Back to Experiments

Viscosity Spectrum: Solid to Gas

Experiment: exp_ns_viscosity_spectrum.sx  |  Domain: Navier-Stokes Regularity  |  Theorems: 21–23

Overview

Matter has three states, governed by viscosity: \(\nu \to \infty\) (solid, infinite damping), \(\nu\) finite (fluid — the NS regime), and \(\nu \to 0\) (gas/inviscid Euler). Instead of treating \(\nu\) as a weak lever, this experiment maps the full spectrum by oscillating viscosity as a probe wave. The system's response — phase, amplitude, and hysteresis — reveals the "material properties" of the blow-up surface.

Key Results

PhenomenonValueSignificance
Resonance period~10,000 stepsEnstrophy response peaks at this oscillation period
Hysteresis ratio78×Enstrophy on \(\nu\)-decreasing sweep vs \(\nu\)-increasing sweep
Ratchet effectPresentMax enstrophy increases in consecutive oscillation windows
Weak lever quantified\(\partial \Omega / \partial \nu \sim \nu^{-2}\)Viscosity effect is nonlinear — small changes near \(\nu = 0\) dominate

Spectrum Phases

Regime\(\nu\) RangeBehaviour
Solid\(\nu > 1.0\)All modes damped, enstrophy \(\to 0\) exponentially
Fluid (stable)\(0.01 < \nu < 1.0\)Enstrophy bounded, oscillatory, diagnostics healthy
Fluid (critical)\(0.001 < \nu < 0.01\)Resonance zone: enstrophy response amplified 78×
Gas\(\nu < 0.001\)Stretching dominates, blow-up threshold reachable
Interpretation. The 78× hysteresis ratio reveals that the blow-up surface is highly asymmetric: reducing viscosity destabilises the system far more than increasing it restabilises. The ratchet effect — where each oscillation cycle leaves the system at higher peak enstrophy — explains why time-varying viscosity can be more dangerous than a constant low viscosity. The resonance at period 10,000 identifies a natural timescale of the coupled enstrophy dynamics.

Analysis

  • Resonance at period 10,000. The system has a characteristic response time. Oscillation periods near this value maximise enstrophy amplification, while faster or slower oscillations are less effective.
  • Hysteresis (78×). The path matters: the system's response to the same viscosity depends on whether viscosity was recently higher or lower. This is a memory effect in the enstrophy dynamics.
  • Chaos/order mapping. The dual agent's chaos/order balance maps directly to the viscosity spectrum: chaos = low viscosity probing, order = high viscosity damping. The oscillation IS the dual agent in action.

Reproducibility

../simplex/build/sxc exp_ns_viscosity_spectrum.sx -o build/exp_ns_viscosity_spectrum.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_viscosity_spectrum.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_viscosity_spectrum \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_viscosity_spectrum

Related