6.1 Newtonian viscosity and momentum transport

Viscosity is the resistance a fluid offers to being sheared. It is the first of three classical transport processes — of momentum, heat, and matter — that this chapter shows to share a single mathematical form. This lesson defines viscosity operationally, then explains it microscopically as the diffusion of momentum.

The shear experiment

Confine a fluid between two parallel plates separated by a gap $h$ . Hold the lower plate fixed and drag the upper one sideways at velocity $U$ . The fluid in contact with each plate moves with it (the no-slip condition), and in steady state the velocity varies linearly across the gap,

u(y) \;=\; U\,\frac{y}{h}.

To keep the top plate moving, a tangential force per unit area — the shear stress $\tau$ — must be applied. A fluid is Newtonian when that stress is proportional to the velocity gradient it sustains:

\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 constant of proportionality is the dynamic viscosity $\mu$ , with units of $\text{Pa}\cdot\text{s}$ . Water at $20^\circ$ C has $\mu \approx 10^{-3}\,\text{Pa}\cdot\text{s}$ ; air has $\mu \approx 1.8\times 10^{-5}\,\text{Pa}\cdot\text{s}$ , some fifty times less. Many common fluids are Newtonian to excellent approximation; polymers, suspensions, and pastes are not, and their nonlinear $\tau(du/dy)$ is the subject of rheology.

Viscosity as a diffusivity of momentum

The shear stress is more than a definition — it is a momentum flux. Each layer of fluid carries $x$ -momentum, and the faster layers hand momentum to the slower ones across the planes between them. The stress $\tau$ is exactly the rate at which $x$ -momentum is transported per unit area in the $y$ -direction. Read this way, $\tau = \mu\,du/dy$ says momentum flux is proportional to the gradient of momentum density — the same form a diffusion law takes.

The point is made dimensionally by the kinematic viscosity

\nu \;=\; \frac{\mu}{\rho},

which has units of $\text{m}^2/\text{s}$ — the units of a diffusion coefficient. Viscosity is the diffusivity of momentum: a velocity disturbance spreads sideways through a fluid exactly as a drop of dye spreads through still water, governed by $\nu$ .

The microscopic origin

For a gas the mechanism is explicit, and follows from the kinetic theory chapter. Molecules in the faster-moving layer thermally wander into the slower layer carrying their surplus momentum, while slower molecules wander the other way; the net momentum exchanged per unit area per unit time is the viscous stress. Estimating the momentum carried across a mean free path $\ell$ by molecules of mean speed $\langle v\rangle$ gives

\mu \;\sim\; \tfrac13\,\rho\,\langle v\rangle\,\ell.

This estimate has a striking consequence. The mean speed rises with temperature, $\langle v\rangle \propto \sqrt{T}$ , while the product $\rho\,\ell$ is independent of density (a denser gas has more carriers but a shorter free path, and the two cancel). So gas viscosity rises with temperature and is independent of pressure — both counterintuitive if one pictures viscosity as molecular stickiness, both confirmed by experiment, and both correctly predicted by the momentum-transport picture. Liquids, where molecules are caged by their neighbours rather than flying freely, behave oppositely: their viscosity falls steeply as temperature rises.