= Control-volume drag formula for a sphere in a tube
{title2=$F=\pi a^2\Delta P+2\pi a\mu(4UI_1-6qI_2)$}
Take a fluid <control volume> spanning a nearly occluding <sphere>. The <pressure> difference contributes $\pi a^2\Delta P$ upwards, while the outer-wall shear in the <sphere> frame contributes $2\pi a\mu\int u_y(h)\,dz$. Since <sphere-frame flux in a tube> gives $u_y(h)=4U/h-6q/h^2$, the drag on the <sphere> is $F=\pi a^2\Delta P+2\pi a\mu(4UI_1-6qI_2)$. Using the wall shear avoids integrating traction over the strongly curved <sphere> directly. The remaining tube <pressure> drop is fixed by <Hagen-Poiseuille flow>.
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