Radial viscous flow in a wedge ($u_r=g(\theta )/r,\nu (g^{(3)}+4g^{\prime })+2g{g}^{\prime }=0$)

A steady two-dimensional purely radial incompressible Newtonian fluid flow has $u_r=g(\theta )/r$ by the continuity equation. The polar Navier-Stokes equation givesEquality 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 ${\int }_{-\alpha }^{\alpha }gd\theta =Q$. Consequently $g=QG$ leaves the single inertial parameter $ϵ=Q/\nu$ in $G^{(3)}+4G^{\prime }+2ϵG{G}^{\prime }=0$. The origin is a source singularity; the radial ansatz describes the wedge away from that point, not the detailed source apparatus.

On the symmetric small-$Q/\nu$ branch of radial viscous flow in a wedge, the leading Stokes flow solves $G_0^{(3)}+4G_0^{\prime }=0$, zero wall values and unit integral. HenceFor $\pi /2<\alpha <{\alpha }_c$, where ${\alpha }_c\simeq 2.24670$, this is positive in $|\theta |<\pi -\alpha$ and negative near the walls. The normalization denominator tends to zero at the stated root. At the root the homogeneous angular mode has zero net flux, so it cannot be rescaled to carry a fixed nonzero source flux.

Approaching that angle makes large outward and reverse contributions cancel to leave the prescribed flux. The leading amplitude diverges and the nonlinear inertial correction ceases to be perturbative, however small its coefficient is at fixed angle. This is a failure of a regular expansion, not a proof of infinite speed in the full Navier-Stokes equation. Distinct inner scalings or different steady branches require separate analysis.