Pressure-driven channel flow ($\mu {\Delta }_{\perp }w=-G$)

A steady unidirectional incompressible viscous flow in a rigid channel, driven by $G=-dp/dz$, reduces the axial momentum equation to the displayed Poisson equation, with $w=0$ on the cross-sectional boundary. Circular-pipe flow is the Hagen-Poiseuille flow special case; an arbitrary cross section need not have that parabolic radial profile.

In a symmetric wedge $|\theta |<\alpha$, a constant Poisson forcing admits the particular solution $r^2(\cos 2\theta /\cos 2\alpha -1)/4$, except at resonant angles. Zero-wall harmonic corrections have exponents ${\lambda }_n=(2n+1)\pi /(2\alpha )$. If the smallest exponent is below two, the global boundary data make that homogeneous mode dominate the local quadratic forcing. If it is below one, the velocity gradient can be singular although the velocity vanishes. When an exponent equals two, the appropriate radial solution contains $r^2\log r$, since $\Delta (r^2\log r\cos 2\theta )=4\cos 2\theta$.