Viscosity, diffusion, and transport

Newtonian shear stress, Fick’s law, Stokes drag, the Einstein relation.

Three classical transport processes — momentum, heat, and matter — share a single mathematical form. In each one, a flux is proportional to the local gradient of an intensive quantity, with a coefficient (viscosity, thermal conductivity, diffusivity) of microscopic origin. This is the operative shape behind every dissipative term in the Sound and Hearing books, from acoustic absorption to viscous damping of the basilar membrane.

Newtonian viscosity — operationally defined

Imagine a fluid sheared between two parallel plates: the upper plate moves with velocity $U$ , the lower is fixed, the gap is $h$ . In steady state the fluid develops a linear velocity profile $u(y) = U y / h$ . The upper plate must exert a tangential force per unit area — the shear stress $\tau$ — to maintain the motion. A Newtonian fluid is one in which $\tau$ is linear in the velocity gradient:

\tau \;=\; \mu \,\frac{du}{dy}.

U1.00

μ1.00

h1.00

τ = μU/h1.000

Plate velocity U = 1.00 Viscosity μ = 1.00 Gap h = 1.00

The top plate drags the fluid; the bottom plate holds it still. In steady state the velocity is *linear* in y — Couette flow — and the shear stress on either plate is τ = μ ∂u/∂y = μU/h. This is the operational definition of viscosity: the force per unit area required to slide one plate over the other at unit velocity, divided by μ.

The dynamic viscosity $\mu$ has units of $\text{Pa}\cdot\text{s}$ . For water at 20°C, $\mu \approx 10^{-3}\,\text{Pa}\cdot\text{s}$ ; for air, $\mu \approx 1.8\times 10^{-5}\,\text{Pa}\cdot\text{s}$ . The kinematic viscosity $\nu = \mu/\rho$ has units of $\text{m}^2/\text{s}$ — the same as a diffusion coefficient, revealing viscosity’s underlying nature as a diffusivity of momentum.

The microscopic picture (from the kinetic-theory chapter): molecules in the faster-moving layer occasionally cross into the slower one carrying their excess momentum, and vice versa. The net momentum transfer per unit area per unit time is the viscous stress. The kinetic-theory estimate gives $\mu \sim \rho \langle v\rangle \ell / 3$ for a gas, which explains why gas viscosity rises with temperature ( $\langle v\rangle \propto \sqrt{T}$ ) — counter-intuitive if one is thinking of viscosity as molecular sluggishness.

Heat conduction — Fourier’s law

The same shape, with temperature replacing velocity and heat flux replacing momentum flux:

\mathbf{q} \;=\; -k\, \nabla T,

where $k$ is the thermal conductivity ( $\text{W}/(\text{m}\cdot\text{K})$ ). Combining with conservation of energy gives the heat equation

\frac{\partial T}{\partial t} \;=\; \alpha\, \nabla^2 T, \qquad \alpha \;=\; \frac{k}{\rho c_p},

with $\alpha$ the thermal diffusivity. The Math Foundations PDE chapter develops the heat-diffusion equation and its Gaussian Green’s function in full.

Mass diffusion — Fick’s law

For a dilute species with concentration $c(\mathbf{r}, t)$ and mass flux $\mathbf{J}$ :

\mathbf{J} \;=\; -D\, \nabla c, \qquad \frac{\partial c}{\partial t} \;=\; D\, \nabla^2 c.

The diffusion coefficient $D$ is the third member of the trio $\{\nu, \alpha, D\}$ . Their ratios are dimensionless and named: $\nu/\alpha = \mathrm{Pr}$ (Prandtl), $\nu/D = \mathrm{Sc}$ (Schmidt), $\alpha/D = \mathrm{Le}$ (Lewis). For water $\mathrm{Pr} \approx 7$ ; for air $\mathrm{Pr} \approx 0.7$ .

A random walk derives Fick’s law

Diffusion is the macroscopic shadow of a microscopic random walk.

▶ From random steps to Fick's law

Let $p(x, t)$ be the probability density for a single walker. After a small time $\delta t$ , the walker has taken a Gaussian step of variance $\sigma^2 \delta t$ :

p(x, t + \delta t) \;=\; \int p(x', t)\, G(x - x'; \sigma^2 \delta t)\, dx'.

Expanding $p(x', t) \approx p(x, t) + (x' - x) p_x + \tfrac12 (x' - x)^2 p_{xx}$ and using $\int G = 1$ , $\int \Delta x\, G\, d\Delta x = 0$ , $\int \Delta x^2 G\, d\Delta x = \sigma^2 \delta t$ :

p(x, t + \delta t) - p(x, t) \;\approx\; \tfrac12 \sigma^2 \delta t\, p_{xx}.

Dividing by $\delta t$ ,

\frac{\partial p}{\partial t} \;=\; \tfrac12 \sigma^2\, \frac{\partial^2 p}{\partial x^2},

so $p$ obeys the diffusion equation with $D = \sigma^2/2$ .

The mean-squared displacement of a single walker grows linearly in time: $\langle x^2\rangle = 2 D t$ in 1-D, $\langle r^2 \rangle = 4 D t$ in 2-D, $\langle r^2\rangle = 6 D t$ in 3-D. The linearity is the signature of normal diffusion.

D1.00

t0.0

measured ⟨r²⟩—

theory 4Dt0.00

Diffusion constant D = 1.00 Number of walkers = 200

Each walker takes an independent Gaussian step every frame; the cloud spreads, and the mean-squared distance from the origin grows linearly in time with slope 4D (in 2-D; 2D in 1-D, 6D in 3-D). This linearity — not the cloud shape, not the particle paths — is the *signature* of diffusion. Below it lies the Einstein relation D = k_BT / γ that ties the diffusion constant to the microscopic friction.

temperature T

300

M / m (big / bath)

20×

elapsed time: 0.0 s
current |Δr|²: 0
D ≈ |Δr|² / (4t): 0.0

heavier big particle → smaller D; hotter bath → larger D. The slope-4D line should track the curve over enough time.

Stokes drag — viscosity meets a sphere

A sphere of radius $a$ moving with velocity $\mathbf{U}$ through a fluid of viscosity $\mu$ at low Reynolds number experiences a drag force

\mathbf{F}_\text{drag} \;=\; -6\pi \mu a\, \mathbf{U}.

This is the central numerical result of low-Re hydrodynamics. It arises from solving the Stokes equation around a no-slip sphere; the $6\pi$ is exact for a rigid spherical surface.

_drag = 6πμaU = 18.85

Viscosity μ = 1.00 Velocity U = 1.00 Radius a = 1.00

The streamlines around a Stokes-flow sphere are *fore-aft symmetric* — the perturbation to the uniform flow is the same upstream and downstream. The total drag, integrated around the sphere, is exactly F = 6πμaU. Doubling the radius doubles the drag; doubling the velocity doubles the drag — a linear-response regime entirely controlled by viscosity.

The streamlines around the sphere are fore-aft symmetric — a hallmark of Stokes flow, where viscous forces dominate and inertia is irrelevant. The drag is linear in both viscosity and velocity. The friction coefficient $\gamma_\text{drag} = 6\pi\mu a$ has units of $\text{N}\cdot\text{s}/\text{m}$ and connects to diffusion through:

The Einstein relation

The same Brownian particle that feels Stokes drag also diffuses. Einstein’s 1905 insight was that the friction coefficient and the diffusion coefficient are not independent — they are tied by temperature:

D \;=\; \frac{k_B T}{\gamma_\text{drag}}.

▶ Deriving the Einstein relation from drift-diffusion balance

Consider a dilute suspension of Brownian particles subject to a weak external force $F$ (e.g., gravity in the −z direction). In steady state two opposing effects balance:

The force drives a drift current $J_\text{drift} = c\, F/\gamma_\text{drag}$ (each particle moving at terminal velocity $F/\gamma_\text{drag}$ ).
Diffusion drives a Fickian counter-current $J_\text{diff} = -D\, \partial c/\partial z$ .

At equilibrium $J_\text{drift} + J_\text{diff} = 0$ :

\frac{1}{c}\frac{\partial c}{\partial z} \;=\; \frac{F}{D\,\gamma_\text{drag}}.

But by the Boltzmann distribution at equilibrium, $c(z) \propto e^{-Fz/k_B T}$ , so $(\partial \ln c/\partial z) = -F/k_B T$ .

Comparing: $F/k_B T = F/(D\,\gamma_\text{drag})$ , i.e.

D \;=\; \frac{k_B T}{\gamma_\text{drag}}.

F (force)0.50

γ (friction)1.00

k_BT1.00

v_drift = F/γ0.500

l_eq = k_BT/F2.000

D = v_drift · l_eq = k_BT/γ1.000

External force F = 0.50 Friction γ = 1.00 Temperature k_BT = 1.00

At equilibrium, the drift current downward (F/γ per particle) is exactly cancelled by the diffusive current upward (D times the concentration gradient). The equilibrium profile is exponential with decay length k_BT/F (Boltzmann). Equating the two currents gives the **Einstein relation** D = k_BT/γ — fluctuations and dissipation arise from the same microscopic process.

For a spherical Brownian particle this becomes $D = k_B T/(6\pi \mu a)$ — the Stokes–Einstein relation. It is the basis for every technique that infers molecular size from diffusion measurements (dynamic light scattering, fluorescence correlation spectroscopy) and for the hearing-threshold floor imposed by Brownian noise.

The Einstein relation is the simplest instance of the fluctuation-dissipation theorem: a thermal system’s response to a small external force ( $\gamma_\text{drag}$ ) and the spontaneous fluctuations of the corresponding observable ( $D$ ) are tied by $k_B T$ . Dissipation and equilibrium fluctuations come from the same microscopic mechanism — molecular collisions.

Diffusion timescales — when is a process “fast”?

A diffusive process equilibrates over a length $L$ on a timescale

\tau \;\sim\; \frac{L^2}{D},

with $D$ replaced by $\nu$ , $\alpha$ , or $D$ as appropriate.

D1.00e-5 m²/s

L1.00e-4 m

τ = L²/D1.00e-3 s

log₁₀ D = -5.00 log₁₀ L = -4.00

The diffusion timescale scales quadratically with length: doubling L quadruples τ. This is the key fact that makes microscopic processes fast and macroscopic ones slow. Sugar takes seconds to diffuse across a 100 μm cell; minutes across a 1 mm tube; *years* across a meter of unstirred water. The L² scaling is why mixing must usually be done by *convection*, not diffusion alone.

The quadratic scaling makes diffusion extremely sensitive to length: halving $L$ quarters $\tau$ . Two cases the Sound book leans on:

Thermal-diffusion length per acoustic cycle. For air, $\alpha \approx 2\times 10^{-5}\,\text{m}^2/\text{s}$ ; at 1 kHz the period is 1 ms, so heat diffuses $\sqrt{\alpha\, T} \approx 4.5\,\mu\text{m}$ . The acoustic wavelength is 0.34 m. The two scales differ by five orders of magnitude — which is why sound is adiabatic in air, not isothermal.
Polytropic exponent inside an oscillating bubble. The dimensionless ratio $\omega R^2/\alpha_g$ decides whether the gas inside is isothermal or adiabatic. Cavitation Ch 3.2 uses precisely this argument.

Acoustic attenuation — viscous and thermal

The classical absorption coefficient for sound in a fluid is (Kirchhoff–Stokes):

\alpha_\text{abs}(f) \;=\; \frac{(2\pi f)^2}{2 \rho c^3} \left[\frac{4}{3}\mu \;+\; (\gamma - 1)\frac{k}{c_p}\right].

Two terms — viscous shear and thermal conduction — both scale as $f^2$ .

Medium:

Classical absorption scales as f² — doubling the frequency *quadruples* the attenuation per metre. Air absorbs sound much more strongly than water; ultrasonic frequencies (MHz) are essentially absorbed within metres in air but propagate hundreds of metres in water. Below 100 kHz absorption in both is utterly negligible on room scales, which is why ordinary acoustics happily treats sound as undamped.

The viscous term diffuses momentum across the velocity gradient of a sound wave; the thermal term diffuses heat between compressed (hot) and rarefied (cold) regions. The combined coefficient — utterly negligible at audible frequencies in air, dominant at MHz ultrasound in water — is what the Sound book’s attenuation chapter plots against frequency.

⏳ The history — Einstein, Perrin, and the molecular reality of fluids

Einstein’s 1905 paper “On the motion of small particles suspended in a stationary liquid required by the molecular-kinetic theory of heat” was one of his three annus mirabilis papers. He proposed that visible Brownian particles undergo a random walk driven by molecular collisions, that the mean-squared displacement grows linearly in time, and — most consequentially — that the diffusion coefficient is set by Boltzmann’s constant and the macroscopic friction: $D = R T / (6\pi \mu a N_A)$ , expressing $k_B = R/N_A$ in terms of measurable quantities.

The prediction was directly testable. Jean Perrin spent 1908–1910 making the measurement: tracking individual mastic and gamboge grains under a microscope, recording their positions every 30 seconds, computing the mean-squared displacement, and inverting Einstein’s formula for Avogadro’s number. His value, $N_A \approx 7 \times 10^{23}$ , was within 20% of the modern value. The molecular-kinetic theory of heat was no longer hypothetical.

The same Einstein paper also implies what is now called the fluctuation–dissipation theorem: a thermal system’s response to a small external force ( $\gamma_\text{drag}$ ) and the spontaneous fluctuations of the corresponding observable ( $D$ ) are tied by $k_B T$ . This deep connection — that dissipation and equilibrium fluctuations come from the same microscopic mechanism — extends far beyond Brownian motion to the Johnson noise of a resistor, the line widths of spectral resonances, and the thermal-noise floor of a hearing system.

For the cross-book applications — acoustic absorption, Brownian-motion hearing floor, polytropic bubble gas, viscous tip-link damping — see the key examples sub-page.