>[!warning] >This content has not been peer reviewed. # Fluid Dynamics — Results --- ## 1. Viscosity calibration (Phase 1) **Model (liquids):** ν(T) = ν₀ + A · [1/μ(T/T_char, n_fluid) − 1] **Model (gases):** ν(T) = ν₀ + A · [1/μ(T_ref/T, n_gas) − 1] **Benchmarks:** Arrhenius ν = A_arr·exp(E/RT) for liquids (2 params); Sutherland for gases (2 params). RST has 4 params. | Fluid | Phase | ν₀ | A | T_char | n_fluid | R²_RST | R²_bench | Benchmark | |:---|:---|---:|---:|---:|---:|---:|---:|:---| | Water | liquid | 0.222 | 18513 | 130 | 9.56 | **0.9993** | 0.9946 | Arrhenius | | Glycerol | liquid | 0.000 | 3.8×10⁹ | 162 | 20.0 | 0.9916 | **0.9999** | Arrhenius | | Ethanol | liquid | 0.092 | 23.4 | 254 | 6.86 | **0.9999** | 0.9986 | Arrhenius | | Air | gas | 0.000 | 0.071 | 5000 | 0.65 | 0.9972 | **0.9999** | Sutherland | ### Key findings 1. **RST beats Arrhenius for moderate liquids.** Water (R² 0.9993 vs 0.9946) and ethanol (0.9999 vs 0.9986) are better captured by the algebraic μ transition than by the exponential Arrhenius form. 2. **RST loses for extreme Arrhenius behavior.** Glycerol's viscosity spans a factor of 128× over 80 K, with exponential divergence near its glass transition. The μ function (algebraic) cannot match this steepness — n hits the ceiling at 20 and the amplitude A becomes unphysically large (10⁹). 3. **Gases are better served by Sutherland.** Air viscosity follows T^(0.7) closely; the Sutherland formula with its T^(3/2) correction is purpose-built for this. RST captures the shape (R² = 0.997) but with lower precision. 4. **n_fluid for liquids ≈ 7–10.** Water (9.56) and ethanol (6.86) cluster in the same range as n_el (electronic transport, ≈ 5). This suggests a similar "high-dimensional coupling" — the interaction between thermal energy and intermolecular bonds occurs in a combined phase-space. 5. **Liquid-gas inversion.** Liquids use η = T/T_char (temperature as signal), gases use η = T_ref/T (temperature as noise). This is the same fidelity inversion seen between gravity and tensile sectors: whether temperature helps or hinders depends on the noise source. --- ## 2. Reynolds transition (Phase 2) **Model:** f(Re) = f_lam · μ(Re_c/Re, n_turb) + f_turb · (1 − μ(Re_c/Re, n_turb)) **Reference:** Churchill (1977) friction factor formula for smooth pipe (covers all regimes). | Parameter | Value | |:---|---:| | Re_c (fitted) | 983 | | n_turb | 20.0 (ceiling) | | R² | 0.9994 | | RMSE | 0.0037 | ### Observations 1. **Transition captured at R² = 0.999.** The μ-weighted interpolation between laminar (64/Re) and turbulent (Blasius) regimes matches the Churchill reference closely through the transition zone. 2. **Re_c = 983** (below the expected ~2300). The μ interpolation distributes the "blending" differently than Churchill's formula, shifting the effective critical Reynolds number downward. The model starts mixing turbulent contribution earlier. 3. **n_turb = 20 (ceiling).** The laminar-turbulent transition is essentially a **step function** — the sharpest possible in the μ framework. This makes physical sense: the onset of turbulence is an abrupt regime change, not a gradual transition like elastic-plastic yield (n ≈ 0.35) or Debye resistivity (n ≈ 5). 4. **High-Re deviation.** At Re > 200,000, the RST model under-predicts f by ~10–15% compared to Churchill. The Blasius correlation (0.316/Re^0.25) loses accuracy at very high Re; the RST model inherits this. ### Physical interpretation The very sharp n_turb ≈ 20 is the largest n value found across all five RST sectors: | Sector | n | Transition type | |:---|:---|:---| | Tensile (FCC) | 0.35 | Gradual yield | | Thermal | 1.3–1.5 | Diffusive backbone | | Gravity | 1.25 | Percolation backbone | | Electronic | ≈ 5 | High-dimensional coupling | | Fluid (viscosity) | 7–10 | Intermolecular activation | | Fluid (Reynolds) | ≈ 20 | Near-step regime change | The progression from gradual (0.35) to sharp (20) maps to the physical nature of each transition: slip-system activation is gradual; thermal and gravitational backbone paths are moderate; electronic and molecular couplings are sharper; turbulence onset is catastrophic. --- ## 3. Figures ![[fluid_dynamics_viscosity.png]] *Viscosity vs temperature: water, glycerol, ethanol (liquids), air (gas).* ![[fluid_dynamics_reynolds.png]] *Left: Moody diagram with RST interpolation. Right: residual vs Churchill.* --- ## 4. The five n values (updated) | Sector | n | Physical backbone | Channel type | |:---|:---|:---|:---| | Tensile | 0.35 (FCC), 1.0 (BCC) | Slip-system connectivity | Mechanical deformation | | Gravity | 1.25 | Spacetime percolation backbone | Wide-area signal relay | | Thermal | 1.3–1.5 | Heat-carrying lattice backbone | Energy transport | | Electronic | ≈ 5 | Fermi surface × phonon spectrum | Charge transport | | Fluid (viscosity) | 7–10 | Intermolecular bond network | Bulk liquid flow | | Fluid (Reynolds) | ≈ 20 | Inertial-viscous regime boundary | Turbulence onset | The pattern: **n increases with the dimensionality of the coupling phase-space.** Mechanical slip uses low-dimensional pathways (0.35). Percolation backbones are ~1.25-dimensional. Phonon-electron scattering occurs in 5D phase space. Molecular activation involves even higher-dimensional energy surfaces. Turbulence onset is so sharp it resembles a phase transition (effectively infinite n). --- ## 5. Limitations 1. **RST has 4 params vs Arrhenius's 2.** The comparison is not parameter-fair; RST's advantage over Arrhenius for water/ethanol may partly be due to extra degrees of freedom. 2. **Glycerol failure.** The algebraic μ function cannot reproduce exponential (Arrhenius/VFT) divergence. Glass-forming liquids with activation energies E_a ≫ kT are outside the model's natural domain. 3. **Reynolds Re_c shift.** The fitted Re_c = 983 differs from the known ~2300. This is a model artifact from the interpolation structure, not a physical prediction. 4. **No turbulence structure.** The model predicts the friction factor curve but says nothing about vortex sizes, intermittency, or the Kolmogorov cascade. --- ## 6. Links - **Theory:** [[Fluid Dynamics (RST)]] - **Code:** [[Fluid Dynamics - Code]] - **Electronic transport (charge):** [[../Electronic Transport/Electronic Transport Results]] - **Thermal transport (heat):** [[../Thermal Transport/Thermal Transport Results]] - **Fidelity inversion (five sectors):** [[expanded theory/Fidelity Inversion — Gravity and Materials]]