Viscosity, diffusion, and transport

Newtonian viscosity, Fourier and Fick, the random-walk origin of diffusion, Stokes drag, the Einstein relation, and transport timescales.

Three classical transport processes — of momentum, heat, and matter — share a single mathematical form: a flux proportional to a gradient, with a diffusivity of microscopic origin, producing a Laplacian diffusion equation. From that common shape come Newtonian viscosity, Fourier’s and Fick’s laws, the random walk that underlies diffusion, the drag on a slow sphere, and Einstein’s relation tying friction to fluctuation. The chapter closes with the timescales these diffusivities set and their role in the absorption of sound.

• 6.1 Newtonian viscosity and momentum transport — the shear experiment, $\tau = \mu\,du/dy$, and viscosity as the diffusivity of momentum.
• 6.2 Fourier, Fick, and the diffusion equation — heat and mass flux down gradients, the shared diffusion equation, and the trio $\{\nu, \alpha, D\}$.
• 6.3 Diffusion as a random walk — Fick’s law from random steps, $\langle x^2\rangle = 2Dt$, and Brownian motion.
• 6.4 Stokes drag — the $F = 6\pi\mu a U$ law, the friction coefficient, and terminal velocity.
• 6.5 The Einstein relation and fluctuation–dissipation — $D = k_B T/\gamma$, the Stokes–Einstein relation, and the fluctuation–dissipation theorem.
• 6.6 Diffusion timescales and acoustic attenuation — $\tau \sim L^2/D$, the adiabatic criterion, and the Kirchhoff–Stokes absorption of sound.