Nonlocal and nonlinear effects in shock waves l 5 NOVEMBER 1991

The formulation of generalized transport laws leading to causal hyperbolic hydrodynamical equations gives rise to a critical Mach number M, at which a singularity develops in the shock wave layer. This happens in kinetic theory of gases {Grad's 13 moments method} and extended irreversible thermodynamics as well. In this paper, it is shown that if nonlinear and nonlocal e6'ects are taken into account, M, significantly increases. Specifically, nonlocal e6'ects raise M, from 2.01 up to 3.03. Further consideration of some nonlinear term. s raise M, from the latter value up to 4.67.


I. INTRODUCTION
The analysis of high-frequency and short-wavelength phenomena has fostered the progress of nonequilibrium statistical mechanics and nonequilibrium thermodynam- ics.In the mentioned conditions, the classical transport laws, namely, Fourier's law of heat conduction and Newton-Stokes law of viscous pressure, must be substituted by more general expressions taking into account nonlocal effects, both in time and space.Furthermore, if the amplitude of the perturbation is high enough, non- linear dissipative e6'ects must be also considered.The derivation and interpretation of such general equations is one of the main topics of research in nonequilibrium phe- nomena.
At high frequencies, one cannot take as independent hydrodynamical variables only the conserved, or slow, variables, as mass, momentum, and energy, but one must include in the set of basic variables those whose relaxation time is of the order of the inverse of the frequency.Thus, some nonconserved variables, as, for instance, dis- sipa, tive 6ows (heat liow, viscous pressure tensor, and so on) may become independent variables at high frequencies.This idea is the basis of some microscopic develop- ments, as the 13-moments method in kinetic theory of gases [1], some formulations of generalized hydrodynam- ics based on projection-operator techniques [2], and the macroscopic theory known as extended (or transient) ir- reversible thermodynamics (EIT), [3 -10].All these de- ve1opments inc1ude as independent variables the usual dissipative flows.
The evolution equations for the flows play in these theories the role of generalized transport equations.Due to the enormous complexity of real systems, the simple evolution equations proposed in these theories are no more than plausible caricatures which allow us to appre- ciate qualitative e6'ects beyond the classical theory as, for instance, the high-frequency behavior of thermal waves [3,11] or of shear waves, the nonequilibrium corrections to hydrodynamic noise at high frequencies [12], none- quilibrium modifications of the equations of state and their consequences on the critical point [13], nonequili- brium contributions to the entropy [3 -10], etc.Such modifications are of interest in the relativistic context too, where the infinite speed of thermal and viscous sig- nals predicted by the classical transport equations are no longer admissible [14].
Shock waves have been used as a testing ground of  [16] and 10 [17], respectively.This has prompted the comment that the EIT is currently a theory in crisis [18].
Heuristically, this feature is quite general in hyperbolic systems where smooth signals cannot propagate at speeds higher than the highest characteristic speed [1].In con- trast, the usual parabolic (classical) theory does not yield any critical Mach number but it allows us, in principle, to evaluate the shock thickness up to arbitrarily high Mach numbers.Accordingly, it seems wise to examine the ex- isting formulations of EIT in order to improve, as far as possible, its connection with experiments, i.e. , to increase the critical Mach number predicted by them.
The purpose of this paper is to reexamine this interest- ing problem from the point of view of the more recent formulation of BIT.Indeed, the simple older models of EIT include as additional independent variables the clas- sical dissipative flows only.However, at frequencies comparable to the inverse of the collision time an in6nite number of variables should be included in the theory as, for instance, higher and higher-order moments of the dis- tribution function.In fact, the relaxation times of these nonconserved variables cannot be shorter than the collision time, so that they should be considered as indepen- dent variables in these conditions (a development of EIT with the inclusion of higher-order flows has been already undertaken [19]).Since the departure with respect to equilibrium is important in shock waves nonlinear eFects are expected to be very relevant.Here we wiH show that their inclusion substantially increases the critical Mach number.
The plan of the paper is as follows.The next section is devoted to a brief review of the analysis of shock waves in the framework of the older model of EIT.In Secs.III and IV spatial nonlocalities and dissipative nonlinearities are considered, respectively.Finally, in Sec.V, the main conclusions are presented.tion of the fiows.When Eqs. (2.3) and (2.4) together with (2.5) -(2.7) are used to study thermal and shear waves one obtains for the high-frequency limit of the propagation speed of the corresponding disturbances II.SHOCK WAVES IN THE OLD VERSION OF EIT c) =(Klpcr, )' (2.8a) In this section we briefly synthesize the main ideas of the old version of EIT.The subject has been extensively reviewed t3, 8] where the reader interested in details is re- ferred to the bibliography.Eiere it is sufBcient to state that EIT, in its barest version, rests on the following hy- potheses: (1) It exists as a generalized entropy function which depends not only on the classical variables but on the dissipative Aows as well.The latter are considered in- dependent variables on the same foot as the classical ones.(2) Such an entropy function is a maximum at equi- librium state.Its How may depend on all dissipative Aows and its production rate is positive definite.
For unicomponent simple fluids the dissipative Aows are taken to be the heat Aow q and the traceless viscous pressure tensor P' (the bulk viscous pressure vanishes identically in ideal monoatomic nonrelativistic gases).
The generalized Gibbs equation, the generalized entropy How and the evolution equation for the Rows are c =(r)/pr Note that ~and g are assumed to be known, so that Eqs.
(2.8) can be used, in principle, to obtain the effective values of ~& and ~2 from the experimental determination of c, and c2.Since actual measurements are dificult, one often identifies r& and r2 from kinetic theory (13-moment method, relaxation time approximations) or from second-order contributions to the dispersion relations of ultrasonic waves.
For one-dimensional shock wave, Eqs. (2.3) -(2.7) reduce to ds =0 'du+8 'II d8qdq -P":dP', (2.3) where q=q", P'=P"', and v =u .These relationships must be supplemented by the state equations p= -pV-v, pv= -Vp -V P'+pF, 0 psi = -V q -pV.v -P':V, (2.6) provide us with a complete set of equations for the evolu- with x and q the thermal conductivity and shear viscosi- ty, ~& and v. 2 the relaxation times of q and P', respective- ly.0 and II denote the generalized absolute temperature and thermodynamic pressure, which reduce to T and p, the usual local-equilibrium absolute temperature and thermodynamic pressure for small values of q and P".
stands for the specific volume p ' and V is the traceless symmetric part of the velocity gradient.
When r&=rz=P=O, (2.1) reduces to the usual Gibbs equation of the local-equilibrium theory, (2.2) becomes the stan- dard entropy fiow, and (2.3) and (2.4) reduce to the classi- cal Fourier and Newton-Stokes laws, respectively.The evolution equations (2.3) and (2.4), together with the classical balance laws of mass, momentum, and ener- in which we have written u (the internal energy) and p in terms of the Laplace velocity of sound c and the adiabatic coeKcient y inasmuch as these expressions are the most suitable ones for the analysis of shock waves.where M*, P *, and Q * are integration constants.
The dissipative Aows play an important role in the determination of the spatial structure of the shock layer, i.e. , the spatial dependence of the thermodynamic quanti- ties and the velocity across the layer.We next sketch the analysis of Anile and Majorana [15].
(2.20a) (2.20b)This system has two real positive solutions P =(co, P ), P+ =(co+, P+) whenever -, ' &a*& -, 'y (y -1 ) .These solutions characterize the equilibrium states at both sides of the shock layer and are given by P+E 7+1 '   (2.21a) Here, R is the ideal gas constant, and m the mass of the molecules.Note that a* may be written in terms of the properties of the equilibrium state upstream to the shock layer as B* '= (y+1) '[(y -1)(a/c)+(4/3)yq] M* with K and g the transport coefficients evaluated for (()=(y+ I) '.Note that the relaxation coefficients r"rz, and r3 do not appear in (2.23) so that for weak shock waves the width and structure of the shock layer is not influenced explicitly by the relaxation terms.Such an inQuence appears, however, in higher-order approxima- tions where numerical methods are needed to study the shock profile.In Table I we show the numerical results obtained by Anile and Majorana for the width of the shock layer defined as CO CO+ L, =- (dc@/dx )," (2.24) in terms of the mean upstream mean free path I .This is usually taken as l 16' The most conspicuous consequence of the relaxation terms in the present version of BIT is to put some limits on the Mach numbers for which the shock layer shows a regular structure.The existence of critical Mach num- bers in the extended description is linked to the mathematical property that a regular unique solution to (2. 19) can only exist if the determinant D(co, g) differs from zero.Otherwise dao/dx and dgldx, as given by Eq.
(2.19), will diverge since generally at the points where D(co, g) vanishes, b, i and bz do not.A necessary condi- tion to avoid the occurrence of D(co, g)=0 is that D(coz, gz) and D(co"g, ) bear the same sign.If they do not there will be an intermediate couple   I.The ratio between the mean free path l and the shock width L is shown for different Mach numbers for three models.
In the next section we show that a possible way out for this problem could be to "renormalize" the transport coefficients in order to take into account higher-order nonlocal and nonlinear effects, ignored in the old version of EIT sketched above.We shall consider three models.(i) The classical one, with the Newton-Stokes and Fourier transport laws.For this model one has r, =rz=r3=0.(ii) The Grad's 13- moments model of kinetic theory [1,15] (G I model) where r, =r2=1, r3= -', .(iii) Model G II with the values r, =r3=2(Sm) ' and r2=0, obtained by Carrassi and Morro [20] by contrasting the dispersion relation that follows from the set of equations (2.9) -(2.13) with the exper- imental data on the phase speed of ultrasounds.
Bearing in mind that according to (2.20b), co=1 -P, Eq. (2.25) reduces to the following.Classical model, ( (2.26c) Note that the physically significant domain of variation of P is, according to (2.21), - where the latter is still successful.
In EIT the explicit form of the evolution equations compatible with the second law is obtained as follows.
One assumes a generalized entropy and a generalized en- tropy Aow of the form [19] and ds = 8 'du +8 'II d8 -g a"8q'"'d q'"' n=1 The entropy production can be derived from the equation of evolution for s, (3.4) ps = -V.J'+o' . (3.5) Upon introducing (3.3) and (3.4) into (3.5) and taking into account the balance equations (2.5) and (2.7) it follows cr'=q"'(V T'+a)q'") y ~(n) (& q(n) P V. q(n+)) P Vq(n -))) n=1 (3.6) with J' the flow of z and o' the production rate of z per unit volume.The flow of a quantity describes how it spreads in space as response to inhomogeneities, i.e. , it describes nonlocal effects.
When q and P' are included in the theory, their respec- tive flows and the flows of these flows should also be in- cluded as independent variables, at least in principle.If the relaxation times of the higher-order flows are much shorter than the inverse of the frequency, these variables should not be considered as independent variables, but they should be considered as so otherwise.In monoatom- ic ideal gases, the relaxation times of the higher-order flowswhich correspond to higher-order moments of the distribution function from a microscopic point of view [21] are all of the same order.Consequently, when q and P' have to be considered as independent vari- ables, all their higher-order flows should be considered as independent variables as well.This point of view has been explored in EIT theories [19].
For the sake of simplicity, we shall restrict ourselves to q (the analysis for P' is analogous).We denote q as q" ', q' ' is a second-order tensor describing the flow of q'" and, in general, q'"' is the nth order tensor corresponding to the How of q'" ",an (n -1)th order tensor.
The evolution equation for q'"' may be written as The simplest form of the evolution equation for the flows satisfying (3.1) is pq'"= -(q"'+a V T)+ V q"', p») An asymptotic expression for Eq.(3.9) can be obtained with the use of the scheme proposed by Gianozzi et al. [22].It yields ~"( with r"and l " the values of r"and i" in the high-n lim- it.From the last equation we have Since q'"+" is the flow of q'"' one has a"= -pP".
When written in the Fourier space this set of equations allows us to define a thermal conductivity which depends on the frequency m as well as on the wave vector k as We shall take l"=l /2, and ~"=~, in accordance with Hess [21].Thus, l "=3k~T r/4m where r denotes the collision time.If one takes into account that in kinet- ic theory y= -5k& Tr, /3m one has r"(r=4r, /5. A similar development for the speed of shear waves leads to (3.15b) If one considers l "=3k~Tr /4m and ri=pr one has ~z.(r= ~z/4. By replacing r, /v as well as rz/ri by r),(r/x and r~,(r/ri, respectively, in expressions (2.17a) for r, and r3 and then inserting these values in Eq. (2.25) for D(co, g) one is led to the new values for the critical Mach number.
In model GII [r, =(2/5m.)0.80, rz =0, and r3=(2/ 5n)0.25] upon setting D(co, g)=0 one obtains the equa- Unfortunately, the normalization of I"2, the coefficient related to the entropy flow, appearing in model GI, re- quires more involved arguments based on fluctuation theory and we will not deal with it here but in a future work which is currently in progress.

IV. NONLINEAR EFFECTS: FOURTH-ORDER TERMS IN THE ENTROPY
In the previous section only linear nonlocal effects were considered.Since a shock wave drives the medium far away from thermodynamic equilibrium with rather high values for the heat flow and the viscous pressure it seems quite natural to take into account nonlinear effects.In this section we examine a particular kind of nonlinear effect related to an expansion of the nonequilibrium en- tropy function up to fourth order in the heat flow and viscous pressure.
We now consider an expression for the nonequilibrium entropy analogous to the integrated form of (2.1), The dispersion relation for thermal waves reads ps =ps,q -P":P', with y=K/pc.Hence it follows for the speed of thermal waves ps=ps, -q 1+ q 2 2a I I P'.P' 1+, P'.P' The coefficients P and P' are new and related to the fourth-order contributions to the entropy.By defining the effective nonlinear relaxation times w& and ~z according to r (4.3a)Thus, Eqs. (4.7) and (4.8) enable us to obtain the none- quilibrium corrections to the effective relaxation times as a function of the Mach number.Here we will take for l /L the value I /L -= 1/4 which is the observed value for that ratio for Mach numbers ranging from 2 to 5 (see Fig. 48 in Cercignani, Ref. [1]) and we will evaluate ri and r 3 in the equations.For model G II we get 1+, P":P' the relationship (4.1b) can be written as (4.1a) but with ri and ~2 replaced by ~& and ~z, respectively.By ~"z and ~2,z we denote the effective relaxation times, which take into account nonlocal effects only, evaluated in the previ- ous section.v& and ~z include both nonlocal corrections (corning from ri,z and r2,&) and nonlinear corrections (corning from the terms in large parentheses).
The coefficient a, P as well as a' and P' can be comput- ed from Auctuation theory, in terms of the second-and fourth-order moments of the Quctuations around an equi- librium state [23].

V. CONCLUDING REMARKS
The ratio 2/5~is the value of r, and r3 according to model G II of Anile and Majorana [15], whereas 0.80 and 0.25 are the nonlocal corrections of the last section and the terms in large parentheses are the nonlinear corrections.
The method to follow is an iterative one.We compute the value of the nonlinear corrections [the terms in large parentheses in (4.9) and (4.10)] for M =3, the critical Mach number in the linear nonlocal theory, and obtain r& =0.0540 and r3 =0.0303.Upon introducing these values back in (2.25) the nontrivial roots of D(co, g}=0 happens to be Pi =0.0365 and $2=0.0274.According to these values the critical Mach number is 3.9797.Then using again the expressions (4.9) and (4.10) we compute the nonlocal-nonlinear corrections for M =3.9797 and obtain r & =0.03598 and r3 =0.2682, respectively.The roots of D(co, g)=0 are now Pi=0.0298 and $2=0.0201 and the critical Mach number increases up to 4.4202.In a subsequent step we get r', =0.009527, r3 =0.02503, $, =0.0267, and Pz=0.005 65.The corresponding criti- cal Mach number now is 4.67.At this point we stop the process for the next step leads to a negative value for r &, which is clearly unphysical.
Thus the nonlinear effects considered in this section have raised the critical Mach  ~(q, P") =aC(y), g(q, P') =qC(y), (5.1)The nonlinearities considered in last section are by no means the only ones conceivable.Other nonlinearities may arise.For instance, Eu has proposed in the frame- work of a modified moment method the following non- linear corrections to the phenomenological transport coefficients [7]: where T* and v" are short for T+/T and v /v+, re- spectively.For monoatomic ideal gases (y =-, ') one has where C(y)=&y/sinh&y and y denotes the dimension- co', P' with coz & co' & m» (()z & P' & Pi such that D(co', P') =0.

1 2QT
the obvious generalization of (4.la) up to fourth order in the flows.The coefficients n and a' are those occurring in the second-order entropy and they are given by Comparison of this equation with (2.8a) allows us to define an effective relaxation time as 7 eff %1 c oo(3.15a)   with ~1 and ~2 the relaxation times of the linear theory.
I shows that in the common domain of validity the numerical results for the width of the shock layer are very similar for the classical model, the G I model, and Table the G II model.The trouble is that the models G I and 6 II, which should be expected to be more accurate than the classical one, fail to describe shock waves for M & 2. 1 L, the width of the shock layer and Kp and gp the values taken by K and g, respectively, at the center of the layer.We borrow these values from kinetic theory with