 Shear-dependent settling of a sphere-and-tail body

For a heavy spherical head with a much thinner long rigid tail, let $P$ be its excess weight, $H=6\pi \mu a$ the head translation resistance, and $K=K_{N}L$ the tail normal translation resistance. Neglect tail mass relative to head mass, sphere rotation resistance relative to tail rotation resistance, and $a/L$ in tail lever arms. In shear $\alpha z{\mathbf{e}}_1$, the stable orientation satisfies $\sin \theta =\alpha /(\beta +\sqrt{{\beta }^2+{\alpha }^2})$, $\cos \theta >0$, with $\beta =3P/[L(K+4H)]$. Resolving resistive-force theory force and torque balance gives the displayed signed vertical velocity. Since $0<\gamma <1$, stronger shear increases $U_z$ towards zero and reduces the downward settling speed. The stable head is below the far tail end. Very large shear can make the neglected sphere-rotation contribution significant.