= Lubrication model of a compliant translating plug
{title2=$p_x=-6\mu U/h^2+12\mu Q/h^3$}
For a freely translating compliant body separated from a cylindrical wall by a thin <lubrication> gap $h$, the body frame has wall speed $-U$ and body surface speed zero. The local axial flow is $w=-U(1-y/h)+p_xy(y-h)/(2\mu)$, so its flux per circumference is $-Uh/2-p_xh^3/(12\mu)=-Q$. Rearranging proves the displayed pressure gradient. A linear compliance $p=p_0+\alpha(r_0-r)$ couples this flow to deformation. Specified end pressures and zero net axial body force close the problem.
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