the Stokes equations

Imagine a bacterium swimming, or a tiny dust grain settling through honey: motion so slow and sticky that inertia simply does not matter. If you stop pushing, the thing stops at once — there is no coasting. The Stokes equations describe exactly this regime of very slow, very viscous flow, where viscosity utterly dominates and the troublesome nonlinear inertia of Navier-Stokes can be thrown away.

You get them by taking the incompressible Navier-Stokes equations and dropping both the time-derivative and the nonlinear advection term — that is, the whole left-hand side rho(u_t + (u dot grad) u) goes to zero. What remains is gloriously linear: mu Laplacian u = grad p - f, together with div u = 0. This is what happens when the Reynolds number is tiny (Re much less than 1). Because the equations are now linear, all the heavy machinery of linear PDE theory applies, solutions can be superposed, and the flow is reversible: run the forcing backwards and the fluid retraces its path exactly. The famous Taylor-Couette demonstration, where dye blobs in viscous fluid un-mix when the shearing is reversed, is Stokes reversibility made visible.

Stokes flow is the everyday physics of the microscopic world: microfluidic chips, sediment settling, the motion of microorganisms, lubrication films, and the slow ooze of glaciers and the Earth's mantle. The reversibility has a deep consequence — Purcell's scallop theorem — that a microswimmer making a reciprocal (back-and-forth, time-symmetric) stroke goes nowhere, which is why real microorganisms use corkscrews and beating flagella instead of simple flaps.