= Solution
Continue the expansion established in part (b), with $p=\epsilon^2p_2+\cdots$ after removal of the background <pressure>. The radial <momentum> equation at order $\epsilon^2$ gives
$$
p_{2y}=-D u_0v_{1x}=D\frac{u_0^2}{r}A_{xx}.
$$
Integrating across the core, then differentiating in $x$, gives
$$
p_{2x}(1)-p_{2x}(0)=\sigma A_{xxx},\qquad\sigma=D\int_0^1\frac{u_0^2}{R+y}\,dy.
$$
This is the dispersive contribution from transverse inertia.
At second order the axial <momentum> equation is
$$
u_0u_{2x}+v_2u_0'=-p_{2x}-\beta u_{1t}-u_1u_{1x}-v_1u_{1y}.
$$
For the <displacement> fields just obtained,
$$
u_1u_{1x}+v_1u_{1y}=AA_x\left[\frac{(u_0')^2-u_0u_0''}{r^2}+\frac{u_0u_0'}{r^3}\right].
$$
Use $u_0=0$ on each wall. The lower and upper regular core limits respectively give
$$
\begin{aligned}
p_{2x}(0)&=-\gamma_0v_2(0)-\beta\frac{\gamma_0}{R}A_t-\frac{\gamma_0^2}{R^2}AA_x,\\
p_{2x}(1)&=\beta\frac{\gamma_1}{R+1}A_t-\frac{\gamma_1^2}{(R+1)^2}AA_x,
\end{aligned}
$$
where $v_2(1)=0$ at the fixed outer wall.
At the moving inner wall $y=\epsilon F$, no penetration requires $v=\epsilon\lambda St F_t+\epsilon uF_x$. Expanding about $y=0$, the axial edge <velocity> is $\epsilon\gamma_0(F+A/R)+O(\epsilon^2)$ and $v_{1y}(0)=-\gamma_0A_x/R$. Thus
$$
\epsilon^2\left[v_2(0)-\frac{\gamma_0}{R}FA_x\right]=\epsilon^2\left[\beta F_t+\gamma_0(F+A/R)F_x\right]+o(\epsilon^2),
$$
or
$$
v_2(0)=\beta F_t+\gamma_0FF_x+\frac{\gamma_0}{R}(AF)_x.
$$
The thin wall layers do not change this normal-flux condition at the retained order, because their flux correction is $o(\epsilon^2)$.
Subtract the two axial pressure-gradient limits and use the radial-pressure difference. With $\alpha_1=\gamma_0/R+\gamma_1/(R+1)$ and $\alpha_2=\gamma_0^2/R^2-\gamma_1^2/(R+1)^2$, the result is the <forced annular displacement equation>
$$
\boxed{\sigma A_{xxx}-\beta\alpha_1A_t-\alpha_2AA_x=\beta\gamma_0F_t+\gamma_0^2FF_x+\frac{\gamma_0^2}{R}(AF)_x}.
$$
Every coefficient follows from the given profile and wall slopes; no explicit base-flow solution has been used. The derivation is conditional on the regular core limits and wall layers remaining asymptotically thin.
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