Scaling laws for stationary Navier-Stokes-Fourier flows and the unreasonable effectiveness of hydrodynamics at the molecular level

Hydrodynamics provides a universal description of the emergent collective dynamics of vastly different many-body systems, based solely on their symmetries and conservation laws. Here we harness this universality, encoded in the Navier-Stokes-Fourier (NSF) equations, to find general scaling laws for the stationary uniaxial solutions of the compressible NSF problem far from equilibrium. We show for general transport coefficients that the steady density and temperature fields are functions of the pressure and a kinetic field that quantifies the quadratic excess velocity relative to the ratio of heat flux and shear stress. This kinetic field obeys in turn a spatial scaling law controlled by pressure and stress, which is inherited by the stationary density and temperature fields. We develop a scaling approach to measure the associated master curves, and confirm our predictions through compelling data collapses in large-scale molecular dynamics simulations of paradigmatic model fluids. Interestingly, the robustness of the scaling laws in the face of significant finite-size effects reveals the surprising accuracy of NSF equations in describing molecular-scale stationary flows. Overall, these scaling laws provide a novel characterization of stationary states in driven fluids.

💡 Research Summary

This paper presents the discovery and validation of universal scaling laws governing the stationary solutions of the compressible Navier-Stokes-Fourier (NSF) equations for uniaxial flows far from equilibrium. The work bridges fundamental hydrodynamic theory and large-scale molecular simulations, revealing a profound simplification in the structure of nonequilibrium steady states and the unexpected accuracy of continuum hydrodynamics at the molecular scale.

The authors start from the NSF equations, the cornerstone of continuum fluid dynamics. They focus on a canonical setup: a d-dimensional fluid driven into a steady state by thermal walls at different temperatures and tangential velocities, creating combined heat and shear flow. Through analytical integration of the stationary NSF equations, they demonstrate that for general transport coefficients and equation of state, the steady-state density ρ(x) and temperature T(x) fields can be expressed solely as functions of the fluid’s constant pressure P and a newly defined “kinetic field” ω(x) = 1/2