Annular inviscid-core displacement ($u_1=A{u}_0^{\prime }/(R+y),v_1=-A_x{u}_0/(R+y)$)

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 $u_0{u}_{1x}+v_1{u}_0^{\prime }=0$. Together with $u_{1x}+(r{v}_1{)}_y/r=0$, this yields ${\psi }_1=A{u}_0$, hence the displayed displacement fields. Thin viscous boundary layers enforce tangential wall conditions separately.

A distinguished annular indentation limit has ${\lambda }^{-2}=O(ϵ)$, $\lambda St=O(ϵ)$ and $Re\gg \lambda {ϵ}^{-3}$. Define $\beta =\lambda St/ϵ$, $\sigma =({\lambda }^2ϵ{)}^{-1}{\int }_0^1{u}_0^2/(R+y)dy$, ${\alpha }_1={\gamma }_0/R+{\gamma }_1/(R+1)$ and ${\alpha }_2={\gamma }_0^2/R^2-{\gamma }_1^2/(R+1{)}^2$. Radial inertia supplies the pressure difference $\sigma A_{xx}$ between the walls. The axial momentum limits and the moving-wall condition supply its derivative, giving the displayed forced KdV equation. This reduction assumes the viscous boundary layers remain thin and their displaced flux is smaller than $O({ϵ}^2)$.

With no forcing, the linear dispersion relation has positive phase velocity $\sigma k^2/(\beta {\alpha }_1)$ and group velocity three times this. A decaying permanent wave with $\xi =x+ct$ instead obeys $\sigma (A^{\prime }{)}^2=A^2(\beta {\alpha }_{1}c+{\alpha }_{2}A/3)$. For positive coefficients a nonzero homoclinic wave requires $c>0$ and is negative, travelling upstream. Its inverse width is $k_s=\frac{1}{2}\sqrt{\beta {\alpha }_{1}c/\sigma }$. The opposite directions reflect the nonlinear amplitude-dependent propagation speed.