6.2
Physics Foundations
§6 · Viscosity, diffusion, and transport

6.2 Fourier, Fick, and the diffusion equation

Momentum is not the only quantity that flows down its own gradient. Heat and matter do the same, and with the same mathematics. This lesson states the two remaining transport laws — Fourier’s for heat, Fick’s for matter — derives the diffusion equation common to both, and collects the three diffusivities into the dimensionless ratios that organise transport problems.

Fourier’s law of heat conduction

Heat flows from hot to cold at a rate proportional to the temperature gradient. The heat flux q\mathbf{q} (energy per unit area per unit time) obeys

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

where kk is the thermal conductivity, in W/(m⋅K)\text{W}/(\text{m}\cdot\text{K}), and the minus sign sends heat downhill in temperature. Combining this with conservation of energy — the rate of change of stored heat ρcp ∂T/∂t\rho c_p\,\partial T/\partial t equals minus the divergence of the flux — gives the heat equation:

∂T∂t  =  α ∇2T,α  =  kρcp.\frac{\partial T}{\partial t} \;=\; \alpha\,\nabla^2 T, \qquad \alpha \;=\; \frac{k}{\rho c_p}.

The thermal diffusivity α\alpha again carries units of m2/s\text{m}^2/\text{s}. The full solution of this equation — its Gaussian Green’s function and the way an initial temperature spike spreads — is developed in the Foundations heat-and-diffusion chapter.

Fick’s law of diffusion

A dissolved species spreads from high concentration to low at a rate proportional to the concentration gradient. With concentration c(r,t)c(\mathbf{r},t) and mass flux J\mathbf{J},

J  =  −D ∇c,\mathbf{J} \;=\; -D\,\nabla c,

and conservation of the species gives the diffusion equation,

∂c∂t  =  D ∇2c,\frac{\partial c}{\partial t} \;=\; D\,\nabla^2 c,

with DD the diffusion coefficient, once more in m2/s\text{m}^2/\text{s}. The mathematical identity with the heat equation is exact: temperature and concentration spread in precisely the same way, and any solution of one is a solution of the other.

One equation, three diffusivities

The three transport laws — Newton’s, Fourier’s, Fick’s — share the same skeleton: a flux proportional to a gradient, producing a Laplacian diffusion equation for the transported quantity. Their coefficients form a trio,

{ ν,  α,  D },all with units m2/s,\{\,\nu,\;\alpha,\;D\,\}, \qquad \text{all with units } \text{m}^2/\text{s},

for the diffusion of momentum, heat, and matter respectively. Because they share units, their ratios are pure numbers, and these dimensionless groups decide which transport process is faster in a given medium:

Pr=να  (Prandtl),Sc=νD  (Schmidt),Le=αD  (Lewis).\mathrm{Pr} = \frac{\nu}{\alpha} \;\text{(Prandtl)}, \qquad \mathrm{Sc} = \frac{\nu}{D} \;\text{(Schmidt)}, \qquad \mathrm{Le} = \frac{\alpha}{D} \;\text{(Lewis)}.

For water the Prandtl number is about 77 — momentum diffuses several times faster than heat — while for air it is about 0.70.7, the two nearly matched. These ratios govern the relative thickness of the velocity, thermal, and concentration boundary layers in any flow, and recur throughout heat- and mass-transfer engineering. The unifying lesson is that a single diffusion equation, with one coefficient swapped for another, describes how momentum, heat, and matter all relax toward uniformity.