= Faxén's first law
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= Faxén relation
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{synonym}
= Faxén translation law
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{synonym}
<Faxén's first law> gives the external <force> needed to translate a rigid <sphere> with the <no-slip boundary condition>, of <radius> $a$ with <velocity> $\mathbf U$ in an ambient <Stokes flow> $\mathbf u_\infty$. In an unbounded <Newtonian fluid> of <dynamic viscosity> $\mu$, with the ambient <Stokes flow> smooth and free of <body forces> throughout the sphere's ball,
$$
\mathbf F_{\mathrm{ext}}=6\pi\mu a\left\{\mathbf U-\left[\mathbf u_\infty+\frac{a^2}{6}\nabla^2\mathbf u_\infty\right]_{\mathbf x=\mathbf x_c}\right\}.
$$
The hydrodynamic <force> on the <sphere> is $-\mathbf F_{\mathrm{ext}}$. This sign convention gives the <Stokes drag law> when the ambient <velocity> vanishes and gives a holding <force> opposite to a uniform ambient <velocity> when the <sphere> is fixed.
The <Lorentz reciprocal theorem for Stokes flow>, using a <translating sphere in Stokes flow> as the auxiliary solution, reduces the ambient contribution to its surface average. The <spherical mean of a biharmonic function> makes that average exactly $\mathbf u_\infty(\mathbf x_c)+(a^2/6)\nabla^2\mathbf u_\infty(\mathbf x_c)$. The finite-size correction therefore measures spatial curvature of the ambient <velocity>; it vanishes in a uniform or linear <Stokes flow>. A <force-free> <sphere> translates with this corrected ambient <velocity>.
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