Stokes flow at low Reynolds number

When \(\mathrm{Re} \ll 1\) the inertial terms are negligible compared with viscosity, and the steady Navier-Stokes equations reduce to the Stokes equations \[ -\nabla p + \mu\Delta\mathbf{u} = 0, \qquad \nabla\cdot\mathbf{u} = 0 \] (with pressure measured again in true units, since we are balancing forces directly). They are linear, and they contain no time derivative: the flow at each instant is determined by the boundary motion at that instant. Taking the divergence gives \(\Delta p = 0\), so the pressure is harmonic; taking the curl gives \(\Delta\boldsymbol{\omega} = 0\).

The most famous solution is the flow past a sphere of radius \(a\) moving at speed \(U\), solved by Stokes in 1851. Its drag is \[ F = 6\pi\mu a U. \] It is linear in speed and in size, unlike high-Reynolds-number drag, which grows like \(U^2\). Balancing it against weight minus buoyancy gives the settling speed of a small particle, \[ 6\pi\mu aU = \tfrac43\pi a^3(\rho_s - \rho)g \quad\Longrightarrow\quad U = \frac{2a^2(\rho_s - \rho)g}{9\mu}. \] A grain of sand of radius 25 micrometres sinks through water at about 2 mm/s. Millikan used this law in his oil-drop measurement of the electron charge. Always check the answer is self-consistent: the Reynolds number \(2aU/\nu\) of the result must indeed be small.

Linearity plus the absence of time has striking consequences. Reverse the boundary motion and the whole flow reverses: G. I. Taylor's demonstration unmixes a blob of dye in syrup by turning the handle back. A swimmer whose stroke looks the same played backwards (a scallop opening and closing) makes no net progress, which is Purcell's scallop theorem (1977); bacteria swim with rotating corkscrew flagella instead. Energy methods also give uniqueness: if two Stokes flows share boundary values, their difference \(\mathbf{w}\) has \(\int\mu|\nabla\mathbf{w}|^2 = 0\), so \(\mathbf{w} = 0\). Helmholtz (1868) showed that the Stokes flow dissipates less energy than any other divergence-free field with the same boundary values.

The approximation has a subtle edge. In two dimensions there is no Stokes solution for flow past a cylinder that matches a uniform stream at infinity (the Stokes paradox), because far from the body inertia is never negligible. Oseen (1910) fixed this by keeping a linearised inertial term; matched asymptotic expansions (Kaplun, and Proudman and Pearson, 1957) later explained why this works.

Picture it: a teaspoon moved through honey leaves no swirl behind; stop the spoon and everything stops at once. Think it: the Stokes operator is minus the Leray projection of the Laplacian, acting on divergence-free fields, and Stokes flow is its elliptic boundary value problem; the time-dependent Navier-Stokes equations are that operator plus the nonlinear term.