Forced annular displacement equation ($\sigma A_{xxx}-\beta {\alpha }_1{A}_t-{\alpha }_{2}A{A}_x=\beta {\gamma }_0{F}_t+{\gamma }_0^{2}F{F}_x+({\gamma }_0^2/R)(AF{)}_x$)

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.