Derivation of regularized Grad's moment system from kinetic equations: modes, ghosts and non-Markov fluxes

Abstract

Derivation of the dynamic correction to Grad’s moment system from kinetic equations (regularized Grad’s 13 moment system, or R13) is revisited. The R13 distribution function is found as a superposition of eight modes. Three primary modes, known from the previous derivation (Karlin et al. 1998 Phys. Rev. E 57, 1668–1672. (doi:10.1103/PhysRevE.57.1668)), are extended into the nonlinear parameter domain. Three essentially nonlinear modes are identified, and two ghost modes which do not contribute to the R13 fluxes are revealed. The eight-mode structure of the R13 distribution function implies partition of R13 fluxes into two types of contributions: dissipative fluxes (both linear and nonlinear) and nonlinear streamline convective fluxes. Physical interpretation of the latter non-dissipative and non-local in time effect is discussed. A non-perturbative R13-type solution is demonstrated for a simple Lorentz scattering kinetic model. The results of this study clarify the intrinsic structure of the R13 system.

This article is part of the theme issue ‘Hilbert’s sixth problem’.

1. Introduction

Grad, in his seminal paper of 1949 [1], derived moment systems by projecting the Boltzmann equation onto an ansatz for the distribution function. Grad considered two sets of moments, and what will be referred to as G13 and G20, with the number indicating how many fields are included. In G13, these are the locally conserved density, momentum and energy plus the non-equilibrium stress tensor and energy flux vector, while G20, a more symmetric system, includes the full third-order flux of the pressure tensor instead of the energy flux vector. Grad’s method was influential in many ways, far beyond applications to rarefied gas dynamics. It was the touchstone for numerous developments in non-equilibrium thermodynamics (see [2–4] and references therein) and, more recently, served as an avenue for the development of modern computational fluid dynamics in the framework of the lattice Boltzmann method [5]. However, the problem with almost any projection on a preselected (often simple) submanifold is that it is not invariant with respect to the detailed dynamics. In Grad’s context, Grad’s distribution function (a polynomial around the local Maxwellian) is not invariant with respect to the Boltzmann equation. This is certainly not just a feature of Grad’s distribution per se. The classical local Maxwellian is also not invariant of the Boltzmann equation, and dynamic correction, well known through the Chapman–Enskog method, delivers the dissipative Navier–Stokes–Fourier (NSF) terms, missing in the projection on the local Maxwell manifold (compressible Euler equations).

A systematic method to derive the dynamic correction to Grad’s G13 projection from kinetic equations was introduced in [6] (KGDN hereafter), and has been realized for near-equilibrium conditions. Taking a different route via a superset moment system with 26 moments, Struchtrup & Torrilhon [7] proposed a nonlinear extension of the dynamic correction, and coined the name of regularized Grad’s system, or R13, which we use here. R13 theory was further refined and extensively studied by Struchtrup, Torrilhon and co-workers in a number of contributions [8–16] that dissect the R13 equations carefully, and show that they can describe all relevant rarefaction phenomena, such as jump and slip at boundaries, Knudsen boundary layers, transpiration flow, thermal stresses, non-temperature-gradient heat flux induced by stresses, damping and dispersion of ultrasound waves and shock structures ( for limited Mach numbers). A good summary of this work is referenced and discussed in a recent review [12].

Nonlinear R13 is a complex system featuring a large number of terms coupling the stress and heat flux to the gradients of basic fields and of themselves. It is the goal of this paper to revisit the derivation of nonlinear R13 from kinetic equations, to clarify the structure of R13 and to give interpretation of various couplings. To that end, the approach of KGDN already partially revealed the structure of the R13 through identifying the three primary modes of the R13 distribution function in the linear approximation. Here, we extend the KGDN approach to the full nonlinear R13. The main findings are as follows.

Result 1.1. —

Structure of the R13 distribution function. The leading-order dynamic correction to Grad’s distribution function is written using relaxation time τ as follows:

1.1

where is Grad’s distribution and , the correction, is the superposition of eight modes,

1.2

Each mode has the form

1.3

where is the local Maxwellian (or the mode’s amplitude), P⁽ⁱ⁾ is the velocity tensor (or the mode’s direction), F⁽ⁱ⁾ is the mode’s frequency, while • stands for convolution of tensors. Tensors P⁽ⁱ⁾ are dimensionless and depend only on C, the particle’s velocity relative to the flow u, reduced by thermal speed, C=(v−u)/v_T, where . The dimension of the mode’s frequencies is the inverse of time, [F⁽ⁱ⁾]∼1 s⁻¹. Modes of R13 are collected in table 1. The three primary modes , and were already identified by KGDN [6], and here their frequencies are extended into the nonlinear domain. They are accompanied by three nonlinear secondary modes , and . Finally, two modes and are ghost modes: while they contribute to the R13 distribution function, their projection onto R13 fluxes vanishes, and they are not visible in the R13 balance equations.

Table 1.

Modes of R13. Angular brackets denote symmetrized traceless tensors of rank two, three and four. D_t=∂_t+u⋅∇ is the material derivative along the streamline; u is the flow velocity, p=ρRT is the pressure, R is the gas constant, ρ is the density and T is the temperature.

	P	F
1	〈CCC〉
2	(C2−7)〈CC〉
3	C4−10C2+15
4	(C2−7)〈CCC〉
5	〈CCCC〉−sym(〈CC〉U)
6	(C4−14C2+35)CC
7	(C2−9)〈CCC〉
8	(C4−14C2+35)C

Result 1.2. —

Structure of R13 equations. The R13 equations for the non-equilibrium stress tensor σ and the heat flux q are compactly written in vector notation as follows:

1.4

and

1.5

where D_t is the material time derivative along the streamline,

1.6

and indicates Grad’s contribution. The rank-three symmetric trace-free tensor and the rank-two symmetric tensor are the R13 fluxes,

1.7

1.8

1.9

1.10

Here angular brackets indicate symmetrized traceless tensors of rank two and three, the overline indicates transposition, U is the unit tensor, is the strain and . For expanded notation, see (8.6), (8.16) and (8.17).

Brackets in (1.7) and (1.9) help to discern contributions of two different types. The first bracket in (1.9) and (1.7) is the dissipation flux. The first part of dissipation is the linear thermodynamics dissipation flux ( first term in (1.7) and (1.9)). The linear dissipation fluxes are the only contributions in (1.7) and (1.8) that survive linearization around equilibrium, as was already shown by KGDN. The rest of the terms in the first bracket form the nonlinear dissipation flux, driven by non-uniformity of the macroscopic velocity field and of the temperature. Both types of dissipative fluxes are associated with the kinematic viscosity ν=τRT, here in the relaxation time approximation.

The second bracket in (1.7) and (1.9) is a remarkably distinct streamline convective flux, which we term this way because of the material time derivative (1.6) participating in their formation. Streamline convective fluxes are nonlinear and their contribution is non-dissipative in nature. They are characterized by the relaxation time τ rather than by the viscosity. From table 1, it can be seen that the streamline convective flux is contributed by primary modes only. Finally, the trace (1.10) also reproduces the said structure. Together with the balance equations of density, momentum and energy,

1.11

1.12

1.13

and with Grad’s contribution,

1.14

and

1.15

where Π=pU+σ is the pressure tensor, (1.4) and (1.5) build up the structure of the R13 system.

The outline of the paper is as follows. We begin in §2 with defining Grad’s fields and deriving balance equations. The material presented in §2 is standard, and is merely for setting up notation and definitions. Similarly, in §3, we review Grad’s G13 system and offer some discussion. The main tool for deriving R13 in KGDN [6] and in the present paper is the method of invariant manifold (MIM) [17]. We recall the basics of the MIM in §4 and consider a familiar case of a local Maxwellian as an example of MIM analysis. In particular, we highlight the relevance of the mode’s decomposition of the invariance defect (driving mechanism of the dynamic correction understood as a discrepancy between the induced and the actual time shift of the distribution function). The main §§5–7 present a detailed derivation and analysis of the invariance defect of Grad’s G13 manifold. In §5, following KGDN [6], the invariance defect is split into the local and non-local parts, and vanishing of the local part for the special case of Maxwell’s molecules is proved. Section 6 begins with remembering the KGDN results for the linear part of the non-local invariance defect. After that, the nonlinear part of the non-local invariance defect is evaluated as a sum of 12 terms in accord with their type of nonlinearity. This step is intermediate for recognizing the mode’s structure. The eight modes ((1.2), table 1) are extracted in §7, and orthogonality theorems are proven; in particular, the orthogonality of ghost modes to R13 fluxes. We also show in detail how various modes contribute to the two types of fluxes, the dissipative and the streamline convective. In §8, we perform a straightforward evaluation of the R13 fluxes directly from the invariance defect, in the relaxation time approximation. This shortcut to the structure of R13 in the collision-dominant regime leads immediately to the R13 equations (1.4) and (1.5), revealing their structure. The result is compared with the work of Struchtrup & Torrilhon [7], and it is demonstrated that R13 equations here and in [7] become identical once the type of collision model is properly adjusted. This is reconsidered in §9, where we discuss the evaluation of R13 for Boltzmann collision models. First, we argue that the local defect of invariance is a mere matter of choice of the initial Grad’s ansatz for the distribution function, and thus it is decoupled from the corrections of the non-local part. We propose some generalizations of Grad’s distribution functions tailored to a given particle’s interaction. Second, we explain how the evaluation of transport coefficients of R13 is reduced to solving linear integral equations, similar to the Chapman–Enskog computation of viscosity and thermal conductivity [18]. While R13 appears as only a leading-order invariance correction, in §10 we consider a solvable example—the three-dimensional Lorentz scattering model—where we find a non-perturbative solution of the invariance equation. First, we show how the well-known result of Hauge [19] for the exact Chapman–Enskog solution of hydrodynamics is recovered by MIM. Second, we solve the MIM iteration equation for the Grad-like ansatz and show that it indeed improves the quality of approximation. The paper is concluded in §11, where we offer physical interpretation of non-locality emerging in the nonlinear R13 system, discuss other R13 variants and propose some further generalizations.

2. Balance equations

(a). Fields and fluxes

With f(v,x,t) being the particle’s distribution function, v the particle’s velocity, x the space and t the time, the density ρ, momentum ρu, full pressure tensor P and energy flux Z are defined as follows:

2.1

2.2

2.3

2.4

Here m is the particle mass, and we proceed with Cartesian coordinates for clarity; Greek subscripts denote coordinates and summation convention is always understood. Identifying pressure p as

2.5

the pressure tensor (2.3) and the energy flux (2.4) are decomposed as follows when measuring a particle’s velocity relative to the flow velocity u:

2.6

and

2.7

where the non-equilibrium stress tensor σ and heat flux q are defined as

2.8

and

2.9

The rank-two symmetric tensor σ is trace-free. Finally, temperature T is defined by the equation of state

2.10

where R is the gas constant,

2.11

Density ρ, flow velocity u, temperature T, non-equilibrium stress σ and heat flux q are the 13 fields of Grad’s G13 system.

(b). Balance equations for 13 fields

Introducing the material derivative along the streamline,

2.12

we write the kinetic equation in the co-moving reference frame,

2.13

where Q is Boltzmann’s collision integral or a relaxation term in the case of kinetic models. Multiplying (2.13) by 1, v_α, v_αv_β and v_αv² and integrating over velocities, we come to the set of exact balance equations for the 13 Grad’s fields. The balance equations for the locally conserved fields (mass–momentum–energy) imply

2.14

2.15

2.16

The non-equilibrium stress and heat flux are the only non-equilibrium fluxes engaged in the balance equations (2.15) and (2.16), and we continue with writing down the exact balance equations for σ and q,

2.17

and

2.18

Here, the symmetric rank-three tensor Q and the symmetric rank-two tensor T are

2.19

and

2.20

For brevity, we refer to them as Q-flux and T-flux, respectively. The Q-flux is engaged as a divergence in the stress balance (2.17) and as a source term in the heat flux balance (2.18), while the T-flux contributes as a divergence in the heat flux balance only. The heat flux and the Q-flux are connected through

2.21

Furthermore, the relaxation terms are defined as rates over collisions,

2.22

and

2.23

The balance equations are identities and cannot be addressed unless a constitutive relation is provided for both the Q- and T-fluxes, as well as for the collision rates.

Before closing this section, we note that, by introducing tensor h,

2.24

the balance of the heat flux (2.18) is compactly written using the balance of momentum (2.2),

2.25

Tensor h may be termed the enthalpy tensor since is the enthalpy of the ideal gas.

3. Grad’s distribution function, closure and system

(a). Grad’s distribution function

We introduce thermal speed v_T as

3.1

and use it to reduce the velocity of the particle in the co-moving reference frame,

3.2

Following Grad [1], the distribution function providing the closure for the balance equations (2.17) and (2.18) is written as

3.3

and

3.4

where is a local Maxwellian (n=ρ/m is the number density),

3.5

and is the non-equilibrium part,

3.6

and

3.7

Without engaging in a discussion of possible violations of positivity, we consider Grad’s function as a submanifold in the space of distribution functions, parametrized with the values of 13 fields. Grad’s system is the natural projection of the kinetic equation onto this submanifold.

(b). Grad’s closure

Grad’s projection amounts to evaluating everything that spoils the closure in the balance equations (2.17) and (2.18) with Grad’s distribution (3.3). For the Q-fluxes (2.19), we get, separating the local equilibrium and non-equilibrium contributions,

3.8

3.9

3.10

In other words, Grad’s closure for the Q-flux amounts to reducing the symmetric rank-three tensor to its trace (3.10). For the T-flux (2.20), one finds

3.11

3.12

3.13

For the collision rates (2.23) and (2.22), there are several realizations depending on the choice of the collision or relaxation model, which we list here in order of increasing complexity.

• (i) A ‘poor man’s’ approach is to use the relaxation time approximation,

  3.14

  then one simply gets

  3.15

  and

  3.16

  Similar results are obtained when using most of the relaxation kinetic models that are available (with more than one relaxation time). While certainly far from realistic, relaxation time approximation is useful for the analysis of complex situations in order to understand the structure of an otherwise involved result. We shall use relaxation time approximation for most of this paper.
• (ii) The Boltzmann’s collision integral is written as

  3.17

  where conventional notation is used for the scattering of a pair of particles with velocities v and v₁ into v′ and v₁′, and where α is the differential scattering cross-section. Substituting Grad’s distribution function, and taking into account detail balance, , one gets

  3.18

  where L is the linearized Boltzmann collision integral,

  3.19
• (iii) Computation of matrix elements of the operator (3.19) greatly simplifies for a special case of repulsive potential decaying as the inverse of the fourth power of distance (Maxwell’s molecules), U=κ/r⁴, where κ is the strength of Maxwell’s potential. This happens because functions and are eigenfunctions of the linearized Boltzmann operator in that case,

  3.20

  and

  3.21

  with μ the viscosity coefficient of Maxwell’s molecules,

  3.22

  where A₂(5) is a number, A₂(5)≈0.436 [18]. With (3.20) and (3.21), evaluation of the matrix elements reduces to the same integrals as in the relaxation time approximation, and we get

  3.23

  and

  3.24

  It should be noted that, for Maxwell’s molecules, evaluation of the relaxation rates for various moments can be done in closed form without specifying the distribution function, also for the nonlinear collision operator [15,20], and relaxation rates (3.23) and (3.24) are valid for the full nonlinear case; more on this in §11.
• (iv) For other particle collision models such as hard spheres or other power law potentials, functions and are not eigenfunctions of the linearized collision integral any longer, and evaluation of the matrix elements (3.23) and (3.24) gives instead

  3.25

  and

  3.26

  where μ₀ is not an exact viscosity coefficient but rather the first approximation thereof. For hard spheres of diameter d, for example [18],

  3.27

  It is well known that first approximation μ₀ is reasonably close to the exact value, in particular for hard spheres [1,18]. This fact, however, does not imply that corresponding eigenfunctions of the linearized Boltzmann collision integral are in any sense ‘close’ to those for Maxwell’s molecules, and it is therefore misleading to judge on the quality of Grad’s approximation for hard spheres on the basis of viscosity coefficient alone. We shall return to this discussion later.

We now continue with finalizing Grad’s G13 approximation by writing down closed-form equations.

(c). Grad’s system

Substituting Grad’s closure relations for the Q- and T-fluxes (3.8) and (3.11) into balance equations (2.17) and (2.18), and also using any of the above realizations of the relaxation terms, one arrives at Grad’s equations. For later use, it proves convenient to partition Grad’s equations into four parts, three of which relate to the non-local in space terms plus the relaxation term. For the stress, we write

3.28

3.29

3.30

3.31

A comment on the genesis of various terms in Grad’s equation (3.28) is in order. The first term, (3.29), is designated NSF because it linearly depends on the strain tensor and gives rise to the Navier–Stokes stress in the first-order approximation to G13. This term appears to be purely kinematic, that is, it shows up already in the balance equation for the stress before any closure assumption. The second term, (3.30), is solely produced by the closure relation for the Q-flux (3.10). It is indicated as ‘linear’ since it depends linearly on the gradient of the heat flux but not on any gradient of locally conserved fields. Consequently, the term survives linearization around a global equilibrium state. Finally, the nonlinear term (3.31) is again purely kinematic and independent of Grad’s closure assumption.

Grad’s heat flux equation is decomposed in a similar manner,

3.32

3.33

3.34

3.35

Here, the term in the balance equation (2.18) conspired with the local equilibrium part of the closure (3.12) to produce the NSF contribution, (3.33). This gives rise to the Fourier law in the first approximation, through balancing the relaxation term. The term (3.34) appears with the right sign owing to Grad’s closure approximation of the non-equilibrium part of the T-flux (3.13). We term it ‘linear’ for the reason explained above, even though it is nonlinear through multiplication with the temperature. Similarly to (3.30), the contribution of does not vanish under linearization. Finally, the nonlinear part of Grad’s heat flux equation, (3.35), contains a mixture of terms both already present in the balance equation (2.18) and resulting from the closure assumption. Note that the ‘most nonlinear’ term, (1/ρ)σ_αβ∂_γσ_γβ, is purely kinematic and is not affected by Grad’s closure. Even when linearized, Grad’s equations (3.28) and (3.32) remain strongly coupled through their respective linear terms, (3.30) and (3.34), and this gives rise to a rather involved Chapman–Enskog series expansion (see the contribution by Marshall Slemrod 21).

(d). Need for dynamic correction to Grad’s system

Nothing tells us that Grad’s closure relations for both the Q- and T-fluxes, as well as the closure relation for the relaxation term, will stay invariant under kinetic equation (2.13). One thus needs, first, to quantify the discrepancy between the proposed projection and the real dynamics due to the kinetic equation, to understand the physical mechanisms arising from this discrepancy and which were neglected by the projection, and, second, to correct the closure on the basis of the kinetic equation. This programme has been initiated by KGDN [6] for the G13 approximation, and shall be extended to the full nonlinear case below. Before continuing with Grad’s G13 system, we shall consider a familiar case of a local Maxwellian. This will serve as a reference to highlight similarities and differences when considering a more involved case of G13.

4. Invariance correction to a local Maxwellian

(a). Method of invariant manifold

We begin with briefly remembering the main steps of the method of invariant manifold (MIM) [17]. For a given set of macroscopic fields (or variables) M, the function is said to provide invariant closure if the time derivative of due to that closure (macroscopic derivative) is identical to its time derivative due to the kinetic equation (2.13) (microscopic derivative). This invariance principle can be expressed as follows:

4.1

where D/DM denotes the variation of F(M) in some appropriate sense. The left-hand side can be recognized as the evaluation of the time derivative by the chain rule. A practical realization of the said principle is to find the invariance correction to an a priori given initial approximation . The invariance correction is subject to iteration applied to (4.1) with as the initial guess and as the correction,

4.2

Here is the variation of the closure,

4.3

The iteration equation (4.2) is solved subject to the orthogonality condition,

4.4

which reflects the fact that the values of the macroscopic fields M are fixed by the input manifold and only their fluxes are corrected.

The left-hand side of the linear equation (4.2) is referred to as the invariance defect (ID) of the approximation ,

4.5

Analysis of the invariance defect is the first step towards revealing the physical mechanisms leading to correction.

(b). Invariance defect of a local Maxwellian

We proceed with a local Maxwellian (3.5) as the input approximation, remembering that it depends on flow velocity u and temperature T through the reduced velocity C (3.2). Balance equations (2.14)–(2.16) become a compressible Euler system upon Maxwellian closure, and ,

4.6

4.7

4.8

In order to compute the macroscopic time derivative of with respect to Euler’s equations by the chain rule, we need derivatives of the Maxwellian with respect to ρ, u and T,

4.9

4.10

4.11

With this, the invariance defect of the local Maxwellian is written as

4.12

Here we have taken into account and have computed the derivative using the chain rule once again. Substituting everything from (4.6)–(4.11), we find

4.13

After some transformations, the invariance defect of the local Maxwellian takes its final shape,

4.14

The invariance defect satisfies an important orthogonality condition: for any functions u and T, projection of the invariance defect onto the fields ρ, u and T vanishes,

4.15

(c). Invariance correction

Proceeding with the correction, we write and the iteration equation (4.2) becomes

4.16

where δσ and δq are corrections to the closure relations provided by , and the last term in (4.16) is the variation of the closure; it corresponds to the first term on the right-hand side of (4.2). Physically, (4.16) means that the invariance defect of the local Maxwellian is balanced by the joint action of collisions and propagation on the new manifold, . If one assumes that collisions are dominant in the balance (4.16), then we get a much simpler linear integral equation,

4.17

This is nothing but the first Chapman–Enskog equation, the study of which is exhaustively described in [18]. For the sake of completeness, we summarize here some major steps.

• (i) The orthogonality condition (4.15) implies that only the special solution of the linear integral equation (4.17) needs to be addressed (Fredholm alternative).

• (ii) This special solution has the form

  4.18

  where two scalar functions, A and B, can depend only on the magnitude of the reduced velocity, temperature and density, while A satisfies

  

  which corresponds to the condition (4.4). Chapman–Enskog functions A and B are found from two integral equations,

  4.19

  and

  4.20

• (iii) Apart from Maxwell’s molecules, for which A and B do not depend on C, the appropriate technique for solving (4.19) and (4.20) is based on Sonine polynomial expansion [18]. Note that, again apart from Maxwell’s molecules, the Chapman–Enskog problem (4.19) and (4.20) is not an eigenvalue problem.

With the space derivative terms included, iteration equation (4.16) is non-perturbative in nature, and places itself between the full summation of the Chapman–Enskog expansion and finite-order truncations thereof [17]. We shall present an example of solving the iteration equation for a Grad-like input approximation, both exactly and perturbatively, for a simple model in §10.

(d). Revelations from the invariance defect: the mode structure

While solving iteration equations for the invariance correction is a self-consistent problem, we note that the invariance defect (4.5) by itself reveals valuable information about the nature of macroscopic equations. The invariance defect of the local Maxwellian (4.14) prompts that the physical mechanism away from the local equilibrium is due to the strain and the non-uniformity of the temperature. That two independent non-equilibrium processes are involved becomes especially clear if the invariance defect is written as a superposition of two independent modes,

4.21

4.22

4.23

The different tensorial structure of the shear mode and of the heat conduction mode suggests that two transport coefficients should be computed as the result of the invariance correction. Furthermore, since the dimension of the invariance defect is a rate,

4.24

and because the invariance defect satisfies the orthogonality condition (4.15), we can immediately supply a surrogate invariance correction, known as the relaxation time approximation, by simply multiplying the invariance defect by −τ,

4.25

and proceed with this to compute the non-equilibrium components of the stress and of the heat flux, which then reveal the structure of the resulting hydrodynamic equations as compressible NSF. In that regard, the Chapman–Enskog problem can be understood as a transformation of the input superposition of modes into the output Chapman–Enskog solution where the modes are modulated by the collision integral, whereas the surrogate relaxation time approximation keeps the input modes intact at the output. Nevertheless, the structure of the resulting hydrodynamic equations is the same in both cases. The above discussion of the familiar case of a local Maxwellian provides a guide and comparison with a more complex case of Grad’s approximation which we now proceed with.

5. Invariance defect of Grad’s approximation

Evaluation of the invariance defect of Grad’s approximation is rather straightforward, and has been already reported by KGDN [6] in the linear approximation. However, algebraic details in the nonlinear case are somewhat involved, and we shall split the computation into a number of steps for transparency. We begin by evaluating the derivatives of Grad’s distribution function with respect to all 13 fields, while separating the contributions of the Maxwellian already available from (4.9)–(4.11) from that of the newly added non-equilibrium part,

5.1

5.2

5.3

where

5.4

and

5.5

Finally, the derivatives with respect to the non-equilibrium fields are

5.6

and

5.7

Using these, we write down the invariance defect of Grad’s 13-moment approximation separating contributions of the propagation terms from those of the collisions. Following KGDN convention, the former are termed non-local and the latter local,

5.8

The local piece reads,

5.9

Neglect of the quadratic terms here is consistent with their neglect already made in Grad’s closure of the relaxation terms.

The first observation, already exposed in KGDN [6], is about the vanishing of the local invariance defect in certain cases of collision models, as follows.

Theorem 5.1. —

Let functions and be eigenfunctions of the linearized collision integral,

5.10

and

5.11

Then the local invariance defect of Grad’s approximation vanishes,

5.12

Indeed, with the conditions (5.10) and (5.11), the relaxation terms in Grad’s approximation become and . Using these in (5.9), we prove (5.12).

Non-vanishing of the local invariance defect for Grad’s distribution function is the first apparent difference from the Maxwellian case. In Grad’s case, the relaxation has to be aligned with the direction dictated by the eigenfunction of the linearized collision integral in order to annihilate the local invariance defect. This is more demanding than the local Maxwellian, which annuls the local terms in the invariance defect independently of the collision model used. Vanishing of the local invariance defect unifies the relaxation time approximation (and any similar kinetic model) with Maxwell’s molecules and is a consequence of the simple fact that all these have the same eigenfunctions of the form (5.10) and (5.11). Vanishing of the local invariance defect has no relation to the values of transport coefficients, and is non-vanishing for any other model, such as hard spheres. We shall return to the discussion of the local invariance defect and the corresponding correction in §9; for now, we proceed with the analysis of the non-local part of the invariance defect which is independent of the choice of the collision model.

6. Non-local invariance defect of Grad’s approximation

(a). Natural partition of the non-local invariance defect

The non-local part of the invariance defect shall be written as the sum of three pieces, where we distinguish the NSF, the linear and the nonlinear contributions,

6.1

Each type of contribution is dictated by the partition of Grad’s equations (3.28) and (3.32), and we have

6.2

6.3

6.4

We shall now proceed with evaluating the three parts of the non-local invariance defect of G13.

(b). Navier–Stokes–Fourier contribution

Theorem 6.1. —

The NSF part of the invariance defect of Grad’s approximation vanishes,

To validate this statement, we note that the first three terms in (6.2) assemble to the defect of invariance of the local Maxwellian already computed, see equation (4.14); evaluating the remaining two terms, we get,

6.5

This observation has already been made in KGDN [6], and it is not surprising. Indeed, since the NSF approximation is fully contained in Grad’s equations, there is nothing to correct in Grad’s dynamics with respect to the NSF fluxes.

(c). Linear part of the invariance defect and three primary modes

Below, we shall use shorthand notation for symmetric traceless velocity tensors of rank two, three and four,

6.6

6.7

6.8

Derivation of the linear part of the non-local invariance defect has already been done in KGDN [6], thus we only quote the result and offer some discussion. Function (6.3) splits up into three pieces,

6.9

where

6.10

6.11

6.12

Note that (6.12) can be written in a more suggestive form,

6.13

where we immediately recognize the quotient of the fifth-order Hermite polynomial,

6.14

Functions (6.10)–(6.12) initiate three primary modes of the non-local invariance defect, , and , promoted by the rank-three traceless velocity tensor 〈CCC〉, the rank-two traceless tensor (C²−7)〈CC〉 and the scalar quotient of the fifth-order Hermite polynomial (C⁴−10C²+15), respectively. Corresponding mode frequencies are the only part of the invariance defect that survive linearization around equilibrium [6], and thus contribute the linear thermodynamics dissipation to R13. Below, the primary modes shall be extended into the nonlinear domain.

(d). Nonlinear part of the invariance defect

We now proceed with the nonlinear part of the non-local invariance defect. First, we shall group various terms in accord with the coupling of the fields σ_αβ, q_α to the gradients ∂_αρ, ∂_βu_α, ∂_αT, ∂_γσ_αβ and ∂_βq_α. Each individual contribution is proportional to either gradient of locally conserved fields and/or is at least quadratic in the non-equilibrium fields σ and q. We shall use nomenclature for these individual contributions as follows:

6.15

where a, b or c is the symbol of the field, ρ, p, u, T, σ or q, where the last symbol (b in the two-index and c in the three-index contribution) indicates the field which contributes to the gradient.

Various contributions to the nonlinear part of the non-local defect of invariance are split into two groups according to their parity under reflection of the molecular velocity, . Six skew-symmetric contributions are as follows:

6.16

6.17

6.18

6.19

6.20

6.21

These are complemented by six symmetric terms,

6.22

6.23

6.24

6.25

6.26

6.27

Summarizing, the nonlinear part of the invariance defect is obtained by putting together 12 terms distinguished by the type of nonlinearity, (6.16)–(6.27):

6.28

(e). Acquisition of modes

Once the invariance defect of Grad’s approximation is computed, we can proceed to R13 equations, e.g. by adopting the relaxation time shortcut, multiplying the non-local invariance defect (6.28) by −τ and evaluating the R13 fluxes. However, due to the large number of terms with different symmetry in (6.28), this makes it difficult to discern any structure behind the resulting equations. Therefore, it is much more instructive to first rewrite as a superposition of modes, in the same way as we have it for the Maxwellian. We therefore defer evaluation of fluxes to §8, after we clarify the modes. To achieve that, some further transformation of the nonlinear contributions is necessary. Specifically, by extracting the irreducible part of the rank-three tensor, and taking into account the fact that σ is trace-free, we transform (6.18) as follows:

6.29

A similar transformation of (6.19) yields

6.30

Furthermore, one immediately recognizes that (6.22) and (6.23) are composed of two terms with the structure of the second and third primary modes,

6.31

and

6.32

Finally, the term (6.26) can be transformed as follows:

6.33

and

6.34

7. Invariance defect as a superposition of eight modes

After transforming various nonlinear contributions, it now becomes easier to recognize the structure of the invariance defect as a superposition of eight modes,

7.1

which we specify below.

(a). Primary mode

The mode

7.2

is associated with the rank-three tensor , and was already identified from the analysis of the linear part of the invariance defect (see (6.10)),

7.3

Collecting pertinent contributions from (6.10), (6.16) and (6.17), and taking into account that tensor (7.3) is symmetric and traceless, we get the extension of the frequency associated with this mode into the nonlinear domain,

7.4

where we have introduced shorthand notation 〈A_αβγ〉 for the symmetrized trace-free generic tensor of rank three. The above can be rewritten in a more compelling way, remembering the transport of velocity along streamline (2.15), re-introducing the material time derivative (1.6) and using the equation of state (2.10),

7.5

When written in the form (7.5), the main mode features contributions of three types. The first term, already discussed (see (6.10)), is the linear dissipation term. The second term is nonlinear dissipation. Finally, the third term couples the non-equilibrium stress to the transfer of the velocity along the streamline.

(b). Primary mode

The mode

7.6

is associated with the rank-two tensor ,

7.7

which at the same time is the velocity polynomial of order four. Collecting relevant contributions from (6.11), (6.24), (6.25) and (6.31)–(6.33), the respective frequency reads,

7.8

Again, using symmetry of the velocity tensor and introducing notation 〈A_αβ〉 for a symmetrized trace-free tensor of rank two, and also remembering the transfer along the streamline for velocity and temperature, we decode the structure of this primary mode as follows:

7.9

It is interesting to note that linear dissipation and streamline convection are present in this mode but no nonlinear dissipation terms.

(c). Primary mode

The scalar mode which was identified by linear approximation,

7.10

is built on the quotient of the fifth-order Hermite polynomial P⁽³⁾,

7.11

and its frequency comprises contributions from (6.12), (6.31) and (6.32),

7.12

It is easy to recognize the terms with the coupling to the streamline:

7.13

The three primary modes are accompanied by the three modes which vanish under linearization; we term them secondary modes.

(d). Secondary mode

The mode

7.14

is defined by rank-three tensor ,

7.15

This is an essentially nonlinear mode which vanishes close enough to equilibrium, and is contributed by the first terms in (6.30),

7.16

(e). Secondary mode

Another secondary mode,

7.17

features the highest rank-four velocity tensor ,

7.18

Only the term (6.34) contributes to the frequency of this mode,

7.19

(f). Secondary mode

Finally, the third secondary mode is established by the invariance defect (6.27),

7.20

where the symmetric rank-two tensor ,

7.21

is associated with the frequency,

7.22

Modes (7.2), (7.6), (7.10), (7.14), (7.17) and (7.20) are contributing modes, their projection onto the R13 fluxes does not vanish. Apart from that, there are two other modes in the invariance defect, which do not contribute to the R13 fluxes.

(g). Ghost mode

The first term in (6.29) contributes to yet another mode, ,

7.23

and is defined by the rank-three traceless tensor ,

7.24

with the frequency

7.25

Note that mode features the same symmetric trace-free rank-three tensor 〈σ_αβ∂_γT〉 as already appears in the primary mode ; see (7.5). As we shall prove below, the projection of mode onto the flux Q_αβγ vanishes. For that reason, we term the ghost mode: while it contributes to the distribution function of R13, its projection onto the R13 system disappears.

(h). Ghost mode

Finally, the mode ,

7.26

is defined by the vector ,

7.27

and the corresponding frequency acquires contributions from pertinent terms of (6.30), (6.20), (6.21) and (6.29),

7.28

Despite many seemingly interesting couplings in the frequency (7.28), this mode is also a ghost, as we shall prove below.

(i). Orthogonality theorems

An important feature of the invariance defect of the Maxwellian was its orthogonality to basic fields (4.15). This is the solvability condition in the Chapman–Enskog method, which defines unique solutions of the corresponding linear integral equations. In general, the invariance defect of any approximation in the MIM has just the same feature: it is orthogonal to the fields building up the approximate manifold we are willing to correct. Here, we can verify a somewhat stronger statement: that each mode is separately orthogonal to Grad’s 13 fields.

Theorem 7.1 (orthogonality of modes). —

Any integral of the form

vanishes, I≡0.

For the main modes , and , this statement was verified in KGDN. For the rest, unless the integral vanishes automatically by symmetry, the statement is verified by direct evaluation of the corresponding Gaussian integrals.

We now turn our attention to modes and to prove that they will not contribute to the R13 fluxes. For that, we need the following proposition.

Theorem 7.2. —

For any indexes α, β, γ, μ, ν, λ, the following Gaussian integrals vanish:

7.29

and

7.30

Indeed, taking into account spherical coordinates, any integral of the form (7.29) is written

where a_αβγλμν is some function of the angles. However,

Similarly, the proof of (7.30) reduces to evaluating the integral

With this, we prove that modes and are ghosts, as follows.

Theorem 7.3. —

Modes and are orthogonal to the flux Q_αβγ.

Indeed, contributions of both these modes, , vanish by the integral theorem 7.2.

Summarizing, the non-local invariance defect of Grad’s approximation is a superposition of eight modes. It is interesting to note that the coupling to the streamline terms is brought about only by the primary modes, whereas the essentially nonlinear secondary modes do not feature this. Six modes are contributing; their projection onto the R13 fluxes does not vanish. The two other modes are ghosts.

8. From modes into fluxes: relaxation approximation and R13

Similar to the invariance defect of the local Maxwellian, the non-local invariance defect of Grad’s approximation is independent of the collision integral and is universal in that sense. Hence, after being able to identify the modes we can proceed towards the simplest R13 equations by using the relaxation time shortcut. Certainly, as in the Maxwellian case, this will only reveal the basic structure of the R13 equations rather than specify it for a given collision model. Nevertheless, what we are aiming for in this paper is to reveal that structure rather than to evaluate it for specific collision models (see the discussion in §9). Introducing the relaxation time τ, we write

8.1

and proceed with evaluating the R13 fluxes. The flux in the equation for the non-equilibrium stress is written as

8.2

where the first part is the contribution of the primary mode and the last part is the contribution of the secondary mode ,

8.3

where we have introduced the viscosity in the relaxation time approximation,

8.4

Similarly, the contribution of mode gives

8.5

Combining both contributions, we finally get (see also (1.7) for the coordinates-free version),

8.6

The second flux,

8.7

is evaluated similarly. It is convenient to consider the trace-free and the diagonal parts of this tensor separately,

8.8

8.9

8.10

where the traceless part is formed by the primary mode and the secondary modes and while the scalar primary mode and the secondary mode contribute to the trace . Proceeding with evaluation of integrals, we obtain

8.11

8.12

8.13

8.14

8.15

Put together, the second R13 flux reads (see also (1.9) and (1.10)),

8.16

and

8.17

Equations (8.6), (8.16) and (8.17) conclude our derivation of the structure of R13 equations. As we already mentioned in the Introduction, one can easily recognize contributions of two different types, the dissipative (both linear and nonlinear) and the streamline convection. In order to see this more clearly, it is instructive to write, for example, equation (8.6) in a non-dimensional form, reducing the variables by reference pressure p₀ and reference thermal speed v₀ at reference temperature T₀. Then, for we get, retaining the terms to second order,

8.18

where reduced quantities are , , and , and where l=ν/v₀ is the mean free path. From (8.18), it is apparent that all three first terms belong to the same category where the mean free path reduces the derivative in space, while in the last term it is the relaxation time that reduces the time derivative along the streamline. We further note that the streamline convective fluxes are contributed solely by the primary modes, whereas the secondary modes only add a part of the nonlinear dissipation.

We now proceed with comparing the above R13 equations with the ones proposed in [7]. To that end, we first match the definition of the fluxes here and in [7] by introducing , and . Second, expanding the streamline derivatives, we write the R13 fluxes (1.7), (1.9) and (1.10) in the form consistent with the nomenclature of [7],

8.19

8.20

8.21

Furthermore, comparison with [7] requires taking into account the difference in collision models (relaxation-time approximation used above and Maxwell’s molecules in [7]). The relation between the two is achieved by changing numerical prefactors, as explained, for example, in [11] (eqn (42) and formula displayed below it in [11]) and summarized in table 2. With this, the following observations can be made.

• (i) The nonlinear R13 fluxes (8.19)–(8.21) match those of [7] term by term.

• (ii) Terms shown underlined in (8.19), (8.20) and (8.21) are regarded as being of higher order in light of a scaling, σ∼q∼∇∼ϵ, and are dropped in many works [8–15]. In our case, the underlined terms are produced by the non-equilibrium part of the streamline convective fluxes, and are cancelled by replacing the full streamline derivatives D_tu and D_tT with Euler approximation thereof,

  

  elsewhere in (1.7), (1.9) and (1.10).

Table 2.

Coefficients for the relaxation approximation and Maxwell’s molecules [7,11].

flux	relaxation approximation	Maxwell’s molecules, τ=μ/p
m	−3τ[…], equation (8.19)	−2τ[…]
R	, equation (8.20)
Δ	−8τ[…], equation (8.21)	−12τ[…]

We shall continue the discussion of other variants of R13 in §11.

9. Beyond relaxation approximation

(a). Decoupling of local correction

Above, we have analysed in some detail the mode structure of R13, which is driven by inhomogeneities of the macroscopic fields. Let us now return to the local part of non-invariance of Grad’s system. The mechanisms that explain why local defect of invariance arises are that velocity polynomials 〈CC〉 and C(C²−5) used in Grad’s approximation are eigenfunctions of the linearized collision integral for Maxwell’s molecules only. Historically, Grad’s G13 approximation is understood as a truncated Hermite tensor-polynomial expansion of the distribution function. However, one need not be necessarily bound to Hermite polynomials when choosing the starting approximation of the distribution function. In that regard, a family of possible Grad-like approximations can be described using two arbitrary functions, A and B, instead of the canonical (3.6) and (3.7) as follows:

9.1

and

9.2

where

9.3

9.4

9.5

It is easy to verify that the choice A=1, B=1 returns the conventional Grad’s (3.6) and (3.7). With generalized Grad’s functions (9.1) and (9.2), the only change in the closure for Q- and T-fluxes is the non-equilibrium part of the T-flux where we get, instead of (3.13),

9.6

where

9.7

Again, it is easy to verify that for the conventional B=1 in (3.13). Using (9.6) implies the following changes in the coefficients of Grad’s terms in (3.34) and (3.35):

9.8

and

9.9

The main change occurs to the relaxation part of Grad’s closure, where matrix elements (3.23) and (3.24) become tunable depending on the choice of functions A and B, and we have a generalization of theorem 5.1, as follows.

Theorem 9.1. —

Let functions A and B in the generalized Grad’s distribution (9.1) and (9.2) correspond to the eigenvalue problem

9.10

and

9.11

Then the local invariance defect of the corresponding Grad’s approximation vanishes,

9.12

Thus, the local invariance defect of Grad’s approximation simply reflects a poor choice of Hermite polynomials to represent the eigenfunctions of the linearized collision integrals except for Maxwell’s molecules, rather than it bearing any physical relevance. Ideally, for realistic collisional models, one should start with a tailored Grad’s closure so that the local invariance defect vanishes. However, to the best of our knowledge, the pertinent eigenfunctions of generic collision integrals are not available in closed form. In that regard, it was shown in KGDN [6] that the first local correction results in Grad’s system with exact Chapman–Enskog transport coefficients (viscosity and thermal conduction) rather than the first approximation thereof in the original Grad’s closure (see §3). Similarly, any other local improvement of Grad’s closure, such as taking into account quadratic terms, is fully decoupled from the non-local part of the problem considered above. Generalized Grad’s closures, tailored to specific molecular models, can be addressed from the inset by proposing a more appropriate distribution function (9.1) and (9.2) and repeating the computation of the non-local invariance defect. Various proposals for the interaction-specific functions A and B (9.2) and (9.1) can be found in the literature: Reinecke & Kremer [22] proposed using the Sonine polynomial expansion of the Chapman–Enskog method; Struchtrup [15] developed self-consistent moment equations for arbitrary collision integral; and Gorban & Karlin [23] considered moments of the linearized collision integral instead of moments of the distribution function.

(b). R13 transport coefficients from the Boltzmann equation

The relaxation time approximation applied above highlights the overall structure of R13 directly from the analysis of the invariance defect. However, this does not substitute by itself the derivation from the Boltzmann equation. In the collision-dominant regime, the correction is found from the linear integral equations similar to those in the Chapman–Enskog problem (4.17), as already indicated in KGDN [6]. For simplicity, let us consider the primary modes only; owing to orthogonality property (7.1), the correction to the distribution function is written as

9.13

where functions A⁽¹⁾, A⁽²⁾ and A⁽³⁾ depend on C² and satisfy orthogonality conditions,

9.14

9.15

9.16

and are found from linear integral equations,

9.17

9.18

9.19

The problems (9.17)–(9.19) can be addressed by the same methods as in the case of transport coefficients computation in the Chapman–Enskog method (cf. (4.19) and (4.20)). When function (9.13) is used instead of its relaxation time surrogate (8.1), the evaluation of R13 fluxes proceeds as in §8. This will only amend the coefficients in front of expressions like (8.3), but nothing will change in the structure of R13. Specifically, all of the mode frequencies will remain intact and demonstrate the same partition into various types of contributions (dissipation and streamline convection) as before. Note that, even for Maxwell’s molecules, computation of the perfactors of the Q- and T-fluxes will require knowledge of functions A⁽¹⁾, etc., from solutions to integral equations (9.17). In that case, computation is relatively easy though [20]. Summarizing, the leading-order R13 correction can be found systematically from the Boltzmann equation, and reduces to solving linear integral equations familiar from the Chapman–Enskog problem.

10. R13 beyond leading order: Lorentz model

(a). The Lorentz model

We shall consider here an example of the non-perturbative R13 solution, beyond the leading order, for a simple kinetic equation, the classical Lorentz model for a gas of light particles in a host medium of random scatterers. We consider the three-dimensional Lorentz model for a dilute gas [19]. Let f(r,v,t) be the one-particle distribution function with r the spatial vector, v the particle’s velocity and t the time. Velocity v takes values on the two-dimensional sphere of radius v. The evolution of f is according to the kinetic equation

10.1

where integration is carried out over the unit sphere. Owing to linearity, it is convenient to Fourier transform in space; equation (10.1) is rewritten for the Fourier image as

10.2

where k is the wavevector. The locally conserved variable is the number density,

10.3

Balance equations of interest are

10.4

and

10.5

Here

10.6

is the Fourier image of the momentum flux, and

10.7

is that of the stress tensor. Whereas the hydrodynamic description assumes the closure of the continuity equation (10.4), i.e. an expression of the momentum flux in terms of the density , the Grad-type description instead requires a closure of the second equation (10.5), i.e. to express the tensor in terms of and .

(b). Invariance principle: hydrodynamics

Before addressing the Grad case, we shall demonstrate how the exact Chapman–Enskog solution found by Hauge [19] is derived with the MIM. Exact closure of the continuity equation (10.4) is provided by a function,

10.8

which is linear in the macroscopic parameter n, and which satisfies

10.9

Knowing function (10.8), the Fourier image of the momentum flux can be computed as

10.10

and the continuity equation (10.4) is closed to give the hydrodynamic equation

10.11

where

10.12

is the relaxation rate. Exact closure satisfies the invariance principle: computing the time derivative of (10.8) by the chain rule due to the closure (10.4),

10.13

and, on the other hand, computing the time derivative due to the kinetic equation (10.2),

10.14

we require that both give the same result independently of ; this yields the invariance equation (4.1),

10.15

whereupon

10.16

Introducing θ as the angle between vectors k and v, integrating (10.16) over the unit sphere and using the consistency condition (10.9), we get an equation for the closure ,

10.17

The latter can be transformed as follows, using :

10.18

Finally, using the Euler identity,

10.19

we find

10.20

which is valid for τkv≤π/2. This is the exact Chapman–Enskog closure relation for the Lorentz model [19].

(c). Invariance principle: Grad’s case

Let us now consider the analogue of the R13 problem. Grad’s moment approximation adapted to the Lorentz model reads,

10.21

Computing , we get

10.22

With (10.22), momentum equation (10.5) becomes Grad-like,

10.23

After computing the derivatives

10.24

and

10.25

the invariance defect of approximation (10.21) is evaluated as

10.26

As in the R13 case, we proceed with the dynamic correction to Grad’s approximation (10.21). Here we address the iterated version of the MIM (4.2), with (10.21) as the input, and consider one iteration of the method. The correction is sought as

10.27

where function is found from (4.2),

10.28

10.29

10.30

and satisfies additional conditions (4.4),

10.31

After simplifications owing to (10.31), equation (10.28) becomes

10.32

where

10.33

is the correction to Grad’s stress tensor (10.22). Integral equation (10.32) is linear, unlike the nonlinear integral equation (10.15) in the case of the exact hydrodynamics. This means that, with (10.32), we address here only the first iteration of the exact invariance condition for Grad’s system. However, even with this limitation, equation (10.32) is non-perturbative in nature since no ‘small parameters’ have been used so far. Yet, if relaxation is strong again, the first term on the right-hand side is dominant, and we get to leading order

10.34

This leading-order correction to Grad’s function (10.21) is precisely the analogue of the R13 correction considered in the main part of the paper. Higher-order corrections can be found based on the recurrent procedure applied to the iteration equation (10.28), , with

However, such a ‘perturbative’ approach is not encouraging. Instead, similar to the above treatment of the exact hydrodynamics, we can solve the iteration equation (10.32),

10.35

and find the correction to the stress tensor (10.33) by integrating (10.35) with vv and solving the resulting linear algebraic system. We quote here only the result for a special flow set-up to be used below: introducing the z-axis along the wave vector k, the longitudinal component of the correction to the stress tensor component of the stress tensor reads,

10.36

With this, we have found the first non-perturbative correction to the Grad approximation from the invariance condition.

In order to compare various approximations, we considering plane wave solutions of the form and find dispersion relation ω(k), where the real part, Re ω, is the attenuation rate and the imaginary part, Im ω, is the frequency. In figure 1, we show the reduced dimensionless functions Re τω(ζ) and Im τω(ζ), where ζ=τkv, for Grad’s approximation (10.21), for the first correction (10.36) and for the leading-order R13-type correction (10.34). The exact Chapman–Enskog result (10.20) is also shown for comparison. Grad’s closure (10.21), the non-perturbative first correction (10.36) and the leading-order R13 (10.34) exhibit a crossover from the diffusion-like to the wave-like propagation of the density, and thus are consistent with the corresponding crossover of the Lorentz model [19]. In addition to that, the leading-order R13 undergoes the inverse crossover at sufficiently short wavelength. This inverse crossover is the result of the approximation done while solving the invariance equation, and is not present in the complete solution (10.36). Finally, comparison with the hydrodynamic branch (10.36) clearly shows that both the corrections to Grad’s approximation agree better with the exact hydrodynamics; thus, the correction is an improvement of Grad’s closure indeed.

Figure 1.

Dispersion relations for the Lorentz model. Top: reduced frequency Im τω(ζ); bottom: reduced attenuation rate Re τω(ζ) as a function of reduced wave vector ζ=τkv. Dots, Grad’s approximation (10.21); solid line, non-perturbative correction (10.36); long dash, leading-order R13-like correction (10.34); dash, exact Chapman–Enskog solution (10.20).

11. Discussion, conclusions and outlook

In this paper, we have extended the derivation of the dynamic correction to Grad’s 13-moment projection from kinetic equations initiated in [6] to the full nonlinear case. We found that the R13 distribution function is a superposition of eight modes, out of which the most important are the three primary modes, accompanied by three essentially nonlinear secondary modes, and that there are also two ghost modes which do not contribute to R13 fluxes. It is interesting to note that the R13 distribution function (1.1) does not reduce to Grad’s 26-moment G26 approximation (see §6.1.3, eqn (6.16) in [15]): unlike the eight-mode structure of (1.3), the latter features only three velocity tensors (7.3), (7.7) and (7.11), corresponding to the primary modes. The presence of ghost modes in the invariance defect of G13 makes it very different from the familiar case of the Maxwellian: all modes of the latter are contributing. Revealing ghost modes signals a hidden constraint of G13, and may trigger interesting questions about finding Grad’s moment system, which would be closed with respect to its modes. The answer need not be unique and is left for future studies.

Mode decomposition reveals a structure of nonlinear R13 transport equations which we write here symbolically for the sake of discussion:

11.1

and

11.2

Here the brackets indicated with G13 are Grad’s contribution, and nlin indicates the nonlinear R13 contribution to dissipation. The nonlinear convective contribution (underlined) is in the form of divergence of a flux which contains both the instantaneous (Grad’s) contributions σu in (11.1) and qu in (11.2) as well as the non-local in time parts due to the R13. Sticking to (11.1), integration over the control volume provides the flux of non-equilibrium stress through the boundary. The R13 convective flux provides an additional contribution to the in/out flux of the non-equilibrium stress and samples the flow velocity along the streamline for the time period τ (see figure 2). To the best of our knowledge, such a non-locality is not present in any instantaneous flux familiar from continuum mechanics (the Hamiltonian structure of propagation terms of G13 was discussed recently in [3,4]), and is thus a Knudsen-layer effect. A similar mechanism is also valid for the flux of heat flux in (11.2).

Figure 2.

Cartoon illustrating the concept of streamline convection. The stress tensor σ, pictured by the principal axes, is carried along the streamline into the control volume for time τ. (Online version in colour.)

To conclude, the NSF approximation is quite intuitive from the physical standpoint. Beyond that, reporting a physically compelling picture of fluxes is difficult, and until now escaped a qualitative interpretation for the most part. Recently, higher-order hydrodynamic fluxes were interpreted as dispersive non-locality of Korteweg’s type (see the contribution by Marshall Slemrod 21 and a recent discussion in [24]). While dispersive non-locality in space is also a part of the G13 system, and is featured by the exact hydrodynamic component of the latter [24], the R13 demonstrates yet another different type of non-locality through finite-time (non-Markov) convective transport of non-equilibrium stress and heat flux. The convective non-locality is essentially a nonlinear effect, unlike the dispersive non-locality. From a more practical perspective, the flux structure can be beneficial for numerical realizations since these can be evaluated directly from the streamline Lagrangian data of basic fields, without the need for a numerical evaluation of their gradients.

Turning now to the outlook, we first remark that it is not surprising that the resulting R13 equations derived above from the kinetic equation are identical to those proposed by Struchtrup & Torrilhon [7], who instead chose to work with the larger G26 moment system. Indeed, both [6,7] proceeded under the same restrictive assumptions of linearity of the collision term and collision dominance in the invariance iteration. However, the corresponding R13 distribution functions are different: when the moments (8.19)–(8.21) are used in Grad’s 26-moment distribution function, the result does not match (1.1).

Several other nonlinear R13 variants were proposed by Struchtrup and Torrilhon and co-workers in order to overcome these and other restrictions [8–15] (see, for example, table 1 in [13]). In order to review these, we follow [8–15] to note that the R13 fluxes can be compactly written using the Navier–Stokes and Fourier approximation for the stress and the heat flux; in the relaxation approximation these are

11.3

and we can write in (8.19)–(8.21)

11.4

11.5

11.6

Here we have dropped higher-order terms by the Euler approximation to the streamline, for simplicity. While R13 fluxes (11.4), (11.6) and (11.5) are written for the relaxation time approximation, the same structure holds also for Maxwell’s molecules, and differs only in the numerical coefficients (cf. table 2). That form for Maxwell’s molecules was amended in the later R13 variants, including the following operations:

• — adding nonlinear local terms with the structure ∼σσ to the counterpart of (11.5) and (11.6) for Maxwell’s molecules, featuring the nonlinear relaxation due to the Boltzmann collision integral [15] (see below),

• — replacing σ^NSF and q^NSF in (11.4), (11.5) and (11.6) with their local values σ and q in order to retain the boundary conditions formulation of [14] also for nonlinear cases, and

• — including the higher-order terms (underlined in (8.19)–(8.21)),

and combinations thereof [8]. Systematic derivation and discussion of these later R13 variants is well documented [7–15]; generalizations to other types of collisions are also available.

In recent work [13], many R13 variants were compared in the shock wave setting. It was found that, while all R13 variants behave similarly and well for weaker shocks (Mach number Ma≤2), the later R13 generalizations degrade significantly when extending into the hypersonic domain. This probably indicates that the development of R13 theory is still ongoing, as may also be inferred from [12]. In that regard, the following path to including the strong nonlinearity of the collisions can be explored: let us write down the pertinent part of the G26 system for Maxwell’s molecules, retaining nonlinearity in the relaxation rates [15],

11.7

11.8

11.9

where , and depend on space derivatives of the displayed fields. The scaling proposed in [15] to treat nonlinearity in the relaxation rates, τ→ϵτ, σ,q,∇→ϵσ,ϵq,ϵ∇, retains the expansion point, m⁽⁰⁾=0, R⁽⁰⁾=0, Δ⁽⁰⁾=0, already used in the linear relaxation case (when nonlinearity ∼σσ is neglected in the relaxation terms of (11.8) and (11.9)). This leads to simply adding the terms ∼σ:σ/ρ and ∼〈σ⋅σ〉/ρ to the Δ- and R-fluxes, respectively, already known from the linear relaxation analysis, while omitting terms of higher orders (order of magnitude method [15]). However, this may be insufficient for the strong shock wave problems. A different route is to shift the expansion point,

11.10

and to compute the invariance correction accordingly,

11.11

11.12

11.13

where the time derivative is understood (as above in the paper) in the sense of the chain rule,

11.14

and

11.15

with (1.14). In other words, the shift of the expansion point (11.10) generates a different R13 branch (11.11)–(11.13). This may be beneficial for strongly nonlinear problems: for example, it is well known that one of the most successful theories of a strong shock wave, the Tamm–Mott–Smith bimodal approximation [25], does not match the NSF limit. Another related example of how a shift in the expansion point helps to advance the simulation of supersonic flow can be found in the context of the lattice Boltzmann method [26]. Analysis of the new R13 branch outlined above could be an interesting further research direction.

As a final comment, Hilbert proposed to derive ‘mathematically the limiting processes … which lead from the atomistic view to the laws of motion of continua’ (see, for example, [24,27] and references therein and the contribution by Marshall Slemrod 21). The issue Hilbert was addressing, namely whether to use the atomistic theory of his day, represented by Boltzmann’s kinetic theory of gases and a passage via a limiting process to the continuum theory of the compressible Euler system as the Knudsen number approaches zero, or to use the NSF system, if small corrections are allowed. The problem is, however, that without a priori knowledge about solutions to Euler or NSF equations, one cannot in general prove that Boltzmann kinetics converges to these continuum models [27]. The resolution probably requires other continuum theories to be addressed: one current candidate is Korteweg-like [24]. Alternatively, one can try to derive continuum theories from Boltzmann. That one attempt is R13, and since only the relevance to experiment and comparison to Boltzmann is the ultimate validation, R13—as a continuum theory—appears to be a viable candidate.

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Competing interests

I declare I have no competing interests.

Funding

Funding by SNF grant no. 200021-149881 is gratefully acknowledged.