 Faxén's first law

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,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.