
&=ik\left(\frac{5}{3}k^{2}B+k^{2}A\right)u_{k}+k^{2}\left(A-k^{2}B\right)p_{k}.\end{split}$
(57)

The microscopic time derivative should coincide with the macroscopic time
derivative for all values of $u_{k}$ and $p_{k}$. This is the invariance
equation:

$\boxed{\upartial_{t}^{\rm macro}\sigma_{k}=\upartial_{t}^{\rm
micro}\sigma_{k}.}$ (58)

For the kinetic system (54), it reduces to a system of two quadratic equations
for functions $A(k^{2})$ and $B(k^{2})$:

$-A-\frac{4}{3}-k^{2}\left(\frac{5}{3}B+A^{2}\right)=0,\;\;-B+A\left(1-k^{2}B\right)=0.$
(59)

Solving the system (59) for $B$, and introducing a new function,
$X(k^{2})=k^{2}B(k^{2})$, we obtain an equivalent cubic equation:

$-\frac{5}{3}(X-1)^{2}\left(X+\frac{4}{5}\right)=\frac{X}{k^{2}}.$ (60)

We need the real-valued functions $A(k^{2})$ and $B(k^{2})$ (55). The real-
valued root $X(k^{2})$ of (60) is unique and negative for all finite values
$k^{2}$. Moreover, the function $X(k^{2})$ is a monotonic function of $k^{2}$
and

$\lim_{|k|\rightarrow
0}X(k^{2})=0,\quad\lim_{|k|\rightarrow\infty}X(k^{2})=-0.8.$ (61)

Therefore, both $B=k^{2}X$ and $A=B/(1-X)$ are negative.

For the given $A(k^{2})$ and $B(k^{2})$ the dispersion relation for the
hydrodynamic modes are (just use the stress tensor $\sigma$ (55) in the first
two equations of system (54) and express $A$ and $B$ through $X$) :

$\lambda_{\pm}=\frac{X}{2(1-X)}\pm
i\frac{|k|}{2}\sqrt{\frac{5X^{2}-16X+20}{3}},$ (62)

where $X=X(k^{2})$ is the real-valued root of equation (60). Since
$0>X(k^{2})>-1$ for all $|k|>0$, the attenuation rate, ${\rm
Re}(\lambda_{\pm})$, is negative for all $|k|>0$, and the exact acoustic
spectrum of the reduced equations is stable for arbitrary wave lengths (Fig.
10, solid line). In the short-wave limit, from (61), (62) we obtain the
saturation of dissipation:

$\lim_{|k|\rightarrow\infty}{\rm Re}\lambda_{\pm}=-\frac{2}{9};\
\;\;\lim_{|k|\rightarrow\infty}\frac{{\rm Im}\lambda_{\pm}}{|k|}=\pm\sqrt{3}.$
(63)

Thus, we found the invariant hydrodynamic manifold in two steps:

  1. 1.

We used the invariance equation, Chapman–Enskog procedure and the symmetry
properties to find a linear space where the hydrodynamic invariant manifold is
located. This space is parameterized by two functions of one variable (55);

  2. 2.

We used the invariance equation again and defined an algebraic manifold in
this space. For the simple kinetic system (39), (54) this manifold is given by
the system of two quadratic equations which depend linearly on $k^{2}$ (59).

##  5 Van der Waals capillarity energy in ideal gas

###  5.1 The energy formula and ‘capillarity’ of ideal gas

Let us look on the stress tensor (55). Traditionally, $\sigma$ in kinetics of
gases is considered as a viscous stress tensor but the second term,
$B(-\upartial_{x}^{2})\upartial_{x}^{2}p(x)$, is proportional to second
derivative of $p(x)$ and it does not meet usual expectations
($\sigma\sim\nabla u$). Slemrod [28, 29] noticed that the proper
interpretation of this term is the capillarity tension rather than the viscous
stress. This is made clear by inspection of the energy equation. Let us derive
the energy equation for the simple model (39). Find the time derivative of the
kinetic energy due to the first two equations (39):

$\begin{split}\frac{1}{2}\upartial_{t}\int_{-\infty}^{\infty}u^{2}\,{\mathrm{d}}x&=\int_{-\infty}^{\infty}u\upartial_{t}u\,{\mathrm{d}}x=-\int_{-\infty}^{\infty}u\upartial_{x}p\,{\mathrm{d}}x-\int_{-\infty}^{\infty}u\upartial_{x}\sigma\,{\mathrm{d}}x\\\
&=-\frac{1}{2}\upartial_{t}\frac{3}{5}\int_{-\infty}^{\infty}p^{2}\,{\mathrm{d}}x+\int_{-\infty}^{\infty}\sigma\upartial_{x}u\,{\mathrm{d}}x.\end{split}$
(64)

Here we used integration by parts under the standard assumptions at infinitiy.
Note, that $\frac{1}{2}\upartial_{t}(p^{2})=-\frac{5}{3}p\upartial_{x}u$.

In $x$-space the energy equation has the standard form (7):

$\frac{1}{2}\upartial_{t}\left(\frac{3}{5}\int_{-\infty}^{\infty}p^{2}\,{\mathrm{d}}x+\int_{-\infty}^{\infty}u^{2}\,{\mathrm{d}}x\right)=\int_{-\infty}^{\infty}\sigma\upartial_{x}u\,{\mathrm{d}}x.$
(65)

Note that the usual factor $\rho$ in front of $u^{2}$ is absent because we
work with linearized equations and dimensionless variables.

Let us use in (65) the representation (52) for $\sigma$ and notice that
$\upartial_{x}u=-\frac{3}{5}\upartial_{t}p$:

$\int_{-\infty}^{\infty}\sigma\upartial_{x}u\,{\mathrm{d}}x=\int_{-\infty}^{\infty}(\upartial_{x}u)(A(-\upartial_{x}^{2})\upartial_{x}u)\,{\mathrm{d}}x-\frac{3}{5}\int_{-\infty}^{\infty}(\upartial_{t}p)[B(-\upartial_{x}^{2})\upartial_{x}^{2}p]\,{\mathrm{d}}x.$

The operator $B(-\upartial_{x}^{2})\upartial_{x}^{2}$ is symmetric, therefore,

$\int_{-\infty}^{\infty}(\upartial_{t}p)[B(-\upartial_{x}^{2})\upartial_{x}^{2}p]\,{\mathrm{d}}x=\frac{1}{2}\upartial_{t}\left(\int_{-\infty}^{\infty}p[B(-\upartial_{x}^{2})\upartial_{x}^{2}p]\,{\mathrm{d}}x\right).$

The quadratic form,

$U_{c}=\frac{3}{5}\int_{-\infty}^{\infty}p(B(-\upartial_{x}^{2})\upartial_{x}^{2}p)\,{\mathrm{d}}x=-\frac{3}{5}\int_{-\infty}^{\infty}(\upartial_{x}p)(B(-\upartial_{x}^{2})\upartial_{x}p)\,{\mathrm{d}}x$
(66)

may be considered as a part of the energy. Moreover, the function $B(y)$ is
negative, hence, this form is positive. Finally, the energy formula in
$x$-space is

$\boxed{\begin{split}\frac{1}{2}\upartial_{t}\int_{-\infty}^{\infty}\left(\frac{3}{5}p^{2}+u^{2}-\frac{3}{5}(\upartial_{x}p)(B(-\upartial_{x}^{2})\upartial_{x}p)\right)\,{\mathrm{d}}x\\\
=\int_{-\infty}^{\infty}(\upartial_{x}u)(A(-\upartial_{x}^{2})\upartial_{x}u)\,{\mathrm{d}}x.\end{split}}$
(67)

It is crucially important that the functions $A(k^{2})$ and $B(k^{2})$ are
negative, indeed, despite of the fact that some of the Taylor coefficients may
be positive and, therefore, the truncation of the formula at some powers of
$\upartial_{x}$ may not work. We have to use either the whole series or
special approximations which preserve negativity of $A$ and $B$.

Slemrod [28] represents the structure of the obtained energy formula as

$\boxed{\begin{split}\upartial_{t}({\rm MECHANICAL\
ENERGY})+\upartial_{t}({\rm CAPILLARITY\ ENERGY})\\\ ={\rm VISCOUS\
DISSIPATION}.\end{split}}$ (68)

###  5.2 Matched asymptotics: from $k^{2}=0$ to $k^{2}=\infty$

For large values of $k^{2}$, an analogue of the Chapman–Enskog expansion at an
infinitely-distant point is useful. Let us rewrite the algebraic equation for
the invariant manifold (59) in the form

$\frac{5}{3}B+A^{2}=-\varsigma(\frac{4}{3}+A),\;\;AB=\varsigma(A-B),$ (69)

where $\varsigma=1/k^{2}$. For the analytic solutions near the point
$\varsigma=0$ the Taylor series is:
$A=\sum_{l=1}^{\infty}\alpha_{l}\varsigma^{l}$,
$B=\sum_{l=1}^{\infty}\beta_{l}\varsigma^{l}$, where
$\alpha_{1}=-\frac{4}{9}$, $\beta_{1}=-\frac{4}{5}$,
$\alpha_{2}=\frac{80}{2187}$, $\beta_{2}=\frac{4}{27}$, … . The first term
gives for the frequency (62) the same limit:

$\lambda_{\pm}=-\frac{2}{9}\pm i{|k|}{\sqrt{3}},$ (70)

and the higher-order term give some corrections.

Let us match the Navier–Stokes term and the first term in the $1/k^{2}$
expansion. Find rational functions $A\approx\tilde{A}(k^{2})$ and
$B\approx\tilde{B}(k^{2})$ such that $\tilde{A}(0)=\tilde{B}(0)=-\frac{4}{3}$
(the Navier–Stokes limit) and $k^{2}\tilde{A}(k^{2})\to-\frac{4}{9}$,
$k^{2}\tilde{B}(k^{2})\to-\frac{4}{5}$ when $k^{2}\to\infty$ (the short wave
limit). Solution with the minimal powers of $k^{2}$ is:

$A\approx-\frac{4}{3+9k^{2}},\;\;B\approx-\frac{4}{3+5k^{2}}$ (71)

and

$\sigma_{k}=ikA(k^{2})u_{k}-k^{2}B(k^{2})p_{k}\approx-\frac{4ik}{3+9k^{2}}u_{k}+\frac{4k^{2}}{3+5k^{2}}p_{k}.$
(72)

This simplest non-locality captures the main effects: the Navier–Stokes
approximation for small Knudsen and Mach numbers (small $k^{2}$) and the
proper asymptotic for short waves (large $k^{2}$) with the saturation of
dissipation. This saturation is a universal effect [61, 63, 62, 60, 64] and
hydrodynamics that do not take it into account cannot pretend to be an
universal asymptotic equation.

For the matched asymptotic (71) we obtain from (53)

$\boxed{\begin{split}&\upartial_{t}p=-\frac{5}{3}\upartial_{x}u,\\\
&(1-3\upartial_{x}^{2})\left(1-\frac{5}{3}\upartial_{x}^{2}\right)\upartial_{t}u=-\upartial_{x}p\\\
&\;\;\;\;+\frac{4}{3}\upartial_{x}\left[\left(1-\frac{5}{3}\upartial_{x}^{2}\right)\upartial_{x}u+(1-3\upartial_{x}^{2})\upartial_{x}^{2}p\right],\end{split}}$
(73)

These equations give us a clue about the proper asymptotic of the continuum
mechanic equations for rarefied non-equilibrium gas: we can expect the
appearance of several factors of the form $(1-\alpha\Delta)$, where $\Delta$
is the Laplace operator.

##  6 Other approaches: conclusion and outlook

We presented the main continuum mechanics equations for compressible fluids,
from the Euler to the Navier–Stokes–Fourier and Korteweg equations. The
problem of deduction of the proper equations for highly non-equilibrium fluxes
was formulated. The essential part of Hilbert’s sixth problem is model
reduction from kinetics to continuum mechanics [29, 59]. We demonstrated two
classical approaches: solution of the invariance equation (29), (31) by the
Chapman–Enskog series and the Ehrenfests coarse graining. We solved the
reduction problem exactly for a simple kinetic system (39) . This system
provided us with a benchmark for comparison of various methods and for the
explicit demonstration of the van der Waals capillarity of ideal gases (67),
(68).

There are many attempts to solve the reduction problem and deduce the
continuum mechanics equations for non-vanishing Knudsen number from the
Boltzmann equation. We can solve the invariance equation by the direct Newton
(or Newton–Kantorovich) method [60]. The Newton iterations for the invariance
equations provide much better results than the Chapman–Enskog expansion. The
first iteration gives the Navier–Stokes asymptotic for long waves and the
qualitatively correct behavior with saturation for short waves. The second
iteration gives the proper higher order approximation in the long wave limit
and the quantitatively proper asymptotic for short waves.

Another idea is extension of the set of independent variables. Grad proposed
to write the equations for higher moments [65]. His method in combination with
the Maximum Entropy approach got the name ‘Extended irreversible
thermodynamics’ (EIT) [58]. We can start from any equation of EIT and apply
the method of invariant manifold: write the invariance equation (29), find the
first terms of the Chapman–Enskog expansion, etc. [66]. There appears also a
group of methods, which take account of some of the higher order terms in the
lower order truncation of the Chapman–Enskog expansion [67, 68, 69]. These
terms may regularize the singularities in the lower orders.

A rich family of mesoscopic lattice Boltzmann methods was developed for
applications in fluid dynamics and beyond [54]. They can be successfully
applied to microfluidics and various other problems between fluid dynamics and
kinetics.

Most of these methods lead beyond continuum mechanics. The Korteweg equation
is the first post Navier–Stokes–Fourier equation, which remains inside
continuum mechanics but captures some nonequilibrium kinetic phenomena like
the capillarity of ideal gas. Exact solutions of the reduction problem (from
kinetics to hydrodynamics) give us hints of how the post Navier–Stokes
equations may look.

Finally, it should be noted that the dissipation and the capillarity terms in
the energy equation resulting from the exact summation are of same order if
the gradients are not small. Thus, Korteweg’s term is not just a small
correction to dissipation but rather a contribution to the energy balance on
the same scale. Korteweg’s equations were originally introduced in relation to
non-ideal gas equation of state to capture the effect of surface tension
between different phases.

So, after all, why capillarity emerges in ideal gas? The answer to this
question is in the nature of the interface of the brick of matter in Cauchy
stress construction. Whenever the gradients of the hydrodynamic fields become
commensurable with the mean free path, there is an energy price to be paid for
their maintenance. The highly idealized conventional picture of continuous
media assumes an almost impenetrable elastic interface (Euler) with only a
small smearing (Navier–Stokes–Fourier) of the order of a mean free path.
However, when the gradients increase, also the dispersion effects come into
play which is precisely what the surface energy is responsible for in
Korteweg’s picture. It is clear that the non-locality associated with this
effect is essential since no truncation of the Chapman–Enskog series is
possible.

We demonstrated how the van der Waals capillarity appears in dynamics of ideal
gas. The above assertion is based on the linearized model equations, the only
case so far which was amenable to the exact solution. The nonlinear case still
requires work but the useful clues are provided by Korteweg’s equations and
kinetic models with exactly solvable reduction problem.

## Acknowledgements

ANG was supported by the EPSRC grant EP/N022653/1, IVK acknowledges support by
the ERC grant 291094-ELBM and SNSF grant 200021_149881.

## Notes on contributors

Alexander N. Gorban holds a personal chair in Applied Mathematics at the
University of Leicester since 2004. He had worked for Russian Academy of
Sciences, Siberian Branch (Krasnoyarsk, Russia) and ETH Zürich (Switzerland),
had been a visiting professor and research scholar at Clay Mathematics
Institute (Cambridge, US), IHES (Bures–sur-Yvette, Île de France), Courant
Institute of Mathematical Sciences (NY, US) and Isaac Newton Institute for
Mathematical Sciences (Cambridge, UK). Main research interests: Dynamics of
systems of physical, Chemical and biological kinetics; Biomathematics; Data
mining and model reduction problems.

Ilya Karlin is Faculty Member at the Department of Mechanical and Process
Engineering, ETH Zurich, Switzerland. He was Alexander von Humboldt Fellow at
the University of Ulm (Germany), CNR Fellow at the Institute of Applied
Mathematics CNR “M. Picone” (Rome, Italy), and Senior Lecturer in Multiscale
Modeling at the University of Southampton (England). Main research interests:
Exact and non-perturbative results in kinetic theory; Fluid dynamics; Entropic
lattice Boltzmann method; Model reduction for combustion systems.

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