= Stokes's law
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{title2=$\mathbf F_D=-6\pi\mu a\mathbf U$}
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= Stokes drag law
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{synonym}
= Stokes drag
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{synonym}
= Stokes law
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{synonym}
For an isolated rigid <sphere> of radius $a$ moving at <velocity> $\mathbf U$ relative to a distant uniform <Newtonian fluid>, the <drag force> exerted by the <fluid> is
$$
\mathbf F_D=-6\pi\mu a\mathbf U,\qquad |\mathbf F_D|=6\pi\mu aU.
$$
The assumptions are steady <incompressible flow>, constant <dynamic viscosity> $\mu$, an unbounded surrounding <fluid>, the <no-slip boundary condition>, and a small diameter-based <particle Reynolds number>, $\mathrm{Re}_d=2a\rho_fU/\mu\ll1$. Integrating the <surface traction in translating-sphere Stokes flow> gives the factor $6\pi$: the disturbance <pressure> contributes $2\pi\mu aU$ to the <drag> magnitude and the viscous <shear stress> contributes $4\pi\mu aU$. See https://farside.ph.utexas.edu/teaching/336L/Fluid/node142.html[Richard Fitzpatrick, Axisymmetric Stokes Flow Around a Solid Sphere, equations 10.120–10.129].
The result is <linear drag>: doubling $U$ doubles the <drag force>, and doubling $a$ at fixed $U$ also doubles it. The required <power> is $6\pi\mu aU^2$, all supplied to <viscous dissipation>. <Density> enters the check of the <particle Reynolds number>, although it does not appear in the leading <drag> formula.
Nearby walls or other particles change the <hydrodynamic resistance matrix> and invalidate the isolated-<sphere> coefficient, even at very small <Reynolds number>. Finite <Reynolds number> introduces <inertia> corrections; the <Oseen approximation> resolves the distant region where the leading <Stokes flow> approximation ceases to be uniform. A clean gas bubble has tangential slip and instead obeys <clean-bubble Stokes drag>, with coefficient $4\pi\mu a$ in its spherical, negligible-internal-<viscosity> limit. See https://www.damtp.cam.ac.uk/user/hkm2/PDFs/Moffatt_1977_GordonandBreach_Slogfdatohdt_149.pdf[H. K. Moffatt, Six Lectures on General Fluid Dynamics, section 3.4] for the <sphere> solution and its small-<Reynolds number> limitation. The <continuum mechanics> and boundary assumptions must also remain valid as the particle becomes very small.
Balancing this <Stokes drag law> against <gravity> and <buoyancy> gives the <Stokes settling velocity> of an isolated rigid <sphere>.
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