Stokes Flow | mysimulator.uk

A world without momentum

The full Navier-Stokes equations balance three kinds of force on a fluid parcel: pressure, viscosity, and inertia (mass times acceleration). Which term dominates is set by the Reynolds number, Re = ρUL/μ — the ratio of inertial to viscous forces, for a flow of speed U past an object of size L in a fluid of density ρ and viscosity μ. A cruising whale has Re in the billions; a bacterium swimming in water, or a glass bead settling through honey, has Re around 10⁻⁵ or lower. At that extreme, inertia is not just small — it is negligible enough to drop from the equations entirely, leaving what is called Stokes flow or creeping flow.

live demo · slow, viscous flow around an obstacle● LIVE

The Stokes equations

Dropping the inertial (nonlinear advection) term from Navier-Stokes leaves a linear balance between pressure gradient and viscous friction, together with incompressibility:

μ∇²u = ∇p          (viscous force balances pressure gradient)
∇·u = 0             (incompressibility)

There is no time derivative of velocity at all — the flow field at any instant depends only on the boundary conditions at that instant, with no memory of how it got there. Stop moving a paddle in this regime and the fluid stops moving with it almost immediately; there is no coasting, no wake that lingers, no vortex shedding of the kind that makes a flag flap in the wind.

Stokes' law and the drag on a sphere

George Gabriel Stokes solved this linear system exactly for a sphere of radius r moving at speed v through an unbounded viscous fluid in 1851, giving the drag force:

F_drag = 6πμrv

This is the basis of the falling-ball viscometer (measure a sphere's terminal velocity to infer μ) and of Millikan's oil-drop experiment, which used it to measure the charge of the electron. Because the drag is linear in velocity rather than quadratic as it is at high Re, doubling the speed exactly doubles the resisting force — there is no drag crisis, no separation, no turbulent wake to complicate the picture.

Linearity buys reversibility

Because the Stokes equations are linear and contain no time derivative, they are invariant under time reversal: reverse every boundary velocity and the entire flow field reverses too, tracing its path backward exactly. This is the physics behind G. I. Taylor's famous demonstration film — a blob of dye injected into glycerin between two concentric cylinders is smeared into an invisible thread by several rotations of the inner cylinder, and then, rotating the cylinder back the same number of turns, the thread reassembles into a compact blob. At high Reynolds number this is impossible: turbulence stirs the dye through countless uncontrolled degrees of freedom that cannot be undone by reversing one macroscopic control.

Why microorganisms cannot coast

Living at low Re has consequences for how organisms move. Edward Purcell's scallop theorem (1977) observes that reversibility forbids any net displacement from a reciprocal motion — a stroke that looks identical played forward and backward, like a scallop opening and closing a single hinge, produces zero net swimming in Stokes flow, because reversing the stroke exactly undoes whatever the forward stroke achieved. This is why bacteria evolved rotating helical flagella and cilia sweep in asymmetric, non-reciprocal patterns — they need a stroke that is genuinely different forwards and backwards to make any progress at all.

Frequently asked questions

What exactly makes a flow low-Reynolds-number?

The Reynolds number Re = ρUL/μ compares inertial forces to viscous forces. It becomes small when the object is tiny (L), the motion is slow (U), or the fluid is very viscous (μ large). A bacterium swimming in water has Re around 10⁻⁵; a sphere dragged slowly through glycerin or honey can also reach Re well below 1, which is why both are described by the same Stokes equations.

Why can't a bacterium just coast to a stop?

Because inertia is negligible at its scale, viscous drag kills its momentum almost instantly — a swimming bacterium that stops paddling halts within a fraction of a nanometre. This is the content of Purcell's "scallop theorem": at low Re, any propulsion stroke that looks the same forwards and backwards produces zero net displacement, which is why microorganisms evolved corkscrew flagella instead of oars.

Does true reversibility require zero viscosity?

The opposite — it requires the viscous (Stokes) regime, not zero viscosity. The reversibility comes from inertia being negligible, so the instantaneous velocity field depends only on the instantaneous boundary motion, with no momentum carried over from the past. Reverse the sphere's rotation and the entire flow field reverses too, undoing the smear. In real experiments a small amount of molecular diffusion means the un-mixing is never perfect, but it can come strikingly close.

Try it live

Everything above runs in your browser — open Stokes Flow and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Stokes Flow simulation

Stokes Flow: Creeping, Reversible Fluid Motion