Force flux at infinity in Stokes flow ($\mathbf{F}={\int }_{S_R}\mathbf{\sigma }\mathbf{n}dS$)

In a three-dimensional exterior Stokes flow with no body force, $\nabla \cdot \mathbf{\sigma }=0$. The divergence theorem on the fluid between a rigid body and a large sphere $S_R$ gives the displayed force on the body, where $\mathbf{n}$ points outward from the sphere. The body's outward normal points oppositely to the inner normal of this fluid domain. A nonzero resultant force produces a leading Stokeslet, so velocity is $O(R^{-1})$, gauge-fixed pressure is $O(R^{-2})$ and rate-of-strain tensor is $O(R^{-2})$. The finite force is therefore compatible with pointwise decay of the stress tensor. Force-free bodies may have faster decay.