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

Abstract

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.

Figures (4)

• Figure 1: (Color online) NSF scaling for $\text{HD}_{2d}$ fluids. (a) Sample of measured average profiles for the density (top row), $y$-velocity (middle row) and temperature (bottom row) for $N=1927$, $\phi=0.5$, $T_1^*=1$ and (from left to right), $T_0^*=2,~10,~20$. Each plot shows profiles for 10 different values of $v_0^*=-v_1^*\in[1,10]$. Colors codify $T_0^*$, while symbols represent different $v_0^*$. Gray bands near $x=0,1$ signal the boundary layers. (b) Scaling plot of the density $\rho^*(x)$ as a function of the reduced kinetic field $\omega^*(x)/P^*$ before (light gray) and after (color symbols) the shifts $\xi$. Inset: The same but for the reduced temperature $T^*(x)/P^*$. All data for different $N$, $\phi$, $T_0^*$ and $v_0^*$ collapse on a pair of master curves, as predicted by the NSF scaling laws \ref{['rhoThd']} for $\text{HD}_{2d}$.
• Figure 2: (Color online) Spatial scaling for $\text{HD}_{2d}$ fluids. (a)-(b) Scaling plot of the reduced kinetic field $\omega^*(x)/P^*$ as a function of the scaled spatial variable $\frac{\sigma^*}{P^*}x$ and the shifts $\xi$ obtained from the density (and temperature) scaling, see Fig. \ref{['figHD1a']}. All kinetic field profiles collapse onto a universal master surface ${\cal W}_\xi(\pm \frac{\sigma^*}{P^*}x+\zeta)$ after an appropriate spatial shift $\zeta$. This spatial scaling is inherited by the density [(c)-(d)] and reduced temperature [(e)-(f)] fields, which collapse onto two master surfaces $\bar{\cal R}\left(\pm \frac{\sigma}{P}x + \zeta, \frac{\omega}{P}+\xi \right)$ and $\bar{\cal T}\left(\pm \frac{\sigma}{P}x + \zeta, \frac{\omega}{P}+\xi \right)$, respectively, as predicted by the NSF scaling laws. Panels (b), (d) and (f) are zooms over the corresponding shaded areas.
• Figure 3: (Color online) NSF scaling for $\text{LJ}_{3d}$ fluids. (a) Sample of density (top row), $y$-velocity (middle row) and temperature (bottom row) profiles measured for $N=10^4$ LJ particles, $T_1^*=1$, target pressure $P^*=3$ and (from left to right), $T_0^*=2,~10,~20$. Each plot shows profiles for 5 different values of $v_0^*=-v_1^*\in[2,10]$. Colors codify $T_0^*$, while symbols represent different $v_0^*$. Gray bands near $x=0,1$ signal the boundary layers. Bottom panels: Scaling plot of $\rho^*(x)$ vs $\omega^*(x)$ for pressures (b) $P^*=1$ and (c) $P^*=5$, before (light gray) and after (color symbols) the shifts $\xi$ along the abscissa. The insets show the equivalent temperature scalings. For each $P^*$, all data for different $N$, $T_0^*$ and $v_0^*$ collapse on a pair of master curves, as predicted by the NSF scaling laws \ref{['rhow']} for $\text{LJ}_{3d}$. The resulting master function ${\cal R}_P(\omega+\xi)$ and ${\cal T}_P(\omega+\xi)$ depend nontrivially on pressure.
• Figure 4: (Color online) Spatial scaling for $\text{LJ}_{3d}$ fluids. Top row: Scaling plot of $\omega^*(x)$ for different $P^*$ as a function of $\sigma^* x$ and the shifts $\xi$ obtained from density (and temperature) scalings, see Fig. \ref{['figLJ1']}. All kinetic field profiles collapse for each pressure onto a universal master surface ${\cal W}_{P,\xi}(\pm \sigma x+\zeta)$ after an appropriate spatial shift $\zeta$. This spatial scaling is inherited by the density (middle row) and temperature (bottom row) fields, which collapse onto two universal master surfaces $\bar{\cal R}_P\left(\pm \sigma x + \zeta, \omega+\xi \right)$ and $\bar{\cal T}_P\left(\pm \sigma x + \zeta, \omega+\xi \right)$, respectively, as predicted by the NSF scaling laws \ref{['rhow']}. Left column [(a)-(c)] corresponds to $P^*=2$, and right column [(d)-(f)] to $P^*=4$. Gray curves projected in the $\rho^*-\omega^*$ and $T^*-\omega^*$ planes of middle and bottom panels correspond to the $P^*=5$ scaling curves, for comparison.