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The weakly nonlinear, unsteady and dispersive core terms must enter at the same order. Axial nonlinearity first contributes at $O({ϵ}^2)$, while unsteadiness contributes $O(ϵ\lambda St)$ and transverse-inertia pressure variation contributes $O(ϵ{\lambda }^{-2})$. Thus take the distinguished limitEquivalently, $\lambda =O({ϵ}^{-1/2})$, $St=O({ϵ}^{3/2})$ and the wall-layer condition becomes $Re\gg {ϵ}^{-7/2}$. These orderings imply the initially stipulated long wavelength, low Strouhal number and large Reynolds number. In the core $\lambda /Re=o({ϵ}^3)$, so perturbation viscous terms are beyond the retained order.

Subtract the undisturbed fully developed state and scale pressure by $\rho {\hat{U}}^2$. The inviscid core equations arePut $u=u_0+ϵu_1+{ϵ}^2{u}_2+\cdots$ and $v=ϵv_1+{ϵ}^2{v}_2+\cdots$. At first order $p_{1y}=0$ andThe wall's normal movement first enters at second order in this scaling, so $v_1=0$ at both walls. Taking the regular wall limits, where $u_0=0$ and $u_0^{\prime }\ne 0$, gives $p_{1x}=0$. The arbitrary constant pressure perturbation can be set to zero.

Define the first-order axisymmetric stream function by $u_1={\psi }_{1y}/r$ and $v_1=-{\psi }_{1x}/r$. The axial equation becomes $u_0{\psi }_{1yx}-u_0^{\prime }{\psi }_{1x}=0$, or $({\psi }_{1x}/u_0{)}_y=0$. Consequently ${\psi }_1=A(x,t)u_0(y)$ after absorbing an $x$-independent profile into the background. This proves the annular inviscid-core displacementIt gives exactly the stated core expansion without requiring an explicit calculation of $u_0(y)$. The factors $1/(R+y)$ arise from cylindrical mass conservation.