5.6 Stokes flow and boundary layers

The Reynolds number sets the balance between inertia and viscosity, and the two extremes of that balance are worlds apart. At very low $\mathrm{Re}$ inertia vanishes and the flow becomes linear and reversible; at very high $\mathrm{Re}$ inertia dominates everywhere except in thin layers hugging solid walls. This lesson takes each limit in turn.

Stokes flow: the inertia-free limit

When $\mathrm{Re} \ll 1$ the inertial term $\rho\,D\mathbf{u}/Dt$ is negligible against the viscous term, and the Navier–Stokes equation reduces to the linear Stokes equation:

The nonlinearity is gone, and three consequences follow at once:

• Linearity. The flow depends linearly on its boundary conditions: double the boundary velocity and the entire flow field doubles.
• Time-reversibility. The equation has no time derivative. Reverse the boundary motion and the flow retraces itself exactly, every parcel returning along its original path.
• Stokes drag. A rigid sphere of radius $a$ moving at speed $U$ through the fluid feels a drag $F = 6\pi\mu a U$ — linear in speed, in size, and in viscosity.

The time-reversibility has a vivid consequence known as the scallop theorem: a microscopic swimmer whose stroke is identical to its own time-reverse can make no net progress. A scallop that simply opens slowly and closes quickly moves backward as much as forward over a full cycle and stays put. To swim at low Reynolds number a body must execute a stroke that is not its own time-reverse — a wave, a rotation, a sequence with a definite handedness.

The hinged scallop opens and closes along the same path; its cycle is its own time-reverse, and the net displacement is zero. The rotating corkscrew traces a different shape going forward than backward, breaks time-reversal symmetry, and advances. Flagellated bacteria swim by exactly this rotating-helix trick, precisely because the world looks the same forward and backward at the scale where viscosity rules.

Boundary layers: viscosity confined to the wall

At the opposite extreme, $\mathrm{Re}\gg 1$, the viscous term is small almost everywhere and the flow is well described by Euler’s equation — except that Euler cannot satisfy the no-slip condition at a solid surface. Viscosity, however weak, must reassert itself in a thin boundary layer adjacent to the wall, where the velocity drops from its free-stream value $U$ to zero over a short distance.

The thickness of this layer follows from a balance of the two surviving effects: momentum diffuses outward from the wall at the rate set by $\nu$, while the flow carries fluid downstream at speed $U$. In a time $t$ momentum diffuses a distance $\sim\sqrt{\nu t}$, and in that time the fluid has travelled $x \sim Ut$ downstream. Eliminating $t$,

The layer thickens downstream as $\sqrt{x}$. Outside it the flow is effectively inviscid and moves at the free-stream speed; inside it the no-slip condition pins the velocity to zero at the wall, and momentum is fed in from the wall diffusively. Almost all the viscous action in a high-Reynolds-number flow — and almost all the drag — is concentrated in these thin layers and in the wakes they shed when they separate.

A closely related thin-film approximation, lubrication theory, applies when a flow is confined to a narrow gap between nearby surfaces. There the geometry makes the cross-gap viscous term dominate, and the full Navier–Stokes equation collapses to a simple relation between pressure gradient and flow rate — the basis of how a thin film of fluid can support a large load, as in a journal bearing or any lubricated contact.

These two limits — reversible Stokes flow at small $\mathrm{Re}$, thin boundary layers at large $\mathrm{Re}$ — bracket the whole range of incompressible behaviour, and between them lies every flow regime catalogued in the previous lesson.