= First-order mobility of a nearly spherical particle
{title2=$\mathbf U_1=\frac{3}{8\pi a^3}\int_{r=a}f(\mathbf n)(\mathbf n\mathbf n-\mathbf I)\mathbf U_0\,dS$}
A force-driven rigid particle with surface $r=a+\varepsilon f(\mathbf n)$ has a leading <translating sphere in Stokes flow> with $\mathbf U_0=\mathbf F/(6\pi\mu a)$. Expanding the <no-slip boundary condition> gives the first-order surface velocity $\mathbf u_1=\mathbf U_1+\boldsymbol\Omega_1\times\mathbf x+\mathbf g$, where $\mathbf g=3f(\mathbf I-\mathbf n\mathbf n)\mathbf U_0/(2a)$. The first-order resultant <force> and <torque> vanish when the applied load is fixed. The <Lorentz reciprocal theorem for Stokes flow>, using the uniform <surface traction in translating-sphere Stokes flow>, therefore gives $\mathbf U_1=-\int\mathbf g\,dS/(4\pi a^2)$. For $f=a\mathbf n\cdot\mathbf D\mathbf n$ with symmetric traceless $\mathbf D$, the isotropic fourth moment gives $\mathbf U_1=\mathbf D\mathbf U_0/5$. A rotating-sphere test similarly gives $\boldsymbol\Omega_1=-3\int\mathbf n\times\mathbf g\,dS/(8\pi a^3)$, which vanishes for an inversion-symmetric deformation.
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