= Symmetric viscous interaction in an indented Poiseuille channel
At large <Reynolds number> $\mathcal R$, a long indentation of relative height $\epsilon$ has nonlinear near-wall interaction when the streamwise scale ratio satisfies $\lambda/\mathcal R=\epsilon^3$. The regime $\mathcal R^{1/7}\ll\lambda\ll\mathcal R$ gives a thin viscous layer and an effectively common <pressure> across the core. The core displacement is $u_1=A(x)U'_0(y)$ and $v_1=-A'(x)U_0(y)$. A lower wall displaced to $\epsilon F(x)$ produces layer displacement $F+A$, while the upper wall produces $-A$. Uniqueness for their common <pressure> forcing and upstream data makes the two layer displacements equal, so $A=-F/2$.
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