Jeffery–Hamel flow

Example

Let $0 < \alpha < \pi$ and consider the plane wedge $\set{(r, \theta) : r > 0, \ \abs{\theta} < \alpha}$ in polar coordinates, filled with a fluid of density $1$ and kinematic viscosity $\nu > 0$. Fluid flows radially out of or into the apex, as from a source or a sink. Jeffery–Hamel flow is the steady flow with purely radial velocity $$u_r = \frac{\nu}{r} F(\theta), \qquad u_\theta = 0,$$ where $F$ solves $$F'' + 4 F + F^2 = C, \qquad F(\pm \alpha) = 0,$$ for a constant $C$, and the flux $\nu \int_{-\alpha}^{\alpha} F(\theta) \, d\theta$ is prescribed.

Check: $\frac{1}{r} \partial_r (r u_r) = 0$, so the field is divergence-free. The azimuthal Navier–Stokes equation gives $p = 2 \nu^2 F(\theta) / r^2 + P(r)$. Putting this into the radial equation gives $\nu^2 (F'' + 4F + F^2) = r^3 P'(r)$. The left side depends only on $\theta$ and the right side only on $r$, so both are a constant $\nu^2 C$. The conditions $F(\pm \alpha) = 0$ are the no-slip condition on the walls.