= Stokes flow
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= Creeping flow
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<Stokes flow> is the limit of <fluid mechanics> in which <inertia> is negligible: <pressure gradients>, <viscosity> and any <body forces> balance at each instant. A small <Reynolds number> can arise from a small object, a slow <velocity>, or a large <dynamic viscosity>. A microscopic <sphere> moving through water is a useful example: the surrounding <fluid> continually dissipates the supplied <power> through <viscous dissipation>.
\Image[/stokes-sphere-streamlines.png]
{title=Exact Stokes streamlines around a stationary no-slip sphere in a uniform incident flow}
The picture is a meridional slice of the three-dimensional <axisymmetric flow> around a <sphere>, with the <sphere> held fixed and the distant <fluid> moving rightward at <speed> $U$. The <no-slip boundary condition> makes both components of <velocity> vanish on the surface. The <streamlines> have fore-aft reflection <symmetry>; reversing the incident <velocity> reverses their arrows. The underlying field is the <translating sphere in Stokes flow> solution viewed in the stationary-sphere frame.
For an <incompressible flow> of a <Newtonian fluid> with constant <dynamic viscosity> $\mu$, the steady <Stokes equation> is
$$
-\nabla p+\mu\nabla^2\mathbf u+\mathbf f=\mathbf 0,\qquad \nabla\cdot\mathbf u=0.
$$
Here $p$ is <pressure>, $\mathbf u$ is <velocity>, and $\mathbf f$ is <body force> per unit <volume>. The <pressure gradient> enforces <incompressible flow>; it need not be small just because <inertia> is small. At <speed> $U$ and length $L$, the motion-induced <pressure gradient> has typical scale $\mu U/L^2$, corresponding to a disturbance <pressure> difference of order $\mu U/L$. See https://notes.shreyasmandre.com/fluid-dynamics-notes/Ch9.html[Shreyas Mandre's fluid dynamics notes, sections 9.1.1–9.1.2].
The controlling <Reynolds number> is
$$
\mathrm{Re}_L=\frac{\rho UL}{\mu}=\frac{UL}{\nu},\qquad \nu=\frac{\mu}{\rho},
$$
where $\rho$ is <density> and $\nu$ is <kinematic viscosity>. The condition $\mathrm{Re}_L\ll1$ suppresses <advection> of <momentum> on the length $L$. Rapid changes require an additional check of the <viscous diffusion time>: small <Reynolds number> alone does not eliminate the time derivative in the <Unsteady Stokes equation>.
Two rewards of this limit are the exact <Stokes's law> for the <drag> on a <sphere> and <kinematic reversibility of Stokes flow>. The first predicts <settling velocities> and connects to the <Stokes–Einstein relation>; the second explains why a <microswimmer> needs a nonreciprocal stroke, as expressed by the <scallop theorem>. Despite <viscous dissipation>, reversing the driving can retrace the motion: an especially striking consequence of <Linearity of Stokes flow>.
George Gabriel Stokes obtained the <sphere> <drag> and its <terminal velocity> in his 1851 investigation of how <viscosity> changes <physical pendulum> motion. He also used the slow fall of small water droplets to discuss clouds. The paper was read on 9 December 1850; 1851 is its publication date. See https://archive.org/details/b22464074[Stokes, On the effect of the internal friction of fluids on the motion of pendulums, original 1851 printing], especially articles 42–43; the formulas are also accessible in https://mural.uv.es/daroig/documentos/stokes1850.pdf[the collected-paper reprint, articles 42–43].
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