Kinetic modelling of non-equilibrium flow of hard-sphere dense gases

A kinetic model is proposed for the non-equilibrium flow of dense gases composed of hard-sphere molecules, which significantly simplifies the collision integral of the Enskog equation using the relaxation-time approach. The model preserves the most important physical properties of high-density gas systems, including the Maxwellian at rest as the equilibrium solution and the equation of state for hard-sphere fluids, all the correct transport coefficients, namely the shear viscosity, thermal conductivity and bulk velocity, and inhomogeneous density distribution in the presence of a solid boundary. The collision operator of the model contains a Shakhov model-like relaxation part and an excess part in low-order spatial derivatives of the macroscopic flow properties; this latter contribution is used to account for the effect arising from the finite size of gas molecules. The density inhomogeneity in the vicinity of a solid boundary in a confined flow is captured by a method based on the density functional theory. Extensive benchmark tests are performed, including the normal shock structure, the Couette, Fourier and Poiseuille flow at different reduced densities and Knudsen numbers, where the results are compared with the solutions from the Enskog equation and molecular dynamics simulations. It is shown that the proposed kinetic model provides a fairly accurate description of all these non-equilibrium dense gas flows. Finally, we apply our model to simulate forced wave propagation in a dense gas confined between two plates. The inhomogeneous density near the solid wall is found to enhance the oscillation amplitude, while the presence of bulk viscosity causes stronger attenuation of the sound wave. This shows the importance of a kinetic model to reproduce density inhomogeneity and correct transport coefficients, including bulk viscosity.