Correlated fluctuating hydrodynamics. II. Scale-dependent Reynolds numbers

Many chemical and biological processes in molecular, soft-matter, and living systems are modeled using the low-Reynolds-number (Re ≪ 1) linearization of the incompressible Navier-Stokes equations. This approximation is justified by the assumption that viscous dissipation dominates nonlinear inertial effects across spatial scales. However, many soft-matter and biological fluids possess internal structure that modifies momentum transport across scales, potentially altering the balance between inertial and viscous effects. In Part I [J. Chem. Phys. 165, 064509 (2026)] of this series, we introduced a thermodynamically consistent fluctuating-hydrodynamic framework for structured fluids and showed that spatial correlations render viscous dissipation scale dependent. Here, we investigate the consequences of this scale dependence for the validity of the classical low-Re linearization. We show that scale-dependent viscous dissipation alters the balance between inertia and viscosity across scales, thereby invalidating the conventional low-Re justification for linearization. Direct numerical simulations in one and two dimensions confirm these predictions. In one dimension, nonlinear mode coupling accelerates the relaxation of high-wavenumber Fourier modes relative to the linearized dynamics. The same mechanism is reflected in particle transport in two dimensions: the particle velocity autocorrelation decays more slowly under the linearized dynamics, leading to diffusion coefficients that differ from the nonlinear prediction by up to 90%. These results demonstrate that a single Reynolds number is no longer sufficient to determine the validity of linearization in spatially correlated fluctuating fluids; instead, it depends on a scale-dependent spectrum of effective Reynolds numbers.