c 2005 Society for Industrial and Applied Mathematics

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1 MULTISCALE MODEL SIMUL Vol 3 No 1 pp c 2005 Society for Industrial and Applied Mathematics DERIVATION OF 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW TO SECOND ORDER ACCURACY FOR ARBITRARY INTERACTION POTENTIALS HENNING STRUCHTRUP Astract A recent approach to derive transport equations for rarefied gases from the Boltzmann equation within higher orders of the Knudsen numer H Struchtrup Phys Fluids 16 (2004) pp is used to derive a set of 13 moment equations for aritrary molecular interaction potentials It is shown that the new set of equations is accurate to second order while Grad s original 13 moment equations are of second order accuracy only for Maxwell molecules and Bhatnagar Gross Krook models Key words Boltzmann equation Chapman Enskog expansion Grad s moment method regularized 13 moment equations AMS suject classifications 76P05 82C40 DOI / Introduction While it is well known that the Navier Stokes Fourier (NSF) equations fail to descrie processes in rarefied gases for Knudsen numers ove 005 it is still a matter of deate how a set of meaningful macroscopic equations for larger Knudsen numers can e derived from the Boltzmann equation The NSF equations are routinely derived from the Boltzmann equation y means of the Chapman Enskog expansion which yields the NSF equations in first order of the Knudsen numer However the Chapman Enskog expansions to second and third order yield the Burnett and super-burnett equations 3 4 which are known to e unstle in transient processes 5 and to give unphysical results in steady state prolems 6 Thus one has to conclude that the Chapman Enskog expansion fails at higher orders The reasons for that failure however remain unclear In recent years some authors proposed ad hoc improvements of the Burnett equations which yield stility ut are somewhat difficult to justify since they are not linked well to the Boltzmann equation; see 11 6 for a discussion Another well-known method for otaining equations for rarefied gas flows is Grad s method of moments which provides stle equations at any level that is for any set of moments considered as the asic variles of the theory There are two major points of criticism against Grad s method namely that Grad s equations fail to descrie smooth shock structures for Mach numers ove a critical value 14 and that the equations are not related a priori to the Knudsen numer as a smallness parameter The failure to descrie smooth shock structures is related to the fact that the Grad equations are of hyperolic type and thus have a maximum signal velocity 14 Only recently Struchtrup and Torrilhon introduced a regularization for Grad s 13 moment equations which is ased on a Chapman Enskog expansion around a nonequilirium state 15 11; see also 16 for a similar approach The regularized Grad equations Received y the editors January ; accepted for pulication (in revised form) April ; pulished electronically January This work was supported y the Natural Sciences and Engineering Research Council (NSERC) of Cana Department of Mechanical Engineering University of Victoria PO Box STN CSC 3055 Victoria BC V8W 3P6 Cana (struchtr@meuvicca) 221

2 222 HENNING STRUCHTRUP produce smooth shock structures for all Mach numers and thus overcome this particular prolem In a series of papers Reinecke and Kremer were le to show that NSF and Burnett equations can e derived from certain sets of Grad s moment equations y means of a Maxwellian iteration a method that is essentially equivalent to a Chapman Enskog expansion of the moments 19 Struchtrup and Torrilhon showed that a Chapman Enskog expansion of the regularized 13 moment equations yields the Burnett and super-burnett equations for Maxwell molecules 15 so that this set of equations is of third order The first attempt to derive Grad s equations y means of arguments on the Knudsen numer is due to Müller Reiteuch and Weiss 20 who considered the infinite system of coupled moment equations of the Bhatnagar Gross Krook (BGK) equation 21 From these they determined the order of magnitude of moments in terms of orders in powers of the Knudsen numer and declared that a theory of order λ needs to consider all terms in all moment equations up to the order O ( ε λ) At first these are the moment equations for all moments of order β λ under omission of higher order terms However these equations split into two independent susystems and only a smaller numer of equations (and variles) remain as equations of importance 20 In the method developed elow we do not ask for the order of terms in all moment equations ut of the order of magnitude of their influence in the conservations laws ie on heat flux and stress tensor This is quite different For example in order to compute the heat flux with third order accuracy as is necessary in a third order theory other moments are needed only with second order accuracy while others can e ignored completely Müller Reiteuch and Weiss s consistent order extended thermodynamics 20 on the other hand would require higher order accuracy for these moments and a larger numer of moments The new method was applied to the special cases of Maxwell molecules and the BGK model in 22 and it could e shown that it yields the Euler equations at zeroth order the NSF equations at first order Grad s 13 moment equations (with omission of a nonlinear term) at second order and Struchtrup and Torrilhon s regularization of these at third order An important feature of the new method is that the equations at any order are stle other than in the Chapman Enskog method where the second and third approximation Burnett and super-burnett equations are unstle The restriction to Maxwell molecules or the BGK model leads to a comparatively simple application of the new method and what has to e done for other models of molecular interaction was only sketched roughly in 22 The present paper gives a detailed account of the method applied to aritrary interaction models up to second order accuracy It will e shown that again the first order approximation agrees with the NSF equations as they are derived y means of the Chapman Enskog expansion while the theory of second order accuracy yields a set of 13 moment equations which differ from Grad s original 13 moment equations In particular the new equations have different numerical coefficients and some additional terms All coefficients depend on the interaction model used The latter feature is well known from the Burnett equations where the coefficients depend on the interaction model used as well 1 18 In the present paper the method is outlined in detail ut numerical values of the coefficients are not computed Indeed this paper serves to set the stage for detailed calculations since it carefully analyzes the prolem for the most general setting aritrary interaction potentials of particles This allows us to identify exactly what quantities mostly coefficient matrices for moments of the collision term must e determined for a theory of second order Thus the present paper presents the

3 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW 223 necessary work that needs to e done efore numerical values for coefficients can e computed Detailed calculations of the numerical values of coefficients for the generalized 13 moment equations and tests of the new equations are planned for the future Accordingly the present paper is quite technical in nature The reader interested in more ackground to the new method used here is referred to 22 where the new method is discussed in detail including careful comparisons with the classical methods of Chapman Enskog and Grad The remainder of the paper is organized as follows In section 2 the moments are defined as asic irreducile tensors and the infinite set of moment equations is derived The moments of the collision term of the Boltzmann equation are discussed in section 3 Their computation requires an expression for the particle distriution function in terms of the moments and the Reinecke Kremer Grad method is suggested to determine that function In section 4 the order of magnitude of the moments is determined and the minimum numer of moments of first order is discussed The results are then used in section 5 to develop the proper sets of equations for zeroth to second order accuracy The paper ends with our conclusions 2 Basic equations 21 Boltzmann equation Our starting point is the Boltzmann equation 1 2 which we write as (21) Df Dt + C f k = 1 ε S (f) where f denotes the phase density C k = c k v k denotes the peculiar velocity with c k as the velocity of a particle and v k denotes the center of mass velocity of the flow D Dt = t + v k denotes the material time derivative ε is a formal smallness parameter which stands for the Knudsen numer This parameter will e used for monitoring the order of magnitude of the moments and the order of magnitude of terms within equations At the end of all calculations ε will e set equal to unity In fact if proper dimensionless quantities were introduced the dimensionless Boltzmann equation would read as (21) with the Knudsen numer instead of ε and the Knudsen numer could e used as the smallness parameter for the procedure elow Reinserting of the dimensions would then remove the Knudsen numer this corresponds to setting ε = 1 at the end of the computations Thus the use of ε removes the necessity of introducing dimensionless quantities S (f) is the collision term that accounts for the change of f due to collisions and is given as 1 2 (22) S (f) = (f f 1 ff 1 ) σg sin ΘdΘdεdc 1 where c and c 1 are the velocity vectors of two colliding particles efore the collision c and c 1 are the velocities after collision g = c c 1 is the relative velocity and g = g σ is the collisional cross section Θ is the collision angle and ε is an angle that defines the direction of the collisional plane Finally f = f (xtc ) etc The collision term has the following properties: (a) conservation of mass momentum and energy so that (23) m {1ci C 2} S (f) dc =0

4 224 HENNING STRUCHTRUP () In equilirium the phase density is the Maxwellian S (f) =0 = f = f M = ρ 3 1 (24) exp C2 m 2πθ 2θ where ρ is the mass density and θ = k mt is the temperature in energy units where T is the temperature m is the particle mass and k denotes Boltzmann s constant (c) The Boltzmann equation leads to a positive entropy production 22 Moments We define the general irreducile moments of the phase density as (25) u a i 1 i n = m C 2a C i1 C i2 C in fdc where indices in angular rackets denote the symmetric and trace-free part of a tensor Note that the moments u a i 1 i n that are used in this paper are trace-free y definition u a i 1 i n = u a i 1 i n and that this is not made explicit with rackets in order to avoid confusing notation involving several pairs of rackets Appendix A gives a short introduction to trace-free tensors Some of the moments have a particular interpretation namely u 0 = ρ u 0 i =0 u 1 =2ρe =3ρθ =3p u 0 ij = σ ij u 1 i =2q i Here we introduced the specific internal energy e = 3 2θ of the ideal gas the pressure p the irreducile part of the pressure tensor σ ij and the heat flux q i The values of the moments in equilirium (E) when y (24) the phase density is a Maxwellian are given y u a E =(2a + 1)!!ρθa u a i 1 i n E =0 n 1 where (2a + 1)!! = a s=1 (2s+1) The moments of the collision term of the Boltzmann equation (21) are defined as = m C 2a C i1 C i2 C in S (f) dc P a i 1 i n and will e discussed in section 3 Due to the conservation conditions (23) we have (26) P 0 = P 0 i = P 1 =0 23 Generic moment equation Multiplication of the Boltzmann equation with mc 2a C i1 C i2 C in and susequent integration over velocity space yields after some rearrangement the general equation for the moments (25) (27) Du a i 1 i n Dt +2au a 1 Dv k i 1 i nk Dt + n 2n +1 (2a +2n Dv in +1)ua i 1 i n 1 + ua i 1 i nk Dt + n u a+1 i 1 i n 1 +2au a 1 i 2n +1 x 1 i nkl in n +2a 2n +1 ua k i 1 i n 1 + nu a v in k i 1 i n 1 + n (n 1) 4n 2 1 n v in 1 (2a +2n +1)ua+1 i 1 i n 2 n +2a n +1 x l 2n +3 ua i 1 i n + u a i 1 i n = 1 ε Pa i 1 i n

5 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW 225 Note that all moments are trace-free y definition and additional trace-free tensors are made explicit y means of angular rackets This implies that each of the terms in the ove equation is trace-free and symmetric in the indices i 1 i n The derivation of the equation ove requires multiple use of relation (A2) 23 C 2a C i1 C i2 C in C k = C 2a C i1 C i2 C in C k + n 2n +1 C2a+2 C i1 C i2 C in 1 δ in k The set of infinitely many moment equations (a n ) is equivalent to the Boltzmann equation We are interested in limits of the Boltzmann equation given y orders of the Knudsen numer ε and due to this equivalence we can perform the limiting process on the moment equations rather than on the Boltzmann equation itself 24 Conservation laws First we consider the conservation laws that is those equations which y (26) have a vanishing collision moment For a = 0 n = 0we otain the mass alance Du 0 Dt + v u0 k =0 and for a =1n = 0 we find the alance of internal energy as Du 1 Dt + u1 k +2u 0 kl + 5 v x l 3 u0 k =0 Note that y introducing ρ θ σ ij q i and some simple manipulations these two equations can e rought into their usual textook form: Dρ Dt + ρ (28) =0 3 2 ρdθ Dt + ρθ + q k (29) + σ kl =0 x l For the choice a =1n = 1 we otain the alance of momentum ρ Dv i Dt + θ ρ + ρ θ + σ ik (210) =0 25 Scalar moments For scalar moments (n = 0) the general equation (27) reduces to Du a Dt +2aua 1 k Dv k Dt + ua k +2au a 1 kl + x l 2a +3 u a = 1 3 ε Pa Next we introduce the difference etween the scalar variles and their equilirium values as (211) w a = u a u a E and rewrite the scalar equations for these new variles where all time derivatives of ρ θ v i are replaced y means of the conservation laws This yields Dw a Dt 2a 3 (2a + q k 1)!!θa 1 2a 3 (2a + 1)!!θa 1 σ kl +2au a 1 kl x l 2au a 1 k θ 2au a 1 k θ ln ρ 2a ua 1 k ρ σ kl x l + ua k + x l 2a +3 w a = 1 3 ε Pa Of course for a = 0 and a = 1 the equations are identically fulfilled so that the ove equation makes sense only for a 2 Note that w 0 = w 1 = 0 and u 0 i =0

6 226 HENNING STRUCHTRUP 26 Vector moments For vectors the general equation (27) reduces to Du a i Dv k Dt +2aua 1 ik Dt + 4a 5 ua i 2a +3 + u a Dv i 3 Dt + ua ik a 3 ua k + u a v i k + u a i u a+1 = 1 ε Pa i +2au a 1 ikl x l We introduce the quantities w a in the vector equation and replace the time derivatives of the velocity y means of the momentum alance (210) to otain (212) Du a i Dt a (2a + 3)!! + ρθ a θ 3 2a +3 w a θ ln ρ w a+1 3 2a ua 1 ik ρ 2a +3 w a θ 3 σ kl x l 2au a 1 θ ik 2au a 1 ik 2a +3w a σ ik 3 ρ θ ln ρ (2a + 3)!! θ a σ ik + ua ik 3 +2au a 1 2a +5 ikl + u a 2a +5 i + u a v i k + 2a x l ua k = 1 ε Pa i This equation is relevant for a 1 Note that w 1 = 0 and u 0 i =0 27 Rank-2 tensor moments After replacing the time derivatives of velocity y (210) the equations for tensors of rank 2 read (213) Du a ij Dt 2a ua 1 ijk ρ ( σkl x l + ua ijk + 2 u a+1 i 5 + θ ρ + ρ θ ) 2 ( 5 (2a +5)ua i σj k + θ ρ + ρ θ ) ρ +2au a 1 ijkl + 6a x l 7 ua ij v j + 4a 5 ua k i +2u a k i + u a ij (2a +5)wa (2a + v 5)!!ρθa+1 i = 1 ε Pa ij This equation is relevant for a 0 Note that all terms in this equation are trace-free and symmetric in the index pair ij 28 General equation For moments of order higher than two the general equation reads Du a ( i 1 i n 2a ua 1 i 1 i nk σkl + θ ρ + ρ θ ) Dt ρ x l n ( 2n +1 (2a +2n +1)ua i 1 i n 1 σin k + θ ρ + ρ θ ) + ua i 1 i nk ρ n n + n u a+1 i 1 i n 1 +2au a 1 i 2n +1 x 1 i nkl in n +2a 2n +1 ua k i 1 i n 1 + nu a v in k i 1 i n 1 + n (n 1) 4n 2 1 n v in 1 (2a +2n +1)ua+1 i 1 i n 2 n +2a n +1 x l 2n +3 ua i 1 i n + u a i 1 i n = 1 ε Pa i 1 i n This equation is relevant for a 0 Note that all terms in this equation are trace-free and symmetric in the indices i 1 i n

7 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW Moments of the collision term The moments of the collision term of the Boltzmann equation (22) are defined as (31) = m C 2a C i1 C i2 C in S (f) dc P a i 1 i n For the method used susequently it is necessary to express the Pi a 1 i n through the moments Since the collision term depends explicitly on the phase density this is tantamount to having a relation etween the moments and the phase density Only in few special cases can the collisional moments e computed without explicit knowledge of the phase density In particular this is the case for BGK models 21 and Maxwell molecules For these models the method was discussed in detail in 22 For general interaction potentials the situation is more difficult and we suggest the Reinecke Kremer Grad method Essentially this method uses the Grad method to compute the phase density as a function of the moments and the microscopic velocity c i That phase density is then used to compute the production terms (31) One prolem that arises here is that this method requires the restriction to a finite numer of moments The ansatz for the Grad phase density reads f G = f M (1+Φ)=f M 1+ A n (32) λ a i 1 i n C 2a C i1 C i2 C in n a=0 where λ a i 1 i n are expansion coefficients and A n denotes the numers of moments of tensorial rank n that are taken into account in the approach The A n introduce some degree of freedom here and we shall later give an argument on how they must e determined to ensure sufficient accuracy For the informed reader we mention already here that the numers A n correspond to the numer of Sonine polynomials that is stanrd in the Chapman Enskog expansion; see eg 3 The expansion coefficients follow from plugging the Grad phase density (32) into the definition of the moments (211) (25) u j 1 j r where w = m = m A 0 C 2a (f G f M ) dc = ρ a=0 A r C 2a C j1 C j2 C jr f G dc = r!ρ B (r) a++r = k=r+1 θ a+ B (0) a λa ( =0A 0 ) a=0 (2k +1) (a 0A r ) θ a++r B (r) a λa j 1 j r ( =0A 0 ) The relations etween moments and the expansion coefficients are linear and we find y inversion (33) λ a j 1 j r = B (r) 1 ū j 1 j r r!ρθ a++r with ū a = w a ū a i 1 i n = u a i 1 i n so that the Grad phase density is explicitly given calculations we refer the reader to 24 For more details on the ove

8 228 HENNING STRUCHTRUP P a i 1 i n With the Grad phase density the collision moments are given y ) = m C 2a C i1 C i2 C in (f G f G1 f Gf G1 σg sin ΘdΘdεdc 1 dc ( ) = m C 2a C i 1 C i 2 C i n C2a C i1 C i2 C in f G f G1 σg sin ΘdΘdεdc 1 dc We use f G = f M (1 + Φ) to otain P a i 1 i n = m (C 2a C i1 C i 2 C i n ) (Φ 1 +Φ Φ Φ 1 +Φ Φ 1 ΦΦ 1 ) = m f M f M1 σg sin ΘdΘdεdc 1 dc ( ) C 2a C i 1 C i 2 C i n C2a C i1 C i2 C in (Φ 1 +Φ+ΦΦ 1 ) f M f M1 σg sin ΘdΘdεdc 1 dc Without evaluating the integral further we see from the equations ove that P a i 1 i n is at most quadratic in the moments and thus must e of the general form (34) P a i 1 i n = C (n) ū i 1 i n τθ a rm c ū Y nrm j 1 j r i 1 i m ū c i m+1i m+2 i n j 1 j r τρθ +c+r a where the matrices C (n) the mean free time τ as Ynrm 1 τ = m ρ contain pure numers Aove we have introduced f M f M1 σg sin ΘdΘdεdc 1 dc In the present paper we shall not compute the matrices C (n) Ynrm which depend on the particularities of the molecular interaction as they are reflected in the collisional cross section σ Oviously their computation is cumersome and one will try to compute only those matrices that are necessary indeed One of the goals in the following will e to give arguments in order to determine which matrices must e considered for a theory of second order For now we shall just assume that the matrices C (n) are invertile which is the case as long as the numers A n remain finite 4 The order of magnitude of moments We shall now assign orders of magnitude to the moments and then construct new sets of moments such that at each order of magnitude we have the minimal numer of variles We ase the discussion on a Chapman Enskog like expansion of the moments with ε as the smallness parameter All moments are expanded according to u a i 1 i n = β=0 ε β u a i 1 i n β = ε0 u a i 1 i n 0 + ε1 u a i 1 i n 1 + ε2 u a i 1 i n 2 + ε3 u a i 1 i n 3 + and a similar series for the w a We shall say that u a i 1 i n is of leading order λ if u a i 1 i n β = 0 for all β<λ We emphasize that we are not interested in computing the expansion coefficients u a i 1 i n β ut only in finding the leading order of the moments

9 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW Zeroth and first order expansion For the evaluation of the order of magnitude it is important to note that the production terms are multiplied y the factor 1 ε If the ove expansion is inserted into the moment equations for the nonconserved quantities it ecomes immediately clear that w a 0 = ua i 1 i n 0 =0 for all moments This follows from alancing the factors of 1 ε on oth sides of the equations there are none of these on the left and thus we have the ove result In other words all quantities that are not conserved are at least of first order In the next step we alance the factors of ε 0 in the equations and find (41) 0= a (2a + 3)!! ρθ a θ = (2a + v 5)!!ρθa+1 i = 0= C (0) w 1 τθ a u i 1 τθ a u ij 1 τθ a C (n) u i 1 i n 1 τθ a It follows that the leading order of vectors and rank-2 tensors u a i and u a ij is the first order while the nonequilirium parts of the scalar moments w a and the higher moments u a i 1 i n (n 3) are at least of second order Note that the nonlinear terms in the productions do not contriute at the zeroth and the first order 42 Second order Next we have a look at the second order quantities Since the vectors and rank-2 tensors are already known to e of first order in ε we have to consider only the other moments We make the equations for tensors of rank 3 and 4 explicit Keeping only factors of ε 1 in the equations (note that eg u a i is a O (ε) contriution) yields 2a 3 (2a + q k 1)!!θa 1 2a 3 (2a + 1)!!θa 1 σ kl +2au a 1 kl x l θ 2au a 1 k 2au a 1 k θ ln ρ + ua k x l = C (0) w 2 τθ a c Y 010 u j uc j τρθ +c+1 a c Y 020 u jk uc jk τρθ +c+2 a so that the w a are of second order For the moments of rank 3 and 4 we otain 3 7 (2a +7)ua ij θ ln ρ + θ u a+1 ij (2a +7)ua+1 i v j = C (3) u ijk 2 τθ a c Y 302 u ij uc k τρθ +c a

10 230 HENNING STRUCHTRUP (2a +9)ua+1 ij = x l C (4) u ijkl 2 τθ a c Y 402 u ij uc kl τρθ +c a so that the u ijk and the u ijkl are also of second order For all higher moments we find 0= C (n) u i 1 i n 2 n 5 a τθ It follows that the nonequilirium parts of the scalar moments w a and the tensors of rank 3 and 4 are second order quantities All higher moments are at least of third order We will not go further ut it is evident that tensors of rank 5 and 6 are third order tensors of rank 7 and 8 are fourth order etc 43 Minimal numer of moments of order O (ε) From our first order result for the scalar and 2-tensors (41) we see that the first order terms u i 1 and u ij 1 are related to the gradients of temperature and velocity respectively and thus they are linearly dependent We otain u i 1 = κ τρθ θ u ij 1 x = µ τρθ 1+ v i (42) i where κ and µ are pure numers given y (43) κ = a=1 a 1 a (2a + 3)!! µ 3 = a=0 a 1 2 (2a + 5)!! 15 The first few values of these coefficients for Maxwell molecules (MM) and the BGK model are 22 (44) (45) MM: κ 1 =15/2 κ 2 = 105 and µ 0 =2 µ 1 =14 BGK: κ 1 =5 κ 2 =70 and µ 0 =2 µ 1 =14 In particular the pressure deviator and heat flux are given to first order as (46) q i 1 = 1 2 u1 i 1 = 1 2 κ 1τρθ θ = κ θ σ ij 1 = u 0 ij 1 = µ 0τρθ v i = 2µ v i where we have introduced heat conductivity and viscosity as κ and µ respectively Thus in first order we otain the laws of Fourier and Navier Stokes Note that the computation of µ and κ involves the inverses of the matrices C(2) It follows from (42) (46) that we can write u i 1 = 2κ θ 1 q i 1 u ij 1 κ = µ θ σ ij 1 1 µ 0 While these equations relate the first order contriutions of the vector and 2-tensor moments it is straightforward to introduce new moments wi awa ij that are of second order only (47) w a i = u a i 2κ a κ 1 θ a 1 q i (a 2) and w a ij = u a ij µ a µ 0 θ a σ ij (a 1)

11 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW 231 so that ( wi a = m C 2a 2 2κ a ( wij a = m C 2a 2 µ a θ a µ 0 ) θ a 1 κ 1 ) C 2 C i fdc (a 2) C i C j fdc (a 1) This means that we can formulate a set of moments where only σ ij and q i are of first order while all other moments are at least of second order (excluding the conserved moments of course) It is in principle possile to go to higher order with this: the second order terms of wi a (say) when expanded will e linearly dependent and again one can use this to otain a minimal set of moments of second order while the remaining ones can e constructed to e of third order etc This is not necessary for the levels of accuracy that are important in this paper and so we shall not pursue this idea further 5 The transport equations with second order accuracy In the previous section we have estlished the order of magnitude of the various moments up to O ( ε 2) Now we ask what equations we need in order to descrie a flow process in a rarefied ideal gas with an accuracy of O ( ε λ) In this section we shall use the smallness parameter ε in a slightly different manner namely as an indicator for the leading order of a quantity Thus in any equation we shall replace u a i 1 i n y ε β u a i 1 i n when β denotes the leading order of u a i 1 i n This will allow for a proper ookkeeping of the order of magnitude of all terms in an equation 51 The conservation laws and the definition of λth order accuracy We start the argument y repeating the conservation laws for mass momentum and energy (28) (210) (29) which read when we assign the factor ε to the first order quantities σ ij and q i (51) 3 2 ρdθ Dt + ρθ + ε Dρ Dt + ρ qk + σ kl ρ Dv i Dt + ρ θ + θ ρ + ε =0 =0 x l =0 σik These equations are not a closed set of equations for ρ v i T ut they contain pressure deviator σ ij and heat flux q i as additional quantities and equations for these are required to otain a closed set of equations We shall speak of a theory of λth order accuracy when oth σ ij and q i are known within the order O ( ε λ) The equations of order O ( ε 0) result from (51) y setting σ ij = q i = 0 that is y ignoring the terms with the factor ε in the alance laws This yields the Euler equations for ideal gases Dρ Dt + ρ =0 3 2 ρdθ Dt + ρθ =0 ρ Dv i Dt + θ ρ + ρ θ =0 For higher order accuracy ie first order and higher we shall need the moment equations for the pressure deviator and heat flux

12 232 HENNING STRUCHTRUP 52 Collision moments for vectors and tensors For these we have to consider the production terms for vectors and 2-tensors up to first order in ε For vectors we otain from (34) y setting n = 1 and making the first two terms of the r-summation explicit 1 ε Pa i = 1 ε 1 ε c Y 111 εu i τθ a 1 ε c Y 101 εu i ε2 w c τρθ +c a εu ij εuc j τρθ +c+1 a 1 ε r=2 c Y 1r1 ε 2r 1 ū ij 1 j r ū c j 1 j r τρθ +c+r a We shall only e interested in terms up to order O (ε) where the ove equation reduces to 1 ε Pa i = 2κ q i κ 1 τθ 1 a ε wi τθ a + Y 111 2µ κ c σ ij q j µ c 0 κ 1 τρθ 2 a For 2-tensors we otain in the same manner 1 ε Pa ik = ε µ σ ik µ 0 τθ a w ik τθ a + c Y 201 4κ κ c q i q k κ 1 κ 1 τρθ 2 a + c Y 211 µ µ c µ 0 µ 0 σ j i σ k j τρθ 1 a 53 Equations for the pressure deviator and heat flux We consider (214) with a = 0 where we introduce the moments (47) and otain after assigning the proper order of magnitude to the various terms (52) ε Dσ ij + 4 Dt 5 + c q i +2σ k i v j + σ ij Y 211 µ µ c 0c σ j i σ k j µ 0 µ 0 τρθ Here we have used that 0 µ = =0 a= ε 2 u 0 ijk + a 1 2 wik 0 τθ + c 15 (2a + 5)!! = a=0 Y 201 4κ κ c q i q k 0c κ 1 κ 1 τρθ 2 σij = ρθ µ +2 v i 2 δ 0a (2a + 5)!! = 2 15 and 1 τµ 0 = ρθ 2µ where µ is the viscosity The last two equations follow directly from (43) and (46) The corresponding equation for the heat flux results from setting a = 1 in (212) where we introduce the second order moments (47) and assign the proper order of

13 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW 233 magnitude to otain (rememer that w 1 =0) (53) ε Dq i 1 Dt + µ 1 θ 1 σ ik σ ik θ ln ρ 1 µ θ σ ik 2 µ 0 2 µ q i q v i k q k ε 2 1 wij w Here we have used that 1 κ = =1 a= u 0 ikl σ ik σ kl + x l ρ x l a w i 1 a (2a + 3)!! = 3 a=1 τθ 1 + c Y 111 µ κ c 1c σ ij q j µ 0 κ 1 τρθ = 5 2 ρθ qi κ + θ δ 1a a (2a + 3)!! 3 = 5!! 3 =5 and 1 = 1 1 τκ 1 2 κ ρθ where κ is the heat conductivity The last two equations follow directly from (43) and (46) We close this section y pointing out that µ and κ are O (ε) as are σ ij and q i so that their respective ratios σij µ qi κ are O ( ε 0) Also τ is O (ε) and wi w ij are O ( ε 2) so that their respective ratio is O (ε) 54 First order accuracy: NSF equations We recall that our goal is to provide the equations for pressure deviator σ ij and heat flux q i within an accuracy of a given order If we are satisfied with first order accuracy we need to consider only the leading terms in (52) (53) those of O ( ε 0) which yields the laws of Navier Stokes and Fourier σ ij = 2µ v i q i = κ θ where viscosity µ and heat conductivity κ are given y (43) (46) The equations of first order accuracy otained here coincide with the first order Chapman Enskog expansion; see 17 where viscosity and heat conductivity are computed in accornce with the ove formulae It is worthwhile to have a look at the corresponding phase density At this order the only relevant moments are the vectors and 2-tensors so that the phase density (32) (33) reduces to f G = f M 1+ A 1 a=0 B (1) 1 u i ρθ a++1 C2a C i + A 2 a=0 B (2) But to first order we have from (42) (46) u i 1 = 2κ κ θ ln θ u ij 1 κ 1 x = 2µ µ θ v i i µ 0 1 u ij 2ρθ a++2 C2a C i C j

14 234 HENNING STRUCHTRUP so that we can write the distriution function for the NSF case as f NSF = f M 1 2κ A 1 1 B (1) κ C 2a C i ln θ ρθ 1/2 κ 1 θ a a=0 θ µ A 2 1 B (2) µ C 2a C ic j v i ρθ µ 0 θ a θ a=0 The terms in square rackets are polynomials in ξ 2 = C 2 /θ and can e rewritten in terms of Sonine polynomials; see eg 25 Thus it is evident that the present method yields the same phase density as the classical first order Chapman Enskog method Heat conductivity κ and viscosity µ are given y A 1 κ = τρθ a=1 1a 1 a (2a + 3)!! A 2 µ = τρθ 6 a=0 0a 1 (2a + 5)!! and thus depend on the numers A 1 A 2 of moments that are considered This gives a good criterion for the determination of these numers: increasing A 1 A 2 gives successive approximations for κ µ and these will converge to the accurate value if A 1 A 2 go to infinity One will choose these numers such that an increase y one will lead only to a sufficiently small change in the values for κ µ This of course corresponds to the growing accuracy when higher order Sonine polynomials are taken into account in the Chapman Enskog method 1 Indeed Reinecke and Kremer computed κ µ essentially in this manner 17 and showed that a fourth order correction is already very accurate Moreover they showed that their values for κ µ to order A = A 1 = A 2 agree with the corresponding values of the Ath degree of approximation in Sonine polynomials as given eg in 1 The difference etween our approach and the one of Reinecke and Kremer lies in the fact that Reinecke and Kremer chose the moments to ase their approach on intuitively (they consider w a u a i and ua ij ) while we reduced the full moment system (w a u a i 1 i n ) y discussion of the order of magnitude of moments and the order of accuracy of equations In particular this makes clear that at first order the scalar moments w a need not e considered Moreover Reinecke and Kremer perform a Maxwell iteration on the full moment equations a method which essentially is equivalent to the Chapman Enskog expansion of the moments 19 For a second order theory they perform the second order Maxwell iteration in 18 which yields the Burnett equations just as in the stanrd Chapman Enskog expansion These as is well known are unstle 5 and therefore should not e used Our approach does not yield the Burnett equations at second order as will ecome clear in the next section 55 Second order accuracy: 13 moment theory In the next order we have to consider all terms in (52) (53) which have the factors ε 1 and ε 0 to otain 15 (54) Dσ ij Dt q i +2σ k i v j + σ ij wik 0 τθ + c Y 201 4κ κ c q i q k 0c κ 1 κ 1 τρθ 2 + c Y 211 µ µ c σ j i σ k j 0c µ 0 µ 0 τρθ σij = ρθ µ +2 v i

15 (55) Dq i Dt + 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW µ µ 0 σ ik θ θσ ik ln ρ + 1 µ µ 0 2 θ σ ik q i q v i k q k + 1 w i 2τθ 1 + c Y 111 µ κ c σ ij q j 1c µ 0 κ 1 τρθ = 5 2 ρθ qi κ + θ Before we proceed we compare the ove set with Grad s famous 13 moment theory In Grad s equations those terms in (54) (55) that are underlined do not appear Moreover our equations contain the general expression µ 1 µ which depends on 0 the interaction potential (through ) while in Grad s equations this value is set to µ 1 µ = 7 Finally Grad s equations contain a term σ ik σ kl 0 ρ x l in the equation for heat flux which does not appear here since it is of order O ( ε 2) It was already shown in 22 that µ 1 µ = 7 for the BGK model and Maxwell 0 molecules where (45) (44) hold In addition the underlined terms do not appear for Maxwell molecules or for the BGK model Thus with the omission of the σ kl x l second order term σ ik ρ Grad s 13 moment equations are the proper equations of second order accuracy for the description of rarefied gas flows for Maxwell molecules and the BGK model only For other interaction models however the ove equations must e used in order to give second order agreement with the Boltzmann equation Together with the conservation laws (51) equations (54) (55) form a set of 13 equations for the 13 variles ρ θ v i σ ij q i This system is not closed since it contains the moments wi awa ij as well and additional equations are required for these quantities The wa i wa ij are of order O ( ε 2) and for the theory of second order accuracy that we are considering here it will e sufficient to know only the leading terms for these quantities The calculations and results for this can e found in Appendices B and C; see (B1) (C2) The final equations for the pressure deviator and heat flux read (56) Dσ ij Dt ( q i v j + ψ 1 +(2+ψ x 2 ) σ k i + ψ j x 2 σ k i ) k 3 ψ 2 σ ij ψ 3 q i ln ρ ln θ q i q k σ j i σ k j ψ 4 q i + ψ x 5 + ψ j µθ 6 µ (57) Dq i Dt + χ θ ln ρ 1σ ik χ x 2 θσ ik + χ k x 3 θ σ ik + 5 k 3 +(1+χ 4 ) q k v i + χ 4 q k + χ 5 σ ij q j µ = χ 6ρθ σij = ψ 7 ρθ µ +2 v i ( χ 4 µ0 q i κ 1 µ + θ ) q i In the ove equations the coefficients ψ i and χ i are pure numers that must e computed according to (58) ψ 1 = 4 1 0d 5 E 1 1 κ a+1µ 0 κ 1 µ da a ψ 2 = da 0d E 1 4a 7

16 236 HENNING STRUCHTRUP ψ 3 = 4 5 ψ 4 = 4 5 ψ 5 = µ 0 2 ψ 6 = µ 0 2 da da 0d E 1 (2a +5)κ aµ 0 κ 1 µ a c c ψ 7 =1 da 0d E 1 (2a +5) κ aµ 0 a κ a+1µ 0 κ 1 µ a κ 1 µ a 4κ κ c Y 201 0c κ 1 κ + 0d E 1 4κ κ c µ0 Y κ 1 κ 1 µ Y201 0c da c a Y 211 µ µ c 0c 0d E 1 + µ 0 µ 0 da 1 µ 0 (2a + 5)!! µ a 15 0d E 1 µ µ c µ0 Y 211 µ 0 µ 0 µ Y211 0c a c and (59) χ 1 = 1 µ 1 1 1d 2 µ D 1 0 da χ 2 =1 da 1d D 1 1 µ 1 1 a κ 1 2 µ 0 κ a 1 a κ 1µ a 1 κ a µ 0 χ 3 = 1 µ µ 0 2 1d D 1 da χ 4 = 2 5 da χ 5 = µ 0 2 c χ 6 = 5 2 da 1d D 1 Y 111 µ κ c 1c 5 1d D a 5 µ 0 κ 1 + da 1 µ µ 0 2 κ 1 κ a 1d D 1 a (2a + 3)!! 6 c κ 1 κ a 1 µ a µ a 1 2 µ 0 µ 0 1 µ a 2 µ 0 µ κ c µ 0 κ 1 (2a + 3)!! 6 κ1 Y 111 κ Y111 1c a The coefficients κ µ are given y κ = 1 a (2a + 3)!! a µ 3 = (510) a=1 a=0 and we have introduced the matrices D = κ 1 (511) 1 C(1) κ a a E = 0 µ 0 µ a 1 2 (2a + 5)!! 15 Equations (56) (511) give the complete set of equations for σ ij and q i Thus we have reduced the prolem to the computation of the matrices ad Y201 Y211 C(1) ad Y111 These follow y using the reduced Grad function (32) A 1 A 2 f G = f M (1+Φ)=f M 1+ λ a i C 2a C i + λ a klc 2a C k C l a=0 a=0

17 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW 237 to compute the collision moments Pi a Pa ij as outlined in section 3 We consider that to e a separate task that will e discussed in a future paper The numers A 1 A 2 must e chosen such that their further increase does not change the values of the coefficients ψ α χ α considerly For Maxwell molecules and the BGK model the coefficients have the values ψ 1 = 4 5 ψ 2 = ψ 3 = ψ 4 = ψ 5 = ψ 6 =0 ψ 7 =1 χ 1 = 5 2 χ 2 =1 χ 3 =1 χ 4 = 2 5 χ 5 =0 χ 6 = 5 2 Furthermore κ1 µ = 15 κ1 0 4 for Maxwell molecules and µ = for the BGK model This choice of coefficients results in the original Grad s 13 moment equations Conclusions From the treatment ove it follows that the conservation laws (51) together with the alance laws (56) (57) form the proper transport equations for rarefied monatomic gases with second order accuracy in the Knudsen numer The coefficients in the alance equations (56) (57) depend on the microscopic interaction potential etween the gas atoms and as was shown already in 22 the well-known 13 moment equations of Grad are of second order accuracy only for Maxwell molecules and the BGK model Indeed a similar ehavior is oserved in the Chapman Enskog method where Burnett coefficients depend on the interaction potential Both the Chapman Enskog method and the method used here ecome simplest when Maxwell molecules or the BGK model are considered In 22 we were le to use our method to derive field equations up to third order accuracy which agree with the regularization of Grad s 13 moment equations proposed efore y Struchtrup and Torrilhon y means of a different argument Going to third order for aritrary interaction models will e more cumersome y far In particular the third order theory will include full alance laws for the minimum numer of vectors wi a and tensors wij a that are of second order and thus the numer of variles will e higher than 13 In the case of Maxwell molecules it is well known that Grad s 13 moment equations are stle with respect to spatial and temporal disturances while the Burnett equations are unstle This points to prolems within the Chapman Enskog method which proposes the Burnett equations as those equations that have second order accuracy Our new method however proposes the 13 moment equations and thus stle equations Whether the 13 moment equations for aritrary interaction potentials that were derived ove are unconditionally stle or not remains to e seen This can only e tested in the future after the coefficients are computed This task is planned for the future and corresponding results will e presented elsewhere Appendix A Symmetric and trace-free tensors A1 Symmetry A tensor S i1i 2 i n of rank n is called symmetric if for each pair of indices i j i k S i1i 2 i j i k i n = S i1i 2 i k i j i n A nonsymmetric tensor A i1i 2i n can e symmetrized y the rule A (i1i 2 i n) = 1 ( Ai1i n! 2 i n 1i n + A i1i 2 i ni + all permutations of indices) n 1

18 238 HENNING STRUCHTRUP Here the indices in round rackets denote the symmetric part of the tensor example for a 2-tensor one finds the symmetric part as For A (ij) = 1 2 (A ij + A ji ) A2 Trace-free tensors A tensor S i1i 2 i n irreducile) if for each pair of indices i j i k of rank n is called trace-free (or (A1) S i1i 2 i j i k i n δ iji k =0 The trace-free part of a symmetric tensor A i1i 2 i n rule 23 can e otained according to the where A i1i 2i n = n 2 k=0 a nk δ (i1i 2 δ i2k 1 i 2k A i2k+1 i n)j 1j k j 1j k a nk =( 1) k n!(2n 2k 1)!! (n 2k)! (2n 1)!! (2k)!! n n 2 n even = 2 n 1 2 n odd n!! = n (n 2) (2 or 1) = n 1 2 j=0 (n 2j) We emphasize that in the ove equations A i1i 2 i n is symmetric which is not made explicit y means of round rackets in order to avoid overly complicated notation For actual computations it is more convenient to use A i1i 2i n = A i1i 2 i n + α n1 (δ i1i 2 A i3 i nkk + permutations) + α n2 (δ i1i 2 δ i3i 4 A i5 i nkkll + permutations) + where only those permutations must e considered that are really different ie that cannot e related y means of symmetry properties Then the coefficients are given as Simple examples are A i = A i α nk = k 1 j=0 A ij = A (ij) 1 3 A kkδ ij ( 1) k (2n 2j 1) A ijk = A (ijk) 1 ( ) A(ill) δ jk + A (jll) δ ik + A (kll) δ ij 5

19 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW 239 As an instructive example let us consider the gradient of a symmetric 2-tensor Aij Its symmetric part is given y ( Aij A (ij = 1 ) 3 and its trace-free and symmetric part is A ij = 1 3 ( Aij A ik x j ( Air x r ( Akr + A jk + A ir x r ) + A rr + A ik x j + A ) jk ) ( Ajr δ jk + x r ) δ ij + A jr x r + A ) rr δ ik x j + + A kr + A rr x r x r Oviously the racket notation allows for a highly condensed notation Note that the moments u a i 1 i n that are used in this paper are trace-free y definition u a i 1 i n = u a i 1 i n and this is not made explicit with rackets This avoids confusing notation involving several pairs of rackets As an example we consider the gradient of u ij made symmetric and trace-free only in two of its three indices: u i j = 1 u ij + 1 u ik 1 u ir δ jk 2 2 x j 3 x r For this quantity we would have to write u i j if we were to make the trace-free properties of the moments explicit The trace-free part of the unit matrix δ ij vanishes δ ij = 0 and this implies A i1i 2i n δ jk =0 A3 Spherical harmonics Trace-free linear cominations of unit vectors n i1 n in are related to spherical harmonics 25 Here the vectors n i are unit vectors in spherical coordinates n i = {sin ϑ sin ϕ sin ϑ cos ϕ cos ϑ} i Spherical harmonics form an orthogonal set of functions in the sense that integration over the solid angle dω = sin ϑdϑdϕ yields 0 n m n i1 n in n j1 n jm dω = 4π n δ i1 i n j 1 j n n = m (2j+1) j=0 where δ k1 k l is a generalized unit tensor defined as ( ) l! δ k1 k l = δ k1k 2 δ kl 1 k l + + elements ( 2)!2 l 2 l An important relation following from this definition is that the product of the ove integral with a tensor A i1i 2i n yields 0 n m A i1i 2i n n i1 n in n j1 n jm dω = 4πn! n A j1 j n n = m (2j+1) j=0

20 240 HENNING STRUCHTRUP Another important relation is (A2) n i1 n in n k = n i1 n in n k + n 2n +1 n i 1 n in 1 δ in k By introducing the direction vector of the microscopic velocity as n i = C i /C the moments (25) can e written as = m C 2a+n+2 n i1 n i2 n in fdcdω u a i 1 i n This indicates that our definition of moments refers to an expansion of the phase density in spherical harmonics n i1 n i2 n in and polynomials in the solute value of the microscopic velocity C Appendix B Moment equations for wi a We consider the moment equation for u a i (212) After introducing the wa i wa ij y means of (47) replacing the time derivative of θ y means of the energy alance (51) 2 and cancelling terms of O ( ε 2) and higher we otain after division y 2κa κ 1 µa 2µ a 1 Dq i Dt + a κ 1 2 κ a µa + κ 1 2κ a µ τ µ 0 κ 1 κ a (2a + 3)!! 3 w i θ τ σ ik θ θ σ ik c θ a 1 a κ 1µ a 1 ln ρ θσ ik κ a µ a + q i + 15 Y 111 µ κ c σ ij q j = µ 0 κ a ρθ 2a +5 v i q k 5 a (2a + 3)!! 6 + 2a 5 q k κ 1 θ ρθ + q i κ a κ From this equation we sutract the alance of the heat flux (55) to otain 1 κ 1 w i 1 2 C(1) κ a τθ 1 = 1+ a κ 1 µ a 2µ a 1 1 µ 1 θ σ ik 2 κ a µ 0 2 µ a κ 1µ a 1 ln ρ 5 θσ ik + κ a µ κ 1 µa (2a + 3)!! 1 µ 1 θ σ ik 2κ a µ µ a 15 q i + 2a 2 v i q k + 2a 2 q k Y 111 µ κ c Y 111 µ κ c σij q j 1c µ c 0 κ a µ 0 κ 1 τρθ ρθ qi κ + θ 5 a (2a + 3)!! 2 6 The ove equation is meaningful for a 2 only The wi follow from inversion of the matrix D = κ 1 1 C(1) as (B1) w d i 2τθ d 1 = a + a D 1 µ 1 κ 1 κ a a µ a 2µ a 1 2 µ 0 2 κ a µ 0 D 1 1 a κ 1µ a 1 ln ρ θσ ik κ a µ 0 σ ik θ κ 1 κ a

21 13 MOMENT EQUATIONS FOR RAREFIED GAS FLOW a + a + a a µ 1 + κ 1 µa µ 0 2κ a µ 0 4 4a 15 q i + 2a 2 v i q k + 2a Y 111 µ κ c Y 111 µ κ c σij q j 1c µ 0 κ a µ 0 κ 1 τρθ 5 D D 1 D 1 D 1 c 5 2 a (2a + 3)!! 6 κ 1 κ a ρθ (2a + 3)!! θ σ ik 3 qi κ + θ q k Appendix C Moment equations for wij a We consider the equation for the 2-tensors u a ij (214) After introducing the wa i wa ij y means of (47) replacing the time derivative of θ y means of the energy alance (51) 2 and cancelling terms of O ( ε 2) and higher we otain after division y µ a µ θ a 0 Dσ ij Dt + 4 κ a+1 µ 0 q i + 6a 5 κ 1 µ a 7 σ ij 4 5 (2a +5)κ aµ 0 ln ρ q i 4 κ 1 µ a 5 + µ 0 1 µ a θ a τ We also have 6a 7 σ ij θa w ik + c + 4a 5 σ k i 2 3 aσ v j ij +2σ k i + σ ij (2a +5) κ aµ 0 a κ a+1µ 0 ln θ q i κ 1 µ a κ 1 µ a Y 201 4κ κ c µ 0 q i q k κ 1 κ 1 µ a τρθ 2 + c σ ij + 4a 5 σ k i 2 3 aσ ij = 4a 7 Y 211 µ µ c σ j i σ k j µ 0 µ a τρθ = µ 0 (2a + 5)!! ρθ µ a 15 v j σik µ +2 v i = 1 3 σ ij σ k i 4 15 σ v r r i ( v j σ k i + σ k i 2 3 σ ij From this we sutract the alance for the pressure deviator (54) to otain after some rearrangements 0 µ 0 + 4a 7 w ik τθ = 4 κa+1 µ κ 1 µ a µ a ( v j σ k i + σ k i 2 ) 3 σ ij 4 5 (2a +5)κ aµ 0 ln ρ q i 4 κ 1 µ a 5 + Y 201 4κ κ c µ 0 Y 201 4κ κ c 0c κ 1 κ 1 µ c a κ 1 κ 1 + Y 211 µ µ c Y 211 µ µ c 0c µ c 0 µ a µ 0 µ 0 q i (2a +5) κ aµ 0 q i q k τρθ 2 σj i σ k j τρθ a κ a+1µ 0 κ 1 µ a κ 1 µ a q i ln θ ) 1 µ 0 (2a + 5)!! σij ρθ µ a 15 µ +2 v i

22 242 HENNING STRUCHTRUP The w ij as follow from inversion of the matrix E = 0 µ 0 µ a (C1) wik d τθ d = E 1 4 κa+1 µ 0 q i κ a 1 µ a a E (2a +5)κ aµ 0 ln ρ q i κ a 1 µ a a + a + a a E 1 E 1 E 1 c c Y 201 Y µ 0 (2a + 5)!! µ a 15 E 1 4a 7 E κ κ c µ 0 Y 201 4κ κ c q i q k 0c κ 1 κ 1 µ a κ 1 κ 1 τρθ 2 µ µ c Y 211 µ µ c σj i σ k j 0c µ 0 µ a µ 0 µ 0 τρθ ρθ σij µ +2 v i ( σ k i v j + σ k i 2 ) 3 σ ij ln θ q i (2a +5) κ aµ 0 a κ a+1µ 0 κ 1 µ a κ 1 µ a Acknowledgment The ideas outlined in this paper were conceived while the author attended the Workshop Invariance and Model Reduction for Multiscale Phenomena ETH Zurich 2003 REFERENCES 1 S Chapman and T G Cowling The Mathematical Theory of Non-uniform Gases Camridge University Press Camridge UK C Cercignani Theory and Application of the Boltzmann Equation Scottish Academic Press Edinurgh Scotland J H Ferziger and H G Kaper Mathematical Theory of Transport Processes in Gases North Holland Amsterm M Sh Shavaliyev Super-Burnett corrections to the stress tensor and the heat flux in a gas of Maxwellian molecules J Appl Math Mech 57 (1993) pp A V Boylev The Chapman-Enskog and Grad methods for solving the Boltzmann equation Soviet Phys Dokl 27 (1982) pp H Struchtrup Failures of the Burnett and super-burnett equations in steady state processes Contin Mech Thermodyn to appear 7 X Zhong R W MacCormack and D R Chapman Stilization of the Burnett Equations and Applications to High-Altitude Hypersonic Flows AIAA X Zhong R W MacCormack and D R Chapman Stilization of the Burnett equations and applications to hypersonic flows AIAA J 31 (1993) pp S Jin and M Slemrod Regularization of the Burnett equations via relaxation J Statist Phys 103 (2001) pp S Jin L Pareschi and M Slemrod A relaxation scheme for solving the Boltzmann equation ased on the Chapman-Enskog expansion Acta Math Appl Sin Engl Ser 18 (2002) pp M Torrilhon and H Struchtrup Regularized 13-moment-equations: Shock structure calculations and comparison to Burnett models J Fluid Mech to appear 12 H Grad On the kinetic theory of rarefied gases Comm Pure Appl Math 2 (1949) pp H Grad Principles of the kinetic theory of gases in Handuch der Physik Vol 12 S Flügge ed Springer Berlin W Weiss Continuous shock structure in extended thermodynamics Phys Rev E (3) 52 (1995) pp

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