Shock wave structure for Generalized Burnett Equations

Transcription

1 Shock wave strctre for Generalized Brnett Eqations A.V. Bobylev, M. Bisi, M.P. Cassinari, G. Spiga Dept. of Mathematics, Karlstad University, SE Karlstad, Sweden, Dip. di Matematica, Università di Parma, Viale G.P. Usberti 53/A, I-434 Parma, Italy, Dip. di Matematica F. Enriqes, Università di Milano, Via Saldini 5, I-33 Milano, Italy o the memory of Carlo Cercignani Abstract Stationary shock wave soltions for Generalized Brnett Eqations (GBE) A. V. Bobylev, J. Stat. Phys. 3, 569 (8) are stdied. Based on reslts of M. Bisi, M. P. Cassinari, M. Groppi, Kinet. Relat. Models, 95 (8), we choose a niqe (optimal) form of GBE, and solve nmerically the shock wave problem for varios Mach nmbers. he reslts are compared with nmerical soltions of Navier Stokes eqations and with the Mott Smith approimation for the Boltzmann eqation (all calclations are done for Mawell molecles), since it is believed that the Mott Smith approimation yields better reslts for strong shocks. he comparison shows that GBE yield certain improvement of the Navier Stokes reslts for moderate Mach nmbers. Introdction We contine in this paper to stdy eqations of hydrodynamics (derived from the Boltzmann eqation) at the Brnett level. he well known instability of classical Brnett eqations became a starting point for several different methods to reglarize these eqations (see, 3, 4 and the references therein for a review). We se in this paper the approach of one of the athors based on the idea of small changes of variables. In other words, we consider the eqations not for tre hydrodynamic variables (density ρ tr, blk velocity tr, and temperatre tr ), bt for slightly modified qantities ρ = ρ tr + O(ε ), = tr + O(ε ), = tr + O(ε ), () for which the standard notations (ρ,, ) are sed. A small parameter ε denotes the Kndsen nmber, i.e. a mean free path divided by a typical macroscopic length. It was shown in 5 how this approach leads to the so called generalized Brnett eqations (GBE). hese eqations are not defined niqely, they depend on two free

2 parameters θ,. A possible choice of the parameters was discssed in 5, 6 on the basis of soltions to some linear problems. he athors of these papers presented several argments in favor of the choice θ =, θ =. herefore we consider the corresponding version of GBE (GBE (,,) in the notation of 5) in the present paper. Or aim is to stdy the shock wave profile for different Mach nmbers in order to see if GBE lead to certain improvement of the Navier Stokes approimation. We shall consider below the case of Mawell molecles, since this is the only model for which the classical Brnett eqations are known in flly eplicit form. We wold like to mention that Carlo Cercignani was always interested in the problem of the shock wave strctre (in particlar, the case of infinite Mach nmber). he papers 7, 8, 9, can be considered as a part of his contribtion to this problem. We regret that it is too late now to discss or reslts with him. he paper is organized as follows. he niqe form of GBE, sed in the paper, is discssed in detail in Section. he statement of the problem for shock wave soltions and transition to dimensionless variables is given in Section 3. he Mott Smith approimation for the Boltzmann eqation is eplained in Section 4. We se this approimation for comparison with or nmerical reslts in Section 5 becase of nmerical evidence that it works relatively good for strong shocks. Another reference model is for s the classical Navier Stokes eqations. he shock profiles for all three models are compared for different Mach nmbers in Section 5. Generalized Brnett Eqations First we define a niqe version of GBE which corresponds to vales of parameters (θ =, θ = ), in 5. Eqalities () for this version read ( ) ρ = ρ tr, = tr, = tr + ε R ρ div, () ρ where R = a( ) (log ρ) + b( ) (log ), (3) with coefficients a( ) and b( ) having for Mawell molecles the following form a( ) = 3 8 η, b( ) = 3 η, η = const. (4) he constant parameter η is given by eqality η = 3 π g(µ) ( µ ) dµ, g(cos θ) = V σ( V, θ), (5) where σ( V, θ) is the differential scattering cross section for Mawell molecles, V is the absolte vale of relative velocity of colliding particles, θ, π is the scattering angle.

3 Generalized Brnett Eqations for variables (ρ,, ) read 5 ρ t + div(ρ ) =, ρ ˆD α + p α + Π αβ β =, 3 ρ ˆD + p div() + Π αβ α β + div(q) =, α, β =,, 3, where the Einstein convention on repeated indices is sed and ˆD = t +, p = ρ, Π αβ = ε π NS αβ + ε (π B αβ δ αβ div Q α = ε q NS α + ε q B α + ε ρ ( R ρ { 3 R β α β + )), ( ) } 3 a + b (div()), α in the case of Mawell molecles. Here Ψ αβ = (Ψ αβ + Ψ βα 3 ) δ αβ Ψ γγ, (6) (7) and the pper indices N S and B denote Navier Stokes and Brnett terms respectively. In particlar, the Navier Stokes terms are π NS αβ = η α β, q NS α = 5 4 η α. (8) he Brnett terms π B αβ and qb α for Mawell molecles can be fond in books, 3, 4, and in the paper. For brevity we present below these and other terms for the simplest case of one dimensional flow: hen, Eqs. (6) read ρ = ρ(, t), = ((, t),, ), = (, t), R. (9) ρ t + (ρ ) =, ρ ( t + ) + p + Π =, () 3 ρ ( t + ) + p + Π + Q =, where lower indices denote partial derivatives, Π and Q denote respectively component of Π αβ and component of Q α. hese fnctions are given by Π = 4 ( ) R 3 η + π B, ρ Q = 5 4 η + q B + ( ) 3 R + ρ a + b, () R = a ρ ρ + b 3, a = a( ) = 8 η, b = b( ) = 3 η, 3

4 where π B = η 9 ρ q B = η 8 ρ 4 + 9( ) 6 ( ) p, ρ 95 6 ρ ρ 4 Note that the Kndsen nmber ε in Eqs. (7) is sed as a formal parameter and we set ε = in the final reslts. hs, we obtain Π = 4 3 η + η 37 3 ρ ρ ρ ρ ρ ρ ρ, (3) Q = 5 4 η + η 3 ρ ( 37 8 ρ 5 3 ρ. () ). (4) Eqs. (), with the constittive eqations (3), (4), are the final one dimensional version of Generalized Brnett Eqations (GBE), which we stdy in this paper. Eqations () for this version read ρ = ρ tr, = tr, tr = η ρ ( 3 ρ + 8 ρ 3 ) ρ where the parameter η is given in Eqs. (5). his parameter is important for comparison with soltions of the Boltzmann eqation. Note that Eqs. (), (3), (4) are simpler than the classical Brnett eqations since they contain only one third derivative ρ. he conservative form of Eqs. () reads (5) ρ t + (ρ ) =, ( ) (ρ ) t + ρ + ρ + Π =, ρ( + 3 ) + ρ ( + 5 ) + ( Π + Q) hese eqations will be sed below for stdy of shock waves. t =. (6) 3 Shock wave soltions Eqations (), (3), (4) are Galilei invariant. In other words, the transformation ρ(, t) = ρ(+ct, t), (, t) = c+ũ(+ct, t), (, t) = (+ct, t), (7) with any constant c R, leads to the same Eqs. (), (3), (4) for ρ(, t), ũ(, t), (, t). If these fnctions do not depend on time t, then ( ρ, ũ, ) are stationary soltions of Eqs. (), (3), (4). he transformation (7) defines in this case travelling waves, moving with constant velocity ( c). It is convenient for or goals to assme that c > and to introdce bondary conditions ρ( ) = ρ, ũ( ) = ũ = c, ( ) =, (8) 4

5 which correspond to npertrbed gas in front of the shock. hs, we consider stationary Eqs. (), (3), (4) for ( ρ, ũ, ) on the whole real line < < with bondary conditions (8). Eqations () can be presented in the conservative form (6). hen, we integrate Eqs. (6) and obtain, omitting all tildas, ρ = J, ρ( + ) + Π = J, ρ ( + 5 ) + ( Π + Q) = J 3, (9) where constant fles J,,3 are given by J = ρ c, J = ρ (c + ), J 3 = ρ c(c + 5 ) () and Π and Q are defined in Eqs. (3) and (4). Note that the speed of sond at (npertrbed gas) is eqal to c = (5 /3) /, therefore c = M 5 3, () where M denotes the Mach nmber. It is well known (at the formal level) that the shock wave soltion eists for M > with limiting vales ρ( ) = ρ +, ( ) = +, ( ) = +, () satisfying Rankine Hgoniot conditions ρ + + = J, ρ + ( ) = J, J ( ) = J 3. (3) hen we obtain (see Eqs. ()) ρ + = 4 ρ c c + 5, + = c c, + = (c + 5 )(3 c ) 6 c, (4) assming that c > (5 /3) /. he density ρ() can be eliminated from Eqs. (9) since hen Eqs. (9) can be written in the form ρ() = J () = ρ c (). (5) where now Π = 4 3 η + ( η 37 3 J Q = 5 4 η + η 3 J J ( + ) + (Π J ) =, J ( + 5 ) + ( Π + Q) J 3 =, ). ( ), (6) (7) 5

6 Eqs. (6) can be transformed to J ( + ) + (Π J ) =, J (3 ) + Q + J J 3 =, where Π and Q are given in Eqs. (7). It is convenient to pass to dimensionless variables. We denote ˆ = ρ + η +, () = + û(ˆ), () = + ˆ (ˆ), S = + +, (9) and omit hats below. hen Π = ρ S Q = ρ ( S ( ). Similarly, we obtain from Eqs. (8) ( ) Π + S + S =, ρ S + Q ρ ( + S) (S + 5) =. ), Finally we denote = w and obtain from Eqs. (3), (3) the following set of three first order ODE: = w, = (37 w 9) 9 w + S S( + S) 3 S + S(S + 5) = A(,, w), w = A 9 A w + w 37 8 S w + S w 9 S 9 S + 9 S( + S) = B(,, w). hese eqations coincide with Eqs. (33) from 6, where the stability of their stationary soltions were stdied. Asymptotic vales of (), (), w() at infinity can be fond from Eqs. (4), (9). We obtain ( ) = ( ) =, w( ) =, ( ) = S S Hence, ρ = ρ +, ( ) = (S + 5)(3 S ) 6 S, w( ) =. (8) (3) (3) (3) (33) 4 S S + 5, c = S S, (S + 5)(3 S ) = +, S = +, (34) 6 S + 6

7 in the notations of Eqs. (8). he parameter S can be epressed therefore throgh the Mach nmber M of the shock wave. We obtain M = 3 c 5 = 3(S + 5) 5(3 S ) S = 5(M + 3) 3(5 M ). (35) he parameter S decreases from S = 5/3 to S = /3, when M increases from M = to M =. A convenience of sch nits is that all terms of Eqs. (3), (33) remain finite in the formal limit M = (infinitely strong wave). Note that w = A(,, w) = B(,, w) =, if = (± ), = (± ), w = w(± ) from Eqs. (33). Hence, these vales of (,, w) correspond to singlar points of the dynamical system = w, = A(,, w), w = B(,, w). (36) he shock wave soltion is a heteroclinic orbit of this dynamical system, which begins at = at the spersonic singlar point (provided M > or, eqivalently, S < 5/3) and ends at = at the sbsonic singlar point. It follows from reslts of 6 that the spersonic point is an nstable node, whereas the sbsonic end is a saddle point, which has only one (ot of three) eigenvale with negative real part that corresponds to incoming phase trajectory. o be more precise, the linearization of Eqs. (3) near ( =, =, w = ) in its standard eponential form = + ũ e λ +..., = + e λ +..., w = w e λ +..., (37) where dots denote nonlinear in (ũ,, w) terms, leads for < S < 5/3 to three different eigenvales λ: one real negative and the other two comple conjgate with positive real part (they can also be real positive for small S > ). Actally, this is a saddle point for S < 5/3 and an nstable node for S > 5/3 (S = 5/3 or, eqivalently, M = is a bifrcation point for Eqs. (3)). Similar linearization near (( ), ( ), ) leads for S < 5/3 to three eigenvales λ with positive real part. his means that the dimension d st A of stable manifold near the sbsonic end = is eqal to one, whereas the similar dimension d st B near the spersonic end = (note that <!) is eqal to three. Hence, we have an eqality d st B = dst A +, which is also flfilled for general Discrete Velocity Models of the Boltzmann eqation with arbitrary nmber of velocities 5. his eqality is not valid for Navier Stokes eqations, d st B = dst A + in that case. Hence, GBE (3) yield certain improvement of NSE in a qalitative pictre of soltions (see also 6 for discssion of half space problems). Or goal is to show that Eqs. () improve Navier Stokes approimation for the shock wave problem. In order to do this we need to compare or reslts with eact soltions of similar problem for the Boltzmann eqation. Sch soltions, however, are not known. On the other hand, nmerical data show that lower moments ρ(), (), () of the distribtion fnction, obtained by Monte Carlo method, are very close for strong shocks to similar hydrodynamic moments of approimate Mott Smith soltions of the Boltzmann eqation. herefore we eplain briefly in Section 4 how to constrct sch soltions and compare the soltions of all three models in Section 5. 7

8 4 Mott Smith soltions of the Boltzmann eqations he plane stationary Boltzmann eqation for Mawell molecles for a distribtion fnction f(, v), R, v = (v, v, v 3 ) R 3, can be written in the following weak form d v ψ(v) f(, v) dv = ψ f(, v)f(, w) dv dw, (38) d R 3 R 3 R 3 where ψ(v) is any good test fnction, ( ) ( V ω ψ = ψ U + V ) V ω where S g ( + ψ U V ) ω ψ(v) ψ(w) dω, (39) V = v w, U = (v + w), ω =, g(cos θ) = V σ( V, θ), where σ( V, θ) is a differential cross section, θ π is a scattering angle. connection of f(, v) with hydrodynamic qantities (ρ,, ) is given by ρ() = f(, v) dv, () = v f(, v) dv R ρ 3 R 3 () = v f(, v) dv. 3 ρ R 3 We are interested in this paper in a specific soltion of Eq. (38), which corresponds to the shock wave moving in the negative direction in with the speed c >. his means that we have to consider a soltion f(, v), which tends for and to Mawellians M (v) and M + (v) respectively, where M ± (v) = ρ ± (π ± ) 3/ ep (v ± ) + v y + v z ± he (4). (4) he vales of parameters (ρ ±, ±, ± ) are eactly the same as in Eqs. (8), (4). he Mott Smith soltion (see, for eample, the book ) is an approimate soltion of Eq. (38), having the following form: f MS (, v) = Θ() M (v) + Θ() M + (v), (4) where Θ( ) =, Θ(+ ) =. It is clear that this fnction with any sch Θ() is non negative and satisfies Eqs. (38) for ψ(v) =, v, v. For any other ψ(v) = ψ(v, v ) we obtain from Eq. (38) the following eqation for Θ() where dθ d = λψ Θ( Θ), Aψ = ψ M + (v)m (w) dv dw, R 3 R 3 Bψ = M (v) M + (v) v ψ(v) dv. R 3 8 Aψ λψ = Bψ, (43) (44)

9 In the typical case ψ = v we obtain after some calclations λv = 3 η ρ + ( + ) ( + ), (45) in the notations of Eqs. (5). A weakness of the Mott Smith approimation is that the parameter λψ in Eq. (43) is very sensitive to the choice of the test fnction ψ(v). On the other hand, it is known that the approimation with ψ = v is in very good agreement with DSMC simlation for hard spheres for a wide range of Mach nmbers. his is the main reason why we chose the Mott Smith approimation with ψ = v and the Navier Stokes soltions as two reference soltions for comparison with soltions of Eqs. (3). he general soltion of Eq. (43) reads Θ() = + C ep( λ), (46) where the arbitrary constant C is de to translational invariance in. We choose C = and λ = λv given in Eq. (45). Note that + > and therefore λv <. Hence, Θ() satisfies bondary conditions at = ±. hen Eq. (4) is eqivalent to f MS (, v) = M (v) + M + (v) M (v) ( + tanh y), y = λv = ρ + ( + ) (47) 6 η ( + ) = a(s) ˆ, where (see Eqs. (9), (34)) ˆ = ρ + η +, S = + a(s) = +( + ) 6 ( + ) = S + ( + ) + < 5 3, 6 ( + ) = S(5 3S) 3(S + )(S + 5). (48) Hence, for any good test fnction ψ(v) where C MS ψ = Dψ ( + tanh y), y = a(s)ˆ, (49) C MS ψ = Dψ = R 3 R 3 f MS (v) M (v) ψ(v) dv, M + (v) M (v) ψ(v) dv. For comparison with soltions of Eqs. (3) we need jst fnctions (ρ MS, MS, MS ). aking ψ = in Eq. (49) we obtain (5) ρ MS () ρ = (ρ + ρ ) ( + tanh y). (5) 9

10 hen MS () can be obtained from conservation law whereas MS () can be obtained from Eq. (49) with ψ = v ρ MS ( MS + 3 MS ) ρ ( +3 ) = MS () = ρ ± ± ρ MS (), (5) ρ + ( ) ρ ( +3 ) (+tanh y). (53) Eqs. (5) (53) define (ρ MS, MS, MS ) as fnctions of y = a(s) ˆ, in the notation of Eqs. (48). he net step is to se these fnctions for comparison with soltions of eqations of hydrodynamics. 5 Comparison of reslts for different models Or goal in this section is to solve eqations (3), (33) and to compare the soltions with similar reslts from (a) Navier Stokes eqations and (b) Mott Smith approimation described in Section 4. he Navier Stokes eqations can be obtained directly from Eqs. (3) (3) by omitting Brnett terms. he reslting system of two ODE for () = NS () and () = NS () (lack of third derivative redces dimension of dynamical system) reads S = S +, (54) 5 S + S + 5 = S + ( + S) + 3, S = 5(M + 3), ( ) = ( ) =, 3(5 M ) ( ) = S + 5 (55) (S + 5)(3 S ), ( ) =. 4 S 6 S It is clear that these eqations and Eqs. (3), (33) are invariant nder translations + const. However, we can choose a niqe soltion by demanding that ρ() = ρ( ) + ρ( ), (56) or, eqivalently, () = ( ) + ( ), (57) since ρ() () = const. Note that the Mott Smith soltions (5) (53) are already chosen in sch a way that the above conditions for ρ() and () are satisfied. o keep the same notation in all three cases we need to denote y = S(5 3 S) 3(S + )(S + 5) ˆ, ρ MS() = ρ + ˆρ MS (ˆ), MS () = + û MS (ˆ), MS () = + ˆMS (ˆ), (58)

11 Φ ().5 M =. MS GBE NSE Φ () Φ 3 () Figre : Profiles of the shock-wave and of the Mott-Smith soltion for M =.: GBE (dotted line), NSE eqations (dashed line), MS soltion (solid line). in Eqs. (5) (53) and then omit hats in final reslts. Note that ρ() = ρ NS () = ρ MS () = ρ( ) + ρ( ). (59) It is convenient to choose for comparison three normalized fnctions Φ () = ρ() ρ( ) ρ( ) ρ( ), Φ () = Φ 3 () = () ( ) ( ) ( ). () ( ) ( ) ( ), (6) hen obviosly Φ i ( ) =, Φ i ( ) =, i =,, 3. Moreover, < Φ i () < for all i =,, 3 are monotone fnctions provided ρ(), () and () are monotone fnctions. hs, we fi a Mach nmber M >, solve nmerically Eqs. (3), (33) and Navier Stokes eqations (54), (55), and then present fnctions Φ,,3 () for the three models on Figs. 8. Shock profiles for GBE are not clearly visible on Figs. and 3, becase they are very close to the corresponding crves for NSE when M =. (Fig. ) and for MS approimation when M = (Fig. 3). It is epected that Navier Stokes soltions are good for relatively small M, whereas the Mott Smith approimation is better for large M. If this assmption is tre, then it is clear from or reslts that GBE (3) improve reslts of Navier Stokes approimation for large Mach nmbers (strong shocks). As epected, all three models yield similar reslts for weak shocks (M =., Figs. and ; M =, Figs. 3 and 4). We also present on Figs. and 4 the corresponding phase

12 M =. 3 A B A.97 d/d B Figre : (Left) Shock-wave crve in the phase space for GBE for M =.. (Right) Comparison between the projection of the shock-wave crve in the (, ) plane for GBE (solid line) and the shock-wave for NSE eqations (dashed line). Φ ().5 M = MS NSE GBE Φ () Φ 3 () Figre 3: Profiles of the shock-wave and of the Mott-Smith soltion for M = : GBE (dotted line), NSE eqations (dashed line), MS soltion (solid line).

13 M = d/d..4.6 A B A B Figre 4: (Left) Shock-wave crve in the phase space for GBE for M =. (Right) Comparison between the projection of the shock-wave crve in the (, ) plane for GBE (solid line) and the shock-wave for NSE eqations (dashed line). Φ ().5 M = 5 MS GBE NSE Φ () Φ 3 () Figre 5: Profiles of the shock-wave and of the Mott-Smith soltion for M = 5: GBE (dotted line), NSE eqations (dashed line), MS soltion (solid line). 3

14 M = 5 A.5 B A d/d B Figre 6: (Left) Shock-wave crve in the phase space for GBE for M = 5. (Right) Comparison between the projection of the shock-wave crve in the (, ) plane for GBE (solid line) and the shock-wave for NSE eqations (dashed line). Φ ().5 M = 7 MS GBE NSE Φ () Φ 3 () Figre 7: Profiles of the shock-wave and of the Mott-Smith soltion for M = 7: GBE (dotted line), NSE eqations (dashed line), MS soltion (solid line). 4

15 M = 7 A B.8 A.5 d/d B Figre 8: (Left) Shock-wave crve in the phase space for GBE for M = 7. (Right) Comparison between the projection of the shock-wave crve in the (, ) plane for GBE (solid line) and the shock-wave for NSE eqations (dashed line). M = Figre 9: Comparison between the shock wave profile of tre temperatre (dashed line) and ailiary temperatre (solid line) for M = 5. 5

16 trajectory in the phase space (,, w) of dynamical system (3). he direction of the phase trajectories are always from the spersonic state B ( = ) to the sbsonic state A ( = ). Similar crves for Navier Stokes eqations were stdied in detail in 6. he difference becomes more visible for strong shocks with M = 5 (Figs. 5, 6) and M = 7 (Figs. 7, 8). In all these cases the reslts obtained form GBE (3) are closer to Mott Smith soltions than to soltions of the Navier Stokes system. At the same time the projection of the phase trajectory to the plane (, ) remains relatively close to similar crve for Navier Stokes eqations even for M = 5 (Fig. 6) and M = 7 (Fig. 8). Strictly speaking, we shold se for comparison not the fnction () from Eqs. (3), bt tr () = ( 3 ρ + ) (6) ρ S 8 ρ 3 ρ obtained from transformation (5). We do not do it for two reasons: (a) the difference is negligible for small Mach nmbers and (b) it becomes visible (see Fig. 9) for larger M near the front of the shock, where, generally speaking, no hydrodynamics can be applied becase of large gradients. he shock wave problem is therefore not very good test for eqations of hydrodynamics. he most important reslt, however, is that or eqations (3) are apparently niqely solvable for all Mach nmbers M > (this assmption is spported by linear stability analysis of sbsonic and spersonic states from 6). hey yield qalitatively correct reslts, improving the Navier Stokes approimation for moderate Mach nmbers. We also present some data on thickness of the shock and asymmetry of the density profile. According to the book of Carlo Cercignani 7, in or dimensionless nits the thickness of the shock is defined as δ = Φ () where = is the central point of the shock (Φ () =.5). Vales corresponding to the eamples of or paper are reported in able I. M=. M= M=5 M=7 GBE δ = 35. δ = 3.7 δ = 4.6 δ = 4.8 NSE δ = 34.6 δ =.8 δ = 9.89 δ = 9.86 MS δ = 39.7 δ = 4. δ = 5. δ = 5.6 able I. he asymmetries of the shock profiles can be captred by measring the areas A = Φ () d, A + = + Reslts corresponding to or cases are reported in able II. (6) ( Φ ()) d. (63) 6

17 GBE NSE MS M=. M= M=5 M=7 A = 5.94 A =.36 A =.74 A =.8 A + = 6.5 A + =.36 A + =.45 A + =.48 A = 6.6 A =.96 A =.79 A =.78 A + = 5.9 A + =.7 A + =.47 A + =.46 A = 6.87 A =.43 A =.64 A =.7 A + = 6.87 A + =.43 A + =.64 A + =.7 able II. 6 Conclsions We stdied the shock wave strctre for Generalized Brnett Eqations 5. he previos reslts obtained in 6 sggest a niqe choice of parameters of GBE and therefore we considered precisely this niqe version of GBE in this paper. he advantage of this version is that third derivatives of (, t) and (, t) are absent in the eqations. herefore GBE are even simpler than classical (nstable) Brnett eqations. We confined orselves to the case of Mawell molecles since it is the only case, for which all Brnett terms are known in eplicit form. he shock wave problem redces for GBE to a set of three nonlinear ODE. he shock wave soltion is, in terminology of the theory of dynamical systems 8, a heteroclinic orbit that connects two singlar points of the corresponding vector field. his soltion was stdied for different Mach nmbers and compared with soltions of Navier Stokes eqations and with the Mott Smith approimation for the Boltzmann eqations. he reslts show that GBE yield certain improvement of the Navier Stokes approimation for moderate Mach nmbers. It is also important that GBE have, roghly speaking, the correct difference = of dimensions of stable manifolds near sbsonic and spersonic singlar points (see 5 for details). his means that a qalitative pictre of soltions of GBE is, in some sense, more similar to the Boltzmann eqations than in the case of Navier Stokes eqations. At least two problems in this area remain open: (a) similar one dimensional flows for non Mawell gases (hard spheres, for eample) and (b) bondary conditions and mlti dimensional flows. We hope to retrn to these problems in the ftre. Acknowledgments Work performed in the frame of activities sponsored by INdAM-GNFM, by the Universities of Parma and Milan (Italy), and by the University of Karlstad (Sweden). he spport from the Swedish Research Concil (grant VR-9575) for A.V.B. is also grateflly acknowledged. 7

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