= Shear-dependent settling of a sphere-and-tail body
{title2=$U_z=-\frac{P}{H+\gamma K}+\frac{PK(1-\gamma)\sin^2\theta}{(H+\gamma K)(K+4H)}$}
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 $K=K_NL$ 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.
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