PHYSICAL REVIEW E 100, 023118 (2019) Exact solutions for shock waves in dilute gases F. J. Uribe * and R. M. Velasco † Department of Physics, Universidad Autónoma Metropolitana-Iztapalapa 09340, CDMX, México (Received 5 May 2019; revised manuscript received 15 August 2019; published 28 August 2019) In 1922 Becker found an exact solution for shock waves in gases using the Navier–Stokes–Fourier constitutive equations for a Prandtl number of value 3/4 with constant transport coefficients. His analysis has been extended to study some cases where an implicit solution can be found in an exact way. In this work we consider this problem for the so-called soft-spheres model in which the viscosity and thermal conductivity are proportional to a power of the temperature η, κ ∝ T σ . In particular, we give implicit exact solutions for the Maxwell model (σ = 1), hard spheres (σ = 1/2), and when σ (the viscosity index) is a natural number. DOI: 10.1103/PhysRevE.100.023118 I. INTRODUCTION The history of the problem we face is long, in fact, the subject of shock waves was founded by Stokes (1819– 1903), Earnshaw (1805–1888), Rankine (1820–1872), Hugo- niot (1851–1887), and Riemann (1826–1866), among oth- ers [1]. Two relevant manuscripts on the subject appeared in 1910, one by Rayleigh [2], who addressed the works of the founders of the subject, including his discussions with Stokes on the subject. The other work by Taylor [3], showed the existence of shock-wave solutions for a fluid with con- stant viscosity with no thermal conductivity or with constant thermal conductivity but no viscosity. He also provided a perturbative solution for weak shocks in which both transport coefficients were included. A remarkable result was obtained in 1922 when Becker [4,5] showed an implicit shock-wave solution for all Mach numbers, including constant viscosity and thermal conductivity with the restriction that the Prandtl number is fixed and equal to 3/4. In 1944 Thomas [6] used the hard-sphere model to criticize Becker’s conclusion in the sense that for strong shocks the Boltzmann equation cannot be applied. Later on, in 1949 Morduchow and Libby [7] improved the Becker’s work by showing the existence of a complete integral of the energy equation and a maximum in the entropy at the inflection point in the velocity distribution, both results turned out to be relevant. In 2013 Johnson [8] found analytical shock solutions at large and small Prandtl numbers and a year later he [9] obtained closed-form (explicit) shock solutions for some of them and also for some of Becker’s implicit solutions, the same year Myong [10] found implicit solutions for the Maxwell model and the hard sphere model. Exact solutions for a van der Waals gas with Pr = 3/4 have been reported recently [11]. In the works mentioned previously the research was based on the Navier-Stokes equations of hydrodynamics under the continuum hypothesis. However, by 1920 the atomistic model for fluids was already accepted, in addition it was already * paco@xanum.uam.mx † rmvb@xanum.uam.mx known that the Navier-Stokes equations can be obtained from the Boltzmann equation by means of the Chapman-Enskog method to solve it [12]. Due to the fact that in a shock wave abrupt changes occur in a narrow region, there has been always doubt about the validity of the Navier-Stokes consti- tutive equations in this case. In a classical paper by Gilbarg and Paolucci in 1953 [13] they rejected such statement based on the evidence found by solving the fluid equations. The discussion is not over yet but the emphasis is now on how to improve on the Navier-Stokes hydrodynamics, if feasible, and the search has been rather prolific as we refer briefly below. By 1950 there were already studies to tackle the shock– wave problem from the Boltzmann equation. One work by Mott-Smith in 1951 [14] relied on an assumption that the distribution function is bimodal and another one by Grad in 1952 [15], that is more in the tradition of hydrodynamics, used the moments method (13 moments) to derive relaxation type equations from the Boltzmann equation. Besides, again in the tradition of hydrodynamics, higher order hydrodynamic equa- tions obtained with the Chapman-Enskog method, Burnett, and super-Burnett equations, were already in consideration by Burnett in 1935 [16] and later on, in 1948, by Wang Chang [17]. Shock-wave phenomena still provides a difficult problem to solve from the continuum point of view. The problem, for dilute gases, can be tackled using the Boltzmann equation and different methods to solve it like the Chapman-Enskog method [12], Grad’s moments method, or probabilistic meth- ods, such as the direct simulation Monte Carlo (DSMC) method [18,19]. Molecular dynamics (MD) [20,21] is not restricted to dilute gases and provides a very flexible tool for studying shock waves in many situations. The computational methods like DSMC and MD provide a direct approach to study shock waves, though the search about the true inter- action potential, must be taken into account. A way to go through this problem takes the ab initio potential calculations to obtain thermophysical and transport properties, which can be taken to study specific applications as actually has been done recently [22–24]. The situation with the continuum approach has some points that need to be addressed. For example, it is known that 2470-0045/2019/100(2)/023118(12) 023118-1 ©2019 American Physical Society