Translating sphere in Stokes flow

A sphere of radius $a$ translating with velocity $\mathbf{U}$ through otherwise stationary fluid hasIts surface traction integrates to the Stokes drag law.

A fixed sphere in a slowly varying incident Stokes flow requires an external force $\mathbf{F}=-6\pi \mu a{\mathbf{u}}_{\infty }$ and external couple $\mathbf{G}=-4\pi \mu a^3{\mathbf{\omega }}_{\infty }$ to leading order. These follow from Faxén translation law and Faxén rotation law; the incident Stokeslet supplies ${\mathbf{u}}_{\infty }$ and ${\mathbf{\omega }}_{\infty }$. Finite-radius Laplacian terms enter the translational force at the next nonzero order. The signs here refer to applied holding loads, not hydrodynamic loads.

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.

For a no-slip sphere in a smooth ambient Stokes flow whose body forces lie outside the sphere and are unchanged by insertion, use the translating sphere as a reciprocal auxiliary field. Its uniform surface traction selects the ambient surface average and annihilates a rigid rotation term. The ambient stress has zero net surface force because its divergence vanishes inside the ball. Thus the sphere's hydrodynamic force is $-6\pi \mu a(\mathbf{V}-⟨{\mathbf{u}}_{\infty }⟩)$; mechanical balance gives the displayed formula. The surface-average-only expression requires a force-free inertialess sphere. The spherical mean of a biharmonic function makes this average exactly $[1+(a^2/6){\nabla }^2]{\mathbf{u}}_{\infty }(0)$.

The Rotne--Prager mobility approximates the hydrodynamic coupling between two equal spheres of radius $a$ in an unbounded fluid of dynamic viscosity $\mu$ at negligible Reynolds number. For centre separation $R$ large compared with $a$, put $\hat{\mathbf{R}}=\mathbf{R}/R$. The cross-mobility, defined by the contribution ${\mathbf{U}}_1={\mathsf{M}}_{12}{\mathbf{F}}_2$ to sphere one's translation, isThe prefactor is the isolated-sphere translational mobility. The leading term is the Stokeslet interaction; the second accounts for finite sphere size. Combining the source-sphere correction with Faxén's first law for the receiving sphere gives this correction. The hydrodynamic mobility matrix maps applied force to velocity, so omitting the prefactor loses both its physical normalization and units. This far-separated pair approximation needs further treatment for near-contact or overlapping particle models.

A passing forced sphere advects a distant force-free sphere along a curved trajectory. Its longitudinal displacement can diverge logarithmically even when its transverse displacement approaches a finite value.