= Rotation-clamped reflection between two spheres
If the otherwise <force-free> second <sphere> is held against rotation, its <angular velocity> relative to the incident fluid is $-3a(\mathbf U_0\times\mathbf n)/(4R^2)$. The <rotlet> reflected back to the first <sphere> adds
$$
\Delta\mathbf U_{\mathrm{rot}}=-\frac34\left(\frac aR\right)^4\left[\mathbf U_0-(\mathbf U_0\cdot\mathbf n)\mathbf n\right].
$$
This follows by evaluating the <rotating sphere in Stokes flow> at displacement $-R\mathbf n$ and using the vector triple-product identity. The holding couple does no work because the held <sphere>'s <angular velocity> is zero.
Back to article page
