= Solution
For steady unidirectional <pressure-driven channel flow>, write $G=-dp/dz>0$. The axial <momentum> equation and <no-slip boundary condition> are
$$
\boxed{\nabla_\perp^2w=-G/\mu\quad\text{in }D,\qquad w=0\quad\text{on }\partial D.}
$$
Near an ideal wedge, the constant forcing has an $r^2$ particular solution by scaling. Substitution gives $f''+4f=-1$, with $f(\pm\alpha)=0$. For $\cos2\alpha\ne0$, the symmetric particular solution is
$$
\boxed{f(\theta)=\tfrac14\left(\frac{\cos2\theta}{\cos2\alpha}-1\right).}
$$
This <corner expansion of pressure-driven sector flow> is a particular solution, not necessarily the dominant corner term; a homogeneous corner mode can be larger, as the requested cases below show.
Separation of the symmetric harmonic correction with zero values on the straight walls gives modes $r^{\lambda_n}\cos(\lambda_n\theta)$, where
$$
\boxed{\lambda_n=\frac{(2n+1)\pi}{2\alpha},\qquad n=0,1,\ldots.}
$$
Regularity at the vertex excludes negative powers. <Orthogonality> on $[-\alpha,\alpha]$ has norm $\alpha$. To cancel the particular solution on $r=a$, use its <cosine> coefficients
$$
f_n=\frac1\alpha\int_{-\alpha}^{\alpha}f(\theta)\cos(\lambda_n\theta)d\theta=\frac{2(-1)^n}{\alpha\lambda_n(\lambda_n^2-4)}.
$$
The <integral> follows by product-to-sum and $\sin(\lambda_n\alpha)=(-1)^n$. Thus
$$
\boxed{w=\frac G\mu r^2f(\theta)+\sum_{n\ge0}A_nr^{\lambda_n}\cos(\lambda_n\theta),\qquad A_n=\frac{2(G/\mu)(-1)^na^{2-\lambda_n}}{\alpha\lambda_n(4-\lambda_n^2)}.}
$$
The arc data are satisfied by the <cosine> expansion, and the <Poisson equation> by construction. At the omitted thresholds $2\alpha=\pi/2,3\pi/2$, a mode has $\lambda_n=2$, so the pure $r^2f$ ansatz is resonant. The corresponding finite-limit term in the full sector solution is $(G/\mu)(-1)^nr^2\log(a/r)\cos2\theta/(4\alpha)$; the other modes remain regular. This explains rather than hides the division by $\cos2\alpha$.
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