= Stokes far-field multipoles of a deforming body
{c}
{title2=$u=JF+(G\times y)/(8\pi\mu r^3)+Qy/(4\pi r^3)-3(y\cdot Sy)y/(8\pi\mu r^5)+O(r^{-3})$}
With body-outward normal $n$, put $F=-\int\sigma n\,dS$, $\Sigma_{ij}=\int x_i(\sigma n)_j\,dS$, $U_{ij}=\int u_i n_j\,dS$, and $M=\Sigma-2\mu U$. Taylor expansion of the exterior <boundary integral representation of Stokes flow> gives a <Stokeslet> from $F$, a <rotlet> from $G_j=-\epsilon_{jkl}\Sigma_{kl}$, a source from $Q=\operatorname{Tr}U$, and a <stresslet> from $S=(M+M^T)/2-I\operatorname{Tr}M/3$. Thus $F$ and $G$ are the force and torque exerted on the fluid, while $Q$ is the instantaneous displaced-volume rate. A volume-preserving deforming body has $Q=0$.
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