BRIEF REPORTS Brief Reports are accounts of completed research which do not warrant regular articles or the priority handling given to Rapid Communications; however, the same standards of scientific quality apply. (Addenda are included in Brief Reports.) A Brief Report may be no longer than four printed pages and must be accompanied by an abstract. The same publication schedule as for regular articles is followed, and page proofs are sent to authors. Bobylev’s instability F. J. Uribe, R. M. Velasco, and L. S. Garcı ´ a-Colı ´ n Departamento de Fı ´sica, Universidad Auto´noma Metropolitana-Iztapalapa 09340, Me´xico Distrito Federal, Mexico ~Received 16 February 2000! In 1982 Bobylev @A.V. Bobylev, Sov. Phys. Dokl. 27, 29 ~1982!# made a linear stability analysis of the Burnett equations and showed that beyond a certain critical reduced wave number there exist normal modes that grow exponentially, concluding that the Burnett equations are linearly unstable. We have partially ex- tended his analysis, originally made for Maxwellian molecules, for any interaction potential and argue that his results can be reinterpreted as to give a bound for the Knudsen number above which the Burnett equations are not valid. PACS number~s!: 05.20.Dd, 47.20.2k, 51.10.1y The question regarding the stability of the solutions to the equations of hydrodynamics for given initial and boundary conditions has been of utmost importance @1#. In particular, since hydrodynamic equations for dilute gases are obtained from the Boltzmann equation by seeking, either solutions in power series in terms of Knudsen’s parameter through the Chapman–Enskog method, or as truncated approximations using Grad’s moment method, and in both cases the transport coefficients are in principle obtainable for given intermolecu- lar potentials, the validity of their solution becomes an im- portant question. In 1982 Bobylev @2# claimed that for the case of Maxwellian molecules, whereas the Navier–Stokes approximation yields equations which are stable against small perturbations, for the equilibrium state characterized by constant temperature ( T 0 ), constant mass density ( r 0 ), and zero hydrodynamic velocity ( u50), this is not the case for the next approximation in Knudsen’s parameter, namely, for the Burnett equations. In fact he showed that small per- turbations to the equilibrium solution which are periodic in the space variable with a wavelength smaller than some criti- cal length are exponentially unstable. This fact is now re- ferred to in the literature as Bobylev’s instability. On the other hand, the Burnett approximation of hydro- dynamics has been recently shown to provide substantial im- provement on many features of the flow occurring in several problems in hydrodynamics. This is the case for a plane Poi- seuille flow @3#, and others @4#. But perhaps the most spec- tacular of them arises in the calculation of the profiles of a shock wave at large Mach numbers. There, it has been shown by many workers in the field that the Burnett approximation substantially improves the accuracy of the different profiles in the shock wave when compared with the direct Monte Carlo simulations @5# or molecular dynamics @6#. Neverthe- less, in the study of this problem, it was found that the solu- tions of the Burnett equations do exhibit certain ‘‘instabili- ties’’ that have been associated to Bobylev’s instability @7,8#, in the case of the time dependent code, or to a bifurcation for a Mach number of value ’2.69 in the stationary situation @9#. Without entering here into a detailed analysis of these features, which will be soon published, the question is if the results obtained by Bobylev can be sustained for more gen- eral models and if affirmative can they be casted in terms of Knudsen’s parameter. This means, can we find a critical value for Knudsen’s parameter beyond which the solutions to the Burnett equations are unstable? The purpose of this communication is to show that this is indeed the case and further we will see that the analysis partially holds true independently of the interatomic poten- tial. This answer provides then a rather clear cut significance to the gradient expansion in the Chapman–Enskog method @10# of solving Boltzmann’s equation. To pursue our objective we start from the conservation equations which, for the longitudinal flow, u( r, t ) 5u ( x , t ) i ˆ , are written as ]r ~ x , t ! ] t 1 ] ] x „u ~ x , t ! r ~ x , t ! …50, ~1! ] u ~ x , t ! ] t 1u ~ x , t ! ] u ~ x , t ! ] x 52 1 r ~ x , t ! ] P xx ] x , ~2! ] T ~ x , t ! ] t 1u ~ x , t ! ] T ~ x , t ! ] x 52 2 m 3 k B r ~ x , t ! S P xx ~ x , t ! ] u ~ x , t ! ] x 1 ] q x ~ x , t ! ] x D , ~3! where m is the mass, k B Boltzmann’s constant, P xx ( x , t ) the xx component of the pressure tensor and q x ( x , t ) the x com- ponent of the heat flux. r ( x , t ), u ( x , t ), and T ( x , t ) are the local values of the mass density, the velocity and the tem- PHYSICAL REVIEW E OCTOBER 2000 VOLUME 62, NUMBER 4 PRE 62 1063-651X/2000/62~4!/5835~4!/$15.00 5835 ©2000 The American Physical Society