= Radial viscous flow in a wedge
{title2=$u_r=g(\theta)/r,\quad\nu(g^{(3)}+4g')+2gg'=0$}
A steady two-dimensional purely radial <incompressibility>[incompressible] <Newtonian fluid> flow has $u_r=g(\theta)/r$ by the <continuity equation>. The polar <Navier-Stokes equation> gives
$$
p_r=\rho(g^2+\nu g'')/r^3,\qquad p_\theta=2\rho\nu g'/r^2.
$$
Equality of the mixed pressure derivatives yields the displayed third-order equation. On fixed walls $\theta=\pm\alpha$, <no-slip boundary conditions> require $g(\pm\alpha)=0$. A source with volume flux $Q$ per unit out-of-plane length requires $\int_{-\alpha}^{\alpha}g\,d\theta=Q$. Consequently $g=QG$ leaves the single inertial parameter $\epsilon=Q/\nu$ in $G^{(3)}+4G'+2\epsilon GG'=0$. The origin is a source singularity; the radial ansatz describes the wedge away from that point, not the detailed source apparatus.
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