Generalized van der Waals equation of state (GWEOS) for in-line use in hydro codes

In many applications one has to deal with the hydrodynamics of matter undergoing a liquid-gas phase transition. The simplest analytic equation of state (EOS), which qualifies for a physically adequate description of such situations, is the well-known van der Waals EOS. As the van der Waals formula proper only prescribes the dependence of pressure P(V,T) on the volume V and temperature T, it must be augmented by the temperature dependence C(T) of the heat capacity C at constant volume, for which usually a constant ideal-gas value is assumed. From the practical viewpoint, however, the van der Waals EOS has too little flexibility for realistic description of specific materials; this shortcoming is often sought to be overcome by constructing its various generalized versions.

The variant of a generalized van der Waals EOS advocated here (GWEOS) has been borrowed from Refs. [1-3]. It has the same simple functional form as the original van der Waals EOS but contains one more free dimensionless parameter, namely, the power exponent n > 1 in the attractive-force correction (the original van der Waals EOS is recovered for n = 2). By varying the value of n one gains a possibility (i) to fit the experimental values of the ratio between the cohesive energy per atom (molecule) and the critical temperature, as well as (ii) to ensure that any expanding isentrope earlier or later crosses the spinodal. At the same time, the analytic simplicity of GWEOS allows straightforward application of the Maxwell rule in the phase coexistence region, thus providing a highly accurate and computationally not very expensive version of a fully equilibrium EOS in the two-phase region suitable for in-line use in hydrodynamic codes.

The detailed analysis of the key properties of GWEOS, accompanied by illustration of its in-line use in the 1D Lagrangian hydro code DEIRA, is presented in Ref. [4]. The Fortran-90 package, containing a collection of all the key subroutines for in-line use of GWEOS, can be downloaded from here.


  1. M. M. Martynyuk. Generalized van der Waals equation of state for liquids and gases. Zh. Fiz. Khim., 65 (1991) 1716–1717.
  2. M. M. Martynyuk. Transition of liquid metals into vapor in the process of pulse heating by current. International Journal of Thermophysics, 14 (1993) 457–470.
  3. M. M. Martynyuk and R. Balasubramanian. Equation of state for fluid alkali metals: Binodal. International Journal of Thermophysics, 16 (1995) 533–543.
  4. M. M. Basko. Generalized van der Waals equation of state for in-line use in hydrodynamic codes. Keldysh Institute Preprints, No. 112 (2018); [PDF file].

Originally composed: 2019.03.10
Latest update: 2019.03.10