PHYSICAL REVIE%' E VOLUME 47, NUMBER 1 JANUARY 1993 Modeling shock waves in an ideal gas: Going beyond the Navier-Stokes level B. L. Holian and C. W. Patterson Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 M. Mareschal and E. Salomons Centre Europeen de Calcu/ Atomique et Moleculaire, Universite Paris-Sud, 91405 Orsay CEDEX, France (Received 28 September 1992) %'e model a shock wave in an ideal gas by solving a modified version of the compressible Navier- Stokes equations of hydrodynamics, where, following an earlier conjecture by Holian [Phys. Rev. A 37, 2562 (1988)], we use the temperature in the direction of shock propagation T, rather than the aver- age temperature T=(T„„+T~~+T„)/3, in the evaluation of the linear transport coefficients. The re- sults are found to agree much better with the molecular-dynamics simulations of Salomons and Mareschal [Phys. Rev. Lett. 69, 269 (1992)] than standard Navier-Stokes theory. PACS number(s): 51. 10. +y, 47.40. Nm, 47.45. — n I. INTRODUCTION Shock waves in dense Auids are remarkably well ap- proximated by solving the compressible Navier-Stokes (NS) equations of hydrodynamics [1, 2]. Nonequilibrium molecular-dynamics (MD) calculations by Holian er al. [2] and by Klimenko and Dremin [3] for shock waves in the Lennard-Jones (LJ) fluid were compared to the NS solution, which requires as input the L3 equation of state (EOS) (the equilibrium pressure P as a function of inter- nal energy F. and density p) and the linear transport coefficients (shear viscosity r)q, bulk viscosity gi, and thermal conductivity x as functions of p and temperature T). The EOS was determined from earlier equilibrium MD and Monte Carlo calculations [4], while the transport coe%cients were obtained either from previous equilibri- um MD simulations, evaluating Green-Kubo Auctuation formulas [5], or from more reliable nonequilibrium MD simulations of Auxes in response to imposed external fields [6]. In subsequent work, Holian [7) noted that any dis- crepancy between NS and nonequilibrium MD shock- wave simulation profiles, such as particle velocity u(x), could be at least qualitatively explained by the thermal dependence of the shear viscosity at the steepest part of the shock front (located approximately at the point where u is halfway between initial and final values), since shear viscosity is the most important of the transport coefticients in the accurate description of a Auid shock-wave profile. In all cases thus far simulated (either fluid or solid), the component of temperature in the direction of shock propa- gation T~ always exceeds the average temperature T =(T„+T~~+T„)/3, even exhibiting a peak T = 1. 3T near the middle of the shock front. T is defined by the x component of the peculiar kinetic energy N 1VkT„„= g p;„/m, (1) i=1 where the local Auid velocity has been subtracted from the momenta of W particles in the thin slab of material (volume V) in this equation, so that the p; are peculiar momenta, i.e. , thermal fluctuations only (m is the atomic mass, k is Boltzmann's constant). At the lower shock strengths [3], the density and temperature states remain in the dense-Auid regime, where the shear viscosity de- creases with increasing temperature; consequently, using T„„rather than T would lower the viscosity and steepen the profile. On the other hand, for the strongest shock wave [2], where the final temperature is sufficiently high that the LJ system begins to approach ideal-gas behavior, using T rather than T would increase the viscosity and broaden the profile. In both cases, the modified NS solu- tion would more closely approximate reality, i.e. , the nonequilibrium MD results. Recently, Salomons and Mareschal [8] performed the first MD simulations of a shock wave in an ideal gas of hard spheres. The ideal gas regime is very diScult for MD, since the mean free path /o can become large com- pared to the molecular size cz. Hence, the simulation re- quires a large system, whose minimum linear dimension is many mean free paths. They also undertook to check the direct-simulation Monte Carlo (DSMC) method of solv- ing the Boltzmann equation [9]; the results literally could not be distinguished from MD, confirming the validity of DSMC. Then, they compared their results (MD and/or DSMC) to the standard Navier-Stokes shock-wave solu- tion, and found rather good agreement, though, as in the strong dense-Auid case, the NS profiles were too steep when compared to the exact solution (MD). They es- timated the Burnett correction to the heat Aux, using the observed MD gradients in velocity, temperature, and pres- sure, and found that it helped explain the observed devia- tions from Fourier's Law. The ideal gas is, in fact, an ideal candidate for testing Holian's conjecture (that T„„should be used in a modifi- cation to Navier-Stokes theory, rather than T), since a self-consistent solution is possible; that is, there is no am- biguity about extracting T„„ from the normal component of the pressure tensor P, the general expression for which is P„V= g (p; /m+F;„x;) =NkT + gF; x; . (2) R24 1993 The American Physical Society