Knowlpedia

Theorem

Pure swirl solution from a radial heat flow

A radial swirl heat profile together with its centrifugal pressure gives an exact unforced Navier–Stokes field away from the axis.

Statement

Suppose a smooth K(r,t)K(r,t) on r>0r>0 satisfies Kt=νLθKK_t=\nu L_\theta K, and assume the following pressure integral and its needed derivatives converge locally uniformly. Define

u=K(r,t)eθ,p(r,t)=−∫r∞K(s,t)2s ds.u=K(r,t)e_\theta,\qquad p(r,t)=-\int_r^\infty\frac{K(s,t)^2}{s}\,ds.

Then (u,p)(u,p) solves the unforced for r>0r>0.

Component check

The field is divergence-free and has no radial or axial velocity. Its radial advection is −K2/r-K^2/r, canceled by pr=K2/rp_r=K^2/r. Its azimuthal equation is exactly the swirl heat equation, and its axial equation vanishes by independence of zz.

An exterior profile with suitable power decay supplies a convergent pressure integral. This construction is independent of the axial coordinate, so a nonzero such field on all of R3\mathbb R^3 does not have finite kinetic energy. Using it only in an exterior region requires a separate joining construction.