 Lubrication model of a compliant translating plug

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 $w=-U(1-y/h)+p_{x}y(y-h)/(2\mu )$, so its flux per circumference is $-Uh/2-p_x{h}^3/(12\mu )=-Q$. 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.