Chapman–Enskog theory provides a framework in which equations of
hydrodynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in ...
for a gas can be derived from the
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G ...
. The technique justifies the otherwise phenomenological
constitutive relations appearing in hydrodynamical descriptions such as the
Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
. In doing so, expressions for various transport coefficients such as
thermal conductivity
The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1.
Heat transfer occurs at a lower rate in materials of low ...
and
viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
are obtained in terms of molecular parameters. Thus, Chapman–Enskog theory constitutes an important step in the passage from a microscopic, particle-based description to a
continuum hydrodynamical one.
The theory is named for
Sydney Chapman and
David Enskog, who introduced it independently in 1916 and 1917.
Description
The starting point of Chapman–Enskog theory is the Boltzmann equation for the 1-particle distribution function
:
where
is a nonlinear integral operator which models the evolution of
under interparticle collisions. This nonlinearity makes solving the full Boltzmann equation difficult, and motivates the development of approximate techniques such as the one provided by Chapman–Enskog theory.
Given this starting point, the various assumptions underlying the Boltzmann equation carry over to Chapman–Enskog theory as well. The most basic of these requires a separation of scale between the collision duration
and the mean free time between collisions
:
. This condition ensures that collisions are well-defined events in space and time, and holds if the dimensionless parameter
is small, where
is the range of interparticle interactions and
is the
number density.
In addition to this assumption, Chapman–Enskog theory also requires that
is much smaller than any ''extrinsic'' timescales
. These are the timescales associated with the terms on the left hand side of the Boltzmann equation, which describe variations of the gas state over macroscopic lengths. Typically, their values are determined by initial/boundary conditions and/or external fields. This separation of scales implies that the collisional term on the right hand side of the Boltzmann equation is much larger than the streaming terms on the left hand side. Thus, an approximate solution can be found from
It can be shown that the solution to this equation is a
Gaussian:
where
is the molecule mass and
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
.
A gas is said to be in ''local equilibrium'' if it satisfies this equation. The assumption of local equilibrium leads directly to the
Euler equations
In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
, which describe fluids without dissipation, i.e. with thermal conductivity and viscosity equal to
. The primary goal of Chapman–Enskog theory is to systematically obtain generalizations of the Euler equations which incorporate dissipation. This is achieved by expressing deviations from local equilibrium as a perturbative series in
Knudsen number
The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. The number is nam ...
, which is small if
. Conceptually, the resulting hydrodynamic equations describe the dynamical interplay between free streaming and interparticle collisions. The latter tend to drive the gas ''towards'' local equilibrium, while the former acts across spatial inhomogeneities to drive the gas ''away'' from local equilibrium. When the Knudsen number is of the order of 1 or greater, the gas in the system being considered cannot be described as a fluid.
To first order in
one obtains the
Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
. Second and third orders give rise, respectively, to the
Burnett equations and super-Burnett equations.
Mathematical formulation
Since the Knudsen number does not appear explicitly in the Boltzmann equation, but rather implicitly in terms of the distribution function and boundary conditions, a dummy variable
is introduced to keep track of the appropriate orders in the Chapman–Enskog expansion:
Small
implies the collisional term
dominates the streaming term
, which is the same as saying the Knudsen number is small. Thus, the appropriate form for the Chapman–Enskog expansion is
Solutions that can be formally expanded in this way are known as ''normal'' solutions to the Boltzmann equation.
This class of solutions excludes non-perturbative contributions (such as
), which appear in boundary layers or near internal
shock layers. Thus, Chapman–Enskog theory is restricted to situations in which such solutions are negligible.
Substituting this expansion and equating orders of
leads to the hierarchy
where
is an integral operator, linear in both its arguments, which satisfies
and
. The solution to the first equation is a Gaussian:
for some functions
,
, and
. The expression for
suggests a connection between these functions and the physical hydrodynamic fields defined as moments of
:
From a purely mathematical point of view, however, the two sets of functions are not necessarily the same for
(for
they are equal by definition). Indeed, proceeding systematically in the hierarchy, one finds that similarly to
, each
also contains arbitrary functions of
and
whose relation to the physical hydrodynamic fields is ''a priori'' unknown. One of the key simplifying assumptions of Chapman–Enskog theory is to assume that these otherwise arbitrary functions can be written in terms of the exact hydrodynamic fields and their spatial gradients. In other words, the space and time dependence of
enters only implicitly through the hydrodynamic fields. This statement is physically plausible because small Knudsen numbers correspond to the hydrodynamic regime, in which the state of the gas is determined solely by the hydrodynamic fields. In the case of
, the functions
,
, and
are assumed exactly equal to the physical hydrodynamic fields.
While these assumptions are physically plausible, there is the question of whether solutions which satisfy these properties actually exist. More precisely, one must show that solutions exist satisfying
Moreover, even if such solutions exist, there remains the additional question of whether they span the complete set of normal solutions to the Boltzmann equation, i.e. do not represent an artificial restriction of the original expansion in
. One of the key technical achievements of Chapman–Enskog theory is to answer both of these questions in the positive.
Thus, at least at the formal level, there is no loss of generality in the Chapman–Enskog approach.
With these formal considerations established, one can proceed to calculate
. The result is
where
is a vector and
a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, each a solution of a linear inhomogeneous
integral equation
In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2 ...
that can be solved explicitly by a polynomial expansion. Here, the colon denotes the
double dot product,
for tensors
,
.
Predictions
To first order in the Knudsen number, the heat flux
is found to obey
Fourier's law of heat conduction,
and the momentum-flux tensor
is that of a
Newtonian fluid
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of cha ...
,
with
the identity tensor. Here,
and
are the thermal conductivity and viscosity. They can be calculated explicitly in terms of molecular parameters by solving a linear integral equation; the table below summarizes the results for a few important molecular models (
is the molecule mass and
is the Boltzmann constant).
With these results, it is straightforward to obtain the Navier–Stokes equations. Taking velocity moments of the Boltzmann equation leads to the ''exact'' balance equations for the hydrodynamic fields
,
, and
:
As in the previous section the colon denotes the double
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
,
. Substituting the Chapman–Enskog expressions for
and
, one arrives at the Navier–Stokes equations.
Comparison with experiment
An important prediction of Chapman–Enskog theory is that viscosity,
, is independent of density (this can be seen for each molecular model in table 1, but is actually model-independent). This counterintuitive result traces back to
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
, who inferred it in 1860 on the basis of more elementary kinetic arguments. It is well-verified experimentally for gases at ordinary densities.
On the other hand, the theory predicts that
does depend on temperature. For rigid elastic spheres, the predicted scaling is
, while other models typically show greater variation with temperature. For instance, for molecules repelling each other with force
the predicted scaling is
, where
. Taking
, corresponding to
, shows reasonable agreement with the experimentally observed scaling for helium. For more complex gases the agreement is not as good, most likely due to the neglect of attractive forces. Indeed, the
Lennard-Jones model, which does incorporate attractions, can be brought into closer agreement with experiment (albeit at the cost of a more opaque
dependence; see the Lennard-Jones entry in table 1). For better agreement with experimental data than that which has been obtained using the
Lennard-Jones model, the more flexible
Mie potential
The Mie potential is an interaction potential describing the interactions between particles on the atomic level. It is mostly used for describing intermolecular interactions, but at times also for modeling intramolecular interaction, i.e. bonds.
T ...
has been used,
the added flexibility of this potential allows for accurate prediction of the transport properties of mixtures of a variety of spherically symmetric molecules.
Chapman–Enskog theory also predicts a simple relation between thermal conductivity,
, and viscosity,
, in the form
, where
is the
specific heat
In thermodynamics, the specific heat capacity (symbol ) of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as massic heat ...
at constant volume and
is a purely numerical factor. For spherically symmetric molecules, its value is predicted to be very close to
in a slightly model-dependent way. For instance, rigid elastic spheres have
, and molecules with repulsive force
have
(the latter deviation is ignored in table 1). The special case of
Maxwell molecules (repulsive force
) has
exactly. Since
,
, and
can be measured directly in experiments, a simple experimental test of Chapman–Enskog theory is to measure
for the spherically symmetric
noble gas
The noble gases (historically the inert gases, sometimes referred to as aerogens) are the members of Group (periodic table), group 18 of the periodic table: helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), radon (Rn) and, in some ...
es. Table 2 shows that there is reasonable agreement between theory and experiment.
[Chapman & Cowling p. 249]
Extensions
The basic principles of Chapman–Enskog theory can be extended to more diverse physical models, including gas mixtures and molecules with internal degrees of freedom. In the high-density regime, the theory can be adapted to account for collisional transport of momentum and energy, i.e. transport over a molecular diameter ''during'' a collision, rather than over a mean free path (''in between'' collisions). Including this mechanism predicts a density dependence of the viscosity at high enough density, which is also observed experimentally. Obtaining the corrections used to account for transport during a collision for soft molecules (i.e.
Lennard-Jones
Sir John Edward Lennard-Jones (27 October 1894 – 1 November 1954) was a British mathematician and professor of theoretical physics at the University of Bristol, and then of theoretical chemistry, theoretical science at the University of C ...
or
Mie molecules) is in general non-trivial, but success has been achieved at applying
Barker-Henderson perturbation theory to accurately describe these effects up to the
critical density of various fluid mixtures.
One can also carry out the theory to higher order in the Knudsen number. In particular, the second-order contribution
has been calculated by Burnett.
In general circumstances, however, these higher-order corrections may not give reliable improvements to the first-order theory, due to the fact that the Chapman–Enskog expansion does not always converge.
(On the other hand, the expansion is thought to be at least asymptotic to solutions of the Boltzmann equation, in which case truncating at low order still gives accurate results.)
Even if the higher order corrections do afford improvement in a given system, the interpretation of the corresponding hydrodynamical equations is still debated.
Revised Enskog theory
The extension of Chapman–Enskog theory for multicomponent mixtures to elevated densities, in particular, densities at which the
covolume of the mixture is non-negligible was carried out in a series of works by
E. G. D. Cohen and others,
and was coined Revised Enskog theory (RET). The successful derivation of RET followed several previous attempt at the same, but which gave results that were shown to be inconsistent with
irreversible thermodynamics. The starting point for developing the RET is a modified form of the Boltzmann Equation for the
-particle velocity distribution function,
where
is the velocity of particles of species
, at position
and time
,
is the particle mass,
is the external force, and
The difference in this equation from classical Chapman–Enskog theory lies in the streaming operator
, within which the velocity distribution of the two particles are evaluated at different points in space, separated by
, where
is the
unit vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
along the line connecting the two particles centre of mass. Another significant difference comes from the introduction of the factors
, which represent the enhanced probability of collisions due to excluded volume. The classical Chapman–Enskog equations are recovered by setting
and
.
A point of significance for the success of the RET is the choice of the factors
, which is interpreted as the pair distribution function evaluated at the contact distance
. An important factor to note here is that in order to obtain results in agreement with
irreversible thermodynamics, the
must be treated as functionals of the density fields, rather than as functions of the local density.
Results from Revised Enskog theory
One of the first results obtained from RET that deviates from the results from the classical Chapman–Enskog theory is the
Equation of State
In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most mo ...
. While from classical Chapman–Enskog theory the
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
is recovered, RET developed for rigid elastic spheres yields the pressure equation
which is consistent with the
Carnahan-Starling Equation of State, and reduces to the ideal gas law in the limit of infinite dilution (i.e. when
)
For the
transport coefficients:
viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
,
thermal conductivity
The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1.
Heat transfer occurs at a lower rate in materials of low ...
,
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
and
thermal diffusion, RET provides expressions that exactly reduce to those obtained from classical Chapman–Enskog theory in the limit of infinite dilution. However, RET predicts a density dependence of the
thermal conductivity
The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1.
Heat transfer occurs at a lower rate in materials of low ...
, which can be expressed as
where
and
are relatively weak functions of the composition, temperature and density, and
is the thermal conductivity obtained from classical Chapman–Enskog theory.
Similarly, the expression obtained for viscosity can be written as
with
and
weak functions of composition, temperature and density, and
the value obtained from classical Chapman–Enskog theory.
For
diffusion coefficients and
thermal diffusion coefficients the picture is somewhat more complex. However, one of the major advantages of RET over classical Chapman–Enskog theory is that the dependence of diffusion coefficients on the thermodynamic factors, i.e. the derivatives of the
chemical potential
In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
s with respect to composition, is predicted. In addition, RET does not predict a strict dependence of
for all densities, but rather predicts that the coefficients will decrease more slowly with density at high densities, which is in good agreement with experiments. These modified density dependencies also lead RET to predict a density dependence of the
Soret coefficient,
while classical Chapman–Enskog theory predicts that the Soret coefficient, like the viscosity and thermal conductivity, is independent of density.
Applications
While Revised Enskog theory provides many advantages over classical Chapman–Enskog theory, this comes at the price of being significantly more difficult to apply in practice. While classical Chapman–Enskog theory can be applied to arbitrarily complex spherical potentials, given sufficiently accurate and fast integration routines to evaluate the required
collision integrals, Revised Enskog Theory, in addition to this, requires knowledge of the contact value of the pair distribution function.
For mixtures of
hard spheres, this value can be computed without large difficulties, but for more complex intermolecular potentials it is generally non-trivial to obtain. However, some success has been achieved at estimating the contact value of the pair distribution function for
Mie fluids (which consists of particles interacting through a generalised
Lennard-Jones potential
In computational chemistry, molecular physics, and physical chemistry, the Lennard-Jones potential (also termed the LJ potential or 12-6 potential; named for John Lennard-Jones) is an intermolecular pair potential. Out of all the intermolecul ...
) and using these estimates to predict the transport properties of dense gas mixtures and supercritical fluids.
Applying RET to particles interacting through realistic potentials also exposes one to the issue of determining a reasonable
"contact diameter" for the soft particles. While these are unambiguously defined for hard spheres, there is still no generally agreed upon value that one should use for the contact diameter of soft particles.
See also
*
Transport phenomena
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mec ...
*
Kinetic theory of gases
The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases. Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small ...
*
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G ...
*
Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
*
Fourier's law
Thermal conduction is the diffusion of thermal energy (heat) within one material or between materials in contact. The higher temperature object has molecules with more kinetic energy; collisions between molecules distributes this kinetic energy ...
*
Newtonian fluid
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of cha ...
Notes
References
The classic monograph on the topic:
*
Contains a technical introduction to normal solutions of the Boltzmann equation:
*
{{DEFAULTSORT:Chapman-Enskog theory
Statistical mechanics