# Beyond Navier--Stokes equations: Capillarity of ideal gas

## TL;DR

This paper explores a new approach to modeling non-equilibrium fluid dynamics beyond classical Navier--Stokes equations, focusing on capillarity effects in ideal gases and proposing a candidate system like the Korteweg equations.

## Contribution

It investigates the role of capillarity in ideal gases and proposes a new modeling framework to better account for non-equilibrium effects beyond traditional fluid equations.

## Key findings

- Capillarity effects are significant in ideal gases at non-equilibrium scales.
- A candidate system, such as the Korteweg equations, can model these effects.
- Analytical solutions connect kinetic theory with fluid dynamics.

## Abstract

The system of Navier--Stokes--Fourier equations is one of the most celebrated systems of equations in modern science. It describes dynamics of fluids in the limit when gradients of density, velocity and temperature are sufficiently small, and loses its applicability when the flux becomes so non-equilibrium that the changes of velocity, density or temperature on the length compatible with the mean free path are non-negligible. The question is: how to model such fluxes? This problem is still open. (Despite the fact that the first `final equations of motion' modified for analysis of thermal creep in rarefied gas were proposed by Maxwell in 1879.) There are, at least, three possible answers: (i) use molecular dynamics with individual particles, (ii) use kinetic equations, like Boltzmann's equation, or (iii) find a new system of equations for description of fluid dynamics with better accounting of non-equilibrium effects. These three approaches work at different scales. We explore the third possibility using the recent findings of capillarity of internal layers in ideal gases and of saturation effect in dissipation (there is a limiting attenuation rate for very short waves in ideal gas and it cannot increase infinitely). One candidate equation is discussed in more detail, the Korteweg system proposed in 1901. The main ideas and approaches are illustrated by a kinetic system for which the problem of reduction of kinetics to fluid dynamics is analytically solvable.