Thermal conductivity at the high-density limit and the levitating granular cluster

The granular Leidenfrost state consists of a dense granular cluster levitating above a hot granular gas. The density of particles inside the cluster can be very high and even close to the density of crystalline packing. To describe this state theoretically, one needs to know the density dependence of constitutive relations (pressure, heat losses, thermal conductivity) at these very high densities. However, the accurate expression for the coefficient of thermal conductivity is lacking. In this work, the constitutive relations were measured at high densities in molecular dynamics simulations in three different settings: a uniform freely cooling dense granulate (to measure heat losses), a uniform ensemble of elastically colliding particles (to measure pressure), and a dense granular medium between two thermal walls under gravity (to measure thermal conductivity). Next, the hydrodynamic equations with the resulting expressions were solved to describe the levitating cluster state in various parameter regimes. Separate molecular dynamics simulations were performed to test the theoretical predictions and measure the density and temperature profiles of the granular Leidenfrost state, and a good agreement with theoretical results was observed.