Symmetric viscous interaction in an indented Poiseuille channel

At large Reynolds number $\mathcal{R}$, a long indentation of relative height $ϵ$ has nonlinear near-wall interaction when the streamwise scale ratio satisfies $\lambda /\mathcal{R}={ϵ}^3$. 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 $u_1=A(x)U_0^{\prime }(y)$ and $v_1=-A^{\prime }(x)U_0(y)$. A lower wall displaced to $ϵF(x)$ 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 $A=-F/2$.

For indentation $F=b{x}^{1/3}$, the common layer displacement is $H=b{x}^{1/3}/2$. The boundary layer stream function $\psi =x^{2/3}G(z{x}^{-1/3})$ gives velocity $U=x^{1/3}{G}^{\prime }$ and pressure $P=\bar{p}{x}^{2/3}$. Substitution into the Prandtl boundary-layer equation yields the displayed equation, with $G(0)=G^{\prime }(0)=0$ and $G^{\prime }\sim 6\eta +3b$. This is a conditional similarity construction: the constant $\bar{p}$ must permit a solution of the boundary value problem.