§6
Physics Foundations
§6 · Viscosity, diffusion, and transport

Key examples — viscosity, diffusion, and transport

Where the chapter’s machinery shows up across the bookshelf.

Example 1: classical viscous-thermal sound absorption

The Kirchhoff-Stokes attenuation αabs(f)∝f2\alpha_\text{abs}(f) \propto f^2 from the chapter gives the classical (Newtonian, Fourier-conduction) absorption of sound. For air at 1 kHz it is ∼10−11 m−1\sim 10^{-11}\,\text{m}^{-1} — far too small to matter on room scales. For water at 1 MHz it is ∼0.025 m−1\sim 0.025\,\text{m}^{-1} — about 0.2 dB/m. For MHz ultrasound used in medical imaging this is significant; for audible frequencies it is utterly negligible. See Sound Ch 10.1.

Example 2: Brownian-motion floor on hearing thresholds

The Sound book’s Brownian-motion lesson uses the fluctuation-dissipation theorem to estimate the irreducible noise floor of any pressure receiver in thermal equilibrium with air. Sivian and White (1933) computed this and found that the predicted floor sits within a factor of 2-3 of the measured human auditory threshold near 1-4 kHz. Hearing is as sensitive as the molecular noise of the air will allow — and the fluctuation-dissipation theorem is what sets the bar.

Example 3: polytropic exponent of bubble gas

For an oscillating Rayleigh-Plesset bubble at radius R(t)R(t) at frequency ω\omega, the gas inside has thermal diffusivity αg\alpha_g. The dimensionless ratio

Π  =  ωR2αg\Pi \;=\; \frac{\omega R^2}{\alpha_g}

decides whether compression is isothermal (Π≪1\Pi \ll 1, slow enough for heat to escape) or adiabatic (Π≫1\Pi \gg 1, too fast for heat to escape). For a 1 μm bubble in air at ω=107 rad/s\omega = 10^7\,\text{rad/s}, Π≈1\Pi \approx 1 — borderline; the effective polytropic exponent is frequency-dependent. See Cavitation Ch 3.2.

Example 4: viscous damping of hair-bundle motion

A stereocilium of radius ∼100 nm\sim 100\,\text{nm} in perilymph experiences Stokes drag F=6πμaUF = 6\pi \mu a U. For a bundle of ∼50\sim 50 stereocilia moving at 1 nm amplitude at 1 kHz, the velocity is U∼10−6 m/sU \sim 10^{-6}\,\text{m/s} and the total drag is ∼1 pN\sim 1\,\text{pN}. The viscoelastic damping of the basilar-membrane / hair-bundle system arises from this dissipation; the cochlear amplifier’s job is to cancel it. See Hearing Ch 4.6.

Example 5: why sound is adiabatic in air

For air, α=2×10−5 m2/s\alpha = 2\times 10^{-5}\,\text{m}^2/\text{s}; at 1 kHz the period is 1 ms, so heat diffuses α T≈4.5 μm\sqrt{\alpha\,T} \approx 4.5\,\mu\text{m} in one cycle. The acoustic wavelength is 340 mm. The thermal-diffusion length is ∼10−4\sim 10^{-4} of the wavelength: heat conducts across the compressed regions over distances vastly smaller than the wavelength scale, so the local temperature stays locked to the local density — the adiabatic condition. This is the operative justification for the adiabatic equation of state Laplace used to correct Newton’s sound speed. See Sound Ch 4.4.