= Annular inviscid-core displacement
{title2=$u_1=A u_0'/(R+y),\quad v_1=-A_xu_0/(R+y)$}
For a long, weak, axisymmetric indentation of an annular <fully developed flow>, use $r=R+y$ and scale normal velocity by the inverse longitudinal scale ratio. The leading <Euler equations for an inviscid fluid> give zero longitudinal pressure gradient and $u_0u_{1x}+v_1u_0'=0$. Together with $u_{1x}+(rv_1)_y/r=0$, this yields $\psi_1=A u_0$, hence the displayed displacement fields. Thin <viscous boundary layers> enforce tangential wall conditions separately.
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