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Viggen, E. M. Acoustic multipole sources from the Boltzmann equation

Proceedings of the 36th Scandinavian Symposium on Physical Acoustics, Geilo February 2–6, 2013.

ACOUSTIC MULTIPOLE SOURCES FROM THE

BOLTZMANN EQUATION

Erlend Magnus Viggen1

1Acoustics Research Center, NTNU

Abstract

By adding a particle source term in the Boltzmann equation of kinetic theory, it is possible

to represent particles appearing and disappearing throughout the fluid with a specified

distribution of particle velocities. By deriving the wave equation from this modified Boltz-

mann equation via the conservation equations of fluid mechanics, multipole source terms

in the wave equation are found. These multipole source terms are given by the particle

source term in the Boltzmann equation. To the Euler level in the momentum equation, a

monopole and a dipole source term appear in the wave equation. To the Navier-Stokes

level, a quadrupole term with negligible magnitude also appears.

Contact author: Viggen, E. M., Department of Electronics and Telecommunications,

NTNU, 7034 Trondheim, Norway, email: erlend.viggen@ntnu.no

Introduction

Acoustic multipoles are oscillating sources that emit acoustic fields of different directivities.

These sources can be either point sources, localized at single points in space, or they can be

distributions throughout the medium. The the first three orders of multipoles are the most well-

known: Monopoles,dipoles, and quadrupoles at zeroth, first, and second order, respectively.

When these three types of multipole sources appear as source terms in the wave equation,

they usually originate from terms in the conservation equations of fluid mechanics. For instance,

monopoles are linked to a source term in the mass conservation equation (also known as the

continuity equation), which represents mass appearing and disappearing throughout the fluid as

a function of time. This can model pulsations of small bodies throughout the fluid [1].

However, an alternative approach is to add a particle source term in the Boltzmann equation,

which is more fundamental than the fluid conservation equations which can be derived from it.

Adding such a source term to the Boltzmann equation allows specifying the velocity distribution

of particles that appear and disappear throughout the fluid. This approach is therefore similar

to, but more general than, the aforementioned method of adding a mass source term, and can

therefore model more general vibrations of small bodies in the fluid.

Such a particle source term was recently examined for the lattice Boltzmann method [2], a

computational fluid dynamics method based on the fully discretised Boltzmann equation. It was

1

Viggen, E. M. Acoustic multipole sources from the Boltzmann equation

Proceedings of the 36th Scandinavian Symposium on Physical Acoustics, Geilo February 2–6, 2013.

found that such an approach results in a wave equation with non-vanishing monopole, dipole,

and quadrupole source terms. This article will similarly examine particle source terms, but in

the classic non-discretised Boltzmann equation.

Acoustic multipole sources

Mathematically, multipoles are related to source terms in the wave equation,

1

𝑐2

0𝜕

𝜕𝑡−∇2𝑝(𝐱,𝑡)=𝑇0(𝐱,𝑡)+ 𝜕

𝜕𝑥𝑖𝑇𝑖(𝐱,𝑡)+ 𝜕2

𝜕𝑥𝑖𝜕𝑥𝑗𝑇𝑖𝑗(𝐱,𝑡)+…. (1)

Here, 𝑝is the pressure and 𝑐0is the speed of sound. The terms on the right-hand side are

multipole source terms.

This article makes use of the index notation commonly used in the field of fluid mechanics.

In this notation, a single index indicates a generic vector element (e.g. 𝑇𝑖could be 𝑇𝑥,𝑇𝑦, or

𝑇𝑧), and multiple indices (as in 𝑇𝑖𝑗 or 𝑎𝑖𝑏𝑗) indicate generic tensor elements. Repeating indices

within a single term implies summation over all possible values of that index. For example,

𝑎𝑖𝑏𝑖= 𝑎𝑥𝑏𝑥+𝑎𝑦𝑏𝑦+𝑎𝑧𝑏𝑧= 𝐚⋅𝐛, and 𝜕𝑇𝑖/𝜕𝑥𝑖= 𝜕𝑇𝑥/𝜕𝑥+𝜕𝑇𝑦/𝜕𝑦+𝜕𝑇𝑧/𝜕𝑧 = ∇⋅𝐓. When

indices repeat in this way, the letter used is arbitrary, so that 𝑎𝑖𝑏𝑖=𝑎𝑘𝑏𝑘.

The general three-dimensional solution to (1) is given by an integral over the entire volume

of the source terms on the right-hand side [1],

𝑝(𝐱,𝑡)= 1

4𝜋𝑇0𝐲,𝑡−|𝐱−𝐲|

𝑐0

|𝐱−𝐲| +𝜕

𝜕𝑥𝑖𝑇𝑖𝐲,𝑡−|𝐱−𝐲|

𝑐0

|𝐱−𝐲| +𝜕2

𝜕𝑥𝑖𝜕𝑥𝑗𝑇𝑖𝑗𝐲,𝑡−|𝐱−𝐲|

𝑐0

|𝐱−𝐲| d𝐲. (2)

Thus, 𝑇0(𝐱,𝑡)indicates the instantaneous monopole strength, 𝑇𝑖(𝐱,𝑡)the 𝑖-dipole strength, and

𝑇𝑖𝑗 the 𝑖𝑗-quadrupole strength.

As mentioned above, monopole sources can be modeled by adding a mass source term to

the mass conservation equation, 𝜕𝜌

𝜕𝑡 +∇⋅(𝜌𝐮)=𝑄, (3)

𝜌being the mass density, 𝐮the fluid velocity, and 𝑄(𝐱,𝑡)the instantaneous mass flux. Dipoles

typically originate from the force term in the momentum conservation equation. To the Euler

level, this equation is 𝜌𝜕𝐮

𝜕𝑡 +(𝐮⋅∇)𝐮=−∇𝑝+𝐅, (4)

where 𝐅represents body forces. Finally, quadrupoles typically originate from the nonlinear

term in (4).

The multipole terms in (1) are usually related to the terms in the conservation equations

as [1] 𝑇0=𝜕𝑄

𝜕𝑡, 𝑇𝑖=−𝐹𝑖, 𝑇𝑖𝑗 ≃𝜌𝑢𝑖𝑢𝑗.

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Viggen, E. M. Acoustic multipole sources from the Boltzmann equation

Proceedings of the 36th Scandinavian Symposium on Physical Acoustics, Geilo February 2–6, 2013.

The Boltzmann equation

The Boltzmann equation describes motion of a gas at a finer level of detail than the fluid conser-

vation equations. In this discussion we shall restrict ourselves to its very basics. More details

can be found in the literature [3, 4, 5]. The equation evolves the particle distribution function,

𝑓(𝐱,𝝃,𝑡), which may be seen as a double density in both physical space and particle velocity

space. Thus, it describes the density of particles with position 𝐱and velocity 𝝃at time 𝑡.

The familiar macroscopic quantities can be recovered as moments of 𝑓, i.e. by weighting

with some function and integrating over the entire velocity space. The mass density and mo-

mentum density are found as the zeroth- and first-order moments,

𝜌(𝐱,𝑡)=𝑓(𝐱,𝝃,𝑡)d𝝃, (5a)

𝜌𝐮(𝐱,𝑡)=𝝃𝑓(𝐱,𝝃,𝑡)d𝝃. (5b)

Neglecting body forces and using the BGK collision operator [6], the Boltzmann equation

is 𝜕𝑓

𝜕𝑡 +𝝃⋅∇𝑓=𝑠−1

𝜏𝑓−𝑓(0).(6)

𝑠(𝐱,𝝃,𝑡)is the aforementioned particle source term which is central to this article. As the left-

hand side of the equation is a standard advection equation, 𝑠(𝐱,𝝃,𝑡)describes the rate at which

particles are added into the 𝑓(𝐱,𝝃,𝑡)distribution. The final term is the BGK collision operator,

which models collisions between particles as a relaxation with a characteristic relaxation time

𝜏to the equilibrium distribution function,

𝑓(0)(𝐱,𝝃,𝑡)=𝜌𝜌

2𝜋𝑝3/2 e−𝜌|𝝃−𝐮|2/2𝑝,(7)

with 𝜌and 𝐮found from (5). In this article, 𝑝is approximated by (8). As the quantities of mass

and momentum are conserved in collisions, substituting 𝑓(0) for 𝑓in (5) must give the same

moments. As a result of this, the BGK collision operator conserves both mass and momentum.

The BGK collision operator is a far simpler model of collisions than Boltzmann's original

and more accurate collision operator. That simplicity comes with drawbacks, chiefly that the

BGK operator slightly mispredicts the Prandtl number [3]. This dimensionless number relates

the transport coefficients in the momentum equation (viscosity) and the energy equation (con-

ductivity). However, this will not matter as we will neglect the effects of conductivity in this

article.

From this point on we will assume that the macroscopic variables fluctuate only slightly

around rest state values 𝜌=𝜌0,𝑝=𝑝0,𝐮=0. This allows us to linearise subsequent equations

in this article, which is in keeping with the usual assumptions in acoustics.

The pressure 𝑝in (7) is approximated using the common isentropic relation [1, 7]

𝑝

𝑝0=𝜌

𝜌0𝛾,(8)

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Viggen, E. M. Acoustic multipole sources from the Boltzmann equation

Proceedings of the 36th Scandinavian Symposium on Physical Acoustics, Geilo February 2–6, 2013.

𝛾=(𝑑+2)/2being the adiabatic index determined by the degrees of freedom 𝑑of the molecules

that make up the gas [7]. In this way, we include the effect of equipartition of energy between

translational and inner (i.e. rotational and vibrational) degrees of freedom, in the limit of rapid

energy transfer. This relation leads to an ideal speed of sound 𝑐0given by

𝑐2

0=𝜕𝑝

𝜕𝜌 =𝛾𝑝0

𝜌0.(9)

It will be useful to introduce an abbreviated notation for the moments of the particle source

term 𝑠,𝑆0(𝐱,𝑡)=𝑠(𝐱,𝝃,𝑡)d𝝃, (10a)

𝑆𝑖(𝐱,𝑡)=𝜉𝑖𝑠(𝐱,𝝃,𝑡)d𝝃, (10b)

𝑆𝑖𝑗(𝐱,𝑡)=𝜉𝑖𝜉𝑗𝑠(𝐱,𝝃,𝑡)d𝝃, (10c)

and so forth. 𝑆0(𝐱,𝑡)represents the instantaneous mass flux of the particles at 𝐱,𝑆𝑖is associated

with odd symmetries of 𝑠in velocity space, and 𝑆𝑖𝑗 is similarly associated with various even

symmetries.

Fluid conservation equations

It is possible to find the conservation equations of fluid mechanics from the Boltzmann equa-

tion (6). To find the Euler equations, we could simply take the zeroth and first moments of (6)

under the assumption that 𝑓≃ 𝑓(0). However, to find the momentum equation to the Navier-

Stokes level, so that it includes the stress tensor term, we must resort to the Chapman-Enskog

expansion. This is a technique used to derive the fluid conservation equations from the Boltz-

mann equation. It is discussed throughout the literature with varying approaches and varying

levels of complexity [3, 4, 8, 5, 9]. In the following derivation, we use a moment-based ap-

proach [9].

Two mathematical techniques are used in this expansion. First, the distribution function 𝑓

is approximated as a perturbation expansion around equilibrium 𝑓(0) in a smallness parameter

𝜖. Second, a multi-scale expansion of time is performed. In mathematical notation,

𝑓=𝑓(0) +𝜖𝑓(1) +𝜖2𝑓(2) +…, 𝜕

𝜕𝑡 =𝜕

𝜕𝑡0+𝜖 𝜕

𝜕𝑡1+… .

The smallness parameter 𝜖is associated with the dimensionless Knudsen number Kn=𝑙mfp/𝐿,

relating the mean free path 𝑙mfp in the gas to a macroscopic length scale 𝐿. Thus, 𝑓(𝑛+1) is of

one order higher in the Knudsen number than 𝑓(𝑛). A dimensional analysis [5] reveals that the

relaxation time is also at first order of smallness, so that 𝜏=𝜖𝜏. As previously explained, the

density and momentum is fully contained in 𝑓(0), so that

𝑓(𝑛) =𝜉𝑖𝑓(𝑛) =0 for 𝑛>0. (11)

4

Viggen, E. M. Acoustic multipole sources from the Boltzmann equation

Proceedings of the 36th Scandinavian Symposium on Physical Acoustics, Geilo February 2–6, 2013.

Expanding the Boltzmann equation in this way and truncating the expansion to 𝒪(𝜖), we

find 𝜕

𝜕𝑡0+𝜖 𝜕

𝜕𝑡1+𝜉𝑖𝜕

𝜕𝑥𝑖𝑓(0) +𝜖𝑓(1)=𝑠− 1

𝜖𝜏𝜖𝑓(1) +𝜖2𝑓(2).(12)

Gathering these terms according to their order of smallness, we find

𝒪(𝜖0)∶ 𝜕

𝜕𝑡0+𝜉𝑖𝜕

𝜕𝑥𝑖𝑓(0) =𝑠−1

𝜏𝑓(1),(13a)

𝒪(𝜖1)∶ 𝜕𝑓(0)

𝜕𝑡1+𝜕

𝜕𝑡0+𝜉𝑖𝜕

𝜕𝑥𝑖𝑓(1) =−1

𝜏𝑓(2).(13b)

To derive the mass and momentum equations to the Euler level, only the 𝒪(𝜖0)terms are needed.

The Navier-Stokes corrections to these equations are found by also including the 𝒪(𝜖1)terms in

the derivation. Similarly, it is possible to find further corrections at 𝒪(𝜖2)(known as the Burnett

corrections) and beyond (super-Burnett), although these further corrections are negligible in

practical cases [4, 8] and have historically been viewed with some suspicion [8].

Deriving the conservation equations from (13) requires the second and third moments of

𝑓(0) [9]. In linearised form, these are

𝜉𝑖𝜉𝑗𝑓(0)d𝝃≃𝑝𝛿𝑖𝑗,(14a)

𝜉𝑖𝜉𝑗𝜉𝑘𝑓(0)d𝝃≃𝑝0(𝑢𝑖𝛿𝑗𝑘 +𝑢𝑗𝛿𝑖𝑘 +𝑢𝑘𝛿𝑖𝑗), (14b)

where 𝛿𝑖𝑗 is the Kronecker delta.

Taking the zeroth to second moments of (13a), using (14), and linearising, we find

𝜕𝜌

𝜕𝑡0+𝜌0𝜕𝑢𝑖

𝜕𝑥𝑖=𝑆0,(15a)

𝜌0𝜕𝑢𝑖

𝜕𝑡0+𝜕𝑝

𝜕𝑥𝑖=𝑆𝑖,(15b)

𝛿𝑖𝑗𝜕𝑝

𝜕𝑡0+𝑝0𝜕𝑢𝑖

𝜕𝑥𝑗+𝜕𝑢𝑗

𝜕𝑥𝑖+𝛿𝑖𝑗𝜕𝑢𝑘

𝜕𝑥𝑘=𝑆𝑖𝑗 −1

𝜏Π(1)

𝑖𝑗 ,(15c)

where Π(1)

𝑖𝑗 =∫𝜉𝑖𝜉𝑗𝑓(1)d𝝃. With the 𝒪(𝜖0)approximation 𝜕/𝜕𝑡0= 𝜕/𝜕𝑡, the two first equa-

tions are equivalent to linearised versions of the mass equation (3) and the Euler momentum

equation (4), with 𝑆0and 𝑆𝑖in the place of 𝑄and 𝐹𝑖, respectively. Taking the zeroth and first

moments of (13a) and linearising, we find 𝒪(𝜖)corrections to the above equations,

𝜕𝜌

𝜕𝑡1=0, (16a)

𝜌0𝜕𝑢𝑖

𝜕𝑡1+𝜕Π(1)

𝑖𝑗

𝜕𝑥𝑗=0. (16b)

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Viggen, E. M. Acoustic multipole sources from the Boltzmann equation

Proceedings of the 36th Scandinavian Symposium on Physical Acoustics, Geilo February 2–6, 2013.

The sum (15a) +𝜖(16a) directly leads to the linearised mass equation, given below in (20a).

Similarly, (15b) +𝜖(16b) leads to the momentum equation, though the unknown tensor Π(1)

𝑖𝑗 .

Π(1)

𝑖𝑗 can be explicitly related to the Navier-Stokes stress tensor using (15c). The pressure

time derivative can be rewritten assuming a nearly isentropic process and using (15a), resulting

in 𝜕𝑝

𝜕𝑡0=𝜕𝑝

𝜕𝜌𝜕𝜌

𝜕𝑡0=𝑐2

0𝑆0−𝜌0𝜕𝑢𝑘

𝜕𝑥𝑘.(17)

Substituting for the speed of sound using (9), the diagonal terms in (15c) become

𝜕𝑝

𝜕𝑡0+𝑝0𝜕𝑢𝑘

𝜕𝑥𝑘=𝑝0−𝜌0𝑐2

0𝜕𝑢𝑘

𝜕𝑥𝑘+𝑐2

0𝑆0=−𝑝0(𝛾−1)𝜕𝑢𝑘

𝜕𝑥𝑘+𝑐2

0𝑆0.(18)

Thus, we find Π(1)

𝑖𝑗 =−𝑝0𝜏𝜕𝑢𝑗

𝜕𝑥𝑖+𝜕𝑢𝑖

𝜕𝑥𝑗+𝛿𝑖𝑗(1−𝛾)𝜕𝑢𝑘

𝜕𝑥𝑘+𝜏𝑆𝑖𝑗 −𝑐2

0𝑆0.(19)

The mass and momentum equations can now be explicitly found as described above,

𝜕𝜌

𝜕𝑡 +𝜌0𝜕𝑢𝑖

𝜕𝑥𝑖=𝑆0,(20a)

𝜌0𝜕𝑢𝑖

𝜕𝑡 +𝜕𝑝

𝜕𝑥𝑖=𝑆𝑖−𝜇

𝑝0𝜕(𝑆𝑖𝑗 −𝛿𝑖𝑗𝑐2

0𝑆0)

𝜕𝑥𝑗+𝜕𝜎

𝑖𝑗

𝜕𝑥𝑗.(20b)

The momentum equation contains a deviatoric stress tensor

𝜎

𝑖𝑗 =𝜇𝜕𝑢𝑖

𝜕𝑥𝑗+𝜕𝑢𝑗

𝜕𝑥𝑖−2

3𝛿𝑖𝑗𝜕𝑢𝑘

𝜕𝑥𝑘+𝜇𝐵𝛿𝑖𝑗𝜕𝑢𝑘

𝜕𝑥𝑘,(21)

with shear viscosity 𝜇=𝑝0𝜏and bulk viscosity 𝜇𝐵/𝜇=(5/3−𝛾). This value of the bulk viscosity

in the limit of rapid transfer of energy between translational and inner degrees of freedom has

previously been found using more rigorous kinetic theory [10].

Comparing (20a) to the classical mass equation (3), 𝑆0appears in the place of the mass

flux 𝑄, which could be expected considering the interpretation of 𝑆0as a mass flux. Compar-

ing (20b) to the Euler-level momentum equation (4), 𝑆𝑖appears in the place of the body force,

which had been neglected from the Boltzmann equation. 𝑆𝑖𝑗 is also present inside a source term

with a single spatial derivative and a small coefficient 𝜇/𝑝0in front.

The wave equation

The wave equation can be found as usual from the mass and momentum equations as 𝜕(20a)/𝜕𝑡−

𝜕(20b)/𝜕𝑥𝑖. Using the isentropic relation (17) in the wave equation operator, and thus neglecting

sound absorption due to thermal conduction and relaxation [11], we find

1

𝑐2

0𝜕2

𝜕𝑡2−∇2𝑝= 𝜕𝑆0

𝜕𝑡 −𝜕𝑆𝑖

𝜕𝑥𝑖+𝜇

𝑝0𝜕2(𝑆𝑖𝑗 −𝛿𝑖𝑗𝑐2

0𝑆0)

𝜕𝑥𝑖𝜕𝑥𝑗+𝜕2𝜎

𝑖𝑗

𝜕𝑥𝑖𝜕𝑥𝑗.(22)

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Viggen, E. M. Acoustic multipole sources from the Boltzmann equation

Proceedings of the 36th Scandinavian Symposium on Physical Acoustics, Geilo February 2–6, 2013.

Comparing with (1) and (2), we find a monopole strength 𝑇0=𝜕𝑆0/𝜕𝑡, a dipole strength 𝑇𝑖=

−𝑆𝑖, and a quadrupole strength which is not fully resolved but involves 𝑆𝑖𝑗,𝛿𝑖𝑗𝑆0, and 𝜎

𝑖𝑗.

The deviatoric stress tensor 𝜎

𝑖𝑗 can be resolved as

𝜕2𝜎

𝑖𝑗

𝜕𝑥𝑖𝜕𝑥𝑗=𝜇(3−𝛾)∇2𝜕𝑢𝑘

𝜕𝑥𝑘=𝜇(3−𝛾)

𝜌0∇2𝑆0−1

𝑐2

0𝜕𝑝

𝜕𝑡.(23)

The last term in the parenthesis contributes to sound absorption and is neglected in line with

previous approximations. The first parenthetical term contributes to the quadrupole strength.

Using the property ∇2=𝛿𝑖𝑗(𝜕2/𝜕𝑥𝑖𝜕𝑥𝑗), we find a fully resolved isentropic wave equation

1

𝑐2

0𝜕2

𝜕𝑡2−∇2𝑝= 𝜕𝑆0

𝜕𝑡 −𝜕𝑆𝑖

𝜕𝑥𝑖+𝜇

𝑝0𝜕2(𝑆𝑖𝑗 −3𝛿𝑖𝑗𝑝0𝑆0/𝜌0)

𝜕𝑥𝑖𝜕𝑥𝑗.(24)

Finally, we find the quadrupole strength as 𝑇𝑖𝑗 =(𝜇/𝑝0)(𝑆𝑖𝑗 −3𝛿𝑖𝑗𝑝0𝑆0/𝜌0).

The coefficient 𝜇/𝑝0in front of the quadrupole strength is typically on the order of 10−10s

in gases. Its small magnitude means that the quadrupoles generated by the Boltzmann equation

source term 𝑠tend to be negligible compared to the monopoles and dipoles.

Summary and conclusion

A source term 𝑠in the Boltzmann equation represents particles appearing or disappearing through-

out the fluid with some distribution of particle velocities. As adding a source term to the mass

equation allows modeling pulsations of small bodies throughout the fluid [1], a source term in

the Boltzmann equation would allow modeling more general vibrations of such small bodies.

From this modified Boltzmann equation, the mass and momentum conservations equa-

tions (20) were derived under the common acoustic approximation of linearity and constant

entropy. These equations gain source terms given by the moments (10) of 𝑠. The mass equation

gains a source term 𝑆0. To the Euler level, the momentum equation gains a source term 𝑆𝑖, and

to the Navier-Stokes level, it gains a source term involving 𝑆𝑖𝑗 and 𝛿𝑖𝑗𝑆0.

The wave equation derived from these two conservation equations contains multipole source

terms. The monopole strength is 𝜕𝑆0/𝜕𝑡and the dipole strength is −𝑆𝑖. The quadrupole strength,

which comes out of the Navier-Stokes level of the momentum equation, involves 𝑆𝑖𝑗 and 𝛿𝑖𝑗𝑆0,

but has a negligible magnitude. That the quadrupole strength is so much smaller than the

monopole and dipole strength could be expected, as the Navier-Stokes level terms are one order

higher in the small Knudsen number than the Euler-level terms.

Similarly, going to the 𝒪(Kn2)Burnett level might lead to a 𝜕2𝑆𝑖𝑗𝑘/𝜕𝑥𝑗𝜕𝑥𝑘term in the mo-

mentum equation, leading to an octupole term in the wave equation. However, since this term

would be at 𝒪(Kn2), it would be even more negligible than the quadrupole term.

Going back to the point of modeling general vibrations of small bodies throughout the fluid,

this analysis indicates that the vibrations of such small bodies can radiate as monopoles and

dipoles, but only very weakly as higher-order multipoles.

7

Viggen, E. M. Acoustic multipole sources from the Boltzmann equation

Proceedings of the 36th Scandinavian Symposium on Physical Acoustics, Geilo February 2–6, 2013.

Comparing this analysis with the analogous analysis for the lattice Boltzmann method [2],

we find that the analogous source term in the lattice Boltzmann equation can only radiate

quadrupoles effectively due to a fortuitous discretisation error that occurs when discretising

the Boltzmann equation (6) in space and time using the first-order rectangle method. Discretis-

ing with the trapezoidal method [9], which results in a scheme fully consistent with (6), would

therefore also lead to a vanishing quadrupole term similarly to what has been shown here.

References

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Cambridge University Press, 3rd edition.

[5] Hänel, D., 2004. Molekulare Gasdynamik. Springer-Verlag.

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